politecnico di milano · 2017-09-12 · 2.4 vessel discretization for interpolation and average...

Politecnico di MilanoSchool of Industrial and Information Engineering

Master of Science in Mathematical Engineering

Master’s thesis in Computational Science and Engineering

Computational modelsfor nanoparticle transport

in the vascular system

Advisor:Prof. Paolo ZUNINO

Co-Advisors:

Prof. Paolo DECUZZIDr. Silvia LORENZANIDr. Alessandro COCLITEDr. Federica LAURINO

Annagiulia TIOZZOMatr. 836483

Academic Year 2016-2017

A Raffaele

Abstract

Nanoparticle-based drug delivery is one of the most promising innovationsof nanomedicine, in particular in the field of the treatment of complex dis-eases, for example cancer. A great contribution to the understanding ofsuch therapies is given by mathematical models and numerical simulations,since they can provide complementary sophisticated and multiscale tools toexperiments.

We build a mathematical model for the transport of nanoparticles in amicrovascular network, for their adhesion to the vessel walls and for the cor-responding release or extravasation of therapeutic agents in the surroundinginterstitial tissue.

All the biological systems share a multiscale structure, since several phe-nomena take place at different time- and space-scales. In order to take intoaccount these effects in the mathematical models, we use the results of sev-eral numerical or microfluidic experiments as values for the parameters inthe model.

Thanks to dimensional model reduction techniques, the blood flow andrelated transport phenomena can be described as a one-dimensional (1D)source within the 3D domain in order to reduce the computational cost ofthe simulations. From the mathematical standpoint, we notice that the high-dimensionality gap (3D/1D) causes an ill-posed formulation. We overcomethis problem by directly exploiting the coupling in the variational finite ele-ment formulation, thanks to suitable restriction operators.

Sommario

Il rilascio di farmaci basato su nanoparticelle e una delle innovazioni piupromettenti della nanomedicina, in particolare nel campo del trattamentodi malattie complesse, ad esempio il cancro. I modelli matematici e le si-mulazioni numeriche possono dare un grande contributo alla comprensionedelle terapie, perche sono in grado di fornire sofisticati strumenti di indaginecomplementari agli esperimenti.

Definiamo un modello matematico che descriva il trasporto di nanopar-ticelle in una rete microvascolare, la loro adesione alle pareti dei vasi e ilcorrispondente rilascio o extravasazione degli agenti terapeutici nel tessutointerstiziale circostante.

Tutti i sistemi biologici condividono una struttura multiscala, poiche di-versi fenomeni avvengono a scale spaziali e temporali differenti. Per tene-re in considerazione questi effetti anche nel modello matematico, utilizzia-mo i risultati di vari esperimenti numerici o microfluidici come valori per icorrispondenti parametri nel modello.

Grazie a opportune tecniche di riduzione dimensionale di modelli, il flussosanguigno e i relativi fenomeni di trasporto possono essere descritti come unasorgente unidimensionale (1D) all’interno del dominio 3D, al fine di ridurreil costo computazionale delle simulazioni. Dal punto di vista matematico,notiamo che il forte salto dimensionale (3D/1D) genera un problema mal po-sto. E possibile superare queste difficolta esprimendo l’accoppiamento tra leequazioni direttamente a livello della formulazione variazionale agli elementifiniti, grazie ad un opportuno operatore di restrizione.

Contents

1 Introduction and motivations 11.1 Mathematical and computational models in nanomedicine . . 2

1.1.1 Nanomedicine . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Mathematical models for nanomedicine . . . . . . . . . 71.1.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Multiscale models in biology . . . . . . . . . . . . . . . . . . . 121.2.1 Multiscale nature of biological systems . . . . . . . . . 121.2.2 Single-scale models and upscaling techniques . . . . . . 121.2.3 Examples of application . . . . . . . . . . . . . . . . . 151.2.4 Nanoparticle delivery in a capillary network . . . . . . 16

2 Mathematical model for particle transport in the microvas-culature 222.1 Three-dimensional model for microvasculature within a tissue 232.2 Model reduction: coupled 3D-1D problem . . . . . . . . . . . 25

2.2.1 Coupling term for the interstitial volume . . . . . . . . 262.2.2 Model reduction for microvascular flow . . . . . . . . . 272.2.3 Governing equations for the coupled problem . . . . . . 28

2.3 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 282.4 Boundary and initial conditions . . . . . . . . . . . . . . . . . 292.5 Junction treatment . . . . . . . . . . . . . . . . . . . . . . . . 302.6 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.1 Weak formulation for the tissue problem . . . . . . . . 312.6.2 Weak formulation for the vessel problem . . . . . . . . 322.6.3 Coupled weak formulation . . . . . . . . . . . . . . . . 342.6.4 Well posedness . . . . . . . . . . . . . . . . . . . . . . 34

2.7 Alternative well-posed weak formulation . . . . . . . . . . . . 352.7.1 Alternative 3D-1D coupling . . . . . . . . . . . . . . . 362.7.2 Alternative weak formulation for the tissue problem . . 362.7.3 Alternative weak formulation for the vessel problem . . 38

vi

CONTENTS

2.7.4 Alternative weak formulation for the coupled transportproblem . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7.5 Well posedness . . . . . . . . . . . . . . . . . . . . . . 392.8 Numerical approximation . . . . . . . . . . . . . . . . . . . . . 392.9 Algebraic counterpart . . . . . . . . . . . . . . . . . . . . . . . 41

3 Well-posedness analysis 443.1 Simplified problem setting . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Geometric setting . . . . . . . . . . . . . . . . . . . . . 443.1.2 Model equation . . . . . . . . . . . . . . . . . . . . . . 453.1.3 Restriction operator . . . . . . . . . . . . . . . . . . . 463.1.4 Weak formulation . . . . . . . . . . . . . . . . . . . . . 46

3.2 Well-posedness analysis . . . . . . . . . . . . . . . . . . . . . . 473.3 Conclusions and further developments . . . . . . . . . . . . . . 49

4 Characterization of the model parameters 514.1 A model for particle adhesion to the vascular wall . . . . . . . 51

4.1.1 Vascular adhesion parameter . . . . . . . . . . . . . . . 524.1.2 Effective vascular adhesion parameter with saturation . 524.1.3 Explicit formula for Pa . . . . . . . . . . . . . . . . . . 54

4.2 Multiscale model for particle adhesion . . . . . . . . . . . . . 574.2.1 Lattice Boltzmann approach for nanoparticle transport 574.2.2 Subscale model for particle adhesion . . . . . . . . . . 59

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Optimal control approach for the prediction of the diffusivity

coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . 634.4.2 Numerical model . . . . . . . . . . . . . . . . . . . . . 644.4.3 Minimization problem . . . . . . . . . . . . . . . . . . 654.4.4 Iterative method . . . . . . . . . . . . . . . . . . . . . 684.4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . 684.4.6 Discussion and conclusions . . . . . . . . . . . . . . . . 75

5 Numerical results 785.1 Nanoparticle transport and adhesion . . . . . . . . . . . . . . 79

5.1.1 Fluid dynamics effects . . . . . . . . . . . . . . . . . . 815.1.2 Explicit formula for Pa . . . . . . . . . . . . . . . . . . 845.1.3 Pa from LB approach . . . . . . . . . . . . . . . . . . . 875.1.4 Effective vascular adhesion parameter and saturation . 915.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2 Dextran transport and extravasation . . . . . . . . . . . . . . 93

vii

CONTENTS

5.2.1 Available data and results . . . . . . . . . . . . . . . . 945.2.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 Drug delivery: combined nanoparticles and Dextran . . . . . . 995.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.2 Parameters and results . . . . . . . . . . . . . . . . . . 1025.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Conclusions and future perspectives 1096.1 Future developments . . . . . . . . . . . . . . . . . . . . . . . 112

A Fluid dynamics model for the vascularized tissue 115A.1 Model set up . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.2 Coupling microcirculation with interstitial flow . . . . . . . . . 117

A.2.1 A reduced model for microvascular flow . . . . . . . . . 118A.2.2 Governing equations for the coupled problem . . . . . . 118

A.3 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 119A.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 120A.5 Junction treatment . . . . . . . . . . . . . . . . . . . . . . . . 121A.6 Variational formulation . . . . . . . . . . . . . . . . . . . . . . 122

A.6.1 Weak formulation of the tissue problem . . . . . . . . . 122A.6.2 Weak formulation of the vessel problem . . . . . . . . . 123A.6.3 Coupled weak formulation . . . . . . . . . . . . . . . . 126

A.7 Numerical approximation . . . . . . . . . . . . . . . . . . . . . 127A.7.1 Discretization of the tissue problem . . . . . . . . . . . 127A.7.2 Discretization of the vessel problem . . . . . . . . . . . 128A.7.3 Discrete coupled weak formulation . . . . . . . . . . . 128

A.8 Algebraic formulation . . . . . . . . . . . . . . . . . . . . . . . 129

B Lattice Boltzmann method 132B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132B.2 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . 133

B.2.1 BGK kinetic model . . . . . . . . . . . . . . . . . . . . 133B.2.2 Time discretization . . . . . . . . . . . . . . . . . . . . 134B.2.3 Choice of the quadrature rule . . . . . . . . . . . . . . 134

B.3 The lattice Boltzmann equation . . . . . . . . . . . . . . . . . 135B.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 137

B.4.1 Standard bounce-back . . . . . . . . . . . . . . . . . . 137B.4.2 Higher-order boundary conditions . . . . . . . . . . . . 138

Bibliography 140

viii

List of Figures

1.1 Effect of 3S parameters on the biodistribution of nanoparticlesin organs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The spectrum of a body-on-chip developed for drug screeningpurposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Integrated microfluidic system for single-cell studies. . . . . . . 61.4 Technical challenges for theoretical and computational scien-

tists in nanomedicine. . . . . . . . . . . . . . . . . . . . . . . . 81.5 Spatial distribution of different sized particles in an inflamed

arterial tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 The multi-phase system within a Representative Elementary

Volume for tumor growth and nutrient evolution. . . . . . . . 101.7 FSI simulation of a VAD in action. . . . . . . . . . . . . . . . 101.8 Multiscale nature of a biological system: length and time scales. 121.9 A classification of methods for biological systems in terms of

their characteristic length- and time-scales. . . . . . . . . . . . 131.10 Different scales within the human circulatory system. . . . . . 151.11 Multiscale modules for vascular and extravascular transport

of nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Microvasculature within a tissue interstitium and reductionfrom 3D to 1D description. . . . . . . . . . . . . . . . . . . . . 23

2.2 Y-shaped bifurcation for the description of the junction treat-ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Sketch of the alternative weak formulation. . . . . . . . . . . . 352.4 Vessel discretization for interpolation and average operators. . 43

3.1 Simplified geometric setting for well posedness analysis. . . . . 45

4.1 Particles trajectory within a Couette flow, solved via LB-IBmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Geometry for the Couette flow and particle transport . . . . . 59

ix

LIST OF FIGURES

4.3 Probability of adhesion computed using the Lattice Boltzmann-Immersed Boundary method. . . . . . . . . . . . . . . . . . . 61

4.4 Geometric model of the Dextran diffusion experiment . . . . . 634.5 Diffusion experiments using 4 kDa Dextran molecules at the

initial and final time. . . . . . . . . . . . . . . . . . . . . . . . 644.6 Initial configuration for the code validation . . . . . . . . . . . 694.7 Reference and optimal concentrations for the code validation . 704.8 Intensity maps of the diffusion experiment using 4 kDa Dex-

tran molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.9 Reference concentrations for 4 kDa Dextran molecules: possi-

ble smooth and non smooth configurations. . . . . . . . . . . . 744.10 Concentration with the optimal diffusivity coefficient: test4A. 74

5.1 Sketch of the three main classes of simulations. . . . . . . . . . 805.2 rat93 geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Pressure field in the vessel network: test1 and test2. . . . . 825.4 Velocity field in the vessel network: test1 and test2. . . . . . 825.5 Wall shear rate in the vessel network: test1 and test2. . . . 835.6 Probability of adhesion in the vessel network with explicit for-

mula: test1 and test2. . . . . . . . . . . . . . . . . . . . . . 855.7 Vascular adhesion parameter in the vessel network without

saturation model: test1 and test2. . . . . . . . . . . . . . . 855.8 Concentration of nanoparticles at the final time in the vessel

network: test1 and test2. . . . . . . . . . . . . . . . . . . . 855.9 Density of nanoparticle adhering per unit surface to the vascu-

lar wall at the final time, without the saturation model: test1and test2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.10 Reynolds number in the vessel network: test3. . . . . . . . . 885.11 Adhesive variables in the vessel network: test3. . . . . . . . . 885.12 Concentration of nanoparticle in the vessel network: test3

and test4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.13 Density of nanoparticle adhering per unit surface to the vas-

cular wall at the final time: test3 and test4. . . . . . . . . . 905.14 Pressure and velocity fields in the vessel network and in the

tissue interstitium: test5. . . . . . . . . . . . . . . . . . . . . 955.15 Concentration in the vessel network and in the tissue intersti-

tium at different simulation time: test5 and test6. . . . . . . 975.16 Mean concentration in the tissue region over time for 40 kDa

and 250 kDa Dextran molecules release: test5 and test6. . . 985.17 Dextran release profile in time: test7, test8 and test9. . . . 104

x

LIST OF FIGURES

5.18 Dextran concentration and density of adhering nanoparticleon the vessel wall in the cases of various molecular weight andsimulation time: test7, test8 and test9. . . . . . . . . . . . 105

5.19 Total amount of Dextran in the tissue region over time for 4kDa, 40 kDa and 250 kDa Dextran molecules release: test7,test8 and test9. . . . . . . . . . . . . . . . . . . . . . . . . . 106

A.1 Y-shaped bifurcation for the description of the junction treat-ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

B.1 D2Q9 lattice configuration . . . . . . . . . . . . . . . . . . . . 136B.2 D2Q9 on-grid bounce-back . . . . . . . . . . . . . . . . . . . . 138B.3 D2Q9 mid-grid bounce-back . . . . . . . . . . . . . . . . . . . 138

xi

List of Tables

4.1 Probability of adhesion computed using the Lattice Boltzmann-Immersed Boundary method for strong ligand-receptor bond. . 60

4.2 Probability of adhesion computed using the Lattice Boltzmann-Immersed Boundary method for mild ligand-receptor bond. . . 60

4.3 Model parameters for code validation . . . . . . . . . . . . . . 694.4 Diffusivity coefficient computed solving the minimization prob-

lem with NLCG for code validation. . . . . . . . . . . . . . . . 704.5 Sensitivity analysis of D∗ with respect to λ for code validation. 714.6 Numerical tests on the Dextran 4 kDa diffusion experiments . 754.7 Numerical tests at different final time T for test4E. . . . . . . 764.8 Numerical tests on the Dextran 40 kDa diffusion experiments . 764.9 Numerical tests on the Dextran 250 kDa diffusion experiments 76

5.1 Sketch of the tests for nanoparticle transport and adhesion. . . 815.2 Physical parameters characterizing the fluid dynamics prob-

lem in test3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Physical parameters characterizing the nanoparticle transport

problem in test3 . . . . . . . . . . . . . . . . . . . . . . . . . 895.4 Characteristic values for the non dimensional analysis in test3 895.5 Sketch of the tests for Dextran transport and extravasation. . 935.6 Physical parameters characterizing the fluid dynamics prob-

lem in test5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.7 Physical parameters characterizing the 40 kDa Dextran molecules

transport problem in test5 . . . . . . . . . . . . . . . . . . . 965.8 Physical parameters characterizing the 250 kDa Dextran molecules

transport problem in test6 . . . . . . . . . . . . . . . . . . . 975.9 Sketch of the tests for nanoparticles transport, adhesion and

Dextran delivery. . . . . . . . . . . . . . . . . . . . . . . . . . 1005.10 Parameters characterizing the release of Dextran molecules in

test7, test8 and test9 . . . . . . . . . . . . . . . . . . . . . 1035.11 Tissue diffusivity coefficient for Dextran delivery experiments . 104

xii

Chapter 1

Introduction and motivations

The final goal of the thesis is to perform simulations of the transport ofnanoparticles in the capillary network and of the drug delivery in the inter-stitial tissue. The use of nanoparticles loaded by therapeutic agents is oneof the most promising innovation in the field of the treatment of complexdiseases, tipically cancer. Moreover, the microvasculature is where the ex-change of nutrients and drugs takes place. Therefore, the need to combinethe two aspects is growing. A great help in the effort for creating a power-ful interconnection among medicine, technology and quantitative sciences isgiven by mathematical and numerical models. In order to provide a preciseformulation of the microcirculation problem, it is necessary to exploit themultiscale nature that characterizes all the biological systems and the inter-actions among results obtained at different time- and space-scales. Withinthis general framework, the specific objective of this work is to encode into themacroscopic model for nanoparticle transport in microcirculation and drugdelivery in the interstitial tissue some detailed information of the complexmicroscopic structure, in particular in terms of diffusion of the extracellularmatrix with respect to nanoparticles and adhesive properties of the nanopar-ticles.

The plan of the thesis is as follows. Chapter 1 is an introductory chapter inwhich an overview of what nanomedicine is and what the role of mathematicaland computational models in medical fields is. This is due to the fact thatnanoparticle-based drug delivery systems fall within the general frameworkof nanomedicine. This chapter also contains an explanation on how themultiscale approach works in biological systems. Chapter 2 proposes themacroscopic model for microcirculation and drug delivery, by means of anetwork of one-dimensional channels immersed in a three-dimensional space.The well posedness of the weak formulation of the macroscopic model is

1

CHAPTER 1. INTRODUCTION AND MOTIVATIONS

addressed in Chapter 3. At the microscale, the estimate of the adhesiveparameters of the nanoparticles as function of the particle properties via aLattice Boltzmann-Immersed Boundary method and the prediction of thediffusivity coefficient of the extracellular tissue by numerically reproducingphysical experiments with the help of an optimization problem are presentedin Chapter 4. Finally, in Chapter 5 we provide some results of the large scalesimulation integrated with the information derived from the analysis at themicroscale.

1.1 Mathematical and computational models

in nanomedicine

1.1.1 Nanomedicine

Nanomedicine is an emerging research branch which combines nanotechno-logical tools and biomedical studies. This research field aims at enhancingclinical diagnosis and providing opportunities for more effective therapeutictreatments. The use of nanotechnology for medical purposes leads to theminiaturization of engineered devices and nanostructures up to the molecu-lar level in order to improve the disease detection and the efficiency of thehealth care [45].

From its very beginning, nanotechnology applied to medicine has been avery fertile field, as many tools with a good therapeutic response have beendeveloped. Applications can be found in many different fields of medicine, forexample drug-delivery systems, nanosensors, imaging agents and implants.

In [5], Bao et al. draw up a list of issues in some biomedical fields thatcan be tackled with mechanical and technological tools, especially consideringdevices at the nanoscale. In particular, the three major challenges that theyoutline deal with (i) the drug delivery by means of injectable nanoparticles,(ii) the design of biomedical devices and (iii) the study of the mechanics ofcellular processes.

(i) Functionalized nanoparticles have been developed for different pur-poses, mainly for cancer treatment [42] and diagnostic imaging [6],but also for atherosclerosis and neurodegenerative diseases, as vectorfor targeted drug or imaging contrast agent delivery or for hypertermiatherapy. Nanoconstructs are widely used as drug carriers, since theycan navigate through the human circulatory system and they can easilyrecognize tumor neovasculature. Nanoparticles are loaded with ther-apeutic or imaging agents that can be released as needed, improving

2


Figure 1.1: Nanoparticle size, shape and surface charge are responsible forthe biodistribution among the different organs, such as lungs, liver, spleenand kidneys. Figure from [8].

the specificity and the personalization of the treatment. Moreover, itis well known that the tumor vasculature and the healthy blood vesselnetwork exhibit several differences. In healthy vessels, blood circulatesat higher mean velocity with respect to the blood that is in the dis-eased vasculature and there is lower interstitial fluid pressure. In thediseased case, the vessels are leakier and more tortuous and the vascu-lature is hyper-permeable [28]. Therefore, many types of nanoparticleswith different sizes, shapes and surface properties have been designedin order to maximize the effectiveness of the treatments, the quantity ofloaded agents, the circulation time and to minimize the sequestrationby other organs. Maximizing the efficacy of the therapy means, in prac-tice, maximizing the accumulation of the agents in the targeted region.As shown in Figure 1.1, the properties of the nanoparticles also dictatethe biodistribution among different organs. By manipulating the phys-ical characteristics of the nanoparticles, it is possible to observe thebehaviour of their dynamics and their adhesion at the diseased walls.The aim of the optimization process is to find which nanoparticle canefficiently marginate and adhere to the diseased region, pushed by thehydrodynamic forces and the adhesive interactions [17].

(ii) Nanoscale technology turns out to be very useful also in the case ofthe design of many biomedical devices. For example, the develop-ment of microfluidic chips that are able to realistically reproduce thefunctionalities of some organs in order to perform the screening of newpharmaceuticals and to recreate biological processes outside the bodyis currently under investigation [30]. Many organs have been designed

3


Figure 1.2: The spectrum of a body-on-chip is being developed for drugscreening purposes. System have been developed for (a) lung, (b) the blood-brain barrier, (c) heart tissue, (d) liver, (e) the gastrointestinal tract, (f)muscle and the (g) microcirculation. Figure from [5].

4


on a microfluidic platform, for example the lung [27], the liver [19] andthe blood brain barrier. In each of these organs, there is a realisticexchange of signals and interaction. The most challenging issue in theframework of the organs-on-chip system is the realistic interaction ofall the single organs in order to reproduce a global body-on-chip. Thiscontribution could be extremely important in the perspective of study-ing possible side effects of therapeutic treatments on organs differentfrom the targeted one. In Figure 1.2, we show a variety of organs onchip and functionalities.

(iii) The third issue proposed by Bao et al. in [5] is about cell mechanics andthe use of microfluidic systems for the study of the behaviour of thecells. Traditional experiments on population of cells that are carriedout by researchers with standard bulk methods usually show hetero-geneity in the results, even though the tests have been performed inidentical environments [3]. This effect is due to the fact that the be-haviour of the cells is different in the case of bulk techniques and single-cell studies. Using the latter type of technique, some specific behavioursof the cells may not be captured, because the experimental ensembleaverage across the population can obscure an important subset withinthe data [31]. Nanoscale devices can be exploited for single-cell studiesas they provide the opportunity of performing in vitro analysis, suchas low-volume sampling, even with the integration of other tools, ifneeded. An example of an integrated microfluidic system for single cellstudies is shown in Figure 1.3.

However, due to the complexity of the human biological system and ofnanomedicines, many nanomedical devices that have been realized in the lastdecade are not yet employed in daily clinical use and their clinical integra-tion is still very challenging. For example, in order to design the process ofa nanomolecule for targeted delivery, it is necessary to study the complexinteractions that take place in the specific biological system. For the correctcharacterization of many tissue models, it is required to include the exchangeof metabolites and the vascular perfusion, which still represent a limitationfor the development. Indeed, some of these steps are currently based ontrial and error. Using microfluidic systems for single-cell studies, it is stillnot possible to perform time-dependent analysis and the sensitivity of thedetection scheme needs for improvement.

Therefore, there is the pressing need to examine in depth the intriguingpossibilities given by the nanomedical technology. By studying them in adifferent light, it may lead to significant guidelines for the solution of thecurrent problems.

5


Figure 1.3: Integrated microfluidic system for single-cell studies. The sistemwould be made of several modules for multi-functions including single-cellmanipulation, isolation, sampling and analysis in an automated and high-throughput manner. Figure from [5].

6


1.1.2 Mathematical models for nanomedicine

All the tools and techniques illustrated so far derive from experimental analy-ses and results. The experiments supply a wide understanding of the physicsof the problems and the potential of the materials used, however in vitro stud-ies are expensive in terms of availability, time and cost. Moreover, as alreadyseen, they not always provide any complete and feasible results. Therefore,mathematical and computational models can address the need and they canbe powerful tools for the future progresses.

Given a physical problem, theoretical scientists may develop a mathe-matical model which describes the problem in a rigorous way and gives aquantitative understanding of its physical behaviour. Some reasonable hy-potesis should be considered and verified in the modelling process. Theymay elaborate some numerical techniques for the discretization of the prob-lem and, togheter with some computational scientists, they may provide someresults as output of the numerical simulations.

Numerical tools should be used in order to lighten the costs associatedwith the physical experiments, especially in terms of availability of resourcesand repeatability of the experiments. Using numerical simulations, theoreti-cal and computational scientists aim at supplying some results which shouldbe useful for the physical experiments. Indeed, the numerical results shouldlimit the range of needed conditions and observations. This is the standardmethod for code validation: simulations are carried out using data for whicha prior knowledge of the result is available as input. The same routine isperformed using all the possible data for which the outputs are accessible. Ifthe numerical results and the physical ones are in some sense similar, then itis possible to state with certainty that the numerical tool is robust. There-fore, if required, in silico experiments can substitute in vitro tests in order toreduce the number of different biological conditions that need to be analysed.

Thanks to the combination of scientific and engineering knowledge, theuse of computational models allows to simulate the effect of a medical treat-ment on an individual patient, leading to the promising field of personalizedmedical therapy, based on their own anatomy and history. Many interest-ing examples that deal with numerical tools for patient-specific treatmentsare available in literature as reported in [5] and [16], in the framework ofboth drug-encapsulated injectable nanoparticles and biomedical devices.

Numerical tools for drug-encapsulated injectable nanoparticles

The delivery of anticancer molecules using nanoparticles is one of the mostwidespread application of nanomedicine, however many challenges that re-

7


Figure 1.4: Technical challenges for theoretical and computational scientistsin nanomedicine. Figure from [16].

quire the intervention of computational studies still need to be addressed(Figure 1.4). Decuzzi in [16] asserts that the first issue that is necessary totackle is the maximization of the loading efficiency, namely the ratio betweenthe mass of encapsulated drug and the total mass, while controlling the re-lease. Simulations for the analysis and optimization are made at molecularlevel by means of Molecular Dynamics, Monte Carlo or other simulations (seefor example [29] and [50]).Moreover, it has been studied that the formation of a protein corona around ananoparticle affects its terapeutic performance [39]. Again, molecular simula-tions are used to model the interaction of blood proteins with the nanoparti-cles in order to predict and control their adsorption and to design the optimalsurface features [18].Lastly, there is the need to maximize the deposit at diseased sites and toavoid non-specific accumulation in organs different from the targeted ones.For this purpose, the description of the journey of nanoparticles within thevascular network can be done by means of different type of analysis, ac-cording to the specific aim, such as continuum mechanics approach [22] andmolecular-based method [52].

An interesting application is about cardiovascular disease, which is themost important cause of death in the USA. Hearth attacks are caused by theblockage of the coronary artery due to the rupture of the so-called vulnerable

8


Figure 1.5: Spatial distribution (cm−2) of different sized particles in an in-flamed arterial tree: (A) dp = 0.1µm, (B) dp = 0.5µm and (C) dp = 2.0µmin terms of nadh/ (ninj × A), where nadh is the number of adhered particles,ninj is the total number of injected particles and A (cm2) is the surfae area.Figure from [26].

plaques that are not always detected by standard imaging techniques. It hasbeen studied that plaque instability is mainly determined by the presence ofplaque inflammation [20] and targeted drug-encapsulated nanoparticles canbe used in the treatment in order to encourage rapid plaque stabilization[9]. For this purpose, in [26] Hossain et al. developed a numerical tool thatwas able to capture essential 3D aspects of the transport and the vasculardeposition of nanoparticles in an inflamed arterial tree (Figure 1.5).

As previously underlined, computational modelling is particularly suit-able for the description and the analysis of cancer treatments. In orderto monitor and predict the abnormal proliferation of mass caused by cancer,numerical tools have been developed for the prediction of the growth of thetumor. Indeed, the most evident consequences of the disease are invasion,metastasis and angiogenesis. Invasion is the capability of the tumor to getinto the tissues, metastasis is its ability to spread to tissues without being di-rectly connected with them and angiogenesis is the unregulated blood vesselsgrowth [21]. The precise description of tumor growth and nutrient evolutioncan be carried out in terms of an extracellular matrix, modelled as a solid,healty cells and tumor cells permeated by several fluids [48]. From the nu-merical point of view, a multi-phase approach (Figure 1.6), as the one justdescribed, does not require some computationally expensive interfaces con-ditions and it is very flexible. Therefore, it is possible to predict the tumorgrowth as function of the nutrient concentration and the initial characteri-zation of the system and it allows to study and monitor the response of a

9


Figure 1.6: The multi-phase system within a Representative Elementary Vol-ume for tumor growth and nutrient evolution. Figure from [5].

Figure 1.7: FSI simulation of a VAD in action. Blood flow velocity during thefill (a) and the eject (b) stages; deformed configuration of the thin structuralmembrane during the fill (c) and the eject (d) stages. Figure from [5].

particular type of tumor to therapies.

Numerical tools for biomedical devices

In the field of design and optimization of biomedical devices, the ventricularassist device (VAD) is an example of biomedical tool which is already clin-ically used. VADs are implanted on a patient in case of heart failure andthey give mechanical support to the heart, in particular to ventricles, fortheir pulsatile aim. There is the need for an improvement in the peadriatricfield, as the devices available for adults are not suitable for children. To thisend, numerical modelling and fluid-structure interaction simulations (Figure1.7) are needed to describe and visualize the behaviour of VAD [7] and someshape-optimization studies for the design of the paediatric device can be per-formed. Other than the fluid-structure interaction models of the device, it is

10


necessary to model the blood coagulation process, since thrombus formationis one of the most important problem in VADs. In this way, it is possibleto understand the source of the problem and to suggest some modificationsin the design. An alternative to the direct modelization of the coagulationprocess that has been exploited in literature, see for example [34] and [35],is the analysis of the regions of blood recirculation and long residence times.Moreover, in order to study the effect of the physiological conditions on thedevice and viceversa, the model should be able to include information andparameters from the physiology of the patient.

1.1.3 Discussions

It is now clear that nanomedicine is a research field where innovative scien-tific discoveries can take place. It is also apparent that nanomedicine needsmathematical modeling, theoretical and computational analyses and simu-lations to rapidly flourish. Indeed, computational modelling is employedin nanomedicine for the optimization of the performances of nanostructures,that is equivalent to the optimization of the efficacy of nanomedicines. More-over, during a series of experiments, mathematical modelling and simulationsallow to limit the required expensive parametric studies which are usuallystrictly dependent on the characteristics of the system. Numerical simula-tions are also less expensive than in vivo tests. Furthermore, computationalstudies provide some hints in the understanding of the physics and mechanicsthat regulate the biological systems. Thanks to the advantages provided bythe modelling, useful guidelines for future improvements in the design pro-cess of devices and in the development of novel experimental techniques arenow available. Theoretical and computational analyses should also help theeffective clinical fruition of nanomedicines, since until now the use of modelsin nanomedicine is not yet widespread.

The most interesting, but also challenging aspect of this new frontierof the scientific knowledge is the need of an interdisciplinary environ-ment. The required coupling among medical, nanotechnological and numer-ical knowledge leads to the essential and stimulating cooperation of scientistscoming from different disciplines. Biologists, chemists, clinicians, pharmacol-ogists, physicists should work side by side with engineers, mathematicians,theoretical and computational scientists, while up to now their knowledgehas been developed in isolation.

11


Figure 1.8: Multiscale nature of a biological system: length and time scales.Figure from [55].

1.2 Multiscale models in biology

1.2.1 Multiscale nature of biological systems

All the biological systems are characterized by a myriad of small elements,typically molecules, that represent the elementary units of any organism.Molecules are assembled in bigger and bigger elements, up to supramulecularstructures, then cells, tissues and organs. The integrated components thatform a living organism interact and mutually depend on each other in orderto play their own role [38]. In Figure 1.8, a sketch of the multiscale structureof the biological system is reported. Biological systems are organized inmultiple scales and each one cannot be considered fully isolate from theothers. Each level of this hierarchical structure has its own time-scale andlength-scale and they are spread over a wide range. Most of the processesthat take place at the largest scale cannot be observed at a smaller scale andviceversa. The events that happen at a given scale can be studied only if theinvestigation scale is the adequate one, nevertheless the effects can be seenat a larger and smaller scale, but under different forms. Moving from thesmallest scale, typically the molecular level, where the elementary blocks areidentifiable, with a larger and larger point of view it is possible to identifyseveral intermediate scales, the so-called mesoscopic levels, up to the largestone, the macroscopic scale that typically is the organism or the organ level.

1.2.2 Single-scale models and upscaling techniques

As previously pointed out, the most effective strategy to understand the be-haviour of the biological systems and to control and predict the effects ofnanotechnologies applied to medicine is the mathematical modelization andsuccessively the computational simulation of the system. Even the mathe-matical model that describes a biological system has to take into account the

12


Figure 1.9: A classification of methods for biological systems in terms of theircharacteristic length- and time-scales. Figure from [55].

intrinsic multiscale structure of the organism. Therefore, to this end, it isnecessary to use several types of methods for the study of the system at dif-ferent time-scales and length-scales. As reported in [55], two main classes ofapproaches can be identified: phenomenological and mechanistic. Theformer approach describes a process using laws based on empirical obser-vations and not from theoretical studies. It also uses lumped parameters,which are usually difficult to understand. In the latter one, a phenomenonis depicted by a set of microstructural models that are integrated exploitingsome functional interdependence rules. A sketch of the two classes of methodis depicted in Figure 1.9 and some examples are listed below:

• Ab initio methods (ABM) work at the atomistic scale. The propertiesof the atoms and the interactions with the surrounding environment donot accept any approximations, therefore they have to be consideredfrom the quantum mechanics point of view [41]. Clearly, this method iscomputationally very expensive when it is applied to problems biggerthan few hundred of atoms.

• The molecular dynamics (MD) approach is based on the generationof the atomic trajectories of a system composed by several particles[2]. The trajectory is obtained by numerical integrating the Newton’slaw. Moreover, the method requires some hypothesis on the interatomicinteractions that need to be given as input, preventing the possibilityto analyse them.

• Coarse grained (CG) technique [56] represents a necessary link betweenthe microscopic and the macroscopic scale. It is similar to the MD

13


method, as it exploits the interactions between single units, but thestructural units are bigger than in MD.

• The discrete models (DM) are based on the description of a singlecell and its interaction with the surrounding environment, which canbe governed by deterministic or probabilistic rules [36]. The position,velocity and internal state of each cell are given as variables and thepossible positions of a cell can be on a regular mesh for the lattice-basedmodel or can be unrestricted for the lattice-free model.

• The continuous models (CM) are characterized by partial differentialequations that describe the evolution in space and time of the variables.A CM is suitable in the case of large scale, since it is based on theassumption of the continuity of the matter [58].

Each one of the single-scale models that have just been detailed is conve-nient for a specific scale. However, the description of a process at differentlevels is not enough, since it is necessary to link the information that de-rives from each single-scale model. This means that a multiscale approachis needed. Indeed, a multiscale approach aims at gathering the models andthe results from different levels and to combine them. Usually, the link-ing between different time-scales and length-scales is performed studying theproblem at a characteristic scale and exporting the results at another scale.Depending on the “direction of study”, the multiscale analysis can be carriedout in a bottom-up way, starting from the lowest scale, where the outputsare used as input conditions for a higher level, or viceversa in a top-downway, where the outputs of a higher scale represent some boundary conditionsfor a lower level [12]. In order to move from one level to another, severaltechniques are available in literature and some examples are summarized in[55]:

• Homogenization techniques are based on the average of the macroscalelaws over the microscale. The method needs some linearity or periodic-ity properties as assumption. For example, the homogenization methodhas been used to determine the macroscopic transport properties of tu-mors in [51] and [43].

• Mixture theory is based on a weighted averaging of the single con-stituents of the matter. An example of application can be found in[4], where the mixture theory has been applied to consider the resid-ual stress caused by the growth and the remodeling of soft biologicaltissues.

14


Figure 1.10: Different scales within the human circulatory system: (a) wholecardiovascular system at the macroscale, (b) artery at the mesoscale and (c)red blood cells at the microscale. Figure from [55].

• Asymptotic expansion method can be used if there is a certain regularityat the microscopic level. It allows to perform an expansion of themacroscopic fields over the microscale. For example, in [46] the effectivediffusivity of the outermost layer of the skin, the stratum corneum, hasbeen obtained through the method of asymptotic expansion.

Therefore, in every biological system, each process at any scale can be anal-ysed provided that a suitable model is considered and that a good approxi-mation technique is used in order to link the scales among them.

1.2.3 Examples of application

In nature, many organisms and many biological systems share an intrinsicmultiscale structure, here a couple of examples taken from [55] are reported.

The cardiovascular system is composed by the heart, blood and bloodvessels and it is divided into the systemic circulation and the pulmunary cir-culation. In the heart, there are four chambers, which are the left atrium,the left ventricle, the right atrium and the right ventricle, each of them occu-pies a volume of a few cubic centimeters (cm3). In the systemic circulation,which provides oxigenated blood to the body, the heart pumps blood fromthe left ventricle to the aorta, the biggest artery (diameter of about 2-3 cm),then the aorta branches in several blood vessels of decreasing size: arteries(cm), arterioles (mm) and arterial capillaries (µm). Since the capillaries are

15


very small, they are able to reach all the peripheral parts of the body andthrough their walls oxigen and the nutrients can extravasate and diffuse inthe intracellular space (nm). At this stage, the deoxigenated blood enters thevenous system in bigger and bigger vessels (venous capillaries, venules andveins) in order to go back to the heart. The deoxigenated blood enters theright atrium flowing in the superior and inferior vena cava. From the rightventricle, the deoxigenated blood is carried by the pulmonary circulationthrough the pulmonary artery (diameter of about 3 cm) to the lungs, whereit is oxigenated and it goes back to the left atrium flowing in the pulmonaryvein. The blood itself shares a multiscale structure, since it contains plasma,red blood cells, proteins, small molecules and ions that are of dimensions ofthe order of few µm. Therefore, it is clear that the human circulatory systeminvolves different characteristic length-scales and time-scales (Figure 1.10),so that a multiscale analysis is needed for a complete description.

A second example is about solid tumors, which are a population of cellsthat abnormally grows and metastasizes in a distant region. The tumor massexploits different properties during its developments, which are related toseveral characteristic scales in length and time. Tissue invasion, migration ofthe cells and creation of metastasis are some peculiar effects of tumor diseaseand they are related to the largest length scale of the human body, tipicallythe organs or the organism level (m). Angiogenesis, which is the unregulatedblood vessels growth, usually occurs at lengths of cm and times of s, while theinteractions among tumor cells and molecules take place at typical scales ofµm and µs [24]. Consequently, also the study and the analysis of the growthof tumor masses require investigations at different time and length scales.

Furthermore, not only the multiscale nature of many biological systemsneeds to be exploited, but it is also necessary to take into account the hierar-chical structure of the system during the investigation of nanotechnologicaldevices or nanoscale medicine. In this sense, a general overview of a multi-scale analysis applied to nanomedicine will be exploited in the next section.In the following chapters, a detailed and complete description of this casewill be reported.

1.2.4 Nanoparticle delivery in a capillary network

In the framework of nanoparticle-based drug delivery system for the treat-ment and early-detection of cancer, an extended multiscale computationalmodel is currently under investigation and in this work we will move in thisdirection. In this regard, the aim of the hierarchical computational modelis the prediction of the vascular and extravascular transport of molecules,

16


Figure 1.11: Multiscale modules for vascular and extravascular transport ofnanoparticles.

nanoconstructs and cells in neoplastic tissues. The complete model (Figure1.11) can be split into several modules at different scales (the tissue module,the vascular module and the extravascular module) in order to better specifyand understand the processes that are involved.

• At a macroscopic level, the tissue module focuses on the temporal andspatial distribution of injected agents within vascular and extravascu-lar compartments of a capillary network. Transport phenomena at thelevel of the microcirculation play a key role in the propagation of thecharacteristic effect of cancer and mass transport is also at the basisof most of the cancer pharmacological treatments. The model endsup to be a system of partial differential equations integrated by meansof a finite element method. The system takes into account a two-waycoupling between the capillary network and the surrounding environ-ment. This connection is exploited because the circulatory system isrepresented by means of a network of one-dimensional channels thatact as a concentrated source of flow immersed into the interstitial vol-ume. Therefore, as a result of the 3D/1D coupling between the networkand the external volume, the tissue module itself shares a multiscalenature. However, a multiscale approach needs to be set up among the

17


modules, as the characterization of some parameters of the model, suchas the vascular adhesion parameter of the nanoparticles to the vesselwall, their diffusivity in the extravascular space or the permeability ofthe vessel walls with respect to some solutes, is still problematic at thetissue level.

• Moving to a smaller scale, in the vascular module the marginationdynamics and vascular adhesion of injected agents are predicted con-sidering the actual blood rheology, transporting a dense suspensionof deformable cells and particles by means of a Lattice Boltzmann-Immersed Boundary method. The adhesion of transported carriers tothe targeted vessel site strongly depends on the particle geometry, onother particle properties and on the flow-related parameters, such asthe red blood cells dynamics. In order to study the near wall dynam-ics of circulating agents and to take into account all these effects, aLattice-Boltzmann methods is used for the description of the fluid dy-namics on a fixed lattice and an Immersed Boundary method is usedfor the presence of moving and deforming bodies.

• At the microscale, the extravascular module deals with the extrava-sation of the nanoconstructs and the following migration toward themalignant tissue. From a numerical point of view, these issues can beapproached with a suitable hybrid version of the Cellular Potts model[47]. The extravascular dynamics is determined by a stochastic mini-mization of a discrete effective energy that contains several terms. Themodel is said to be hybrid in the sense that the nanostructure migra-tion is determined by the kinetics of some environmental chemical vari-ables, by their possible absorption and by the resulting directional cues.Therefore, it is possible to determine the influence in the nanoparticleextravasation and following migration of both external and internal de-terminants: it is possible to analyze dimensions, elasticity and remod-eling ability of the nanoconstructs in order to pass in between capillarywalls, without disruptive consequences for the vasculature. Moreover,with this model it is possible to link the biophysical mesoscopic prop-erties of tumor masses to the kinetics of some biochemical variables,specifically the kinetics of the drugs carried by the nanoparticles.Some interesting studies for the extravascular module can be also per-formed by considering some experiments. More precisely, it is possibleto determine the interstitial properties by means of inverse problems:by knowing part of the solution of a physical problem, we recover whichparameters have generated that particular solution. In the specific case,

18


the diffusion of solutes and nanoparticles in an interstitial matrix isexperimentally observed and some images of the phenomenon are cap-tured. From the numerical point of view, the diffusion is simulated andthe diffusive properties of the solutes or nanoparticles in the matrix arederived by looking for the parameters that minimize the discrepancybetween the experimental and numerical observations.

As can be seen in Figure 1.11, the two-way connections among the mod-ules are very efficient towards the achievement of the final aim, that is thedevelopment of a multiscale integrated approach for a new class of nanocon-structs for more effective cancer detection and treatment. In this respect, thecoupling between the tissue and the extravascular module can be exploitedin one way (from extravascular to tissue) as looking for a tissue equivalentdiffusion. Indeed, the CPM or the control approach can provide a completedescription of the nanoparticle behaviour in the extracellular matrix and itcan be summarized in a single lumped parameter, such as the diffusion of thenanoconstruct in the tissue, that can be easily included in the macroscopicmodule. On the other way, the tissue module can supply some values thatcan be used as initial/boundary conditions in the study of the extravascularspace, such as the value of concentration of chemoattractant. Also the rela-tion between the tissue and the vascular module has two directions: from thetissue to the vascular module and viceversa. On one hand, some concentra-tion values or other biophysical conditions are given as boundary conditionfor the vascular module. On the other hand, the vascular model can provideinformation in the form of lumped parameter to the tissue model, such asthe vascular adhesion parameter.

In this work, we will focus on some particular aspects of the multiscalestructure that has just been detailed. In particular, the work will be based onthree different levels of the multiscale system and the final aim is to specifythe general model of microcirculation in the case of transport of nanoparticle,exploiting the hierarchical structure. Here a general overview of the modelat the adequate scales is proposed, then their complete description will beprovided in the following chapters.

Macroscale: microcirculation and drug delivery

At the macroscopic scale, the mathematical model for microcirculation anddrug delivery is analysed. The model has the ability to capture many phe-nomena that take place in the delivery of drugs and that represent a possiblebarrier to the final aim of reaching a target site, such as blood perfusion,particle transport and interaction with the tissue. The mathematical model

19


is able to combine many important aspects and mechanisms for nano-basedtreatments, such as realistic vasculature, coupled capillary and interstitialflow, coupled capillary and interstitial nanoparticle transport and coupledcapillary and interstitial drug transport. Blood flow and mass transport isdescribed by means of advection-diffusion equations in one dimensional chan-nels, immersed in a 3D space that represents the interstitial tissue. The mainadvantage of this approach is that the computational grids needed for the ap-proximation of the equations on the microvasculature and on the interstitialvolume are completely independent. Therefore, the geometry of the capillarynetwork can be arbitrarily complex and this effect will not affect too muchthe global computational cost. This work will be specifically addressed inChapter 2.

Microscale: adhesion of a nanoparticle in microvessels

At the microscale, in the analysis of the vascular module, the study of thevascular adhesion parameter of nanoparticles is exploited. The model is basedon a Lattice Boltzmann-Immersed Boundary method, through which it ispossible to analyse the effect of size, shape, surface properties and mechanicalstiffness of a nanoparticle on some parameters related to the marginationdynamics, such as the probability of adhesion to the vessel wall. The particlesare treated as Lagrangian solid domains immersed in the fluid. Fluid andstructures interaction is considered, imposing no-slip boundary condition onthe surface of the solids and evaluating the hydrodynamic and interactionforces on the moving bodies. The idea is to build a map for the vascularadhesion parameter as function of physical properties of the nanoparticlesand then use it in the macroscopic model. The results of this work will beshown in Sections 4.1, 4.2 and 4.3.

Microscale: diffusion of drug in the extracellular matrix

Again at the microscopic scale, considering the extravascular computationalmodule, a characterization of the interaction of delivered drugs with the in-terstitial tissue is performed by means of a predictive numerical model forsome parameters. Many parameters can be estimated with this model, in par-ticular we will focus on the diffusivity coefficient of the extracellular matrixwith respect to the chemical species. For the computation of the parameters,it is possible to take advantage of some available experiments in which thetransport of molecules and nanoparticles is studied in a collagen matrix. Theprediction of the parameters is made by means of a control problem in whichthe discrepancy between the particle concentration in the numerical and the

20


physical experiment is minimized, provided that suitable diffusion equationsare satisfied in the numerical device. The idea is to provide the optimalvalue of the diffusivity coefficient of the extracellular matrix with respect tothe chemical species from specific numerical experiments and, subsequently,include these results as lumped parameters in the macroscopic model. Theresult of this work will be presented in Section 4.4.

21

Chapter 2

Mathematical model forparticle transport in themicrovasculature

The aim of this work is to provide tools for the investigation of a drug deliverysystem in the microcirculation based on injectable nanoparticles. The intentof the forthcoming study is to develop a model able to capture the distri-bution of nanoparticles injected in the microcirculation, accounting for theirtransport, their adhesion to the capillary wall and material extravasationfrom the capillary walls.

In Section 2.1 we state the problem in the microvasculature immersed inthe interstitial tissue as a system of PDEs. Then, in Section 2.2 we exploitthe embedded multiscale approach to couple the two problems leaving onseparate scales, while the dimensionless formulation of the coupled problemis derived in Section 2.3. In Section 2.4 a set of suitable boundary andinitial conditions for both the tissue and vessel variables is specified, whilean overview of compatibility conditions at junctions between multiple vesselsis given in Section 2.5. In order to approximate the coupled differentialproblem with the mixed finite element method, firstly, a dual mixed weakformulation of both the vessel and tissue problem is provided in Section 2.6and successively, an alternative primal mixed weak formulation is derivedin Section 2.7. The latter formulation is the foundation for the upcominganalysis as well as for the numerical approximation strategy presented inSection 2.8 and the corresponding algebraic formulation in Section 2.9.

22

CHAPTER 2. MATHEMATICAL MODEL FOR PARTICLETRANSPORT IN THE MICROVASCULATURE

Figure 2.1: On the left, microvasculature within a tissue interstitium; in thecenter, the interstitial tissue slab with one embedded capillary; on the right,the reduction from 3D to 1D description of the capillary vessel.

2.1 Three-dimensional model for

microvasculature within a tissue

We define a mathematical model for mass transport in a permeable biologicaltissue perfused by a vessel network. The domain Ω ∈ R3 where the modelis defined is composed by two parts, Ωt and Ωv, representing the interstitialvolume (tissue) and the capillary bed (vessel), respectively. Assuming thatthe capillary vessels can be described as a set of cylinders, we denote with Γthe outer surface of Ωv and with Λ the one-dimensional line that describes thecenterline of the vessel. In general, the vessel radius R can change along thearc length of Λ (see Figure 2.1). In particular, the forthcoming model aimsat describing the transport of nanoparticles that are injected in the capillarynetwork and their behaviour in the blood stream and in the interstitial tis-sue. The physical quantities of interest are the concentration of transportednanoparticles in the capillary network and in the tissue interstitium, cv andct, respectively. The variables have to be intended as function of space, beingx ∈ Ω the spatial coordinates, and time t.

Concerning the interstitial volume Ωt, it can be considered as an isotropicporous medium with velocity field ut and pressure pt. We assume that theparticles are advected by the fluid and diffuse in all Ωt. An important effectfor tissue perfusion is the lymphatic drainage. Excess of fluid extravasatedfrom the blood circulation is drained by lymphatic vessels and returns tothe blood stream. Since a geometrical description of the lymphatic vesselsis not available, following [53], the lymphatic drainage is modeled as a sinkterm for the interstitial flow. Therefore, the distribution of particles in theinterstitial tissue is also affected by the lymphatic drainage. Accounting forthese effects, the equation for mass transport in the interstitium is

∂ct∂t

+∇ · (ctut −Dt∇ct) + LLFpS

V(pt − pL) ct = 0 in Ωt × (0, T ), (2.1)

23


where Dt is the particle diffusivity in the interstitium, assumed to be con-stant, LLFp is the hydraulic conductivity of the lymphatic wall, S

Vis the surface

area of lymphatic vessels per unit volume of tissue and pL is the hydrostaticpressure within the lymphatic channels.

For the blood capillary, given the flow velocity vector uv and the pressurefield pv, the particles are advected by the fluid and diffuse in Ωv, similarlyto the previous case. Moreover, a peculiarity of the injected particles is theirability to adhere to the vessel wall. This effect is due to the presence ofligand molecules on the surface of the particles and of receptor molecules onthe endothelial layer. The adhesion of particles to the wall is described as asink term along the capillary network. The model accounting for the particletransport in the blood stream and their adhesion to the wall results in thefollowing equation:

∂cv∂t

+∇ · (cvuv −Dv∇cv) + Πeffcv = 0 in Ωv × (0, T ), (2.2)

where Dv is the particle diffusivity in the capillary network, assumed to beconstant in Ωv, Πeff is the effective vascular adhesion parameter which coulddepend on time.

In order to couple the two problems, it is necessary to impose some condi-tions at the interface Γ = ∂Ωv ∩ ∂Ωt. In particular, we describe the capillarywall as a semipermeable membrane allowing for the leakage of the fluid andfor selective filtration of particles. A good model for mass transport acrosssemi-permeable membranes is the Kedem-Katchalsky equation. Accordingto this equation, the flux of particles per unit surface across the capillarywalls is

(cvuv −Dv∇cv) · n = (ctut −Dt∇ct) · n = (2.3)

= (1− σ)Lp [(pv − pt)− σ (πv − πt)] cavg + P (cv − ct) on Γ× (0, T ) .

In particular, Lp is the hydraulic conductivity of the vessel wall and πv − πtis the difference in osmotic pressure, where π = RgTc is the osmotic pressuregiven by a concentration c of a given solvent, Rg is the universal gas constantand T is the absolute temperature. Indeed, because of osmosis, the pressuredrop across the capillary wall is affected by the difference in concentrationof the substances dissolved in blood. However, only the large molecules caninduce a significant effect, for this reason we only consider the presence ofproteins, therefore from now on πv and πt will represent the concentration ofproteins in the vessel network and in the tissue, respectively. The reflectioncoefficient σ quantifies how different a semi-permeable membrane is fromthe ideal permeability. The term accountig for the pressure difference is

24


multiplied by the average concentration within the capillary vessel

cavg := wct + (1− w) cv

and it is defined as a suitable combination of cv and ct. In particular, 0 < w <1 is a weight that depends on the Peclet number of the particle transportthrough the wall, however, for the sake of simplicity, we will use w = 1

2.

Moreover, the flux depends on the concentration gradient across the capillarywalls in terms of P , the permeability of the vessel wall with respect to theparticle.

Therefore, these modelling assumptions lead to the following mass trans-port problem in the entire domain Ω:

∂ct∂t

+∇ · (ctut −Dt∇ct) + LLFpSV

(pt − pL) ct = 0 in Ωt × (0, T ),∂cv∂t

+∇ · (cvuv −Dv∇cv) + Πeffcv = 0 in Ωv × (0, T ),

(cvuv −Dv∇cv) · n = (ctut −Dt∇ct) · n =

= (1− σ)Lp [(pv − pt)− σ (πv − πt)] cavg + P (cv − ct) on Γ× (0, T ).

(2.4)Denoting with f the flux per unit area released by the surface Γ, the fluxcontinuity between the capillary network and the tissue is guaranteed bymeans of

(ctut −Dt∇ct) · n = f (ct, cv) on Γ× (0, T ).

In our case, f (ct, cv) = (1− σ)Lp [(pv − pt)− σ (πv − πt)] cavg + P (cv − ct).

To ensure the uniqueness of the solution of problem (2.4), it is necessaryto provide some initial conditions and some suitable boundary conditionson ∂Ωv and ∂Ωt. The prescription of these conditions significantly dependson the particular features of the problem, as well as on the available data,therefore the discussion will be postponed to Section 2.4.

We observe that the expressions for the velocity and pressure fields ut,uv, pt and pv have not been detailed yet. Indeed, a complete description ofthe governing equations for the fluid-dynamical variables of problem (2.4) isgiven in Appendix A.

2.2 Model reduction: coupled 3D-1D

problem

The fully three-dimensional model (2.4) is able to capture the phenomenawe are interested in. However, in order to face some technical difficulties

25


that arise in the numerical approximation of the coupling between a complexnetwork with the surrounding volume, a multiscale approach [15, 14, 13]based on the Immersed Boundary Method (IBM) [59, 33] can be exploited.

To avoid resolving the complex 3D geometry of the vascular network, thecombination of the IBM and the assumption of large aspect ratio betweenvessel radius and capillary axial length can be convenient. Precisely, withthis approach, a suitable rescaling of the equations is applied and the cap-illary radius is let going to zero (R→ 0) (Figure 2.1). In this way, the 3Ddescription of the vessels is reduced to a simplified 1D representation andthe immersed interface and the related interface conditions are replaced byan equivalent mass source.

2.2.1 Coupling term for the interstitial volume

Thanks to the IBM, the action of f on Γ can be represented as an equivalentsource term F distributed on the entire domain Ω:

F (ct, cv) = f (ct, cv) δΓ.

More precisely, F is the Dirac measure concentrated on Γ, having density f ,defined by:∫

Ωt

F (ct, cv) v dΩ =

∫Γ

f (ct, cv) v dσ ∀v ∈ C∞ (Ωt) .

As specified in [10, 15], when R → 0 the mass flux per unit area canbe replaced by an equivalent mass flux per unit length, distributed on thecenterline Λ. Let γ (s) be the intersection of Γ with a plane orthogonal to Λ,located at s and denote by (s, θ) the local axial and angular coordinates onthe cylindrical surface generated by Γ with radius R. It is possible to applythe mean value theorem in order to represent the action of F on v by meansof an integral with respect to the arc length on Λ. To be precise, there existsθ ∈ [0, 2π] such that∫

Ωt

F (ct, cv) v dΩ =

∫Λ

∫γ(s)

f (ct (s, θ) , cv (s, θ)) v (s, θ)Rdθ ds = (2.5)

=

∫Λ

|γ (s)| f(ct

(s, θ), cv

(s, θ))

v(s, θ)ds, ∀v ∈ C∞(Ωt).

Then, assuming that the capillaries are narrow with respect to the character-istic dimension of the surrounding volume, namely, R |Ωt|1/d, with d = 3

26


the space dimension of the model and assuming that f is a linear function,it is possible to have

limR→0

f

(ct

(s, θ)∣∣∣

γ(s), cv

(s, θ)∣∣∣

γ(s)

)= f (ct (s) , cv (s)) , (2.6)

h (s) :=1

|γ (s)|

∫γ(s)

h (s, θ)Rdθ,

where the bar operator is defined as the averaging operator on the circle ofradius R laying on the cylindrical surface Γ and normal to the line Λ. Inconclusion, combining (2.5) with (2.6), we recover the following expressionfor the distributed source term:∫

Ωt

F (ct, cv) v dΩ '∫

Λ

|γ (s)| f (ct (s) , cv (s)) v (s) ds, ∀v ∈ C∞(Ωt). (2.7)

2.2.2 Model reduction for microvascular flow

In order to exploit the one-dimensional representation of the vessel, the gen-eral cylindrical coordinate system (R, θ, s) aligned with the centerline Λ canbe reduced using uniquely the arc length s and the tangent unit vector λthat accounts for an arbitrary orientation. To this purpose, it is possibleto assume the concentration cv to be constant on a section orthogonal to Λlocated at s. Therefore, the derivative with respect to the variables r and θare negligible and an average of the quantities of interest on the section canbe performed. For this reason, the pointwise and average concentrations inthe channel coincide, namely cv (s, θ, r) = cv (s). Thanks to the tangent unitvector, differentiation is defined as ∂s := ∇ · λ.

Thanks to (2.7), the one dimensional representation of equation (2.4)(b)with the coupling forcing term f reads:

∂cv∂t

+∂

∂s

(cvuv · λ−Dv

∂cv∂s

)+

2πR

πR2Πeffcv = −2πR

πR2f (cv, ct) , on Λ× (0, T ).

(2.8)Notice that we have introduced the adhesion term as a flux per unit length(2πRΠeffcv

)and the forcing term is again multiplied by 2πR. Successively,

we had to scale the obtained flux per unit length with the cross section,πR2, since the nanoparticle concentration inside the vessel cv is measured asnumber of particles per unit volume [#/m3].

Moreover, as a consequence of the geometric assumptions, the vessel ve-locity has a fixed direction, i.e. uv = uv λ, so the vessel problem could beformulated with the scalar variable uv only.

27


2.2.3 Governing equations for the coupled problem

At this stage, thanks to (2.7), equation (2.4)(a) can be expressed with theequivalent mass source term f . It is now possible to reformulate the flowproblem (2.4) in differential form in terms of coupled equations in a three-dimensional space for the interstitial tissue and in a one-dimensional spacefor the capillary network.

The coupled problem for the transport of particles from the microvascu-lature to the interstitium consists to find the concentrations ct and cv suchthat∂ct∂t

+∇ · (ctut −Dt∇ct) + LLFpSV

(pt − pL) ct = 2πRf (cv, ct) δΛ on Ω× (0, T )∂cv∂t

+ ∂∂s

(cvuv −Dv

∂cv∂s

)+ 2πR

πR2 Πeffcv = −2πRπR2 f (cv, ct) on Λ× (0, T )

(2.9)

where

f (cv, ct) =

(1− σ)Lp [(pv − pt)− σ (πv − πt)]

(1

2cv +

1

2ct

)+ P (cv − ct)

and we also recall the average operator

gt (s) =1

2πR

∫ 2π

0

gt (s, θ)Rdθ. (2.10)

Notice that the distinction between the subregion Ωt and the entire domainΩ is no longer meaningful, since the one-dimensional Λ has zero measure inR3, therefore, for notational convienience, from now on Ωt will be identifiedwith Ω and Ωv with Λ.

2.3 Dimensional analysis

The first step toward the application of the model is to perform a dimensionalanalysis of equations (2.9), in order to highlight the relative magnitude ofall the phenomena that affect mass transport, such as diffusion, convection,ligand-receptor interactions and lymphatic effects. For this purpose, the pri-mary variables that have been chosen for the analysis are length, velocity,pressure and concentration. We choose the average spacing between capil-lary vessels, d, as characteristic value for the length, the average velocity inthe capillary bed, U , for the velocity, the typical pressure drop along theextrema of the vessel network, δP , for the pressure and the maximal admis-sible value of concentration at the systemic level, C, for the concentration.Correspondingly, the dimensionless groups that characterize the equationsare:

28


• R′ = Rd

the non dimensional radius,

• Av = Dv

Udthe ratio of diffusion and transport in the vessel network,

• At = Dt

Udthe ratio of diffusion and transport in the tissue interstitium,

• M = Πeff

Uthe magnitude of vascular deposition,

• Q = LpδP

Uthe hydraulic conductivity of the capillary walls,

• QLF = LLFpSVdδPU

the non dimensional lymphatic drainage,

• Υ = PU

the magnitude of leakage from the capillary bed.

Therefore, the dimensionless form of (2.9) is∂ct∂t

+∇ · (ctut − At∇ct) +QLF (pt − pL) ct = 2πR′fadim (cv, ct) δΛ on Ω× (0, T )∂cv∂t

+ ∂∂s

(cvuv − Av ∂cv∂s

)+ 2πR′

πR′2Mcv = −2πR′

πR′2fadim (cv, ct) on Λ× (0, T ),

(2.11)

where

fadim(cv, ct) = (1− σ)Q [(pv − pt)− σ (πv − πt)](

1

2ct +

1

2cv

)+ Υ(cv − ct).

For the sake of simplicity, the same symbols for the standard and dimension-less variables and for the standard and dimensionless operators have beenused. Notice that the dimensionality of δΛ is [length]−2 and that the dimen-sionless Dirac distribution is again called δΛ.

2.4 Boundary and initial conditions

As previously mentioned, in order to guarantee the uniqueness of the solu-tion of problem (2.11), it is necessary to specify some boundary conditions(BCs) on both the tissue and vessel boundary, that is ∂Ω and ∂Λ, respec-tively. Generally speaking, the prescription of adequate boundary conditionsis a consequence of the variational formulation of the problem. The sameoccurs for the BCs for problem (2.11), indeed the choice will be driven bythe particular integration by parts that will be performed or, alternatively,by the need to model some special behaviour of the problem unknowns onthe boundary. However, the choice of the BCs also depends on the availabledata.

29


Concerning the capillary flow, we refer to the collection of inflow andoutflow tips of the vessel network as ∂Λ ≡ Λin∪Λout, i.e. non junction pointswhere the tangent unit vector is inward-pointing and outward-pointing. Onthe inflow boundary of the network, a given nanoparticle concentration cinj isinjected in the blood stream, while on the outflow boundary a homogeneousNeumann boundary condition is enforced, letting the particles to freely leavethe system, namely:

cv = cinj on Λin × (0, T ), (2.12)

∂cv∂s

= 0 on Λout × (0, T ). (2.13)

We require cinj ∈ H1/2 (Λ) .For the interstitial volume Ω, we enforce on all the artificial interfaces

of the tissue, ∂Ω, boundary conditions that mimic the resistance of the sur-rounding material. In particular, a fixed value for the normal diffusive fluxwill be imposed, namely:

− At∇ct · n = βcct on ∂Ω× (0, T ), (2.14)

where βc quantifies the conductivity of the outer tissue with respect to theparticle transport.

We observe that the transport equations depend on time, therefore someinitial conditions are needed, in particular the system is assumed to be freeof particles at time zero, namely

ct(t = 0) = 0 in Ω,

cv(t = 0) = 0 in Λ.

2.5 Junction treatment

The enforcement of boundary conditions is necessary, but not sufficient toclose the problem (2.11). Indeed, the domain splitting approach requiresthe imposition of suitable compatibility conditions at the branching points(junctions) of the vessel tree. The aim of this section is to propose a suitableset of compatibility conditions at junction points.

Firstly, we assume that the concentration in the vessel is continuousin all Λ and automatically at the junction points. Typically, the continu-ity of cv is guaranteed thanks to the choice of the functional framework(cv ∈ H1 (Λ) ⊂ C0

(Λ))

. In order to guarantee the conservation of mass, weneed to ensure the continuity of the flux at the junctions. Let us introduce

30


Figure 2.2: On the left, a simple network made by a single Y-shaped bifur-cation, arrows show the flow orientation. On the right, the domain has beensplit into branches with inflow and outflow variables at the junction point.

this condition in a simple Y-shaped network (see Figure 2.2). The continuityof flux φ at the junction point xJ is equivalent to require that the inflowflux φ+

1 is equal to the sum of the outflow fluxes φ−2 + φ−3 . Explicitly, theconvective and diffusive flux is φ = uvcv − Av ∂cv∂s . However, it is interestingto observe that the convective and diffusive fluxes are continuous also con-sidering them separately. Indeed, for the convective term, the conservationof mass (equivalently of the velocity) and the continuity of the concentrationlead to u+

v,1c+v,1 = u−v,2c

−v,2 + u−v,3c

−v,3. Consequently, the conservation of the

diffusive flux holds.

2.6 Weak formulation

For complex geometrical configurations, explicit solutions of problem (2.11)are not available. The only way to apply such a model to real cases is torefer to a more general variational framework and consequently to numericalsimulations. In particular, in this section we will propose a dual mixed weakformulation of both the tissue and vessel problems.

2.6.1 Weak formulation for the tissue problem

To obtain a variational formulation of the particle transport problem in theinterstitial tissue, the test space for the concentration is

Bt := H1 (Ω) .

31


Let us proceed multiplying (2.11)(a) with a sufficiently smooth function btand integrating over Ω:∫

Ω

∂ct∂tbt dΩ +

∫Ω

∇ · (ctut − At∇ct) bt dΩ +

∫Ω

QLF (pt − pL) ctbt dΩ =

=

∫Ω

2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)]1

2−Υ

δΛctbt dΩ+

+

∫Ω

2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)]1

2+ Υ

δΛcvbt dΩ.

By applying the Green’s theorem and the boundary condition (2.14) on thediffusive term and by re-writing the advection term in non conservative form,the formulation becomes:∫

Ω

∂ct∂tbt dΩ +

∫Ω

ut · ∇ctbt dΩ +

∫Ω

∇ · utctbt dΩ +

∫Ω

At∇ct · ∇bt dΩ+

+

∫∂Ω

βcctbt dσ +

∫Ω

QLF (pt − pL) ctbt dΩ =

=

∫Ω

2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)]1

2−Υ

δΛctbt dΩ+

+

∫Ω

2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)]1

2+ Υ

δΛcvbt dΩ, (2.15)

where the divergence of the velocity field can be re-written in terms of thepressure field, according to the flow equations.

2.6.2 Weak formulation for the vessel problem

In order to derive a variational formulation of the particle transport problemin the capillary network, the test space for the vessel concentration is

Bv := H1 (Λ) .

We approach the weak formulation in a standard way, by means of multiply-ing equation (2.11)(b) by a test function bv and integrating over Λ:∫

Λ

∂cv∂tbv ds+

∫Λ

∂

∂s

(uvcv − Av

∂cv∂s

)bv ds+

∫Λ

2

R′Mcvbv ds+

+

∫Λ

2

R′

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2+ Υ

cvbv ds =

=−∫

Λ

2

R′

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2−Υ

ctbv ds.

32


At this point, it is not possible to simply integrate by parts, because of thepresence of multiple junctions. To tackle this issue, we proceed as follows.

Let us consider the second integral and let us analyze separately the con-vective and the diffusive term. For the convective term, we do not integrateby parts and we adopt a non conservative formulation, namely:∫

Λ

∂

∂s(uvcv) bv ds =

∫Λ

uv∂cv∂s

bv ds+

∫Λ

∂uv∂s

cvbv ds,

where ∂uv∂s

can be written in terms of the pressure field, according to the fluiddynamics of the system. In order to analyze the diffusive term, let us assumethat the vessel radius is constant over each branch Λi, we split the integraland we apply Green’s formula over each segment:∫

Λ

∂

∂s

(−Av

∂cv∂s

)bv ds =

N∑i=1

∫Λi

∂

∂s

(−Av

∂cv∂s

)bv ds =

=N∑i=1

∫Λi

Av∂cv∂s

∂bv∂s

ds−[Av∂cv∂s

bv

]Λ+i

Λ−i

,

where Λ−i and Λ+i are the inflow and outflow extrema of Λi, according to the

orientation λi. Exploiting the continuity of the flux described in section 2.5,we observe that the summation of the diffusive flux for each junction pointcancels out and only the terms at the outer inflow and outflow nodes remain,namely,

N∑i=1

∫Λi

Av∂cv∂s

∂bv∂s

ds−[Av∂cv∂s

bv

]Λ+i

Λ−i

=

∫Λ

Av∂cv∂s

∂bv∂s

ds−[Av∂cv∂s

bv

]Λout

Λin

.

The enforcement of boundary conditions (2.12)-(2.13) is then needed,

that means to make the term[Av

∂cv∂sbv]Λout

Λin explicit. Focusing on Λout, (2.13)implies that

Av∂cv∂s

bv

∣∣∣∣Λout

= 0.

Concerning Λin, in order to enforce non homogeneous Dirichlet boundaryconditions, we should refer to an auxiliary variable and an auxiliary prob-lem with homogeneous Dirichlet BC. For this purpose, we should look for afunction

Rin ∈ H1 (Λ) s.t. Rin|Λin = cinj,

in such a way that the new variable c∗v = cv − Rin ∈ Bv. Thanks to theassumption cinj ∈ H1/2 (Λ) and the trace theorem, Rin exists. Therefore, the

33


boundary term on Λin reads

Av∂cv∂s

bv

∣∣∣∣Λin

= 0.

For the sake of simplicity, we will introduce the new variable c∗v only in theimplementation of the problem.

Globally, combining the previous results, the weak formulation for thevessel problem is∫

Λ

∂cv∂tbv ds+

∫Λ

uv∂cv∂s

bv ds+

∫Λ

∂uv∂s

cvbv ds+

+

∫Λ

2

R′Mcvbv ds+

∫Λ

Av∂cv∂s

∂bv∂s

ds+

+

∫Λ

2

R′

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2+ Υ

cvbv ds =

= −∫

Λ

2

R′

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2−Υ

ctbv ds. (2.16)

2.6.3 Coupled weak formulation

Globally, combining (2.15) with (2.16), the weak formulation for the coupledproblem (2.11) accounting for the particle transport in the vessel networkand in the tissue interstitium is:

to find ct ∈ Bt × (0, T ) and cv ∈ Bv × (0, T ) such that

(∂ct∂t, bt)

Ω+ (ut · ∇ct, bt)Ω + (∇ · utct, bt)Ω + (At∇ct,∇bt)Ω

+(QLF (pt − pL) ct, bt

)Ω

+ (βcct, bt)∂Ω +

+(2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2−Υ

ct, bt

)Λ

+

+(2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2+ Υ

cv, bt

)Λ

= 0 ∀bt ∈ Bt,(∂cv∂t, bv)

Λ+(uv

∂cv∂s, bv)

Λ+(∂uv∂scv, bv

)Λ

+

+(Av

∂cv∂s, ∂bv∂s

)Λ

+(

2R′Mcv, bv

)Λ

+

+(

2R′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2+ Υ

cv, bv

)Λ

+

+(

2R′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2−Υ

ct, bv

)Λ

= 0 ∀bv ∈ Bv,

(2.17)

with ct(t = 0) = 0 and cv(t = 0) = 0.

2.6.4 Well posedness

Existence, uniqueness and regularity of the solution of problem (2.11) havebeen addressed for a similar problem setting in [15]. It is a highly technical

34


3D

1D meanvalue

3D

1D3D

3D

Original model (3D-3D) Reduced model (3D-1D) New reduced model (3D-1D)

Figure 2.3: On the left, the original 3D-3D model for the representation ofthe capillary and the interstitial tissue. In the center, the previous 3D-1Dcoupling with the 1D source concentrated on the centerline of the vessel,Λ. On the right, the new alternative 3D-1D coupling with the 1D sourceaveraged on the surface of the capillary, Γ.

analysis because of the presence of the term δΛ. Hovewer, it is possible tocircumvent the problem by means of the reformulation of the weak problemin such a way that the delta function is no more needed. The details areshown in the next sections.

2.7 Alternative well-posed weak formulation

The main issues in the analysis of problem (2.11) and the correspondingweak formulation (2.17) consist in the lack of regularity of the solution dueto the presence of the Dirac density function concentrated over Λ. In order torecover some results on existence, uniqueness and regularity of the solution,we change the perspective and we derive an alternative weak formulationof the problem whose well-posedness analysis can be achieved within thecontext of the standard Lax-Milgram theory.

The idea of the alternative weak formulation is sketched in Figure 2.3.The 1D source is no more considered as a concentrated source on the cen-terline of the vessel, Λ, but as an averaged source on a virtual interface,which could be the real surface of the capillary, Γ. However, even with thisformulation, the 3D-1D coupling remains valid.

35


2.7.1 Alternative 3D-1D coupling

As we have already underlined, we aim at expressing the interface conditionwithout the use of the delta function on Λ, in order to guarantee some regu-larity estimates. This target is reached by means of exploiting the couplingat the variational level. To this purpose, we refer to the mass transportproblem in the tissue Ωt in dimensionless form, in particular:

∂ct∂t

+∇ · (ctut − At∇ct) +QLF (pt − pL) ct = 0 in Ωt × (0, T ),

(2.18)

− At∇ct · n = βcct on ∂Ωt × (0, T ),(2.19)

− (ctut − At∇ct) · n = (2.20)

= (1− σ)Q [(pv − pt)− σ (πv − πt)] cavg + Υ (cv − ct) on Γ× (0, T ).

On the other hand, the mass transport in the one-dimensional capillary net-work reads

∂cv∂t

+∂

∂s

(uvcv − Av

∂cv∂s

)+

2πR′

πR′2Mcv = −2πR′

πR′2fadim (cv, ct) on Λ× (0, T ),

(2.21)

recalling that

fadim (cv, ct) = (1− σ)Q [(pv − pt)− σ (πv − πt)](

1

2ct +

1

2cv

)+ Υ (cv − ct) .

In the remainder, we will derive a weak form of the coupled problem (2.18)-(2.21), enforcing the interface condition directly in the variational formula-tion.

2.7.2 Alternative weak formulation for the tissueproblem

The natural functional space for the particle concentration in the interstitialtissue is

Bt := H1 (Ωt) .

We multiply equation (2.18) by a test function bt ∈ Bt and integrate overΩt :∫

Ωt

∂ct∂tbt dΩ +

∫Ωt

∇ · (ctut − At∇ct) bt dΩ +

∫Ωt

QLF (pt − pL) ctbt dΩ = 0.

36


Then, we integrate by parts and we use (2.19)-(2.20) as follows:∫Ωt

∇ · (ctut − At∇ct) bt dΩ =

= −∫

Ωt

(ctut − At∇ct) · ∇bt dΩ +

∫∂Ωt∪Γ

(ctut − At∇ct) · n bt dσ =

= −∫

Ωt


∫∂Ωt

ctut · n bt dσ +

∫∂Ωt

βcctbt dσ+

−∫

Γ

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2−Υ

ctbt dσ+

−∫

Γ

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2+ Υ

cvbt dσ.

Now, we split the functions ct and bt in their mean values plus fluctuations,i.e.

ct = ct + ct and bt = bt + bt,

where, by definition, ¯bt = ¯ct = 0. Using the cylindrical coordinates system(s, θ) on Γ, we obtain that∫

Γ

Etctbt dσ =

∫Λ

Et

∫ 2π

0

(ct + ct)(bt + bt

)R′ dθds =

=

∫Λ

2πR′Etctbt ds+

∫Λ

Et

∫ 2π

0

ctbtR′ dθds∫

Γ

Evcvbt dσ =

∫Λ

Ev

∫ 2π

0

cv

(bt + bt

)R′ dθds =

∫Λ

2πR′Evcv bt ds

where

Et :=

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2−Υ

,

Ev :=

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2+ Υ

.

Then, we make the following modelling assumptions:

1. Small residuals ∫ 2π

0

ctbtR′ dθ ' 0. (2.22)

2. Shrinkage of the capillary vessel∫Ωt

dΩ '∫

Ω

dΩ and

∫∂Ωt

dσ '∫∂Ω

dσ. (2.23)

37


Assembling all the terms, we obtain the following expression:∫Ω

∂ct∂tbt dΩ−

∫Ω


∫Ω

QLF (pt − pL) ctbt dΩ+

+

∫∂Ω

ctut · n bt dσ +

∫∂Ω

βcctbt dσ+

−∫

Λ

2πR′Etctbt ds−∫

Λ

2πR′Evcv bt ds = 0.

For the sake of coherence, we would like to re-write the advection term usinga non conservative formulation, as we have done in the previous version ofthe weak form. To this end, it is possible to integrate back by parts theadvection term, namely:

−∫

Ω

ctut · ∇bt dΩ =

∫Ω

∇ · (ctut) bt dΩ−∫∂Ω

ut · n ctbt dσ =

=

∫Ω

ut · ∇ctbt dΩ +

∫Ω

∇ · utctbt dΩ−∫∂Ω

ut · n ctbt dσ,

so that the weak formulation of the tissue problem reads as:∫Ω

∂ct∂tbt dΩ +

∫Ω

ut · ∇ctbt dΩ +

∫Ω

∇ · utctbt dΩ +

∫Ω

At∇ct · ∇bt dΩ+

+

∫Ω

QLF (pt − pL) ctbt dΩ +

∫∂Ω

βcctbt dσ+

−∫

Λ

2πR′Etctbt ds−∫

Λ

2πR′Evcv bt ds = 0. (2.24)

2.7.3 Alternative weak formulation for the vesselproblem

Since (2.21) is exactly the same transport equation in the vessel network ofthe initial formulation (2.11), we do not repeat the derivation of the weakproblem detailed in Section 2.6.2. For the sake of clarity, we rewrite the finalresult: ∫

Λ

∂cv∂tbv ds+

∫Λ

uv∂cv∂s

bv ds+

∫Λ

∂uv∂s

cvbv ds+

+

∫Λ

Av∂cv∂s

∂bv∂s

ds+

∫Λ

2

R′Mcvbv ds+

+

∫Λ

2

R′

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2+ Υ

cvbv ds =

= −∫

Λ

2

R′

(1− σ)Q [(pv − pt)− σ (πv − πt)]

1

2−Υ

ctbv ds. (2.25)

38


2.7.4 Alternative weak formulation for the coupledtransport problem

Assembling (2.25) and (2.24), the whole weak formulation for the coupledtransport problem within the vessel network and the interstitial tissue with-out the delta function (2.18)-(2.21) reads

to find cv ∈ Bv × (0, T ) and ct ∈ Bt × (0, T ) such that

(∂ct∂t, bt)

Ω+ (ut · ∇ct, bt)Ω + (∇ · utct, bt)Ω + (At∇ct,∇bt)Ω

+(QLF (pt − pL) ct, bt

)Ω

+ (βcct, bt)∂Ω +

−(2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2−Υ

ct, bt

)Λ

+

−(2πR′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2+ Υ

cv, bt

)Λ

= 0 ∀bt ∈ Bt,(∂cv∂t, bv)

Λ+(uv

∂cv∂s, bv)

Λ+(∂uv∂scv, bv

)Λ

+

+(Av

∂cv∂s, ∂bv∂s

)Λ

+(

2R′Mcv, bv

)Λ

+

+(

2R′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2+ Υ

cv, bv

)Λ

+

+(

2R′

(1− σ)Q [(pv − pt)− σ (πv − πt)] 1

2−Υ

ct, bv

)Λ

= 0 ∀bv ∈ Bv,

(2.26)

with ct(t = 0) = 0 and cv(t = 0) = 0.

2.7.5 Well posedness

The advantage of the second weak formulation (2.26) with respect to the firstweak formulation (2.17) is the existence of a well posedness analysis in theframework of the Lax-Milgram lemma. In particular, in Chapter 3, a study onthe existence and uniqueness of the solution will be exploited. Accordingly,from now on, the variational formulation (2.26) based on a coupling approachthat does not involve any Dirac measure will be used.

2.8 Numerical approximation

The discretization of problem (2.11) is achieved by means of the finite elementmethod that arises from the variational formulation (2.26) combined witha discretization of the domains Ω and Λ. At the discrete level, one of theadvantages of our formulation is that the partitions of Ω and Λ are completelyindependent.

In order to discretize the domain for the tissue interstitium problem, weintroduce an admissible triangulation T ht of Ω, i.e.

Ω =⋃

K∈T ht

K,

39


which satisfies the usual conditions of a conforming triangulation of Ω, whilewe are implicitly assuming that Ω is a polygonal domain. With a stan-dard notation, h = maxK∈T h

thK , where hK is the diameter of the element

K. The solutions of (2.26)(a) are approximated using continuous piecewise-polynomial finite elements for the concentration. More precisely, we have

Xhk :=

vh ∈ C0

(Ω)

s.t. vh|K ∈ Pk (K) ∀K ∈ T ht

for every integer k ≥ 0, where Pk indicates the standard space of polynomialsof degree ≤ k in the variables x = (x1, . . . , xd).

Concerning the capillary network problem, we adopt the same splittingof the domain described at the continuous level, denoted by

Λh =N⋃i=1

Λhi ,

where Λhi is a partition of the one-dimensional manifold Λi made by segments

S. For the concentration, we define the finite element space over the wholenetwork Λh as

Y hk (Λ) :=

wh ∈ C0

(Λ)

s.t. wh|S ∈ Pk (S) ∀S ∈ Λh

,

for every integer k ≥ 0.The discrete formulation arising from (2.26) is easily obtained by project-

ing the equations on the discrete spaces

Bht = Xh

k (Ω) and Bhv = Y h

k (Λ)

for k ≥ 0 and adding the subscript h to each variable (cht and chv).The space discretization must be complemented with the time advancing

scheme. Let us subdivide the time interval [0, T ] in N timesteps of size∆t > 0, so that tn = n∆t, with n = 0, . . . , N − 1. The equations have beensolved with the backward Euler finite difference scheme:

∂y

∂t= f (y) ⇒ yn+1 − yn

∆t= f

(yn+1

).

Let us denote with ch,nt and ch,nv , the numerical approximation of cht (tn)and chv (tn), respectively, therefore the fully discrete formulation of problem(2.26) reads as follows:

40


∀n = 0, . . . , N − 1, to find ch,n+1t ∈ Bh

t and ch,n+1v ∈ Bh

v such that

1∆t

(ch,n+1t , bht

)Ω

+(uht · ∇c

h,n+1t , bht

)Ω

+(∇ · uht c

h,n+1t , bht

)Ω

+

+(At∇ch,n+1

t ,∇bht)

Ω+(QLF

(pht − phL

)ch,n+1t , bht

)Ω

+(βcc

h,n+1t , bht

)∂Ω

+

−(

2πR′

(1− σ)Q[(phv − pht

)− σ

(πhv − πht

)]12−Υ

ch,n+1t , bht

)Λ

+

−(2πR′

(1− σ)Q

[(phv − pht

)− σ

(πhv − πht

)]12

+ Υch,n+1v , bht

)Λ

=

= 1∆t

(ch,nt , bht

)Ω

∀bht ∈ Bht ,

1∆t

(ch,n+1v , bhv

)Λ

+(uhv

∂ch,n+1v

∂s, bhv

)Λ

+(∂uhv∂sch,n+1v , bhv

)Λ

+

+(Av

∂ch,n+1v

∂s, ∂b

hv

∂s

)Λ

+(

2R′Mh,n+1ch,n+1

v , bhv)

Λ+

+(

2R′

(1− σ)Q

[(phv − pht

)− σ

(πhv − πht

)]12

+ Υch,n+1v , bhv

)Λ

+

+(

2R′

(1− σ)Q

[(phv − pht

)− σ

(πhv − πht

)]12−Υ

ch,n+1t , bhv

)Λ

=

= 1∆t

(ch,nv , bhv

)Λ

∀bhv ∈ Bhv .

(2.27)

with ch,0t = 0 and ch,0v = 0.

2.9 Algebraic counterpart

We aim at studying the algebraic counterpart of the discrete problem (2.27).The number of degrees of freedom of the discrete spaces are defined as

Nht := dim

(Bht

)and Nh

v := dim(Bhv

),

Let us introduce the finite element basis for Bht and Bh

v : ϕitNh

t

i=1 and ϕivNh

v

i=1,respectively. These two sets are completely independent, since the 3D and 1D

meshes do not conform. Let Cnt =

Cj,nt

Nht

j=1, Cn

v = Cj,nv

Nhv

j=1 be the degrees

of freedom of the finite element approximation, using the finite element basisit is possible to set:

ch,nt (x) =

Nht∑

j=1

Cj,nt ϕjt (x) , ∀x ∈ Ω and ch,nv (s) =

Nhv∑

j=1

Cj,nv ϕjv (s) , ∀s ∈ Λ.

Exploiting the linear combinations in the discrete weak form and thelinearity of the inner products, the fully discrete form (2.27) of the modelleads to the linear system[

1∆tMtt + Att + Btt Btv

Bvt 1∆tMvv + Avv + Bvv

] [Cn+1t

Cn+1v

]=

[1

∆tMttC

nt

1∆tMvvC

nv

]. (2.28)

41


Submatrices and subvectors in (2.28) are defined as follows:

[Mtt]i,j :=(ϕjt , ϕ

it

)Ω,

[Att]i,j :=(uht · ∇ϕ

jt , ϕ

it

)Ω

+(∇ · uht ϕ

jt , ϕ

it

)Ω

+(At∇ϕjt ,∇ϕit

)Ω

+(QLF

(pht − phL

)ϕjt , ϕ

it

)Ω

+(βcϕ

jt , ϕ

it

)∂Ω,

[Btt]i,j :=

(−2πR′

(1− σ)Q

[(phv − pht

)− σ

(πhv − πht

)] 1

2−Υ

ϕjt , ϕ

it

)Λ

,

[Btv]i,j :=

(−2πR′

(1− σ)Q

[(phv − pht

)− σ

(πhv − πht

)] 1

2+ Υ

ϕjv, ϕ

it

)Λ

,

[Bvt]i,j :=

(2

R′

(1− σ)Q

[(phv − pht

)− σ

(πhv − πht

)] 1

2−Υ

ϕjt , ϕ

iv

)Λ

,

[Mvv]i,j :=(ϕjv, ϕ

iv

)Λ,

[Avv]i,j :=

(uhv∂ϕjv∂s

, ϕiv

)Λ

+

(∂uhv∂s

ϕjv, ϕiv

)Λ

+

(Av∂ϕjv∂s

,∂ϕiv∂s

)Λ

+

+

(2

R′Mh,n+1ϕjv, ϕ

iv

)Λ

,

[Bvv]i,j :=

(2

R′

(1− σ)Q

[(phv − pht

)− σ

(πhv − πht

)] 1

2+ Υ

ϕjv, ϕ

iv

)Λ

,

where the bar operator corresponds to the average operator as in (2.10). Inparticular, it holds

Mtt ∈ RNht ×Nh

t , Att ∈ RNht ×Nh

t , Btt ∈ RNht ×Nh

t , Btv ∈ RNht ×Nh

v ,

Mvv ∈ RNhv×Nh

v , Avv ∈ RNhv×Nh

v , Bvv ∈ RNhv×Nh

v , Bvt ∈ RNhv×Nh

t .

Concerning the implementation of the exchange matrices Btt, Btv andBvt, it is necessary to introduce a discrete average operator πvt : Bh

t →Bhv that extracts the mean value of a generic basis function of Bh

t and adiscrete interpolation operator πtv : Bh

v → Bht that returns the value of

a basis function of Bht in correspondence of nodes of Bh

v . For every nodesk ∈ Λh, we let Tγ (sk) be the discretization of the perimeter of the vesselγ (sk), assuming that γ (sk) is a circle of radius R defined on the orthogonalplane to Λh at point sk (see Figure 2.4). The set of points of Tγ (sk) is used tointerpolate the basis function ϕit. The average operator πvt is defined in sucha way that qt = πvtqt and each row of the corresponding matrix Πvt ∈ RNh

v×Nht

is defined asΠvt

∣∣k

= wT (sk) Πγ (sk) k = 1, . . . , Nhv ,

where w is the vector of weights of the quadrature formula for the approxi-mation of qt (s) = 1

2πR′

∫ 2π

0qt (s, θ)R′dθ in the nodes belonging to Tγ (sk) and

42


Figure 2.4: Illustration of the vessel with its centerline Λh, a cross section,its perimeter γ (sk) and its discretization Tγ (sk) used for the definition of theoperators πvt and πtv.

Πγ (sk) is the local interpolation matrix that returns the values of each testfunction ϕit on the set of points belonging to Tγ (sk). Omitting some terms,it is possible to analyze the structure of the exchange matrices. Therefore,thanks to these operators, the exchange matrices are implemented as

Bvv ∝Mvv

Btt ∝ ΠTvtMvvΠvt

Btv ∝ ΠTvtMvv

Bvt ∝MvvΠvt.

43

Chapter 3

Well-posedness analysis

The alternative variational formulation that we have presented in the previ-ous chapter has been introduced in order to circumvent the issues related tothe well posedness of a formulation with a high dimensionality gap betweenthe domains. In the present chapter, we aim at proving the wellposedness ofa simplified version of the variational formulation (2.26) within the contextof the Lax-Milgram Lemma.

3.1 Simplified problem setting

We address here a simplified geometrical configuration of the 1D inclusionsimmersed in the 3D domain, in particular only one small inclusion will beconsidered in the tissue domain. Moreover, we will introduce some simplifi-cations in the model equations, but keeping the general structure of secondorder partial differential equations. The way of coupling the two equationswill be simplified as well, by means of solving only the equation in the tissuedomain, while considering the solution of the cylindrical inclusion as given.

3.1.1 Geometric setting

The domain where the model is defined is denoted as Ω ∈ R3 and it iscomposed by two parts, Ωt and Ωv. For the sake of simplicity, we assume Ωv

to be a cylindrical domain, we denote Γ the outer surface of the cylinder andΛ its centerline, parametrized by the arc length s; a tangent unit vector λ isalso defined over it, determining in this way an arbitrary branch orientation.The radius of the cylinder R is for simplicity considered to be constant alongthe axis and the length of the cylinder is L. In this case, the domain Ωv can

44

CHAPTER 3. WELL-POSEDNESS ANALYSIS

Figure 3.1: Simplified geometric setting for well posedness analysis.

be defined as follows:

Ωv = x ∈ R3; x = s+ r,s ∈ Λ =M(Λ′ ⊂ R),

r ∈ DΛ(R) = rnΛ; r ∈ (0, R),

where M is a mapping from a reference domain Λ′ to the manifold Λ ⊂ R3

and nΛ a unit normal vector with respect to Λ. See Figure 3.1 as example.

3.1.2 Model equation

In order to comprehend the core of the proposed weak formulation, we willget rid of many terms of the equations and we will simply consider a diffusionequation in Ωt, with Robin boundary conditions at the interface Γ that is as-sumed to be permeable and a given intensity U on Γ. The strong formulationof the problem consist to find u ∈ Ωt such that

−∆u = f in Ωt,

−∇u · nt = ν (u− U) on Γ,

u = 0 on ∂Ωt \ Γ.

(3.1)

The term ν (u− U) is modelling a mass flux through the interface Γ fromthe inclusion to the tissue or viceversa.

It is possible to introduce a simplified version of problem (3.1), wherethe domain Ωv shrinks to its centerline Λ and the variables defined on Γ areaveraged on the cylinder cross section DΛ(R), as it has already been donein the previous chapter. This new problem setting will be also called thereduced problem.

45


3.1.3 Restriction operator

Let us recall that a central role in the transformation of (3.1) into well posed3D-1D coupled problem is played by the restriction operator (·) that is definedas:

w (s) =1

2πR

∫ 2π

0

w (Λ (s) +RnΛ (s, θ))Rdθ .

The vector nΛ (s, θ) indicates a normal vector perpendicular to λ (s) and itdepends on an angle θ, in such a way that the set nΛ (s, θ) : θ ∈ [0, 2π)describes a unit circle around Λ (s) and perpendicular to λ (s). Taking thesedefinitions into account, it becomes obvious that w represents an averagevalue of w with respect to a circle of radius R around Λ (s) and perpendicularto λ (s). By this, the variable u can be split as sum of its average and somefluctuations from the average:

u = u+ u.

3.1.4 Weak formulation

The weak formulation of problem (3.1) arises similarly to what we have donein Chapter 2 for the alternative weak formulation. Let us briefly recall howit has been derived.

Let us consider V = H10 (Ωt) as natural functional space for the main

variable, then let us multiply the diffusion equation by a test function v ∈ Vand integrate over Ωt:

−∫

Ωt

∆uv dΩ =

∫Ωt

fv dΩ.

Then, we integrate by parts and we use the conditions on Γ and ∂Ωt:∫Ωt

fv dΩ = −∫

Ωt

∆uv dΩ =

∫Ωt

∇u · ∇v dΩ−∫∂Ωt∪Γ

∇u · nv dσ =

=

∫Ωt

∇u · ∇v dΩ +

∫Γ

νuv dσ −∫

Γ

νUv dσ.

Now, we split the functions u and v in their mean values plus fluctuations,i.e.

u = u+ u and v = v + v,

46


where, by definition, ¯u = ¯v = 0. Using the cylindrical coordinates system(s, θ) on Γ, we obtain that∫

Γ

νuv dσ =

∫Λ

ν

∫ 2π

0

(u+ u) (v + v)Rdθds =

=

∫Λ

2πRνuv ds+

∫Λ

ν

∫ 2π

0

uvR dθds,∫Γ

νUv dσ =

∫Λ

ν

∫ 2π

0

U (v + v)Rdθds =

∫Λ

2πRνUv ds.

Then, we make the following modelling assumptions:

1. Small residuals ∫ 2π

0

uvR dθ ' 0. (3.2)

2. Shrinkage of the capillary vessel∫Ωt

dΩ '∫

Ω

dΩ and

∫∂Ωt

dσ '∫∂Ω

dσ. (3.3)

Assembling all the terms, the weak formulation of problem (3.1) reads as:to find u ∈ H1

0 (Ω) such that

a (u, v) + bΛ (u, v) = (f, v)Ω + bΛ (U, v) ∀v ∈ H10 (Ω) , (3.4)

with the following bilinear forms:

a(w, v) = (∇w,∇v)Ω, bΛ(w, v) = 2πR ν(w, v)Λ.

3.2 Well-posedness analysis

Here we aim at analyzing the existence and uniqueness of the weak solutionof problem (3.4) in the framework of the Lax-Milgram lemma and to describethe dependence of the coercivity and continuity constants of the problem withrespect to the radius of the inclusion. It is crucial to know how the constantsdepend on the radius R of the cylindrical inclusion in order to derive suitableupper bounds for the numerical error.

For the sake of simplicity, we rewrite the weak formulation (3.4) in termsof bilinear form and linear form, namely:

to find u ∈ V = H10 (Ω) such that

A (u, v) = L (v) ∀v ∈ V, (3.5)

47


where

A (w, v) :=(∇w,∇v)Ω + 2πR ν(w, v)Λ,

L (v) := (f, v)Ω + 2πR ν(U, v)Λ.

Theorem 1. Let f ∈ L2 (Ω) and ν > 0, the problem (3.5) has a uniquesolution

u ∈ H10 (Ω) ∩H

32−ε (Ω) , ∀ε > 0.

Proof. We prove the existence and the uniqueness of the solution u using theLax-Milgram lemma. More precisely, we prove that the bilinear form A (·, ·) :V × V → R is continuous and coercive and the linear form L (·) : V → R isbounded with respect to the norm || · ||V := ||∇ · ||L2(Ω).

• Continuity of A (·, ·).

|A (u, v)| = |(∇u,∇v)Ω + 2πRν(u, v)Λ| ≤ |(∇u,∇v)Ω|+2πRν |(u, v)Λ| .

Using the Holder inequality, we have:

2πRν

∫Λ

|uv| ds ≤ 2πRν

(∫Λ

u2 ds

) 12(∫

Λ

v2 ds

) 12

.

From the definition of u and applying the Jensen’s inequality, we obtain:∫Λ

u2 ds =

∫Λ

(1

2πR

∫ 2π

0

uR dθ

)2

ds ≤ 1

4π2R2

∫Λ

2π

∫ 2π

0

u2R2dθds =

=1

2πR

∫Γ

u2 dσ =1

2πR||u||2L2(Γ) ≤

1

2πR||u||2V .

Consequently, the combination of all the previous results leads to

|A (u, v)| ≤ (1 + ν) ||u||V ||v||V ,

which proves the continuity of the bilinear form.

• Coercivity of A (·, ·).The coercivity of the bilinear form is trivial:

A (v, v) = (∇v,∇v)Ω + 2πRν(v, v)Λ ≥ (∇v,∇v)Ω = ||v||2V .

48


• Continuity of L (·).Thanks to the Poincare inequality and the Jensen inequality, it followsthat:

|L (v)| = |(f, v)Ω + 2πRν(U, v)Λ| ≤

≤Cp||f ||L2(Ω)||v||V + 2πRν||U ||L∞(Λ)

∣∣∣∣∫Λ

v ds

∣∣∣∣ ≤≤(Cp||f ||L2(Ω) +

√2πRLν||U ||L∞(Λ)

)||v||V .

Therefore, thanks to the Lax-Milgram lemma, the problem (3.5) has a uniquesolution u ∈ H1

0 (Ω). Moreover, we observe that the weak formulation (3.4)could have been formally written in strong form as

−∆u = f − ν (u− U) δΓ in Ω, u = 0 on ∂Ω. (3.6)

Due to the presence of the Dirac source, no H2-regularity can be recoveredand the issue arises to which interspace V with H2 (Ω) ⊂ V ⊂ H1

0 (Ω) the

solution u belongs. Since the right hand side in (3.6) is in H−12−ε(Ω), ∀ε > 0

it can be shown, analogously to Case (iii) in [23, Theorem 2.1], that u ∈H

32−ε(Ω), ∀ε > 0, which ends the proof.

Remark 1. We observe that not only the existence and uniqueness of the weaksolution of problem (3.5) is proved, but also the stability of the solution withrespect to the radius of the inclusion is guaranteed. Hence, the continuityand coercivity constants are independent of R and the effect of the intensityof the inclusion U vanishes as R→ 0.

We could have done the same well-posedness analysis by introducing theenergy norm associated with the bilinear form A:

|||u|||2 = A(u, u) = ||∇u||2L2(Ω) + 2πRν||u||2L2(Λ).

We would prove the same results in terms of uniqueness and existence ofthe solution and in terms of the stability of the constants with respect tothe radius of the inclusion R as the analysis with respect to the norm of thegradient.

3.3 Conclusions and further developments

In conclusion, a simplified version of the coupled problem presented in Chap-ter 2 has been derived in order to prove the well posedness of the alterna-tive weak formulation within the context of the Lax-Milgram Lemma. The

49


simplifications are related to the geometric setting and to the model equa-tions, moreover the coupling between the two equations has not been consid-ered. Under these hypothesis, the existence and uniqueness of the solutionin H1

0 (Ω) ∩H 32−ε (Ω) , ∀ε > 0 have been proved with respect to the norm of

the gradient ||∇ · ||L2(Ω). Furthermore, we have observed that the constantsarising from the analysis based on the norm of the gradient are bounded asthe radius R goes to zero, actually the continuity and coercivity constantsare independent of R. This requirement will be crucial in the study of thenumerical errors and it proves that the eigenvalues of the simplified problemare bounded from above and from below with respect to the radius R. Inthis perspective, the analysis of the spectrum of the Finite Element-discreteoperator would be a desired development, in order to derive ad hoc precon-ditioners. On the other hand, the relaxation of the simplifying assumptionswould lead to interesting extensions.

50

Chapter 4

Characterization of the modelparameters

The 3D/1D coupled equations for the description of the mass transport in avessel network embedded in an interstitial tissue require the explanation ofsome parameters, for which a complete description has not been detailed yet.Therefore, the current chapter aims at characterizing the adhesiveness of thenanoparticles to the vessel wall, in particular the effective vascular adhesionparameter and the probability of adhesion from both a continuum and amultiscale model (Sections 4.1, 4.2 and 4.3) and the diffusion coefficient ofsome molecules in the interstitial space (Section 4.4).

4.1 A model for particle adhesion to the

vascular wall

One reason for which the injection of nanoparticles has been studied as vec-tor for therapeutic agents is their ability to recognize vascular receptors andfirmly adhere to the vessel walls. Therefore, the equation describing the be-haviour of the nanoparticle concentration in the vessel bed differs from astandard advection-diffusion-reaction equation for the presence of the adhe-sive term, defined by the effective vascular adhesion parameter Πeff . For thesake of clarity, we report the model equation (2.8) for the concentration ofnanoparticles in the vessel network Λ:

∂cv∂t

+∂

∂s

(cvuv · λ−Dv

∂cv∂s

)+

2πR

πR2Πeffcv = −2πR

πR2f (cv, ct) , on Λ× (0, T ),

where

f (ct, cv) = (1− σ)Lp [(pv − pt)− σ (πv − πt)] cavg + P (cv − ct) .

51

CHAPTER 4. CHARACTERIZATION OF THE MODELPARAMETERS

4.1.1 Vascular adhesion parameter

The effective vascular adhesion parameter Πeff at first can be estimatedwith a coefficient, denoted as Π and called vascular adhesion parameter,that does not depend on time and that describes the strength whereby thenanoparticle sequestration to the vascular wall take place, as described in[26]. The vascular adhesion parameter Π is estimated using a ligand-receptormodel for the interaction of particles with the endothelial layer. The adhesionparameter is assumed to be directly influenced by three terms in a linear way:the probability of adhesion Pa of the particle to the endothelial wall, whichis a measure of the strength of adhesion, the local wall shear rate γ of thefluid and the nanoparticle diameter dp, i.e:

Π (s) = Pa |γ (s)| dp2. (4.1)

The wall shear rate at the axial coordinate s along the capillary network canbe computed from the flow equation, while the description of the probabilityof adhesion is more challenging, even though several models at different scalelevels are available. In the case of Poiseuille’s flow for the blood in the vesselnetwork, the wall shear rate can be computed as function of the velocity fieldand it reads as

γ (s) = 4uv (s)

R. (4.2)

For a detailed characterization of Pa, we postpone the description to Sections4.1.3 and 4.2.

In order to quantify the effect of the adhesion on the transport of nanopar-ticles in the vessel bed, it is possible to introduce the density of nanoparticlesadhering per unit surface to the wall as

Ψ (s, t) :=

∫ t

0

Π (s) cv (s, τ) dτ. (4.3)

4.1.2 Effective vascular adhesion parameter withsaturation

The adhesive term 2R

Πeffcv in (2.8), with Πeff = Π as in 4.1.1, behaves ina linear way with respect to the nanoparticle concentration cv in the vesselnetwork. As already mentioned, the vascular adhesion parameter acts asa coefficient for the sequestration of nanoparticles from the ligand-receptorbonds. Therefore, it should be natural to design the vascular adhesion param-eter and its relation with the concentration in such a way that this parameter

52


takes into account the number of ligand-receptor bonds on the vascular wall.Indeed, up to now, the adhesive term has been influenced by the probabilityof adhesion of the nanoparticle to the wall, the local wall shear rate and theradius of the nanoparticle. However, we would expect the adhesive term tobe influenced also by the relative number of free ligand-receptor bond. It isstraightforward to see that the adhesive term of the current model does nottake into account these effects, hence it could happen in the case of strongadhesion that the nanoparticles completely adhere to the capillary wall inthe regions next to the inlet section and they do not reach the outlet region,even observing the system for long time. These qualitative observations willbe verified by the numerical experiments in Chapter 5.

In order to prevent this unphysical behaviour, a new model for the ad-hesion of the particle that considers some kind of saturation effects of theligand-receptor bond has been derived. Recalling the definition of the densityof nanoparticles adhering per unit surface to the wall Ψ, we assume to knowa maximum value of Ψ, Ψmax, beyond which the particles cannot adhere tothe wall anymore because of lack of free receptor molecules. We define thesaturation function at time t as

S (s, t) :=

(Ψmax −Ψ (s, t)

Ψmax

)+

, (4.4)

where (g)+ = max0, g is the positive part of g and we introduce the effectivevascular adhesion parameter Πeff as

Πeff (s, t) := Π (s)S (s, t) = Π (s)

(Ψmax −Ψ (s, t)

Ψmax

)+

. (4.5)

Notice that if Ψ = 0, then the effective adhesion parameter is equivalent toΠ; if Ψ ≥ Ψmax, then Πeff = 0, while an intermediate effect is obtained forvalues of Ψ in between 0 and Ψmax. Consequently, the expression for Ψ readsas

Ψ (s, t) =

∫ t

0

Πeff (s, τ) cv (s, τ) dτ =

∫ t

0

Π (s)S (s, τ) cv (s, τ) dτ (4.6)

and the adhesive term becomes

2

RΠeff (s, t) cv (s, t) =

2

RΠ (s)S (s, t) cv (s, t) . (4.7)

Time discretization

Some observations on the numerical discretization of the adhesion term needto be reported. Indeed, it is worth noting that the adhesive term (4.7) is non

53


linear with respect to cv (s, t) because of the contribution of the integral upto time t in the definition of Ψ (s, t).

In order to preserve the linearity of the coupled problem, it is useful tocompute Πeff in an explicit way. In particular, we refer to the time advancingscheme presented in Section 2.8 in which we subdivide the time interval [0, T ]in N timesteps of size ∆t > 0, so that tn = n∆t, with n = 0, . . . , N − 1.Exploiting this time discretization, we approximate

Ψ (s, t) =

∫ t

0

Π (s)

(Ψmax −Ψ (s, τ)

Ψmax

)+

cv (s, τ) dτ

with an explicit expression, such as

Ψh,n+1 =n∑k=0

Πh

(Ψmax −Ψh,k

Ψmax

)+

ch,kv ∆t = Πh

(Ψmax −Ψh,n

Ψmax

)+

ch,nv ∆t+Ψh,n,

(4.8)where the apices h and n refer to the space and time discretization, respec-tively. Consequently, the adhesive term (4.7) can be approximated at the(n+ 1)-th time step as

2

RΠh

(Ψmax −Ψh,n+1

Ψmax

)+

ch,n+1v .

We observe that the summation in (4.8) stops at the n-th time step, thereforeΨh,n+1 can be computed in an explicit way and the linearity of the adhesiveterm with respect to the concentration at the current iteration is preserved.

4.1.3 Explicit formula for Pa

The adhesion of a nanoparticle to the vessel wall is determined by the balancebetween the adhesive interactions and the hemodynamic forces that act onthe particle. In particular, the adhesive strength is governed by the presenceof receptor molecules on the target surface (vessel wall) and their conjugatedmolecules (ligands) on the particle surface. A possible ligand-receptor bondis characterized by many biological factors such as the surface density mr

of receptors on the endothelial layer, the surface density ml of ligands onthe particle surface and the affinity constant Ka of the ligand-receptor bond.On the other hand, the fluid dynamic forces, which are responsible for thedislodging of the particle away from the target surface, are influenced byphysiological factors such as the wall shear rate γ.

In [44], Pa has been defined as the probability of having at least oneclosed ligand-receptor bond. From [17], the probability of adhesion of a

54


particle can be expressed as a function of its geometry (size and shape) andsurface properties (ligand density, ligand type). For a spherical particle ofdiameter dp in point contact with the wall, the mathematical parameter Pacan be computed as

Pa (s) = mlK0amrπr

20 exp

−βNP µ |γ (s)|

α2

. (4.9)

In (4.9), ml is the surface density of ligand molecules that decorate thenanoparticle surface, K0

a is the affinity constant of the ligand-receptor in-teraction at zero mechanical load and mr is the surface density of receptormolecules, while the contribution of mr is also taken into account in theparameter α2:

α2 = mr

[1−

(1− ∆

dp/2

)2].

The parameter ∆ is the separation distance between the particle and thesubstrate at equilibrium and r0 is the radius of adhesion point. We recallthat the adhesion point is the maximum distance of the particle from thewall at which specific ligand-receptor bond can occur. Given µ the blood(plasma) viscosity, µ |γ| is the wall shear stress and the parameter βNP isdefined as

βNP =6Fλ

kBT,

where F is the coefficient of the hydrodynamic drag force on a sphericalparticle, λ is a characteristic length of the ligand-receptor bond and kBT isthe Boltzmann thermal energy.

We can notice that globally Pa is proportional to a negative exponentialfunction: the more the velocity, the less the adhesive strength.

Let us now present some advantages and disadvantages of formula (4.9).

Advantages

First of all, the formula for Pa has been derived from detailed studies thatinvolve both the probabilistic formulation of bond formation at the molecularlevel and the hydrodynamic analysis, therefore, we can say that (4.9) includesseveral aspects that affect the adhesion strength.

Moreover, (4.9) is an explicit formula. Given the value of the parameters,the probability of adhesion is uniquely determined by knowing the value ofthe local wall shear rate in the network. Therefore, the computation of Pais straightforward, since γ can be easily calculated from the local velocityprofile of the blood in the vessel network.

55


Lastly, the previous formula can be helpful for an optimization processthat aims at finding the 3S parameters (shape, size and surface properties)of a nanoparticle that maximize the probability of adhesion. Formula (4.9)clearly shows that the strength of adhesion is directly proportional to theproduct of ml, mr and K0

a , therefore it increases as the surface densitiesmr and ml grow and as the affinity constant increases. A similar formulacan be derived for non-spherical particle (for example, spheroidal) and ananalogous analysis can be presented. The shape and size of the particles canbe also optimized. Indeed, in [17], the authors have demonstrated that existan optimal volume for each aspect ratio as function of the wall shear stressand the receptor density for which the adhesion probability has a maximum.However, the optimization of the 3S parameters is beyond the scope of thiswork.

Disadvantages

On the contrary, there are some aspects of (4.9) that make it difficult to use.By definition, Pa is the probability of having at least one closed bond,

therefore Pa ∈ [0, 1]. However, by construction, it is not guaranteed that Pais in the desired interval. Unless a renormalization of the result is made, theonly way to have Pa ≤ 1 is to accurately choose the parameters. For example,it would be possible to set mlK

0amrπr

20 = 1, so that the adhesion probability

would always be less than one. On the other hand, Pa would be exactly oneonly in the case of zero wall shear rate, that means zero velocity; it is clearthat the implication Pa = 1 ⇐⇒ γ = 0 is not physically correct. Anotherpossibility is to set mlK

0amrπr

20 in such a way that Pa = 1 in the case of

minimum wall shear rate, however it is not possible to know a priori theminimum value of γ and, more importantly, it could be different for differentproblems. We can conclude that this strategy is not convenient. Moreover,it would never happen to have zero probability of adhesion, even in the caseof very high wall shear stress, except for trivial cases. Physically, we wouldexpect to have some kind of saturation as γ increases.

The choice of the value of the parameters is a challenging aspect alsobecause it requires the knowledge of the molecular and chemical behaviourof the ligand-receptor bond.

Finally, since Pa has the form of an exponential function, it is very sen-sitive to variation of the parameters. Indeed, a small variation of the valueof one parameter, expecially µ, βNP or α2, would dramatically change thevalue of Pa.

56


4.2 Multiscale model for particle adhesion

In the light of the above analysis of formula (4.9), we need to modify theexpression for Pa. We will now provide an estimate of the probability ofadhesion of the nanoparticles to the vessel wall from a combined LatticeBoltzmann-Immersed boundary method.

4.2.1 Lattice Boltzmann approach for nanoparticletransport

Vascular flow and nanoconstruct transport have been modeled with differenttechniques, both with discretization and integration of the Navier-Stokesequations and with particle-based methods. As we are interested in subscaletechniques, we focus on the particle-based approaches and we refer to thework of Coclite et al. [11] on a numerical model that combines the LatticeBolzmann Method (LBM) and the Immersed Boundary Method (IBM). TheLattice Boltzmann method allows to describe the fluid dynamics of the flownext to a rigid wall by means of N discrete distribution functions thay obey tothe Boltzmann equation. The presence of the boundary is taken into accountas forcing term in the Boltzmann equation using a moving least squaresalgorithm. With the Immersed Boundary method, the presence of a particlein the flow is described as a collection of Lagrangian markers superimposed tothe fluid lattice and it is again considered in the governing equation thanksto the moving least squares method. The methods have been applied forpredicting the near wall dynamics of particles with different shapes in alaminar flow: transport of circular, elliptical, squared and triangular particlesare analyzed in a Couette flow at a given Reynold number. As a result, thetrajectories of the centroids of the particles have been depicted consideringdifferent particle shapes and different release positions (Figure 4.1) and theequilibrium position of the particles, their oscillation around the equilibriumin terms of linear and angular velocity can be determined. The authorshave demonstrated that the proposed combined method is able to predictcomplex particle dynamics and the same approach could be used for thedescription of the adhesion of the particles to the vessel walls. The describedstrategy provides high flexibility of application and can be translated intoparallel computing, in such a way to handle long simulations. Please refer toAppendix B for an extensive discussion of the Lattice Boltzmann method.

57


(a) Circular particles (b) Squared particles

(c) Elliptical particles (d) Triangular particles

Figure 4.1: Trajectory of the particles in the channel, initially located atdifferent heights within the flow domain. H : height of the channel, H/4 :characteristic dimension of each particle. Figure from [11]

58


Figure 4.2: Geometry for the Couette flow and particle transport

4.2.2 Subscale model for particle adhesion

Some interesting studies on the transport of nanoparticles covered by lig-and molecules in linear laminar flow next to a rigid wall covered by receptormolecules for low Reynolds numbers have been performed by our collabora-tors, A. Coclite and P. Decuzzi, using the techniques we have just detailed. Inparticular, the vascular flow and particle transport have been solved combin-ing the Lattice Boltzmann method, with the Immersed Boundary method.This approach can reproduce and predict the lateral drifting and vascularadhesion of nanoparticles at the microcirculation level and it can provide thevalue of the probability of adhesion under different flow conditions. In thenumerical experiment, we consider a two dimensional channel Ω of height Hand a circular particle of diameter dp, as shown in Figure 4.2. The particleis immersed in a Couette flow, therefore the velocity field is governed by

u(y) = umaxy

H, (4.10)

where umax is the velocity at the top of the channel.The particle surface is decorated with uniformly distribuited ligand molecules

(ρl is the percentage of particle surface covered by ligands) and the lower wallof the channel is covered by receptor molecules (ρr is the percentage of chan-nel surface covered by receptors). The blockage ratio of the channel is 4(H/dp = 4), the strenghtness of the ligand-receptor bond has been set to twodifferent values (strong or mild) and ρr = 1, therefore the channel wall iscompletely covered by receptor molecules. Observe that the relevant valueis the ratio between the density of ligands on the particle surface and thedensity of receptors on the channel wall.

The probability of adhesion Pa is estimated as the ratio between thenumber of closed ligand-receptor bonds and the number of possible bonds. Pais one of the output of this numerical tool and the most important parameters

59


Pa ρl = 0.3 ρl = 0.5 ρl = 0.7 ρl = 0.9Re = 0.01 0.3 1 1 1Re = 0.1 0 0.33 0.64 0.8Re = 1 0 0.28 0.56 0.7

Table 4.1: Probability of adhesion computed using the Lattice Boltzmann-Immersed Boundary method under different experimental conditions for acircular particle for strong ligand-receptor bond

Pa ρl = 0.3 ρl = 0.5 ρl = 0.7 ρl = 0.9Re = 0.01 0 0.56 1 1Re = 0.1 0 0.05 0.64 0.8Re = 1 0 0 0.56 0.7

Table 4.2: Probability of adhesion computed using the Lattice Boltzmann-Immersed Boundary method under different experimental conditions for acircular particle for mild ligand-receptor bond

that affect the results are the Reynolds number Re and the percentage ofparticle surface covered by ligands, ρl.

Let us now report in Tables 4.1 and 4.2 and in Figure 4.3 the values of Paunder different experimental conditions for a circular particle. In particular,the cases Re ∈ 0.01, 0.1, 1, ρl ∈ 0.3, 0.5, 0.7, 0.9 and the bondstrength, σ ∈ strong, mild, have been considered.

We can make some considerations on the results. First of all, the rangefor the probability of adhesion is between 0 and 1, indeed, by definition, Pais guaranteed to be a probability. Moreover, the particle transport has beenstudied in a limited range of low Reynolds numbers, however, it is possibleto obtain zero probability of adhesion in the case of low ligand density forthe highest Reynolds number and, on the contrary, Pa = 1 with high liganddensity in the case of the lowest Re. These observations suggest that thehydrodynamic forces and the adhesive interactions do not have the sameinfluence for different Reynolds numbers. Indeed, we can state that thereexist a spectrum of models according to the balance bewteen the two kindof forces. For low Reynolds number, Pa reaches 1, we have firm adhesionand the adhesive forces dominate over the hydrodynamics; for high Reynoldsnumber, Pa = 0 and the particle transport is governed by the fluid dynamicforces and there exist an intermediate case in which the adhesion and thefluid dynamics have the same role. Notice that these three behaviours canreproduce the dynamics of the vascular flow and particle transport in differentbiological ranges: in the capillary flow there is only adhesion, the arteriole

60


Re

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of adhesion, Strong bond

ρl = 0.3

ρl = 0.5

ρl = 0.7

ρl = 0.9

(a)

Re

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Probability of adhesion, Mild bond

ρl = 0.3

ρl = 0.5

ρl = 0.7

ρl = 0.9

(b)

Figure 4.3: Probability of adhesion computed using the Lattice Boltzmann-Immersed Boundary method.

flow corresponds to the intermediate level and in the arteries the flow isgoverned by the hydrodynamic forces. Furthermore, the results clearly showthat the behaviour of the probability of adhesion is similar to an exponentialfunction, as it was for the explicit formula for Pa. Therefore, it is important tonotice that the new model for the adhesion is qualitalively in accordance withthe previous model. Recall that in formula (4.9), the independent variable isthe wall shear rate γ, while in the new model is Re; however, a qualitativecomparison can be perfomed, indeed both γ and Re are proportional to thevelocity of the fluid.

4.3 Discussion

Let us summarize the outputs of Sections 4.1 and 4.2. A continuum modelto estimate the probability of adhesion of a nanoparticle immersed in a fluidin motion is available. The model provides an explicit formula for Pa asfunction of the local wall shear rate of the blood. It is based on the analysisof balance between the hydrodynamic forces and the adhesive forces thatact on the particle at the molecular level, but we have pointed out severaldifficulties that arise using it. Therefore, an alternative formulation for theprobability of adhesion is needed. We recall that Pa is one of the parametersthat influence the vascular adhesion, which is the reaction term in the particletransport equation in the vessel bed (2.8). In the following, some simulationsfor the transport of nanoparticles in the capillary network will be performedand the results will be shown. As results of the simulations, adhesive maps

61


will be drawn illustrating the adhesive strength in all the regions of the vesselnetwork as a function of the fluid dynamics quantities and the ligand density.

Moreover, two models for the effective vascular adhesion parameter areavailable. The first one does not take into account the time dependence andthe number of closed ligand-receptor bonds, therefore it does not providea correct behaviour of the nanoparticle concentration, while the second ap-proach aims at filling this gap, providing some kind of saturation model forthe bonds. In the following simulations, both approaches will be tested.

Lastly, a model for the probability of adhesion of the nanoparticles tothe vessel wall that involves some simulations at the microscale has beenconsidered, because of the final aim of this work and because of the difficultiesthat arise computing Pa in an explicit way. The model is a numerical routinethat describes the fluid flow and the particle transport in a Couette flow andit has been solved by means of the implementation of the Lattice Boltzmannmethod combined with the Immersed boundary method. The numerical toolgives as output the probability of adhesion for some value of the Reynoldsnumber and of ligand density on the particle.

4.4 Optimal control approach for the

prediction of the diffusivity coefficient

The prediction of unknown parameters can be done using an optimizationapproach, an efficient algorithm for fitting a vector of parameters on a givendataset. Indeed, some results from physical experiments are available andit could be useful to look for the parameters from which those results havebeen generated.

The generic framework for a control problem can be expressed as a systemof equations, whose solution y depends on a variable u that represents aquantity that can be externally controlled. The aim of the control problemis to determine the value of u in such a way that a suitable cost functionalJ , which describes the distance between an observed output variable z anda desired value zd of the observation, is minimum.

In this case, we look at the parameters as control variables of the opti-mization problem, consisting in minimizing a cost functional that representsan estimate of the error made with respect to some observations.

The main advantage of the numerical tools with respect to the physicalexperiments is their ability to provide an estimate of several parametersat a time. Usually, in order to predict the value of several parameters, itis necessary to perform several experiments, thereby increasing the effort in

62


Figure 4.4: Geometric model of the Dextran diffusion experiment

terms of time and costs. Thanks to the optimal control appoach, it is possibleto make the most of one experiment for the analysis of different parameters.

We underline that the numerical studies of physical phenomena are fun-damental for the estimate of some physical quantities that cannot be exper-imentally measured, especially the time-dependent variables.

For the sake of simplicity, we present the theory of the optimal controland the application of the method on a case of just one unknown parameter,even though all the theory can be easily generalized to more complex sets.

4.4.1 Experimental setup

We decided to apply the optimization procedure to a model describing thepure diffusion of Dextran molecules in a channel filled with a gel in order topredict the value of the diffusivity coefficient D of the system. The exper-imental setting we can work with is a channel filled with collagen, while inthe initial configuration a small band of molecules is inserted in it. We areinterested in studying how the molecules diffuse in the channel in a giventime interval.

Let us consider a channel Ω of length L and height H. In the initialconfiguration, a small band of the channel of height h is filled with a gelcontaining a concentration cinj of molecules, as shown in Figure 4.4 and theconcentration c = c(t) of molecules in the time interval [0, T ] is measured.

The physical experiment consists in observing how the green front pro-gresses in time and computing from this quantity the corresponding diffusioncoefficient. The images have been captured by our collaborators, V. Lusi andP. Decuzzi, every 30 seconds for a total observation time of 4 minutes and30 seconds or 5 minutes and the same experiment has been computed for 4kDa, 40 kDa and 250 kDa Dextran molecules. Figure 4.5 shows the captured

63


(a) t = 0 s (b) t = 270 s

Figure 4.5: Diffusion experiments using 4 kDa Dextran molecules at theinitial and final time. Credits to V. Lusi.

images at the initial time and at the end of the experiment in the case of 4kDa molecules.

4.4.2 Numerical model

In order to numerically reproduce the diffusion phenomenon introduced in theabove section, we mathematically describe the experimental setting assumingthat c is the solution of

∂c∂t−D∆c = 0 in Ω× (0, T ],

D∇c · n = 0 on ∂Ω× (0, T ],

c =

cinj for (x, y) ∈ (0, L)× (a, a+ h), t = 0

0 for (x, y) /∈ (0, L)× (a, a+ h), t = 0,

(4.11)

namely, a pure diffusive time-dependent equation with homogeneous diffusiveflux on the boundary of the domain.

Let us now recover the weak formulation of (4.11) that will be useful inthe forthcoming analysis. Let V = H1 (Ω) be the reference Hilbert spaceand let us proceed in a standard way: we choose a test function φ ∈ V ,we multiply equation (4.11) by the test function and we integrate over thedomain Ω: ∫

Ω

∂c

∂tφ dΩ−

∫Ω

D∆c φ dΩ = 0.

We apply the Green’s identity and we exploit the homogeneous boundary

64


conditions:

0 =

∫Ω

∂c

∂tφ dΩ+

∫Ω

D∇c ·∇φ dΩ−∫∂Ω

D∇c·n dσ =

∫Ω

∂c

∂tφ dΩ+

∫Ω

D∇c·∇φ dΩ.

The weak formulation of problem (4.11) reads:to find c ∈ V × [0, T ], with

c =

cinj for (x, y) ∈ (0, L)× (a, a+ h), t = 0

0 for (x, y) /∈ (0, L)× (a, a+ h), t = 0,

such that ∫Ω

∂c

∂tφ dΩ +

∫Ω

D∇c · ∇φ dΩ = 0 ∀φ ∈ V. (4.12)

4.4.3 Minimization problem

Let us now introduce a possible formulation for the minimization problemthat aims at finding the optimal diffusion coefficient D∗ such that the cor-responding concentration of Dextran molecules, computed with (4.11), is assimilar as possible to the concentration given by the physical experiment(Figure 4.5).

The optimal control problem reads:to find D∗ ∈ R+ such that

J (c (D∗) , D∗) = min J (c (D) , D) =1

2||c(T )− c(T )||2L2(Ω) +

λ

2|D|2, (4.13)

subject to the conditions (4.11), called state equation.In this case, the optimization of the cost functional consists in the mini-

mization of two terms. The first term is a measure of the distance betweenthe concentration at the final time T , solution of the state equation with agiven diffusivity coefficient, and its desired value, namely the concentrationmap at the final time (Figure 4.5(b)), c(T ). The second addend λ

2|D|2 is a

regularization term and it guarantees that the value of D does not divergeduring the minimization process.

To analyze the optimal control problem, we adopt the Lagrangian ap-proach as studied in [57], which is complementary to the analysis on optimalcontrol developed by Lions [32]. The idea of the Lagrangian method is theextension of the constrained optimization theory for function in finite dimen-sional space (for example, Rn).

65


Exploiting the Lagrangian approach, the optimal control problem (4.13)with solution (c∗ = c (D∗) , D∗) reduces to find the critical point of the La-grangian functional defined as

L (c,D, p) = J (c,D)−∫ T

0

∫Ω

∂c

∂tp dΩdτ −

∫ T

0

∫Ω

D∇c · ∇p dΩdτ, (4.14)

with c unconstrained and where p is the Lagrange multiplier. We observethat the Lagrangian functional differs from the cost functional for the stateequation in weak form, tested over the Lagrange multiplier and integratedover the time interval. Therefore, we expect that the following optimalityconditions are satisfied:

Lp(c∗, D∗, p)φ = 0 ∀φ ∈ V, (4.15)

Lc(c∗, D∗, p)ϕ = 0 ∀ϕ ∈ V × [0, T ], ϕ(0) = 0, (4.16)

LD(c∗, D∗, p)ψ = 0 ∀ψ ∈ R+, (4.17)

where Lαξ denotes the Frechet derivative of L with respect to α in directionξ.

Let us make the optimality conditions explicit, focusing on (4.16) thatleads to the adjoint equation. For the sake of brevity, we will use the notationϕ(·, t) = ϕ(t) and we will omit the differentials in the integrals. Using thefact that the derivative of a linear mapping is that mapping itself, we obtainthat ∀ϕ ∈ V × [0, T ], ϕ(0) = 0

0 =Lc (c∗, D∗, p) ϕ =

=

∫Ω

(c∗(T )− c(T ))ϕ(T ) dΩ−∫ T

0

∫Ω

∂ϕ

∂tp dΩdτ −

∫ T

0

∫Ω

D∗∇ϕ · ∇p dΩdτ.

Integrating by parts with respect to t in ∂ϕ∂t

and using Green’s first identity,we obtain

0 =

∫Ω

(c∗(T )− c(T ))ϕ(T ) dΩ +

∫ T

0

∫Ω

ϕ∂p

∂tdΩdτ −

∫Ω

ϕ(T )p(T ) dΩ+

−∫ T

0

∫∂Ω

D∗ϕ∇p · n dσdτ +

∫ T

0

∫Ω

D∗ϕ∆p dΩ, ∀ϕ ∈ V × [0, T ], ϕ(0) = 0.

This is a possible weak formulation for the adjoint equation, however it isinteresting to describe the corresponding strong form. To this aim, we choosethe functions ϕ ∈ C∞0 (Ω× (0, T )), so that ϕ(T ) = ϕ(0) = 0 in Ω and ϕ = 0in ∂Ω× (0, T ), from which we have∫ T

0

∫Ω

ϕ∂p

∂tdΩdτ +

∫ T

0

∫Ω

D∗ϕ∆p dΩdτ = 0, ∀ϕ ∈ C∞0 (Ω× (0, T ))

66


and the density of C∞0 (Ω× (0, T )) in L2(Ω× (0, T )) implies that

∂p

∂t+D∗∆p = 0 in Ω× (0, T ).

Next, we let ϕ(T ) vary, while we postulate ϕ = 0 on ∂Ω× (0, T ); for all suchϕ we have∫

Ω

(c∗(T )− c(T ))ϕ(T ) dΩ−∫

Ω

ϕ(T )p(T ) dΩ = 0 in Ω.

Being the set of the possible values of ϕ(T ) dense in L2(Ω× (0, T )), we have

p(T ) = c∗(T )− c(T ) in Ω.

Finally, we make no longer the assumption of ϕ = 0 on ∂Ω×(0, T ), obtaining

−∫ T

0

∫∂Ω

D∗ϕ∇p · n = 0.

This relation is satisfied for every function ϕ by imposing:

D∗∇p · n = 0 on ∂Ω× (0, T ).

In conclusion, we can write the following system for p, called adjoint equation:∂p∂t

+D∗∆P = 0 in Ω, t > 0,

D∗∇p · n = 0 on ∂Ω× (0, T ),

p(T ) = c∗(T )− c(T ) in Ω.

(4.18)

From now on, it is possible to forget about the time dependence of the testfunction ϕ and the weak adjoint equation can be formulated as:

to find p ∈ V × [0, T ], with p(T ) = c∗(T )− c(T ) in Ω, such that∫Ω

∂p

∂tϕ dΩ−

∫Ω

D∇p · ∇ϕdΩ = 0, ∀ ϕ ∈ V . (4.19)

Notice that the adjoint equation turns out to be backward in time.Let us now have a look at (4.17):

LD(c∗, D∗, p)ψ = λD∗ψ −∫ T

0

∫Ω

ψ∇c∗ · ∇p dΩdτ = 0, ∀ψ ∈ R+.

We notice that the derivative LD can be conveniently expressed as functionof the strong derivative of the cost functional by means of the Riesz theorem.More precisely, since the space of admissible controls is R+, the derivative ofthe cost functional turns out to be

JD (c(D∗), D∗) = LD(c∗, D∗, p) = λD∗ −∫ T

0

∫Ω

∇c∗ · ∇p dΩdτ. (4.20)

We have skipped the analysis of (4.15), because it can be easily provedthat Lp(c∗, D∗, p)φ = 0 leads to the state equation in weak form (4.12).

67


4.4.4 Iterative method

At this point, we can use a minimization algorithm to find D∗ such thatthe value of LD is close to zero. Consequently, the derivative of J withrespect to D will be close to zero as well. The control problem is solved byan iterative method, in particular our choice is the Non Linear ConjugateGradient (NLCG) method. More precisely, the algorithm can be sketched as

1. provide a starting value for the control variable D0;

2. solve the state equation (4.12) to find c;

3. given c(T ) and c(T ), compute the value of cost functional J (4.13);

4. given c(T ) and c(T ), solve the adjoint equation (4.19) to find p;

5. given p, evaluate LD;

6. if a suitable stopping criterion is satisfied, then exit (Dk is the optimalvalue);

7. update the control variable to Dk+1 and go back to 2,

where k = 0, 1, 2, . . . is the iteration index.

4.4.5 Numerical results

In order to numerically solve the control problem, we decided to implementa routine to reproduce the iterative procedure that has been detailed in theprevious section. Moreover, the solution of the state and adjoint variablecan be computed through the Galerkin-finite element approximation of theirweak formulation. The algorithm has been implemented and the problemhas been solved using FreeFem++, a software for PDEs that uses the finiteelement method.

Code validation

Before employing the code for the estimate of the parameter under inves-tigation by comparing the numerical results with the experimental images,the code needs to be validated by applying the numerical routine to sometest-cases in such a way that we can verify whether it is reliable and robust.Once the validation is achieved, we shall apply our tool to the interestingcircumstances.

68


H 6 mm L 8 mma 2 mm T 60 sh 2 mm cinj 1 mg/ml

Table 4.3: Model parameters for code validation

Figure 4.6: Initial configuration for the code validation

The code validation that has been performed consists of two steps. Firstly,the reference concentration c has been numerically generated in the case ofa known value of D and succesively, the algorithm has been tested on thatreference concentration. The algorithm is reliable if the estimated diffusioncoefficient is close to the real one.

In our case, we computed the concentration c solving problem (4.11) withD = 0.01 mm2/s on a regular mesh and using the values in Table 4.3 forthe other parameters. Figure 4.6 shows the initial condition that has beenimposed. We use the resulting c shown in Figure 4.7(a) as reference con-centration c to test our algorithm. Therefore, we expect that our algorithmcomputes a diffusivity coefficient close to 0.01 mm2/s.

In Table 4.4 we report the results for D and for the cost functional J ,obtained using λ = 10, the values in Table 4.3 for the other parameters anddifferent starting valuesDstart forD. From the results in Table 4.4, we can saythat the algorithm is reliable, because in each case the estimate of the diffusiv-ity coefficient is close to the reference value D = 0.01 mm2/s. Moreover, wenotice that there is no dependence on the choice of the starting value Dstart.This property is very important for the reliability of the numerical tool, be-cause neither the real value of the coefficient nor its order of magnitude isknown a priori. In Figure 4.7(b), we show the concentration at the finaliteration of the minimization algorithm, in the case of Dstart = 0.2 mm2/s.Comparing Figures 4.7(a) and 4.7(b), we observe that they are very similar,

69


(a) Reference concentration c(T ) (b) Optimal concentration c∗(T )

Figure 4.7: Reference and optimal concentrations for the code validation

Dstart D∗ J Niter

1 0.009989 0.000499 240.2 0.009989 0.000499 26

0.014 0.009989 0.000499 170.005 0.009989 0.000499 440.0001 0.009989 0.000499 56

Table 4.4: Diffusivity coefficient D∗ computed solving the minimization prob-lem (4.13) with NLCG, with λ = 10 for code validation

70


λ Dstart D∗ J Niter

25 1 0.009973 0.001246 3725 0.005 0.009973 0.001246 4425 0.0001 0.009973 0.001246 59

50 1 0.009945 0.002486 3350 0.005 0.009945 0.002486 4250 0.0001 0.009945 0.002486 52

100 1 0.009892 0.004945 27100 0.005 0.009892 0.004945 35100 0.0001 0.009892 0.004945 45

Table 4.5: Sensitivity analysis of D∗ with respect to λ for code validation.

since D∗ is very close to the exact D. Observe that the discrepancy betweenthe exact value of D and D∗ is due to the presence of the regularization termwith λ, indeed the minimization needs to be performed on two terms.

Moreover, we have performed a sensitivity analysis of D∗ with respect toλ, to study how much this parameter influences the results and the compu-tational cost. As we can see in Table 4.5, the number of iterations sligthlydecreases as the value of λ grows, however the range for the computationalcost is almost similar. On the other hand, the gap between D∗ and the de-sired value grows, although the relative error on the estimate of D is alwaysbeneath the 1%. Therefore, it is possible to choose λ in quite wide range ofvalue without affecting too much the results both in term of accuracy of theprediction and in term of computational cost.

In conclusion, we can state that the numerical tool we have built is ableto provide an estimate for the diffusion coefficient D which is close to thereal known value. Moreover, the algorithm is insensitive to the choice of theinitial guess Dstart, that means that our tool is reliable and robust. Lastly, theroutine has been tested for different values of the regolarization parameterλ ∈ [10, 100] and we can state that all of these values are equally valid forthe accuracy of the result and for the computational load. Therefore, we canconclude that the next required step is the application of the minimizationroutine to the cases of interest.

Application to real data

Let us apply the minimization algorithm described in the previous sectionsto the case we are interested in. Two main steps can be identified: the image

71


(a) Initial map c (t = 0) (b) Reference map c (T = 300 s)

Figure 4.8: Intensity maps of the diffusion experiment using 4 kDa Dextranmolecules

processing and the minimization procedure.Since we are working with images, an intensity map has to be dealt with

in place of a concentration map. In particular, it is possible to convert theRGB images (Figure 4.5) into a grey scale image in such a way that theconcentration of Dextran molecules associated to each pixel is representedby the level of grey (in a range from 0 to 255) of that pixel. To this aim, theprocedure is subdivided into several phases:

• conversion of the RGB image into a grey scale image;

• creation of a fine mesh of triangular elements of dimensions as thenumber of pixels of the image (nbx pixels by nby pixels);

• computation of the coordinates of the centre of gravity of each element;

• reading of the grey level of the pixel corresponding to each centre ofgravity;

• definition of a finite element variable that accounts for the grey levelof each triangular element: c(T );

• creation of a coarse mesh of triangular elements of dimensions nbx

pixels by nby pixels;

• projection of the grey image on the coarse grid.

Figure 4.8(b) shows the reference intensity map c (T ) that corresponds toFigure 4.5(b) defined as a piecewise constant finite element function on a

72


mesh of 60092 triangular element. Observe that the color bar value representsthe grey scale level of each element, from 0 to 255, that means from black towhite.

For the sake of convenience, the number of pixels has been used as unitof measure for the domain. Therefore, the resulting diffusivity coefficient is

computed as[

pixel2

s

]and succesively its value is converted in

[m2

s

].

The last remark on the model description is about the initial conditionof (4.11), indeed it should be necessary to specify the value of cinj and thedimension of the stripe h and its position. For the definition of these threeparameters, user’s intervention is still needed, indeed cinj could be set tothe average value of the grey scale level of the stripe at the initial time,while the height and the position of the initial band could be set to thecorresponding height and position observable in the experiment. However,the final result on the diffusivity coefficient that guarantees the minimumcost functional could be strongly biased by the choice of these parameters,therefore an alternative approach could be preferable. Instead of enforcingan artificial initial condition, in the following simulations the experimentalinitial configuration (Figure 4.8(a)) will be directly used as c (t = 0). Thisprocedure will avoid the presence of some parameters that could influencethe numerical experiment.

At this point, all the ingredients for the minimization procedure are avail-able and the iterative method can be applied.

As previously explained, the experimental data are about the pure diffu-sion of 4 kDa, 40 kDa and 250 kDa Dextran molecules. In order to take intoaccount the natural variability of the physical trials, several datasets, whichcorrespond to several sets of images, are available and each of them has beenanalysed and tested with our numerical routine.

Let us focus first on the six experiments available for the diffusion of4 kDa Dextran molecules. In Table 4.6, the values of the parameter usedfor the minimization procedure are reported, in particular the regularizationcoefficient λ and the total simulation time T . The simulations have been per-formed with a time step dt = 5 s, the starting value Dstart = 10 pixel2/s of thealgorithm and a suitable conversion rule for the image resolution. The lasttwo columns show the optimal diffusion coefficient found by the algorithmD∗ and the corresponding experimental diffusion coefficient D, computed byV. Lusi and P. Decuzzi. It can be seen that the results of the minimizationprocedure are mutually consistent and consistent also with the experimentalresults. However, the value of D∗ is biased by the quality of the experiments,in terms of the smoothness and the symmetrical behaviour of the Dextraninjection, as can be observed comparing the results and the images of test4A

73


(a) test4A (b) test4C

Figure 4.9: Reference concentration c (T ), T = 300 s for 4 kDa Dextranmolecules: possible smooth and non smooth configurations

Figure 4.10: Concentration c (T = 300 s) with the optimal diffusivity coeffi-cient D∗ = 1.61 · 10−6 cm2/s for test4A

and of test4C (Figures 4.9(a) and 4.9(b)), where the discrepancy betweenthe numerical and experimental D is the smallest and the largest, respec-tively. Hence, the diffusion model only describes symmetrical phenomenon,in particular the numerical concentration looks like Figure 4.10 (concentra-tion with D∗ = 1.61 · 10−6cm2/s for test4A), therefore the gap between D∗

and D is larger when the experiments show some asymmetric or non standardtrends.

In order to check whether the optimal diffusion coefficient is influencedby the choice of other parameters, a sensitivity analysis has been performedevaluating how the variations of Dstart and cinj affect the final result. Weconsidered test4A for the sensitivity analysis. Moving to the cases withDstart = 1 pixel2/s and Dstart = 100 pixel2/s, while keeping constant theother parameters, we found that a variation of the starting value for the

74


test λ [s2/px2] T [s] D∗ [cm2/s] D [cm2/s]

4A 20 300 1, 61 · 10−6 1, 50 · 10−6

4B 20 300 1, 46 · 10−6 1, 00 · 10−6

4C 20 300 1, 83 · 10−6 1, 00 · 10−6

4D 20 270 1, 63 · 10−6 1, 50 · 10−6

4E 20 270 1, 30 · 10−6 1, 00 · 10−6

4F 20 270 1, 25 · 10−6 1, 50 · 10−6

Mean value 1, 51 · 10−6 1, 25 · 10−6

Standard deviation 2, 00 · 10−7 2, 50 · 10−7

Table 4.6: Numerical tests on the Dextran 4 kDa diffusion experiments

algorithm does not modify the optimal diffusivity coefficient, as we expectedsince we have already proven the insensitiveness of D∗ to perturbation ofDstart in the code validation phase.

The results provided in Table 4.6 are satisfactory comparing the estimatesamong different tests and with the corresponding experimental one, howeverin order to check one more time the reliability of the algorithm, we haveapplied the routine on the same test at each available time step. In particular,we considered test4E and we set c (T ) as reference concentration in thecost functional, where T = 30, 60, 90, 120, 150, 180, 210, 240, 270 s. Weexpected the optimal diffusion coefficient not to vary by changing the finaltime, as the pure diffusive mathematical model should describe correctly thephysical phenomenon. The results shown in Table 4.7 prove our idea, indeedD∗ does not change in a significant way.

We have applied the same numerical routine to experiments related tothe diffusion of 40 kDa and 250 kDa Dextran molecules: the value of theparameters and the results are available in Tables 4.8 and 4.9, respectively,from which we can observe that considerations similar to the 4 kDa case canbe done.

4.4.6 Discussion and conclusions

In conclusion, we have applied the optimal control theory in order to predictthe value of the diffusion coefficient of Dextran molecules in collagen, mini-mizing the discrepancy between the concentration of molecules available interms of images and the concentration that have been numerically simulated.The minimization problem has been solved using an iterative procedure basedon the non linear conjugate gradient method, the code has been validated on

75


T [s] dt [s] D∗ [cm2/s]

30 2 1, 36 · 10−6

60 2 1, 44 · 10−6

90 2 1, 49 · 10−6

120 2 1, 42 · 10−6

150 2 1, 41 · 10−6

180 2 1, 34 · 10−6

210 2 1, 33 · 10−6

240 2 1, 31 · 10−6

270 2 1, 30 · 10−6

Mean value 1, 38 · 10−6

Standard deviation 6, 14 · 10−8

Table 4.7: test4E - Numerical tests at different final time T


40A 20 270 7, 93 · 10−7 1, 00 · 10−6

40B 20 270 3, 98 · 10−7 5, 00 · 10−7

40C 20 270 9, 85 · 10−7 1, 50 · 10−6

40D 20 270 8, 24 · 10−7 3, 00 · 10−7

40E 20 270 1, 34 · 10−6 1, 50 · 10−6

40F 20 270 4, 07 · 10−7 1, 00 · 10−6

Mean value 7, 91 · 10−7 9, 67 · 10−7




250A 20 270 3, 80 · 10−7 2, 00 · 10−7

250B 20 270 4, 43 · 10−7 3, 50 · 10−7

250C 20 270 6, 26 · 10−7 1, 00 · 10−7

250D 20 270 3, 86 · 10−7 2, 00 · 10−7

250E 20 270 3, 23 · 10−7 2, 50 · 10−7

250F 20 270 4, 95 · 10−7 5, 00 · 10−7

Mean value 4, 42 · 10−7 2, 67 · 10−7



76


a test case in order to verify its reliability and some sensitivity analysis onthe starting value for the diffusivity coefficient and on the regularization pa-rameter have been performed. Once the code has been validated, the methodhas been applied to the real cases, namely several experiments on the diffu-sion of 4 kDa, 40 kDa and 250 kDa Dextran molecules, obtaining that thenumerical algorithm is able to predict in an accurate way the value of thediffusion coefficient.

77

Chapter 5

Numerical results

In this chapter, the computational model for mass transport and the esti-mates of some parameters that we have developed in the previous chapterswill be applied to concrete cases and the results will be shown and inter-preted.

The description of the motion of nanoparticles and their adhesive prop-erties in a capillary network, or in a more general way in a vessel network,is the fundamental objective of the current work. Therefore, in Section 5.1the behaviour of the nanoparticles in the vessel network will be simulated,neglecting the possibility for the particle to extravasate in the interstitialtissue and observing the effects of different expressions for the probability ofadhesion and of the effective vascular adhesion parameter on some interestingquantities.

The final aim that guides the need of injectable nanoparticles in nano-medicine is their use for drug delivery systems or as carrier for imaging agents.This is the main reason why there is the need to perform some simulationsthat describe the extravasation of drugs or imaging agents in the interstitialtissue, as described in Section 5.2. To this end, we will consider the Dextranmolecule as agent that needs to be carried and it will be used in place ofany imaging agent or drug. The choice of Dextran molecules is motivatedby the availability of data and parameters from small-scale experiments (asin the previous chapters). Moreover, Dextran molecules are actually used inmany experimental setup since they can be easily made fluorescent. However,for future development, Dextran molecules can be replaced by any othersubstance in the interstitial tissue, if needed.

To complete the discussion and the model about the drug delivery systemby means of nanoparticles, a more realistic situation will be numerically sim-ulated in Section 5.3. The process will account for the transport of nanopar-ticles in the vessel network and their adhesion to the vessel wall according to

78

CHAPTER 5. NUMERICAL RESULTS

the adhesive properties of the nanoparticles. Once the particles are in directcontact with the vessel wall, the drug release in the tissue is modeled as asource term that depends on the drug flux released by a single particle andthe density of particles adhering to the wall and the transport of drug is thensimulated in the interstitial tissue.

For the sake of clarity, we report in Figure 5.1 the sketch of the threemain classes of simulations we have performed:

(A) transport of nanoparticles in the vascular network and their adhesionto the vessel walls, according to different adhesive properties (Section5.1);

(B) transport of Dextran in the capillary network and its extravasation inthe tissue, considering different molecular weights (Section 5.2);

(C) transport of nanoparticles in the network, their adhesion, delivery ofDextran molecules by the NPs and Dextran transport in the intersti-tial tissue (Section 5.3).

5.1 Nanoparticle transport and adhesion

The main objective of the current section is to apply the computational model(2.26) to investigate the transport of nanoparticles in the vessel network andtheir adhesion to the vascular wall. In particular, the influence of the differentexpressions for the probability of adhesion and for the vascular adhesionparameter on the most interesting quantities that characterize the adhesionproblem will be presented.

For the reconstruction of the geometrical model of the vessel network,we use the data for a R3230AC mammary carcinoma in rat dorsal skin flappreparation, available in [49]. We consider a dataset, labelled as rat93, thatrepresents the microvascular structure over a region of dimensions 270×370×200µm, as shown in Figure 5.2.

In the following, the description of the numerical experiments that havebeen performed is presented, detailing which terms of the flow and transportequations have been activated and which values for the parameters have beenused. In particular, in Section 5.1.1 we will shortly focus on the fluid dynam-ics variables of the system and successively, we will move to the transportequations. Section 5.1.2 aims at comparing two test cases under differentconditions using the exponential formula for the probability of adhesion andthe vascular adhesion parameter without saturation. Lastly, the simulationsin Sections 5.1.3 and 5.1.4 will be performed exploiting the values of the

79


Tissue NP

(A)

(1/m2)

Particle adhesion (probability Pa)

Tissue Dextran

Diffusion (Dt)

(g/m3) (g/m3)

(B)

Tissue

NP (C)

Diffusion (Dt)

(1/m2) (g/m3)

Particle adhesion (probability Pa)

Dextran

Figure 5.1: Sketch of the three main classes of simulations: (A) nanoparticletransport and adhesion; (B) transport of Dextran and extravasation; (C)nanoparticle transport, adhesion and Dextran delivery.

80


Figure 5.2: rat93 geometry

test1 test2 test3 test4

PaExplicit Explicit Lattice Latticeformula formula Boltzmann Boltzmann

Pressure0.20 mmHg 1.35 mmHg 5 mmHg 5 mmHg

drop

Ligandlow high - -

density

Radius R 7.64 µm 7.64 µm 30 µm 30 µm

Πeff No saturation No saturation No saturation Saturation

Table 5.1: Sketch of the tests for nanoparticle transport and adhesion (Sim-ulation A)

probability of adhesion provided by the Lattice-Boltzmann simulation, whilecomparing the effective vascular adhesion parameter without and with thesaturation, respectively.

In Table 5.1, we outline the main similarities and differences among thefour tests that are presented in the following sections for the description ofthe transport of nanoparticle in the vessel network and their adhesion to thecapillary walls.

5.1.1 Fluid dynamics effects

In order to deeply comprehend the transport and adhesion properties of thenanoparticles in the vessel network, it is necessary to observe the behaviourof the most important fluid dynamic variables under suitable flow conditions.

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(a) test1 (b) test2

Figure 5.3: Pressure field pv [mmHg] in the vessel network

(a) test1 (b) test2

Figure 5.4: Velocity field uv [µm/s] in the vessel network

The fluid dynamical variables influence the Reynolds number and the wallshear rate that are the most important parameters for the analysis of theprobability of adhesion, for both the formulations that have been proposed.Hence, let us first consider the flow equations (cfr. Appendix A).

The characteristic length of the problem is chosen as d = 50µm. Ac-cording to [10], a vascularized healthy tissue is characterized by an averageinterstitial pressure P = 1 mmHg and by a characteristic flow speed in thecapillary bed of U = 200µm/s. The Reynolds number has been computedas

Re =ρuvR

µv, (5.1)

where uv is the local velocity of the fluid in the vessel network, ρ = 1025 kg/m3

is the density of the blood, µv = 0.004 kg/ms is the plasma viscosity. Theradius of the capillary vessels is assumed to be constant over the whole net-work and set to R = 7.64µm. The other parameters that characterize the

82


(a) test1 (b) test2

Figure 5.5: Wall shear rate γ [1/s] in the vessel network

flow properties of the system are the hydraulic conductivity of the intersti-tium, k = 10−18 m2 and the hydraulic conductivity of the capillary walls,Lp = 10−12 m2s/kg. Accounting for the osmotic effects, we set the reflec-tion coefficient, σ = 1, the average osmotic pressure in the vessel network,πv = 28 mmHg, and in the tissue interstitium, πt = 0.1 mmHg. For thelymphatic term, we set the hydraulic conductivity of the lymphatic walltimes the surface area of the lymphatic vessels per unit volume of tissue,LLFp

SV

= 1.042 · 10−6 kg/s and the hydrostatic pressure within the lymphaticchannels, pL = 0 mmHg. Lastly, we need to specify some boundary condi-tions. At this moment, we introduce two different sets of boundary condi-tions, that bring to two different simulations which can be identified as test1and test2, respectively. For the capillary flow, we aim at enforcing a suitablepressure gradient along the network. Observing that the inflow and outflowsections of the network lay on the lateral side of the tissue slab, we enforce agiven pressure pin = 35 mmHg on two adjacent faces for both the tests and apressure pout = 34.80 mmHg on the opposite ones for test1 and pout = 33.65mmHg for test2. By this way, the pressure drops pin − pout = 0.20 mmHgand pin−pout = 1.35 mmHg are enforced at the tips of the network in the caseof test1 and test2, respectively. For the interstitial flow, we aim at model-ing the in-vivo configuration, where the available tissue sample is embeddedinto a similar environment. To represent this case, we enforce Robin-typeboundary conditions for the interstitial pressure with the far field pressurep0 = 0 mmHg and the coefficient βt = 5 · 10−11 m2s/kg.

Let us show the most important fluid dynamic variables in the vesselnetwork, in order to better understand the global behaviour of the system inboth the cases test1 and test2. In view of a smooth trend of the pressurefield pv (Figures 5.3(a) and 5.3(b)) in Λ, which reproduces the boundaryconditions that have been enforced at the tips of the network, the behaviour

83


of the velocity field uv (Figures 5.4(a) and 5.4(b)) is more heterogeneousover the branches. There exist some almost dead branches where the valueof the velocity is very low, possibly due to the configuration of the network,and some paths in which the blood flows more easily. Figures 5.5(a) and5.5(b) show the local shear rate, computed as (4.2); we recall that this termis involved in the computation of the vascular adhesion parameter Π and inthe expression for Pa.

5.1.2 Explicit formula for Pa

According to the two adopted approaches for the probability of adhesion Padescribed in Section 4.1, we will now move to the mass transport equationsand we will consider the effect of the adhesion term, in particular we willfirstly focus on the explicit formulation for Pa, given by formula (4.9).

We set the characteristic pressure drop along the network δP = 0.20mmHg and δP = 1.35 mmHg for test1 and test2, respectively, and thenominal injected concentration of nanoparticle C = 1 #/m3. The diameterof the nanoparticles that have been injected is dp = 2µm and their diffusivityin the network is Dv = 2.22·10−13 m2/s. For the adhesion properties, we needto specify the value of the parameters involved in formula (4.9), specificallywe set α2 = 3.4 ·109 m−2, βNP = 5.97 ·1010 s2/kg m, while mlK

0ar

20 = 1.2585 ·

10−12 m2 for test1 and mlK0ar

20 = 1.2585 · 10−9 m2 for test2. Therefore we

set test1 to represent a configuration with low pressure drop at the tips ofthe network and low ligand density on the nanoparticle surface and test2 torepresent a configuration with high pressure drop at the tips of the networkand high ligand density on the nanoparticle surface. Moreover, we enforcea nominal value of injected nanoparticle concentration cinj = 1 #/m3 atthe inflow tips of the network during the simulation time [0, T ] = [0, 12 h],while the particles are set free to leave the system from the outflow tips. Asalready underlined, the numerical simulation of nanoparticle delivery aimsat describing the transport and the adhesion of the NP in the vessel bed.On the contrary, it does involve neither the extravasation of the particles inthe intersitial tissue, since the dimensions of the nanostructures do not allowthis process, nor the extravasation of solutes carried by the nanoparticles,since we are not modeling the drug release so far. Therefore, in the followingsimulation the value of the vascular permeability coefficient has been set tozero, P = 0 m/s. As a consequence of the choice of the value of P and σ, noparticles will circulate in the interstitial region in [0, T ], since the boundaryconditions for the tissue slab is of Neumann-type with βc = 10−5 m/s.

Let us show the variables that are involved in the adhesion process, fo-cusing our attention on the first model for the effective vascular adhesion

84


(a) test1 (b) test2

Figure 5.6: Probability of adhesion Pa [-] in the vessel network, given by (4.9)

(a) test1 (b) test2

Figure 5.7: Vascular adhesion parameter Π [µm/s] in the vessel network,computed as (4.1) without saturation model

(a) test1 (b) test2

Figure 5.8: Concentration of nanoparticles cv [#/m3] at the final time t = 12h in the vessel network

85


(a) test1 (b) test2

Figure 5.9: Density of nanoparticle adhering per unit surface to the vascularwall Ψ [1/m2] at the final time t = 12 h, without the saturation model

parameter, that means Π with no time dependence and no saturation, as in(4.1) and the explicit expression for Pa, as in (4.9). We report in Figures5.6(a) and 5.6(b) the value of the probability of adhesion Pa for test1 andtest2, respectively, the results clearly represent the exponential trend of theprobability of adhesion parameter with respect to the wall shear stress. Inorder to compare the adhesion preperties of test1 and test2, it is impor-tant to observe that in the first case the absolute value of the velocity, andconsequently of the wall shear rate, is smaller than in the second case, whilethe ligand density on the particle surface in the first case is three orders ofmagnitude less than the value in the second case. This comparison aims atexploiting the balance between the adhesive and the fluid dynamics proper-ties in the formula of Pa. Hence, it is important to notice that in test1 thetrend of Pa is more homogeneous than in test2, moreover, in the first casePa < 1 even though its order of magnitude is in [10−7, 10−2], while in thesecond case the maximum of Pa is greater than one and its range is widerthan the previous case. These remarks highlight the problems of the explicitformula for Pa that have been exploited in the description of the method.

From Figures 5.7(a) and 5.7(b), it can be observed that the vascularadhesion parameter Π depends on both the probability of adhesion and thewall shear rate induced by the interaction of blood flow with the capillarywalls, even though the most important contribution is given by Pa, since thebranches of minimum Π correspond to the branches of minimum Pa and thesame occurs for the maximum.

Figures 5.8(a) and 5.8(b) show the nanoparticle concentration profile cvafter 12 h of injection at the inflow tips of the network and Figures 5.9(a)and 5.9(b) the corresponding density of nanoparticles adhering to the vesselwall. Also for these variables, the distinction between homogeneity of test1

86


and the non homogeneity of test2 is noticeable, indeed the concentration oftest2 and consequently Ψ (t = 12 h) reflect the high value of Pa in the deadbranch, where the particles are not able to flow through. A part from thiscase, the nanoparticles are able to flow across the network in a smooth wayfor test1 with Ψ (t = 12 h) different from zero in several branches, while fortest2 Ψ (t = 12 h) is significantly different from zero only in the dead branch.However, the non saturated vascular adhesion parameter Π does not implythe problems that have been detailed in the previous section, because of thepresence of Pa that is several orders of magnitude lower than one.

5.1.3 Pa from LB approach

Once we have underlined the problems related to the choice of Pa with theexponential formula, we will move to the second formulation for the prob-ability of adhesion, namely the values provided by the Lattice-Boltzmannsimulations. Besides these observations, the other purpose of the followingcase, called test3, is the analysis of the unphysical behaviour of the effectivevascular adhesion parameter without saturation, that leads to the idea ofintroducing a time-dependent Πeff .

The values of the parameters that have been used for the simulationtest3 can be seen in Table 5.2 and 5.3, for the flow and transport equations,respectively, while in Table 5.4 the characteristic values for the non dimen-sional analysis are reported. Recall the fluid dynamical variables influencethe Reynolds number that is one of the most important parameter for theanalysis of the probability of adhesion. To this end, the parameters have beenset in such a way to obtain Re in an interesting range of values, in particularin the range that corresponds to the arteriole flow, where the values for Pachanges in a more significant way. In particular, the pressure drop acrossthe vessel network has been enhanced and the radius of the vessels has beenartificially increased as well, leading to a 1D configuration that is equivalentto the dataset rat93, while its 3D reconstruction will represent a bigger slabwith a network of arterioles. Indeed, the choice of a bigger vessel radius in thesame tissue domain would have led to a too big vessel network with respectto the dimensions of the 3D domain for which a 1D representation wouldhave been unphysical. For this reason, the characteristic length of the prob-lem has been increased as well, leading to a bigger domain (1 × 1.48 × 0.8mm), in such a way that the ratio between the non-dimensional radius ofthe vessels and the non-dimensional sizes of the domain is acceptable for a1D/3D representation.

For the adhesion properties, we computed the value of the probability ofadhesion Pa from the data obtained in the Lattice-Boltzmann simulation, as

87


Figure 5.10: test3 - Reynolds number Re [-] in the vessel network

(a) Pa [−] (b) Π [mm/s]

Figure 5.11: test3 - Adhesive variables in the vessel network: probabil-ity of adhesion given by the LB approach and vascular adhesion parametercomputed as (4.1) without the saturation model

in Section 4.2, in the case of percentage of particle surface by ligands ρl = 0.5for a strong ligand-receptor bond. Moreover, we enforced a nominal value ofinjected nanoparticle concentration cinj = 1 #/m3 at the inflow tips of thenetwork during the simulation time [0, T ] = [0, 12 h], while the particles areset free to leave the system from the outflow tips. The choice of P , σ andthe Robin-type tissue boundary conditions lead also in this case to the lackof particles in the interstitial region in [0, T ].

First of all, it is interesting to highlight the Reynolds number Re, com-puted as in (5.1), in Figure 5.10. We observe that Re falls into the rangeof values for which non trivial values of probability of adhesion Pa are avail-able, that also corresponds to the range in which the balance between thefluid dynamic forces and the adhesive interactions occurs. Moving to thetransport equations and the adhesion of nanoparticles, we report in Figure5.11(a) the value of the probability of adhesion Pa, whose value is always lessthan one accordingly to the provided data, moreover its value is about 10−1

88


Symbol Parameter Units Value

ρ Blood density kg/m3 1025µv Plasma viscosity kg/ms 0.004R Capillary radius µm 30k Hydraulic conductivity, interstitium m2 10−18

Lp Hydraulic conductivity, vessel wall m2s/kg 10−12

σ Reflection coefficient - 1πv Osmotic pressure, vessel network mmHg 28πt Osmotic pressure, interstitium mmHg 0.1LLFp

SV

Hydraulic conductivity, lymphatic vessel kg/s 1.042 · 10−6

pL Hydrostatic pressure, lymphatic vessel mmHg 0pin Inflow pressure, vessel network mmHg 35pout Outflow pressure, vessel network mmHg 30p0 Far field pressure, interstitial BC mmHg 0βt Coefficient, interstitial BC m2s/kg 5 · 10−11

Table 5.2: Physical parameters characterizing the fluid dynamics problem intest3


Dv Vascular diffusivity m2/s 2.22 · 10−13

dp Nanoparticle diameter µm 2P Vascular permeability m/s 0βc Coefficient, interstitial BC m/s 10−5

cinj Injected concentration #/m3 1

Table 5.3: Physical parameters characterizing the nanoparticle transportproblem in test3


d Characteristic length µm 200P Characteristic pressure mmHg 1U Characteristic velocity µm/s 200δP Characteristic pressure drop mmHg 5

C Characteristic concentration #/m3 1

Table 5.4: Characteristic values for the non dimensional analysis in test3

89


(a) test3: No saturation, t = 12 h

(b) test4: Saturation, t = 6 h (c) test4: Saturation, t = 12 h

Figure 5.12: Concentration of nanoparticle cv [#/m3] at time t in the vesselnetwork

(a) test3: No saturation (b) test4: Saturation

Figure 5.13: Density of nanoparticle adhering per unit surface to the vascularwall Ψ [1/m2] at the final time t = 12 h

90


as order of magnitude, thus we expect to have a more significant influence ofthe adhesive term with respect to the previous tests. Hence, Figure 5.11(b)shows that the vascular adhesion parameter Π takes value that are severalorders of magnitude higher than the previous tests. Let us now focus on theconcentration of nanoparticle in the vessel network and the density of NPadhering to the vessel wall. Figure 5.12(a) shows the nanoparticle concentra-tion profile cv after 12 h of injection at the inflow tips of the network. Thebehaviour of the nanoparticle concentration reflects some observations thathave already been detailed in Section 4.1.2. In particular, even though thelong injection time interval (12 h), the concentration of nanoparticle is nothomogenoeus in the network and the concentration at the outflow is muchless than the concentration at the inflow, indeed we would expect to havea concentration approximately as cinj in the whole vessel bed after a longtime. This effect can be explained by observing that the sequestration termΠ of the nanoparticles does not take into account any upper bound for thenumber of ligand-receptor bonds that can be done. As a consequence, aftera short transient, the nanoparticle profile in the network reaches a steadystate as depicted in Figure 5.12(a) and the density of nanoparticle adheringto the vessel wall Ψ (Figure 5.13(a) at time t = 12 h) grows without reachinga maximum value. As a matter of fact, a similar cv plot can be seen per-forming the corresponding stationary simulation, which aims at describingthe configuration after an infinite injection time.

5.1.4 Effective vascular adhesion parameter andsaturation

The second step towards a more realistic model for the adhesion of thenanoparticles is the introduction of a correction on the previous model thattakes into account the saturation of the ligand-receptor bonds in the struc-ture of the effective vascular adhesion parameter, as detailed in Section 4.1.2.We performed a simulation, called test4, over a time interval of 12 hourswith Πeff as in (4.5) and Ψ as in (4.6). The maximum value for the density ofnanoparticle adhering to the vessel wall that has been used is Ψmax = 7.4 m−2,corresponding to the maximum Ψ of a simulation performed over 2 hoursexploiting the model without saturation. Notice that the concentration ofnanoparticle in the vessel network after 2 hours in the case of linear adhe-sion parameter corresponds to the steady solution. Except for the effectivevascular adhesion parameter, test4 is equivalent to test3, thus the physicalparameters characterizing the simulation can be seen in Tables 5.2, 5.3 and5.4. In Figures 5.12(b) and 5.12(c), the nanoparticle concentration in the

91


vessel bed at time t = 6 h and at the final time t = 12 h is depicted. Theeffect of the saturation model can be clearly seen, indeed the distribution ofthe nanoparticles is more homogeneous than the previous case and the valueof the concentration at the outflow tips of the network is significantly higherthan the model without saturation. Moreover, even though the value of cvis not yet similar to cinj in the whole Λ, comparing 5.12(b) with 5.12(c) itis possible to observe that the network is gradually filling, contrary to whatwas happening in the case without saturation. Figure 5.13(b) shows thatthe density of nanoparticle adhering the vascular wall does not exceed themaximum value Ψmax. The plot reflects the trend of the vascular adhesionparameter, in turn revealing the presence of some dead branches in whichthe adhesion does not occur.

5.1.5 Conclusions

To sum up, we have performed four numerical simulations for the systemthat describes the transport of nanoparticles in a vessel bed and their adhe-sion to the vascular wall, introducing different formulations for the adhesionprobability coefficient Pa and for the effective vascular adhesion parameterΠeff . Hence, in test1 and test2, where the explicit formula for Pa has beenused, the main disadvantages of this formula have been observed, while intest3 and test4, where the values of Pa computed at the microscopic levelhave been included, the physical problems related to the vascular adhesionparameter and the use of a saturation model on the maximum number ofligand-receptor bonds on the vascular wall have been detailed.

Focusing on test1 and test2, we can conclude that Pa does not guaranteethe probability of adhesion to be in the range of a real probability, namely[0, 1], and the exponential formula does not allow the balance between thefluid dynamics quantities and the adhesive interactions to be easily analysed.

Regarding the comparison between test3 and test4, it is possible to inferthat the model for the saturation is needed, because the current model doesnot guarantee the concentration to fill the whole network, even though thelong injection time. A linear saturation model that accounts for the presenceof a maximum value of density of adhering nanoparticles on the wall hasbeen implemented and the results are satisfactory from the physical point ofview. Up to now, the value of the upper bound on Ψ has been set to thecorresponding value of a steady solution of the problem without saturation,nevertheless it could be interesting to develop a more sophisticated estimatefor Ψmax, possibly linked to some physical quantities at the microscale.

92


test5 test6

Dextran40 kDa 250 kDa

molecular weight

Tissue Diffusivity Dt 7.91 · 10−11 m2/s 4.42 · 10−11 m2/sfrom experiments

Permeability P 6 · 10−7 m/s 2 · 10−7 m/s

Table 5.5: Sketch of the tests for Dextran transport and extravasation (Sim-ulation B)

5.2 Dextran transport and extravasation

The current section aims at applying the computational model (2.26) in orderto examine the transport of Dextran molecules in the vessel network and theirextravasation and transport in the interstitial tissue that is represented by acollagen matrix, modeled as a porous medium.

In the following, we will use the dataset rat93 as geometrical setting forthe description of a complete vessel network, available in [49] and shown inFigure 5.2. In particular, a real complex network geometry for the vesselrepresentation has been used and the effect of the presence of the permeablemembrane on the concentration of molecules in the interstitial tissue will bepresented.

The general transport equations, which have been detailed in Chapter2, have been obtained for the description of the nanoparticle transport andadhesion to the vessel wall. However, the equations also include a possibleexchange of materials from the vessel to the tissue and viceversa owing tothe permeability of the vessel wall, even though in the previous case thisproperty has not been exploited. Nevertheless, the transport equations canbe easily extended to the case of interest, by means of neglecting the adhesionterm, namely setting Πeff ≡ 0, while considering the value of the reflectioncoefficient σ and the permeability P different from 1 and 0, respectively.

In Table 5.5, we outline the main similarities and differences among thetwo tests that are presented in the following sections for the description of thetransport of Dextran molecules in the vessel network and their extravasationin the interstitial tissue.

93



ρ Blood density kg/m3 1025µv Plasma viscosity kg/ms 0.004µt Collagen viscosity kg/ms 0.4R Vessel radius µm 30k Hydraulic conductivity, interstitium m2 10−18

Lp Hydraulic conductivity, vessel wall m2s/kg 10−12

σ Reflection coefficient - 0.9πv Osmotic pressure, vessel network mmHg 28πt Osmotic pressure, interstitium mmHg 0LLFp

SV

Hydraulic conductivity, lymphatic vessel kg/s 1.042 · 10−6

pL Hydrostatic pressure, lymphatic vessel mmHg 0pin Inflow pressure, vessel network mmHg 35pout Outflow pressure, vessel network mmHg 30p0 Far field pressure, interstitial BC mmHg 0βt Coefficient, interstitial BC m2s/kg 5 · 10−11

Table 5.6: Physical parameters characterizing the fluid dynamics problem intest5

5.2.1 Available data and results

In the following test, labelled test5, we will consider the transport of 40 kDaDextran molecules in a complex vascular network and their distribution inthe surrounding interstitial tissue. Let us consider the geometry rat93 forthe description of the vessel network immersed in a tissue slab of dimensions1× 1.48× 0.8 mm, with an average capillary radius R = 30µm.

The behaviour of the Dextran concentration in the network and in theinterstitium is governed by coupled advection-diffusion equations, thereforewe firstly need to specify the parameters that have been used in the flowequations and then move to the transport equations. Concerning the fluiddynamics equations, we refer to the formulation given in Appendix A andwe report in Table 5.6 the values for the main parameters. In particular,for our current application, it is important to focus also on the parametersthat characterize the interstitial tissue, since we are interested in observingthe concentration in that region. Numerical results for the pressure field andthe fluid velocity both in the vessels and in the surrounding tissue are givenin Figures 5.14(a) and 5.14(b). As expected, the pressure field in the vesselgradually changes according to the pressure gradient enforced as boundarycondition and the pressure field in the tissue smoothly decreses from itsmaximum to its minimum as well. The velocity is discontinuous at the vessel

94


(a) Pressure

(b) Velocity

Figure 5.14: test5 - Pressure field [mmHg] and velocity field [µm/s] in thevessel network and in the tissue interstitium

junctions, while in the tissue ut is outgoing from the vessels.Let us now focus on the computational model for the transport equations,

without the adhesion term as specified in the previous section. A nominalconcentration of 40 kDa Dextran molecules (cinj = 1 g/m3) is injected at theinflow tips of the vessel network and the other parameters for the transportequations are available in Table 5.7. We notice that the value for the dif-fusion coefficient of 40 kDa Dextran molecule in the collagen, Dt, has beentaken from the experimental and numerical tests that have been performedin Chapter 4, while the value of the diffusivity coefficient in the blood, Dv,has been set to a value that is two order of magnitude higher than Dt. Theexchange of molecules from the vessel to the tissue and viceversa is regulatedby the vascular permeability P that is in this case different from zero and

95




Dt Tissue diffusivity m2/s 7.91 · 10−11

P Vascular permeability m/s 6 · 10−7

βc Coefficient, interstitial BC m/s 10−5

cinj Injected concentration g/m3 1

Table 5.7: Physical parameters characterizing the 40 kDa Dextran moleculestransport problem in test5

that is estimated by the data available in [37]. Moreover, the reflection coef-ficient σ 6= 1 activates also the exchange of molecules by means of pressuredrop and osmotic pressure drop across the vessel wall. The concentrationscv and ct have been computed in a simulation time interval [0, T ] = [0, 6 h],with a time step ∆t = 10 s. For this type of simulation, the concentration ofDextran molecules in the vessel network and in the interstitial tissue, cv andct, are the only non trivial variables for which a visualization is necessary.The transport of Dextran molecules in the arterioles is almost instantaneous,hence at the first time iteration, namely after 10 s, the network is almostcompletely filled by a concentration cv ' 1 g/m3 that corresponds to the in-jected concentration, as shown in Figure 5.15(a). On the other hand, it canbe seen that the transport of molecules in the surrounding tissue is slowerthan in the vessel, in the same figure also ct at the first iteration (t = 10s) is depicted. As expected, the concentration in Ω is different from zero,due to the presence of the permeable vessel wall and it gradually decresesfrom the regions next to the vessel wall to the outer boundary of the tissueslab. Moving further in time, the Dextran molecules progressively diffuse andare advected by the velocity field in the tissue, therefore the concentrationenhances in Ω up to a configuration representing an almost homogeneousmaterial filled by a unit concentration ct, as shown in Figure 5.15(b) at theend of the simulation time (t = 6 h).

In order to be able to compare different trends of Dextran molecules inthe interstitial tissue, the same experiment has been also performed using250 kDa Dextran molecules as tracer. The only parameters that have beenchanged with respect to test5 are the diffusivity of the molecules in thevessels and in the tissue and the permeability coefficient of the membrane.Their values are available in Table 5.8 and we refer to the current test astest6. Figures 5.15(c) and 5.15(d) show the concentration in the vessel andin the tissue at the first iteration (t = 10 s) and at the final iteration (t = 6h).

96


(a) test5: 40 kDa, t = 10 s (b) test5: 40 kDa, t = 6 h

(c) test6: 250 kDa, t = 10 s (d) test6: 250 kDa, t = 6 h

Figure 5.15: Concentration cv and ct [g/m3] in the vessel network and inthe tissue interstitium at different simulation time for 40 kDa and 250 kDaDextran molecules



Dt Tissue diffusivity m2/s 4.42 · 10−11

P Vascular permeability m/s 2 · 10−7

Table 5.8: Physical parameters characterizing the 250 kDa Dextran moleculestransport problem in test6

97


time [h]0 1 2 3 4 5 6

norm

aliz

ed m

ean v

alu

e

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

40 kDa250 kDa

Figure 5.16: Comparison of the mean concentration 1|Ω|

∫ΩctdΩ in the tissue

region over time for 40 kDa and 250 kDa Dextran molecules release (test5and test6)

In view of the final aim of the Dextran molecules injection, namely thedrug delivery in the interstitial tissue, it is useful to observe which is themolecules timecourse in the region of interest for molecules that differs inmolecular weight. In Figure 5.16, we can visualize the timecourse of thenormalized mean concentration available in the tissue, namely the volumetricintegral of 40 kDa and 250 kDa Dextran concentration averaged on the tissuedomain. As expected, the smaller molecules fill the tissue in a shorter timethan the larger molecules, indeed 40 kDa Dextran molecules reach the plateau(98% of the injected concentration) in approximately three hours, while forthe larger molecules the filling takes six hours and a less percentage (93%)of the injected concentration is available as maximum value in the tissue.It is noteworthy that some quantitative information can be extracted fromthe time profile of concentration in the tissue: once defined a tracer and thematerial properties of the fluid and the tissue, given a target concentrationvalue in the tissue, the time interval for the injection is determined by theprofile.

98


5.2.2 Conclusions

To summarize, we have simulated the transport of Dextran molecules in thevessel network, their extravasation through the vascular permeable mem-brane and their transport in the interstitial tissue. The tissue has beenmodeled as collagen in order to be able to exploit the diffusivity coefficientwe have numerically studied in Section 4.4. We have represented the concen-tration of molecules in the tissue in the case of different molecular weights(40 kDa and 250 kDa) and we have visualized the Dextran mean concen-tration value in the interstitial tissue over the simulation time. This plotprovides a quantitative estimate on the injection time needed for reaching acertain amount of a given molecule in the tissue, potentially for therapeuticalpurposes.

5.3 Drug delivery: combined nanoparticles

and Dextran

The purpose of the current section is the application of a modified versionof the computationl model (2.26) to a case that combines the transport ofnanoparticles and their adhesion in the vessel network, as seen in Section5.1 and the transport of Dextran molecules in the interstitial tissue, as seenin Section 5.2. The two equations are no more directly coupled as in theprevious case, but the source of Dextran in the tissue is related to the densityof particles adhering to the vessel wall and the Dextran release rate of a singlenanoparticle.

In the following, we will firstly present the model equations that we willuse for the description of the phenomenon (Section 5.3.1) and then the resultsof the simulations for different type of Dextran molecules (Section 5.3.2).

In Table 5.9, we outline the main similarities and differences among thethree tests that are presented in the following sections for the description ofthe transport of nanoparticles in the vessel network, their adhesion to thevessel walls, the delivery of Dextran molecules and the transport of Dextranin the interstitial tissue.

5.3.1 Model

Let us present the model equations that have been used for the simulationand which modifications have been performed on the global computationalmodel (2.26).

99


test7 test8 test9

NP diameter dp 2 µm 2 µm 2 µm

Vascular diffusivity Dv 2.22 · 10−13 m2/s 2.22 · 10−13 m2/s 2.22 · 10−13 m2/s

Dextran4 kDa 40 kDa 250 kDa

molecular weight

Tissue diffusivity Dt 1.51 · 10−10 m2/s 7.91 · 10−11 m2/s 4.42 · 10−11 m2/sfrom experiments

Table 5.9: Sketch of the tests for nanoparticles transport, adhesion and Dex-tran delivery (Simulation C)

We refer to the standard notation for the domain, where Ω is the three-dimensional domain for the tissue and Λ is the one-dimensional domain forthe vessel network.

Firstly, we need to describe the model accounting for the nanoparticletransport in the blood stream and their adhesion to the wall. It is possibleto exploit the same transport equation that has been used in Section 5.1,namely the concentration of nanoparticles cv needs to satisfy the followingequation:

∂cv∂t

+∂

∂s

(cvuv −Dv

∂cv∂s

)+

2πR

πR2Πeffcv = 0 on Λ× (0, T ), (5.2)

where the concentration cv is expressed as a number of particles per unitvolume

[#m3

], together with suitable boundary conditions on Λin and Λout.

Once the problem for particle transport and adhesion is solved, it is pos-sible to compute the density of nanoparticles adhering per unit surface tothe wall as usual:

Ψ (s, t) =

∫ t

0

Πeff (s, τ) cv (s, τ) dτ.

At this stage, a description of the drug release model is needed. Theparticles decorating the capillary wall are loaded with Dextran moleculesand they are able to release it to the surrounding tissue, in particular we candescribe the action of Dextran loaded nanoparticles as a source term J (t) inthe equation for the concentration of Dextran molecules ct, as follows:

∂ct∂t

+∇ · (ctut −Dt∇ct) + LLFpS

V(pt − pL) ct = 2πRJ (t) δΛ in Ω× (0, T ),

(5.3)

100


where ct is expressed as a mass of molecules per unit volume[gm3

], together

with suitable boundary conditions on ∂Ω, usually Robin boundary condi-tions.

The total drug release rate per unit surface J (s, t) is assumed to beinfluenced by the the combination of the density of nanoparticles adheringto the wall Ψ (s, t) and the flux delivered by a single particle JNP (t), i.e.

J (s, t) = Ψ (s, t) JNP (t) .

In order to determine the expression for JNP , we refer to the problem ofdetermining the release profile of a loaded particle, for which we decided touse a power law model with saturation, as in [40]. In particular, the amountof drug released q (t) is given by

q (t) =tb

tb + χcdexNPVNP ,

where the product of cdexNP , the total Dextran concentration inside the nanopar-ticle, and VNP , the volume of the nanoparticle, turns out to be the totalDextran load of a nanoparticle. The other parameters b and χ are related tosome properties of the delivery system. The drug release rate from a singlenanoparticle is the time derivative of q (t):

JNP (t) =dq (t)

dt.

The weak formulation, the numerical approximation and the algebraiccounterpart of the abovementioned equations can be built the same way weconstructed them for the coupled equations in Chapter 2. For the sake ofclarity, we report the two linear systems that need to be solved:

[1

∆tMvv + Avv

] [Cn+1v

]=

[1

∆tMttC

nt

], (5.4)[

1

∆tMtt + Att

] [Cn+1t

]=

[1

∆tMttC

nt + BtvJn+1

],

where Cnv ∈ RNh

v and Cnt ∈ RNh

t are the vector of degrees of freedom of thefinite element approximation of cnv and cnt , respectively, while Jn ∈ RNh

v isthe set of degrees of freedom of Jn built on the same basis as cnv . Submatrices

101


are defined as follows:

[Mvv]i,j :=(ϕjv, ϕ

iv

)Λ,

[Avv]i,j :=

(uhv∂ϕjv∂s

, ϕiv

)Λ

+

(∂uhv∂s

ϕjv, ϕiv

)Λ

+

+

(Dv

∂ϕjv∂s

,∂ϕiv∂s

)Λ

+

(2πR

πR2Πh,n+1eff ϕjv, ϕ

iv

)Λ

,

[Mtt]i,j :=(ϕjt , ϕ

it

)Ω,

[Att]i,j :=(uht · ∇ϕ

jt , ϕ

it

)Ω

+(∇ · uht ϕ

jt , ϕ

it

)Ω

+(Dt∇ϕjt ,∇ϕit

)Ω

+

(LLFp

S

V

(pht − phL

)ϕjt , ϕ

it

)Ω

+(βcϕ

jt , ϕ

it

)∂Ω,

[Btv]i,j :=(ϕjv, ϕ

it

)Λ,

where the bar operator corresponds to the average operator as in (2.10). Inparticular, it holds

Mvv ∈ RNhv×Nh

v , Avv ∈ RNhv×Nh

v ,

Mtt ∈ RNht ×Nh

t , Att ∈ RNht ×Nh

t ,

Btv ∈ RNht ×Nh

v .

5.3.2 Parameters and results

In the current section, we will describe the behaviour of nanoparticles in thevessel network and Dextran molecules in the interstitial tissue, by means ofsolving the linear systems (5.4). The main idea is to perform simulations forspherical nanoparticles of diameter dp = 2µm loaded with Dextran moleculesthat differs in molecular weight. Since these tests will be the combinationof tests on the nanoparticles in Λ (test4) and on Dextran in Ω (test5 andtest6), we will use the same parameters, when possible, even though, forthe sake of simplicity, we will report some of their values.

The simulations will be performed on the 1D domain described by thenetwork labelled rat93, with vessel radius of 30µm, immersed in a 3D do-main of size 1 × 1.48 × 0.8 mm. For the parameters that characterize thefluid dynamics problem we refer to Table 5.2, where the pressure drop at theboundary tips of the network is ∆p = 5 mmHg. For the transport of spheri-cal nanoparticles with a diameter of 2µm, the vascular diffusivity has beenset to 2.22 · 10−13 m2/s and a nominal injected concentration at the inflow

102



b Power law exponent - 0.88

χ Power law saturation parameter hb 1

cdexNP Total Dextran concentration in a particle g/m3 5.125 · 105

VNP Particle volume m3 4.18 · 10−18

Table 5.10: Parameters characterizing the release of Dextran molecules intest7, test8 and test9

tips of the network of 1 #/m3. In order to describe the adhesion propertiesof the system, we have set the probability of adhesion Pa to the values givenby the Lattice-Boltzmann simulations in the case of ρl = 0.5 and strongligand-receptor bonds, moreover the effective vascular adhesion parameterΠeff with saturation has been exploited. The maximum value for density ofadhering nanoparticle has been set to Ψmax = 7.4 m−2, which has been com-puted as the maximum Ψ in a two-hours simulation without the saturationmodel.

Moving to the tissue problem, let us focus on the total drug release rateper unit surface J , where we need to calibrate the power law release modeland the total drug load of a nanoparticle. The parameter values are reportedin Table 5.10, in particular χ and b have been fixed in such a way 50% ofthe total drug is released in 1 h and 90% of the total drug is released in12 h. This trend can be checked in Figure 5.17, where the Dextran releaseprofile q (t) / cdexNPVNP over time is shown. To determine the total Dextranconcentration inside the nanoparticle cdexNP , measured as

[gm3

], we assume

that (a) the density of the particle is comparable to the blood density (ρ)and (b) the drug mass fraction in each particle is 1/2, which means that halfof the particle mass represents the drug mass inside a particle: cdexNP = 1

2ρ.

Three experiments of drug delivery (test7, test8 and test9) have beenperformed by means of using the abovementioned parameters, while changingthe molecular weight of the Dextran, that basically affect the diffusivitycoefficient in the tissue domain Dt, as seen in Table 5.11. Once again, weused the experimental values we detailed in the previous chapter as diffusivitycoefficient of the Dextran molecules in the interstitial tissue.

We compare the simulations of 4 kDa, 40 kDa and 250 kDa Dextranmolecules delivery for 12 h of nanoparticle injection. As expected, thenanoparticle concentration follows the same trend we have discussed in Sec-tion 5.1, in particular, the saturation model for the adhesion of nanoparticlesallows them to uniformly fill the network. However, under the current flowand adhesive conditions, the complete filling of the vessel network is a process

103


time [h]0 5 10 15 20 25

q(t

) / c

de

x

NP

VN

P [-]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Drug release profile

Figure 5.17: Dextran release profile in time of test7, test8 and test9:power law model with saturation

Test Dextran molecular weight Diffusivity coefficient Dt

test7 4 kDa 1.51 · 10−10 m2/stest8 40 kDa 7.91 · 10−11 m2/stest9 250 kDa 4.42 · 10−11 m2/s

Table 5.11: Tissue diffusivity coefficient for Dextran delivery experiments

104


(a) test7: 4 kDa Dextran, t=1 h (b) test7: 4 kDa Dextran, t=12 h

(c) test8: 40 kDa Dextran, t=1 h (d) test8: 40 kDa Dextran, t=12 h

(e) test9: 250 kDa Dextran, t=1 h (f) test9: 250 kDa Dextran, t=12 h

Figure 5.18: Comparison of Dextran concentration ct [g/m3] and densityof adhering nanoparticle on the vessel wall Ψ [m−2] in the cases of variousmolecular weight and simulation time

105


time [h]0 2 4 6 8 10 12

Tota

l am

ount of D

extr

an in Ω

[g]

×10-18

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

4 kDa40 kDa250 kDa

Figure 5.19: Comparison of the total amount of Dextran∫

Ωct dΩ in the tissue

region over time for 4 kDa, 40 kDa and 250 kDa Dextran molecules release(test7, test8 and test9)

106


that takes more than 12 h, indeed cv at the outflow tips of the network is farfrom the target nominal concentration. The Dextran concentration in thecollagen matrix Ω strongly depends on the density of adhering nanoparticleto a specific region of the vessel network and on the diffusivity coefficientof the molecules in the tissue. For a given Dextran molecular weight, forexample 4 kDa (test7), the concentration ct at different time together withthe corresponding Ψ are reported in Figure 5.18. It is noteworthy that theyprogress in parallel, since no source term for ct is available in regions whereΨ is still close to zero. Moreover, the longer the adhesion time for a nanopar-ticle, the higher the quantity of released Dextran in the neighboring region,thanks to the release profile and to the diffusion and advection mechanismsthat regulate the Dextran transport in the tissue. Figure 5.18 also reportsthe same analysis in the case of 40 kDa and 250 kDa Dextran molecules.From a vertical comparison of the visualizations, namely observing differentDextran molecules at the same time and with the same nanoparticle profile,we can conclude, as expected, that the smallest molecules penetrate the in-terstitial space deeper than the largest. Indeed, by observing that the rangeof variability of ct at the same time becomes wider as the molecular weightof the Dextran grows, it is possible to conclude that the concentration dis-tribution in the tissue is more homogeneous in the case of small Dextranmolecules than in the case of large molecules. Even though the spatial dis-tribution of the Dextran concentration varies with the molecular weight, thetotal amount of drug at each time in the tissue is almost independent of themolecular weight, because it is uniquely defined by the release profile and thetotal amount of drug in a nanoparticle. The timecourse of the total amountof Dextran in the tissue

∫Ωct dΩ (Figure 5.19) supports these observations,

hence it can be seen that the three curves are almost identical. The smalldiscrepancy is caused by the presence of the boundary conditions: due tothe differences in the diffusivity coefficient, the smallest molecules reach theboundary of the domain and leave the slab earlier than the largest ones.

5.3.3 Conclusions

In conclusion, we have merged the computational models we have analizedso far in order to describe a more realistic situation for the drug delivery.We have slightly modified the standard computational model to be able todescribe the nanoparticle transport and adhesion in the vessel network and,since we have assumed to have particles loaded with Dextran molecules, theDextran transport in the interstitial tissue, modeled as a collagen matrix. Itis interesting to notice that the two equations for the network and for thetissue are no more coupled, however the Dextran concentration is directly

107


influenced by the nanoparticle concentration. It is possible to summarize theprocedure by means of identifying several steps for the transport equationsat each time iteration:

1. solve the nanoparticle concentration cv in the vessel network,

2. compute the density of adhering nanoparticles to the vessel wall Ψ,

3. compute the source term J for the Dextran molecules concentration ct,

4. solve the Dextran concentration ct in the intestitial tissue.

Finally, we have shown the nanoparticle concentration in the vessel network,the density of adhering nanoparticles to the capillary wall and the Dextranconcentration in the interstitial tissue in the case of different molecules (4kDa, 40 kDa and 250 kDa). We have observed that the total amount ofDextran in the tissue over the time does not vary with the molecular weight,while the spatial distribution of the drug concentration in the tissue dependson the size of the Dextran molecules.

108

Chapter 6

Conclusions and futureperspectives

The aim of the present work was to propose a mathematical model for thetransport of nanoparticles in the microcirculation and the correspondingdrug delivery in the surrounding interstitial tissue. We implemented a mul-tiscale computational model that provides estimates for the concentrationof nanoparticles or drug in the vessels and in the interstitial tissue at themacroscale, while encoding the values of some parameters coming from nu-merical experiments at the microscale.

As described in Chapter 1, nanomedicine is an emerging research fieldthat combines nanotechnological, biomedical and therapeutic studies withthe purpouse of enhancing clinical diagnosis and of improving the efficiencyof many treatments. Many applications are related, for example, to drug-delivery systems for the treatment of complex diseases as cancer, nanosen-sors, implants and imaging agents. However, since physical experiments areoften expensive in terms of availability, costs and time, mathematical andcomputational models represent useful tools in order to lighten the costs as-sociated with the physical experiments and in order to provide a quantitativeunderstanding of a given physical process. One of the most promising fieldof application of mathematical models to nanomedicine is the framework ofdrug-encapsulated injectable nanoparticles, with the aim of maximizing theloading efficiency, for example. In order to describe in an accurate way thecomplex biological systems, it is necessary to take into account the multi-scale nature that characterizes all the living organisms. Each level of thehierarchical structure has a different time-scale and space-scale and severalprocesses need to be observed at a suitable scale to really comprehend them.Also the mathematical models need to consider the intrinsic multiscale struc-ture of the organism, therefore several methods are available for solving each

109

CHAPTER 6. CONCLUSIONS AND FUTURE PERSPECTIVES

phenomenon at the right scale, from the atomistic to the continuum scale:ab initio method, molecular dynamics approach, coarse grained technique,discrete model and continuous model. Once a particular scale is solved, thenseveral techniques, as homogenization, mixture theory or asymptotic expan-sion method, are exploited in order to link the information from one scaleto the other. The specific study of this work was focused at themacroscale on the nanoparticle transport in a vessel network andin the surrounding interstitial tissue, while at a subscale on theadhesive properties of the nanoparticles with respect to the ves-sel wall and on the diffusive properties of the loaded drug in theexternal tissue.

To this purpose, we started in Chapter 2 with the derivation of theequations that regulate the three-dimensional mass transport in a coupleddomain composed by a network Ωv and the tissue Ωt. In particular, we mod-eled the phenomenon with two time-dependent reaction-advection-diffusionequations, where the reaction term in the tissue represents the effect of thelymphatic vessels, while in the network it describes the adhesion of particlesto the vascular wall. The two equations are coupled thanks to the presenceof a permeable membrane for which the Kedem-Katchalsky model has beenused. Thanks to the assumption of large aspect ratio between the capil-lary axial length and the vessel radius, we reduced the 3D description of thevessels to a 1D representation on Λ, the centerline of the vessel network, byintroducing a non local restriction operator that combines the standard tracewith the mean value of the solution on a low dimensional manifold. This ap-proach leads to a system of coupled 3D/1D PDEs. We proposed a variationalformulation that accounted for the coupling by means of concentrating thesource term (f) that represents the membrane condition on a one dimensionalmanifold (fδΛ). We then moved to another variational formulation in whichwe exploited the coupling directly at the variational level and we decided todiscretize this second weak formulation because its well-posedness could bederived in the framework of the Lax-Milgram lemma. As a conclusion of thestudy of this chapter, we highlight that we developed a mathematicalmodel for the microcirculation that was

• able to represent the physics of the problem in an accurate way;

• easy to solve;

• flexible with respect to computational mesh grids and to different ap-proximation methods.

In Chapter 3, we proposed the well-posedness analysis of a simplifiedversion of our system, which still contains the dimensionality gap that is

110


under investigation. We introduced an analysis with respect to the normof the gradient ||∇ · ||L2(Ω) and we proved the existence and uniqueness of

the solution in H10 (Ω) ∩ H 3

2−ε(Ω), ∀ε > 0. Moreover, we showed that the

continuity and coercivity constants are stable as the radius of the vesselgoes to zero. Thanks to the studies outlined in this chapter, we figuredout that the results of the simulations rely on solid and robustmathematical foundations and they represents the physics of thephenomena in an accurate way, provided that the mesh grid forthe domain discretization is adequately fine.

Chapter 4 was devoted to the characterization of some parameters of themodel at the microscale. We initially focused on the vascular adhesion pa-rameter Πeff of the nanoparticles to the vessel walls with a first formulationin which the strength of the particle sequestration depends on the proba-bility of adhesion of the NPs to the capillary wall, the local wall shear rateand the dimension of the NPs, without taking into account the free spaceavailable on the vessel wall for the adhesion. Then, we proposed a morephysical description in which a saturation model for the density of nanopar-ticles adhering to the wall had been introduced. Within the framework ofthe adhesive properties of the nanoparticles, we firstly determined the valueof the probability of adhesion Pa with an explicit formula at the continuumlevel, for which we listed some problems that made it difficult to use. Forthis reason, a subscale approach for the prediction of Pa was preferred: wereferred to the studies of our collaborators, A. Coclite and P. Decuzzi, on thetransport of nanoparticles covered by adhesive molecules in a Couette flowwith a combined Lattice Boltzmann - Immersed Boundary approach. Themethod gave as output the quantity of interest as function of the Reynoldsnumber and of the percentage of adhesive molecules that covers the NPs.

Another parameter we estimated by using subscale techniques is the dif-fusion coefficient of the loaded drug in the interstitial tissue: in this work, weused Dextran molecules that diffuses in collagen. For the prediction of thediffusivity coefficient we referred to the experimental results of our collabo-rators, V. Lusi and P. Decuzzi, who tested the values in a lab experiment.The images captured during the experiments showed the concentration ofDextran molecules in a channel at different time instants. Based on thoseframes, we implemented a numerical routine for the prediction of the coeffi-cient with an optimal control approach. In the algorithm, we looked for theoptimal diffusion coefficient D∗ that minimized the cost functional, namelythe discrepancy between the concentration given by the numerical routineand the concentration measured in the physical experiment, both at the finaltime. To analyze the control problem, we adopted the Lagrangian approach,

111


in which the state equation - the equation that describes the phenomenon -the adjoint equation and the optimality constraint were the derivative of theLagrangian functional - a combination of cost functional and state equation- with respect to different parameters. Globally, we solved the minimizationalgorithm using the Non Linear Conjugate Gradient method and we wereable to conclude that the method was reliable and the predictions on the dif-fusion coefficient were in accordance with the experimental values. This wasa key chapter. As a result of the work of this chapter, we can conclude thatwe developed several tools to be able to link and integrate the the-oretical mathematical model with the corresponding observationsand experimental results.

Lastly, in Chapter 5 the results of the large scale simulations werepresented, all the strategies for the estimate of the parameters were usedand their main advantages and disadvantages were observed. We performedthree main classes of simulations:

(A) transport of nanoparticles in the vascular network and their adhesionto the vessel walls, according to different adhesive properties;

(B) transport of Dextran in the capillary network and its extravasation inthe tissue, according to different molecular weights;

(C) transport of nanoparticles in the network, their adhesion, delivery ofDextran molecules by the NPs and Dextran transport in the interstitialtissue.

6.1 Future developments

For future researches, several extension of the current model can be consid-ered.

The multiscale interaction has been the beating heart of this work. Inorder to obtain a more realistic and precise description of the phenomena,it would be useful to refine on the integration among the scales. In thiswork, we have used data at the microscale from both numerical and physicalexperiments, therefore it could be possible to improve on both the aspects.Whenever possible, it would be very important to create other microfluidicexperiments able to describe particular physical phenomena. We could con-sider the corresponding experimental data, numerically validate the results,for example through a minimization problem, and characterize in this waynew parameters for the model. From the numerical point of view, at themoment, we include in the model some adhesive maps that depend on two

112


variables only. It would be interesting to incorporate in the macroscopicmodel a more complex response surface, with variables depending onboth the nanoparticle structure and the vessel characteristics.

Regarding the flow model, at the moment, we are supposing to studythe transport of nanoparticles in a Newtonian fluid, however, the blood iscomposed by plasma in which blood cells are suspended. Usually, red bloodcells are bigger than the nanoparticles and their effect in the mass transportcannot be neglected. To this end, a study that combines the transportof nanoparticles and red blood cells, possibly in terms of hematocrit,or other blood cells would be of great interest to deeply comprehend and de-scribe the mass transport in the human circulatory network for the treatmentof many diseases.

From the mathematical standpoint, we have already noticed that thewell-posedness analysis based on the norm of the gradient guarantees to havebounded eigenvalues with respect to the radius R. In this direction, furtherinvestigations on ad hoc preconditioners would be interesting.

It is possible to exploit in an efficient way the mathematical model forthe treatment of complex diseases or for diagnostic imaging by means ofextending the observation scale. Indeed, since organ scale simulations withthe current level of precision would be of great interest, it would be necessaryto modify the structure of the numerical code in view of high performancecomputing approach. The same tool could be used to perform real timesimulations for diagnostics.

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Appendices

Appendix A

Fluid dynamics model for thevascularized tissue

The discussion developed in Chapter 2 about the governing equations forthe mass transport in the tissue interstitium and in the vascular networkneeds to be completed. Indeed, the fluid dynamic behaviour of the blood inthe microcirculation and in the interstitium has not been detailed yet. Inparticular, the knowledge of the governing equations for the velocity fieldsut, uv and the pressure fields pt, pv is required.

In the following, a three-dimensional model for the fluid dynamics of theproblem is presented, together with its coupled 3D/1D version. Then, theweak formulation is derived and a finite element discretization is proposed,up to the final algebraic form.

Concerning the two alternative formulations that have been detailed inChapter 2, only the first version with the Dirac measure δΛ have been imple-mented for the flow problem. Indeed, the main focus of this work has beenon the transport equations. However, thanks to the good properties of theformulation without δΛ, a re-organization of the flow equations towards thealternative formulation is under consideration.

A.1 Model set up

We aim at modeling fluid transport in a permeable biological tissue perfusedby a vessel network. We refer to the same geometrical setup as in Chapter2, where Ω = Ωv ∪ Ωt, we use the same notation as in Figure 2.1 and wereduce the complexity of the model in the same way. Indeed, it is necessaryto recall that the flow problem and the mass transport problem are stronglylinked and it is natural to keep the same notation and setup.

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APPENDIX A. FLUID DYNAMICS MODEL FOR THEVASCULARIZED TISSUE

Concerning the interstitial volume Ωt, it can be considered as an isotropicporous medium, such that the Darcy’s law applies as follows:

ut = −kµ∇pt, (A.1)

where ut denotes the filtration velocity vector in the tissue, κ the tissueconstant permeability, µ the dynamic blood viscosity and pt the fluid pres-sure. The effect of the lymphatic drainage is taken into account in the massconservation equation for the tissue domain, which is guaranteed by meansof

∇ · ut + LLFpS

V(pt − pL) = 0, (A.2)

where LLFp is the hydraulic conductivity of the lymphatic wall, SV

is thesurface area of lymphatic vessels per unit volume of tissue and pL is thehydrostatic pressure within the lymphatic channels.

For the blood flow in capillaries Ωv, we start assuming a Newtonian model,described by the steady incompressible Navier-Stokes equation, namely

ρ (uv · ∇) uv − µ∆uv +∇pv = 0, (A.3)

where uv is the flow velocity vector, ρ the blood density and pv the fluidpressure. Mass conservation in the vessel domain is imposed as

∇ · uv = 0. (A.4)

The continuity of the flow at the interface Γ = ∂Ωv ∩ ∂Ωt is enforced bymeans of

uv · n = ut · n = Lp

((pv − pt)−

∑k

σk (πv,k − πt,k)

), (A.5)

ut · τ = uv · τ = 0, (A.6)

where n and τ are the outward unit normal vector and unit tangent vectorof the surface Γ, respectively. The flux across the capillary wall can be againmodeled using the Kedem-Katchalsky equation, in particular Lp is the hy-draulic conductivity of the vessel wall and πv−πt is the difference in osmoticpressure, where π = RgTc is the osmotic pressure given by the concentrationc of a given solvent, Rg is the universal gas constant and T is the absolutetemperature. Indeed, because of osmosis, the pressure drop across the cap-illary wall is affected by the difference in concentration of the substancesdissolved in blood. However, only the large molecules can induce a signifi-cant effect on it, for this reason we only consider the presence of proteins,

116


therefore we will omit the pedex k: ut ·n = Lp ((pv − pt)− σ (πv − πt)). Thereflection coefficient σ quantifies how different a semi-permeable membraneis from the ideal permeability.

Therefore, these modelling assumptions lead to the following fluid prob-lem in the entire domain Ω:

ut + kµ∇pt = 0 in Ωt,

∇ · ut + LLFpSV

(pt − pL) = 0 in Ωt,

ρ (uv · ∇) uv − µ∆uv +∇pv = 0 in Ωv,

∇ · uv = 0 in Ωv,

uv · n = ut · n = Lp ((pv − pt)− σ (πv − πt)) on Γ,

ut · τ = 0 on Γ.

(A.7)

The description of the boundary conditions on ∂Ωv and ∂Ωt will be ana-lyzed in detail in Section A.4.

A.2 Coupling microcirculation with

interstitial flow

The fully three-dimensional model (A.7) can be simplified exploiting theImmersed Boundary Method, as it has been done for the mass transportproblem. We combine the IBM and the assumption of large aspect ratiobetween vessel radius and capillary axial length and we adopt a rescaling ofthe equations with R → 0. In this way, the 3D description of the vesselsis reduced to a simplified 1D representation and the immersed interface andthe related interface conditions are replaced by an equivalent mass source. Inparticular, the flux continuity between the capillary network and the tissueis guaranteed by means of

ut · n = f (pt, pv) on Γ,

which represents a point-wise constitutive law for the capillary leakage interms of the fluid pressure.

Thanks to the IBM, the action of f on Γ can be represented as an equiv-alent source term F distributed on the entire domain Ω:

F (pt, pv) = f (pt, pv) δΓ.

where F is the Dirac measure concentrated on Γ, having density f , definedby: ∫

Ωt

F (pt, pv) v dΩ =

∫Γ

f (pt, pv) v dσ ∀v ∈ C∞ (Ωt) .

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By exploiting the same techniques used in Chapter 2 for the coupling of thetrasport equation in the tissue with the equation in the microvasculature, werecover the following expression for the distributed source term:∫

Ωt

F (pt, pv) v dΩ =

∫Λ

|γ (s)| f (pt (s) , pv (s)) v (s) ds ∀v ∈ C∞ (Ωt) ,

p (s) : =1

|γ (s)|

∫γ(s)

p (s, θ)Rdθ, (A.8)

where the bar operator is defined as the averaging operator on circles ofradius R laying on the cylindrical surface Γ and normal to the line Λ.

A.2.1 A reduced model for microvascular flow

The one-dimensional representation of the vessel that has been detailed aboveis naturally combined with a 1D reduced model for blood flow, in order toreplace the full Navier-Stokes equations. Indeed, for microcirculation it ispossible to introduce a relevant simplification at the modeling level, therefore,the quasi-static approximation is acceptable [10]. As a result, the blood flowalong each branch of the network can be described by means of Poiseuille’slaw for laminar stationary flow of incompressible viscous fluid through acylindrical tube with radius R.

To this end, the network Λ has been decomposed into individual branchesΛi, i = 1, . . . , N . For each branch, an arc length si and a tangent unitvector λi accounting for an arbitrary branch orientation are defined. Thanksto the tangent unit vector, differentiation over each branch λi is definedas ∂si := ∇ · λi. According to Poiseuille’s flow, conservation of mass andmomentum becomes:

uv,i = −R2

8µ

∂pv,i∂si

λi and − πR2∂uv,i∂si

= gi on Λi, (A.9)

gi (s) = 2πRi (s)Lp ((pv,i (s)− pt (s))− σ (πv,i (s)− πt (s))) ,

where gi is the local restriction of f on Λi and the bar operator is the averageoperator in (A.8). Since the vessel velocity has a fixed direction, i.e. uv,i =uv,iλi, also for the flow equations the vessel problem will be formulated withthe scalar unknown uv only.

A.2.2 Governing equations for the coupled problem

It is now possible to reformulate the flow problem (A.7) in differential formin terms of coupled equations in a three-dimensional space for the interstitial

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tissue and in a one-dimensional space for the capillary network. The reducedmodel for the flow problem in the vessel network Λ is obtained by summingup the local equations (A.9) over all the branches.

The coupled problem for the microcirculation and interstitial flow consiststo find the pressure field pt, pv and the velocity fields ut and uv (uv = uvλ)such that

ut + kµ∇pt = 0 in Ω,

∇ · ut + LLFpSV

(pt − pL)− f (pt, pv) δΛ = 0 in Ω,

uv + R2

8µ∂pv∂s

= 0 on Λ,

πR2 ∂uv∂s

+ f (pt, pv) = 0 on Λ,

(A.10)

where f (pt, pv) = 2πRLp ((pv − pt)− σ (πv − πt)) .As we have already underlined for the mass transport equations, the

distinction between the subregion Ωt and the entire domain Ω is no longermeaningful, since the one-dimensional Λ has zero measure in R3.

A.3 Dimensional analysis

In order to point out the most significant mechanisms that govern the flowbetween microcirculation and interstitial tissue (A.10), it is essential to writethe equation in dimensionless form. To this end, we identify the character-istic dimensions of the problem: the primary variables for the analysis arelength, velocity and pressure. We choose the average spacing between cap-illary vessels, d, as characteristic value for the length, the average velocityin the capillary bed, U , for the velocity and the average pressure in the in-terstitial space, P , for the pressure. The dimensionless groups affecting theequations are:

• R′ = Rd

the non dimensional radius,

• κt = kPµUd

the hydraulic conductivity of the tissue,

• Q = 2πR′LpPU

the hydraulic conductivity of the capillary walls,

• QLF = LLFpSVdPU

the non dimensional lymphatic drainage,

• κv = πR′4Pd8µU

the hydraulic conductivity of the vessels.

119


The dimensionless form of (A.7) readsut + κt∇pt = 0 in Ω,

∇ · ut +QLF (pt − pL)− fadim (pv, pt) δΛ = 0 in Ω,

uv + κvπR′2

∂pv∂s

= 0 on Λ,

πR′2 ∂uv∂s

+ fadim (pv, pt) = 0 on Λ,

(A.11)

where fadim (pv, pt) = Q ((pv − pt)− σ (πv − πt)) .For the sake of simplicity, the same symbols for the standard and dimen-

sionless variables and for the standard and dimensionless operators have beenused.

A.4 Boundary conditions

As previously mentioned, it is necessary to specify some boundary conditions(BCs) on both the tissue and vessel boundary, that is ∂Ω and ∂Λ, respectively.

At this stage, let us introduce the most generic set of boundary conditionsthat can be applied. For interstitial volume Ω, let us assume a partition as

∂Ω = Γp ∪ Γu, Γp ∩ Γu = ∅, (A.12)

where the apex p suggests that a given pressure distribution will be enforcedover Γp and u means that a fixed value for the normal flux will be imposedon Γu, i.e:

pt = gt on Γp, (A.13)

ut · n = −κt∇pt · n = βt (pt − p0) on Γu. (A.14)

In (A.14), p0 represents a far field pressure value and βt can be interpretedas an effective conductivity accounting for layers of tissue surrounding theconsidered sample. Moreover, we require gt ∈ L2 (Γp).

Concerning the capillary flow, we refer to the collection of inflow and out-flow tips of the vessel network as ∂Λ ≡ Λin ∪ Λout, i.e. non junction pointswhere the tangent unit vector is inward-pointing and outward-pointing. How-ever, ∂Λ contains the proper boundary points, i.e. extrema belonging to ∂Ω,but it also contains immersed tips, that are network extrema in Ω. Let usdefine Ep ≡ (Λin ∪ Λout) ∩ ∂Ω and Eu ≡ (Λin ∪ Λout) ∩ Ω that represent adisjoint partition of ∂Λ. The meaning of the apices p and u is the same as inthe case of interstitial tissue. The most generic set of boundary conditionsfor the vessel problem is

pv = gv on Ep, (A.15)

πR2uv = −κv∇pv · n = βv (pv − p0) on Eu. (A.16)

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Figure A.1: On the left, a simple network made by a single Y-shaped bifur-cation, arrows show the flow orientation. On the right, the discretization:the domain has been split into branches with inflow and outflow variables atthe junction point.

The same considerations for p0 and βv as in the treatment of the tissueboundary term can be done and we require gv ∈ L2 (Ep) for the pressuredatum.

Finally, in order to get the dimensionless form of the boundary conditions,the external pressure p0 can be scaled according to the characteristic pressureP. Therefore, assuming that the interstitial pressure decays from pt to p0

over a distance comparable to the sample characteristic size, D, dimensionalanalysis shows that a rough estimate of the conductivity is βt = κt/D.

A.5 Junction treatment

The imposition of suitable compatibility conditions at the branching points(junctions) of the vessel tree is required. It is necessary to underline thatthe vessel velocity has to be discontinuous at the junctions in order to satisfylocal conservation of mass.

Suitable compatibility conditions are (i) the conservation of mass and(ii) continuity of total pressure at junctions. For the sake of simplicity, theseconditions will be detailed in the case of a simple Y-shaped network (see Fig-ure A.1) and we will assume that the cross-section is constant over the wholenetwork. As a consequence, (i) the conservation of flow rate (Qv = πR2uv)is equivalent to require that in correspondence of the junction point xJ theinflow velocity u0

v is equal to the sum of the outflow velocity u1v and u2

v, i.e.u0v (xJ) = u1

v (xJ) + u2v (xJ). (ii) The requirement of continuity of total pres-

sure reads as the need to have constant pressure over each branch, namelyp0v (xJ) = p1

v (xJ) = p2v (xJ). It is important to highlight that such compat-

ibility conditions will be enforced at the level of the variational formulationin a natural way.

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A.6 Variational formulation

Explicit solutions of problem (A.11) are not available, as already seen for thetransport problem. Again, the only way of applying the model to real casesis to exploit numerical simulations.

A.6.1 Weak formulation of the tissue problem

Due to the presence of a Dirac measure in the forcing term in (A.11)(b), thesolution of the tissue interstitium problem does not satisfy standard regular-ity estimates. For this reason, we introduce ad-hoc weighted Sobolev spacesin order to recover some kind of regularity, as suggested in [14].

Given a bounded open set Ω ⊂ Rd with Lipschitz boundary and an im-mersed hyper-surface Λ of dimension ≤ d− 1, let α ∈ (−1, 1) and m > 0, wedefine

L2α (Ω) :=

v : Ω→ R measurable s.t.

∫Ω

v2 (x) d2α (x,Λ) dΩ <∞,

Hmα (Ω) :=

v ∈ L2

α (Ω) s.t. Dv, . . . , D(m)v ∈ L2α (Ω)

,

where d (x,Λ) = dist (x,Λ). L2α (Ω) and Hm

α (Ω) are Hilbert spaces, equippedwith the inner products defined by

(v, w)α,Ω :=

∫Ω

v (x)w (x) d2α (x,Λ) dΩ,

(v, w)α,m,Ω := (v, w)α,Ω + · · ·+(D(m)v,D(m)w

)α,Ω

,

respectively. Moreover, we define an additional weighted Sobolev space thatis peculiar for our formulation:

Hdivα,β (Ω) :=

v ∈ L2

α

(Ω;Rd

)s.t. ∇ · v ∈ L2

β (Ω),

with α, β ∈ (−1, 1) . This is a Hilbert space with respect to the inner product

(v,w)α,β,Ω := (v,w)α,Ω + (∇ · v,∇ ·w)β,Ω .

At this stage, we are able to specify the proper functional setting for thetissue problem. As trial spaces for the tissue velocity and the pressure, wedefine

Vt = Hdivα,β (Ω) and Qt = L2

α (Ω) ,

respectively. It is now possible to obtain the weak formulation of the inter-stitial problem, by multiplying equations (A.11)(a,b) by sufficiently smoothfunctions vt and qt and integrating over the domain Ω, i.e:

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∫Ω

1

κtut · vt dΩ +

∫Ω

∇pt · vt dΩ = 0, (A.17)∫Ω

(∇ · ut) qt dΩ +

∫Ω

QLF (pt − pL) qt dΩ−∫

Ω

Q (pv − pt) δΛqt dΩ+

+

∫Ω

Qσ (πv − πt) δΛqt dΩ = 0. (A.18)

Let us apply the Green’s theorem to (A.17), namely:∫Ω

1

κtut · vt dΩ−

∫Ω

pt∇ · vt dΩ +

∫∂Ω

ptvt · n dσ = 0, (A.19)

split the integral on the boundary according to (A.12) and enforce the bound-ary conditions as in (A.13)-(A.14):∫∂Ω

ptvt · n dσ =

∫Γp

gtvt · n dσ +

∫Γu

1

βt(ut · n) (vt · n) dσ +

∫Γu

p0vt · n dσ.

(A.20)By substituting (A.20) in (A.19), the weak formulation of (A.11)(a,b) readsas ∫

Ω

1

κtut · vt dΩ−

∫Ω

pt∇ · vt dΩ +

∫Γu

1

βt(ut · n) (vt · n) dσ =

=−∫

Γp

gtvt · n dσ −∫

Γu

p0vt · n dσ, (A.21)∫Ω

(∇ · ut) qt dΩ +

∫Ω

QLF (pt − pL) qt −∫

Ω

Q (pv − pt) δΛqt dΩ+

+

∫Ω

Qσ (πv − πt) δΛqt dΩ = 0. (A.22)

A.6.2 Weak formulation of the vessel problem

For the network problem, we start requiring regularity for the vessel velocityand pressure over each branch separately, therefore the first definition fortrial and test spaces reads as

Vv =N⋃i=1

H1 (Λi) and Qv =N⋃i=1

L2 (Λi) .

In order to derive the weak formulation for the vessel problem, the standardprocedure requires to multiply equations (A.11) (c,d) by sufficiently smooth

123


functions vv and qv and integrate over the domain Λ:∫Λ

π2R′4

κvuvvv ds+

∫Λ

πR′2∂pv∂s

vv ds = 0, (A.23)∫Λ

πR′2∂uv∂s

qv ds+

∫Λ

Q (pv − pt) qv ds−∫

Λ

Qσ (πv − πt) qv ds = 0. (A.24)

At this point, it is not possible to simply integrate by parts, since thevessel variable uv and pv may be discontinuous at multiple junctions. Totackle this issue, we assume that the pressure is continuous

(pv ∈ C0

(Λ))

,while for the vessel velocity we proceed as follows. Let us assume that thevessel radius is constant over each branch Λi. We rewrite the second integralof (A.23) as a summation of integrals over each Λi and we apply Green’sformula over each segment, i.e.:∫

Λ

πR′2∂pv∂s

vv ds =N∑i=1

πR′2i

∫Λi

∂pv∂s

vv ds =

=N∑i=1

πR′2i

−∫

Λi

pv∂vv∂s

ds+ [pvvv]Λ+i

Λ−i

=

= −∫

Λ

πR′2pv∂vv∂s

ds+N∑i=1

R′2i [pvvv]Λ+i

Λ−i,

where Λ−i and Λ+i are the inflow and outflow extrema of Λi, according to the

orientation λi. At this point, we can re-organize the local boundary terms inorder to collect contributions of different branches affecting the same junctionpoint. It is useful to define the set of indices of junction points, namely:

J :=j ∈ N s.t. sj ∈ Λ, card

(Psj)≥ 2,

where Psj is the collection of all branches joining at the branching point sj(patch) and it is also useful to introduce the following partition of indices inPsj :

Poutj :=i ∈ 1, . . . , N s.t. Λ+

i = sj, ∀j ∈ J ,

P inj :=i ∈ 1, . . . , N s.t. Λ−i = sj

, ∀j ∈ J .

According to the orientation unit vector λi, the former set represents branchesthat enter the node, whose contribution to mass conservation is positive,while the latter represents branches that leave the node, whose contributionis negative. At this point, the boundary term can be rewritten by separating

124


the terms relative to inner junction nodes, to outer inflow nodes and to outeroutflow nodes, namely:

N∑i=1

R′2i [pvvv]Λ+i

Λ−i=

=[πR′2pvvv

]Λouti

Λini

+∑j∈J

∑i∈Pout

j

πR′2i pvvv|Λ+i−∑i∈Pin

j

πR′2i pvvv|Λ−i

=

=[πR′2pvvv

]Λouti

Λini

+∑j∈J

pv (sj)

∑i∈Pout

j

πR′2i vv|Λ+i−∑i∈Pin

j

πR′2i vv|Λ−i

,where Λin and Λout indicate the collection of inflow and outflow tips of thevessel network.

After this manipulation, the conservation of local flow rate at vessel junc-tions can be naturally expressed as:∑

i∈Poutj

πR′2i vv|Λ+i

=∑i∈Pin

j

πR′2i vv|Λ−i , ∀j ∈ J . (A.25)

Therefore, it is reasonable to enforce the conservation of local flow rate in thevariational formulation by multiplying (A.25) with the pressure test functionqv, which acts as a Lagrange multiplier for this constraint, i.e.:

∑j∈J

∑i∈Pout

j


j

πR′2i vv|Λ−i

qv (sj) = 0.

The last step is to enforce suitable boundary conditions at the vessel

trip, in order to make the term [πR′2pvvv]Λouti

Λini

explicit. As mentioned in

Section A.4, on boundary extrema Ep we enforce a pressure distribution andon immersed extrema Eu we enforce the flux. In particular, since it is usefulto enforce a constant pressure drop ∆Pv = P out

v − P inv , a suitable choice for

the boundary data (A.15) reads as

gv (s) =

P outv , s ∈ Ep ∩ Λout

P inv , s ∈ Ep ∩ Λin.

Moreover, similarly to what it has been done for the tissue problem, we canenforce essential boundary conditions (A.16) in weak form as

pv = p0 +πR′2

βvuv on Eu.

125


Therefore, assembling all the terms that have been detailed in this section,the most generic formulation of the vessel problem is:∫

Λ

π2R′4

κvuvvv ds−

∫Λ

πR′2pv∂vv∂s

ds+1

βv

[π2R′4uvvv

]Eu

+

+∑j∈J

pv (sj)

∑i∈Pout

j


j

πR′2i vv|Λ−i

=

= −[πR′2p0vv

]Eu−[πR′2gvvv

]Ep, (A.26)

∫Λ

πR′2∂uv∂s

qv ds+

∫Λ

Q (pv − pt) qv ds−∫

Λ

Qσ (πv − πt) qv ds+

−∑j∈J

∑i∈Pout

j


j

πR′2i vv|Λ−i

qv (sj) = 0.

(A.27)

A.6.3 Coupled weak formulation

Combining (A.21), (A.22), (A.26) and (A.27), the whole weak formulationof the 3D/1D coupled model of fluid exchange between microcirculation andtissue interstitium (A.11) reads as follows:

find ut ∈ Vt, pt ∈ Qt, uv ∈ Vv and pv ∈ Qv such that

(1

κtut,vt

)Ω

− (pt,∇ · vt)Ω +

(1

βtut · n,vt · n

)Γu

=

= − (gt,vt · n)Γp− (p0,vt · n)Γu

∀vt ∈ Vt,

(∇ · ut, qt)Ω +(QLFpt, qt

)Ω− (Q (pv − pt) , qt)Λ =

= − (Qσ (πv − πt) , qt)Λ +(QLFpL, qt

)Ω

∀ qt ∈ Qt,(π2R′4

κvuv, vv

)Λ

−(πR′2pv,

∂vv∂s

)Λ

+1

βv

[π2R′4uvvv

]Eu

+

− 〈vv, pv〉J = −[πR′2gvvv

]Ep−[πR′2p0vv

]Eu

∀ vv ∈ Vv,(πR′2

∂uv∂s

, qv

)Λ

+ (Q (pv − pt) , qv)Λ + 〈uv, qv〉J =

= (Qσ (πv − πt) , qt)Λ ∀ qv ∈ Qv,(A.28)

126


where the notation 〈·, ·〉J indicates the weak imposition of local mass con-servation over each junction point in J , namely

〈uv, qv〉J := −∑j∈J

∑i∈Pout

j


j

πR′2i vv|Λ−i

qv (sj) .

A.7 Numerical approximation

The discretization of problem (A.11) is achieved by means of the finite ele-ment method that arises from the variational formulation (A.28) combinedwith a discretization of the domain Ω and Λ. At the discrete level, one of theadvantages of our formulation is that the partitions of Ω and Λ are completelyindependent.

A.7.1 Discretization of the tissue problem

In order to discretize the domain for the tissue interstitium problem, weintroduce an admissible triangulation T ht of Ω, i.e.

Ω =⋃

K∈T ht

K,

which satisfies the usual conditions of a conforming triangulation of Ω, whilewe are implicitly assuming that Ω is a polygonal domain. With a standardnotation, h = maxK∈T h

thK , where hK is the diameter of element K.

The solution of (A.28)(a,b) is approximated using discontinuous piecewise-polynomial finite elements for the pressure and Hdiv-conforming Raviart-Thomas finite elements (RT) for velocity, namely

Y hk :=

vh ∈ L2 (Ω) s.t. vh|K ∈ Pk (K) ∀K ∈ T ht

,

RT hk :=wh ∈ Hdiv (Ω) s.t. wh|K ∈ Pk

(K;Rd

)⊕ xPk (K) ∀K ∈ T ht

,

for every integer k ≥ 0, where Pk indicates the standard space of polynomialsof degree ≤ k in the variables x = (x1, . . . , xd). We recall that

Hdiv (Ω) :=v ∈ L2

(Ω;Rd

)s.t. ∇ · v ∈ L2 (Ω)

.

127


A.7.2 Discretization of the vessel problem

Concerning the capillary network problem, we adopt the same splitting ofthe domain described at the continuous level, denoted by

Λh =N⋃i=1

Λhi ,

where Λhi is a partition of the one-dimensional manifold Λi made by segments

S.The solution of (A.28)(c,d) over Λi is approximated using continuous

piecewise-polynomial finite element spaces for both pressure and velocity. Forthe velocity field, we define the finite element space over the whole networkΛh as a collection of the local spaces of the single branches Λh

i , since the vesselvelocity has to be discontinuous at multiple junctions, while a standard finiteelement approximation is used for the pressure, namely

Xhk+1 (Λ) :=

wh ∈ C0

(Λ)

s.t. wh|S ∈ Pk+1 (S) ∀S ∈ Λh

,

W hk+2 (Λ) :=

N⋃i=1

Xhk+2 (Λi) ,

for every integer k ≥ 0.

A.7.3 Discrete coupled weak formulation

The discrete formulation arising from (A.28) is easily obtained by projectingthe equations on the discrete spaces

Vht = RT hk (Ω) and Qh

t = Y hk (Ω) ,

V hv = W h

k+2 (Λ) and Qhv = Xh

k+1 (Λ) ,

for k ≥ 0 and by adding the subscript h to each variable, namely

128


find uht ∈ Vht , pht ∈ Qh

t , uhv ∈ V h

v and phv ∈ Qhv such that

(1

κtuht ,v

ht

)Ω

−(pht ,∇ · vht

)Ω

+

(1

βtuht · n,vht · n

)Γu

=

= −(ght ,v

ht · n

)Γp−(ph0 ,v

ht · n

)Γu

∀vht ∈ Vht ,(

∇ · uht , qht)

Ω+(QLFpht , q

ht

)Ω−(Q(phv − pht

), qht)

Λ=

=(Qσ(πhv − πht

), qht)

Λ+(QLFphL, q

ht

)Ω

∀ qht ∈ Qht ,(

π2R′4

κvuhv , v

hv

)Λ

−(πR′2phv ,

∂vhv∂s

)Λ

+1

βv

[π2R′4uhvv

hv

]Eu

+

− 〈vhv , phv〉J = −[πR′2ghvv

hv

]Ep−[πR′2ph0v

hv

]Eu

∀ vhv ∈ V hv ,(

πR′2∂uhv∂s

, qv

)Λ

+(Q(phv − pht

), qhv)

Λ+ 〈uhv , qhv 〉J =

=(Qσ(πhv − πht

), qht)

Λ∀ qhv ∈ Qh

v ,(A.29)

where ght , pg0, g, vh represent the discrete counterpart of continuous boundarydata.

A.8 Algebraic formulation

We aim at studying the algebraic counterpart of the discrete problem (A.29).The number of degrees of freedom of the discrete spaces are defined as

Nht := dim

(Vht

)and Mh

t := dim(Qht

),

Nhv := dim

(V hv

)and Mh

v := dim(Qhv

).

Let us introduce the finite element basis for Vht ×Qh

t : ϕitNh

t

i=1×ψitMh

t

i=1 and

V hv ×Qh

v : ϕivNh

v

i=1×ψivMh

v

i=1. These two sets are completely independent, since

the 3D and 1D meshes do not conform. Let Ut =U jt

Nht

j=1, Pt =

P jt

Mht

j=1,

Uv = U jv

Nhv

j=1 and Pv = P jv

Mhv

j=1 be the degrees of freedom of the finiteelement approximation, by using the finite element basis it is possible to set:

uht (x) =

Nht∑

j=1

U jtϕ

jt (x) , pht (x) =

Mht∑

j=1

P jt ψ

jt (x) , ∀x ∈ Ω,

uhv (s) =

Nhv∑

j=1

U jvϕ

jv (s) , phv (s) =

Mhv∑

j=1

P jvψ

jv (s) , ∀s ∈ Λ.

129


By exploiting the linear combinations in the discrete weak form and thelinearity of the inner products, the fully discrete form (A.29) of the modelleads to the linear system

Mtt −DTtt O O

Dtt Btt O −BtvO O Mvv −DT

vv − JTvvO −Bvt Dvv + Jvv Bvv

Ut

Pt

Uv

Pv

=

Ft

−Btv∆π + FLF

Fv

Bvv∆π

. (A.30)

Submatrices and subvectors in (A.30) are defined as follows:

[Mtt]i,j :=

(1

κtϕjt ,ϕ

it

)Ω

+

(1

βtϕjt · n,ϕit · n

)Γu

Mtt ∈ RNht ×Nh

t ,

[Dtt]i,j :=(∇ ·ϕjt , ψit

)Ω

Dtt ∈ RNht ×Mh

t ,

[Btt]i,j :=(Qψjt δΛh

, ψit)

ΩBtt ∈ RMh

t ×Mht ,

[Btv]i,j :=(QψjvδΛh

, ψit)

ΩBtv ∈ RMh

t ×Mhv ,

[Bvt]i,j :=(Qψjt , ψ

iv

)Λ

Bvt ∈ RMhv×Mh

t ,

[Bvv]i,j :=(Qψjv, ψ

iv

)Λ

Bvv ∈ RMhv×Mh

v ,

[Mvv]i,j :=

(π2R′4

κvϕjv, ϕ

iv

)Λ

+1

βv

[π2R′4ϕjvϕ

iv

]Eu

Mvv ∈ RNhv×Nh

v ,

[Dvv]i,j :=

(πR′2

∂ϕjv∂s

, ψiv

)Λ

Dvv ∈ RNhv×Mh

v ,

[Jvv]i,j := 〈ϕjv, ψiv〉J Jvv ∈ RNhv×Mh

v ,

[Ft]i := −(ght ,ϕ

it · n

)Γp−(ph0 ,ϕ

it · n

)Γu

Ft ∈ RNht ,

[FLF ]i :=(QLFphL, ψ

it

)Ω

FLF ∈ RMht ,

[Fv]i := −[πR′2ghvϕ

iv

]Ep−[πR′2p0ϕ

iv

]Eu

Fv ∈ RNhv ,

where the bar operator corresponds to the average operator as in (A.8) andwhere ∆π is the osmotic pressure difference across the capillary wall on Λ.

Concerning the implementation of the exchange matrices Btt, Btv andBvt, it is necessary to introduce a discrete average operator πvt : Qh

t →Qhv that extracts the mean value of a generic basis function of Qh

t and adiscrete interpolation operator πtv : Qh

v → Qht that returns the value of

a basis function of Qht in correspondence of nodes of Qh

v . For every nodesk ∈ Λh, we let Tγ (sk) be the discretization of the perimeter of the vesselγ (sk), assuming that γ (sk) is a circle of radius R defined on the orthogonalplane to Λh at point sk. The set of points of Tγ (sk) is used to interpolate

130


the basis function ψit. The average operator πvt is defined in such a way thatqt = πvtqt and each row of the corresponding matrix Πvt ∈ RMh

v×Mht is defined

asΠvt

∣∣k

= wT (sk) Πγ (sk) k = 1, . . . ,Mhv ,

where w is the vector of weights of the quadrature formula for the approxi-mation of qt (s) = 1

2πR

∫ 2π

0qt (s, θ)Rdθ in the nodes belonging to Tγ (sk) and

Πγ (sk) is the local interpolation matrix that returns the values of each testfunction ψit on the set of points belonging to Tγ (sk). The interpolation ma-

trix Πtv ∈ RMhv×Mh

t corresponds to the interpolation operator πtv. Thanks tothese operators, the exchange matrices are implemented as

Btt = ΠTtvBvvΠvt

Btv = ΠTtvBvv

Bvt = BvvΠvt.

In order to implement the junction matrix Jvv, we define a linear operatorthat gives the restriction with sign of a basis function of V h

v over a givenjunction node. In particular, for a given k ∈ J , Rk : V h

v → R is defined as

Rk

(ϕjv)

:=

+πR′2l ϕ

jv (sk) j inΛh

l ∧ l ∈ Poutk

−πR′2l ϕjv (sk) j inΛhl ∧ l ∈ P ink

∀j = 1, . . . , Nhv ,

where the expression j inΛhl means that the j-th degree of freedom is linked

to some vertex of the l-th branch. Observe that Rk may only assume values−πR′2l , 0, πR′2l for some l and in particular Rk

(ϕhv)

= 0 for all the couplesof indices (k, j) that are uncorrelated. Therefore, the generic element (i, j)of Jvv can be computed as

[Jvv]i,j = −∑k∈J

Rk

(ϕjv)ψiv (sk) .

131

Appendix B

Lattice Boltzmann method

The aim of the current Appendix is to describe more in detail the equationsthat govern the Lattice-Boltzmann (LB) method. In particular, in SectionsB.2 and B.3 we will derive the LB model from the BGK-Boltzmann equationand standard bounce-back boundary conditions will be reviewed in SectionB.4.

B.1 Introduction

Historically, the lattice Boltzmann method has been developed for solvingflow problems with a simpler formulation than the Navier-Stokes equations.In particular, the method aims at capturing the physics of the flow at themacroscopic scale by microscale collision models, namely, by simulating theinteractions among a limited number of particles located on the nodes of alattice. The lattice-Boltzmann equation is grounded in the kinetic theory andit is inspired by the Boltzmann equation. In particular, it has been provedin [25] that the lattice Boltzmann equation can be directly derived from thecontinuous Boltzmann equation by means of a suitable discretization in spaceand time.

The LB model that we will present in the following can be generalized todifferent cases. An example of generalization is the Entropic Lattice Boltz-mann method that aims at avoiding the presence of negative values for thedistribution function in the case of large flow velocity for a given lattice, bywriting the equation in terms of the entropy. The simplicity of the formula-tion and its versatily allow further developments of the standard equation tocomplex and multiscale flow; for example, we refer to the application of themethod to turbulent flows, multiphase flows, deformable particles and fibersuspensions [1].

132

APPENDIX B. LATTICE BOLTZMANN METHOD

B.2 The Boltzmann equation

We start the discussion from the Boltzmann equation

∂f

∂t+ e · ∇f = Q(f, f), (B.1)

where f ≡ f (x, e, t) is the single particle distribution function, defined insuch a way that the quantity f (x, e, t) dxde represents the probability that amolecule is located in (x,x+dx) space element with velocity in the (e, e+de)interval. Q(f, f) describes the effect of interparticle binary collisions and itis a nonlinear integral operator, leading to an integro-differential equation.

B.2.1 BGK kinetic model

In order to facilitate the numerical and analytical studies of the Boltzmannequation, the nonlinear collision operator is often replaced by simplified ki-netic models that on average share the same properties of the original op-erator Q(f, f). We consider the simplest kinetic model, the so-called BGK(Bhatnagar - Gross - Krook) collision operator

QBGK (f) = −1

τ(f − f eq) , (B.2)

where f eq is a local equilibrium and τ is a typical time-scale associated withthe collisional relaxation to the local equilibrium. One of the most importantsimplification of the BGK model is that τ does not depend on the molecularvelocity (this is equivalent to lumping the spectrum of relaxation scales into asingle value). The distribution function f has the same density ρ, bulk veloc-ity u and temperature T as the Boltzmann-Maxwellian distribution function

f eq :=ρ

(2πRT )D/2exp

−(e− u)2

2RT

; (B.3)

that is,

ρ =

∫fde =

∫f eqde,

ρu =

∫efde =

∫ef eqde,

ρε =1

2

∫(e− u)2 fde =

1

2

∫(e− u)2 f eqde,

(B.4)

where R is the ideal gas constant and D is the dimension of the space.

133


The internal energy per unit mass ε can be written in terms of the tem-perature T as

ε =D0

2RT,

where D0 is the number of degrees of freedom of a particle.Therefore, the BGK equation reads as:

∂f

∂t+ e · ∇f = −1

τ(f − f eq) (B.5)

B.2.2 Time discretization

Let us observe that (B.5) can be formally written in terms of an ordinarydifferential equation:

df

dt= −1

τ(f − f eq) , (B.6)

by considering the time derivative along the characteristic as

d

dt:=

∂

∂t+ e · ∇.

Let us discretize in time the BGK equation (B.6) by using an explicitEuler forward scheme with a time step of ∆t along the characteristic ∆x =e∆t:

f (x + e∆t, e, t+ ∆t)− f (x, e, t)

∆t= −1

τ[f (x, e, t)− f eq (x, e, t)] . (B.7)

B.2.3 Choice of the quadrature rule

The numerical evaluation of the macroscopic variables, ρ, u and ε, needs theintroduction of an appropriate discretization of the velocity space, in orderto approximate the integrals using suitable quadrature rules, namely:∫

φ (e) f (x, e, t) de =∑α

wαφ (eα) f (x, eα, t) , (B.8)

where φ (e) is a polynomial function of e, wα is the weight of the quadraturerule and eα is the discrete set of velocities chosen as basis for the quadraturerule. Accordingly, the moments can be written as

134


ρ =∑α

fα =∑α

f eqα ,

ρu =∑α

eαfα =∑α

eαfeqα ,

ρε =1

2

∑α

(eα − u)2 fα =1

2

∑α

(eα − u)2 f eqα ,

(B.9)

withfα := wαf (x, eα, t) .

B.3 The lattice Boltzmann equation

Given the discretized BGK-Boltzmann equation

fα (x + eα∆t, t+ ∆t)− fα (x, t) = −∆t

τ[fα (x, t)− f eqα (x, t)] , (B.10)

the derivation of the Lattice Boltzmann-BGK equation (LBGK) [54, 25, 1, 11]requires the definition of

(I) a proper equilibrium distribution function;

(II) a lattice structure and a discrete set of velocities;

(III) some conservation constraints as (B.15).

(I) Equilibrium distribution function

For the definition of the equilibrium distribution function, we consider theBoltzmann-Maxwellian distribution (B.3) and we approximate the velocitywith its truncated small velocity expansion up to order O (u3), namely:

f eq =ρ

(2πRT )D/2exp

− e2

2RT

[1 +

e · uRT

+(e · u)2

2 (RT )2 −u2

2RT

]. (B.11)

(II) Lattice structure and discrete velocities

The choice of the discretization for the velocity space needs to be related tothe lattice configuration, but it also needs to guarantee that the conservationconstraints and some necessary simmetries are exactly preserved.

In particular, let us focus on a particular lattice configuration that be-longs to the so-called DdQq family of schemes, i.e. q discrete velocities in d

135


Figure B.1: D2Q9 lattice configuration

dimensions. We refer to the squared D2Q9 configuration, whose diagram issketched in Figure B.1. Therefore, we consider the set of discrete velocitieseα8

α=0.On the other hand, we observe that the calculation of the moments of f eq

is equivalent to the evaluation of the following integral

I =ρ

(2πRT )D/2

∫φ (e) exp

− e2

2RT

[1 +

e · uRT

+(e · u)2

2 (RT )2 −u2

2RT

]de,

where it is easy to identify the general structure∫e−x

2

φ (x) dx,

which can be numerically calculated with a Gaussian-Hermite quadrature.By using the Cartesian coordinate system and by applying the third-orderHermite formula for the quadrature rule, we define the following discrete setof velocities:

eα =

(0, 0) , α = 0,(

cos(

(α−1)π2

), sin

((α−1)π

2

))c, α = 1, 2, 3, 4

√2(

cos(

(2α−9)π4

), sin

((2α−9)π

4

))c, α = 5, 6, 7, 8,

(B.12)

where c =√

3cs =√

3RT , with cs the reticular sound speed. The corre-sponding weights for the quadrature rule read as

wα =

4/9, α = 0,

1/9, α = 1, 2, 3, 4

1/36, α = 5, 6, 7, 8.

(B.13)

By combining (B.12) and (B.13), we obtain the following equilibrium distri-bution function of the D2Q9 LGBK model:

f eqα = wαρ

[1 +

3eα · uc2

+9 (eα · u)2

2c4− 3u2

2c2

]. (B.14)

136


(III) Conservation constraints

Thanks to the definition of the basis (B.12) and the weights (B.13) for thequadrature rule, the moments can be expressed in an explicit way

ρ =8∑

α=0

f eqα ,

ρu =8∑

α=0

eαfeqα ,

ρε =1

2

8∑α=0

(eα − u)2 f eqα .

(B.15)

B.4 Boundary conditions

The lattice Boltzmann equation requires the definition of some boundaryconditions for selecting solutions that are compatible with the external con-straints. The most widely used and simplest to implement wall boundarycondition is the standard bounce-back.

B.4.1 Standard bounce-back

The bounce-back method that we will consider in the following is typicallyused to implement no-slip boundary conditions on a fixed wall, namely zerofluid velocity at a given solid surface. The bounce-back method is quitesimple and it implies that an incoming fluid particle hitting the solid surfacebuonces back into the flow domain:

fα (x, t) = fα′ (x, t′) , (B.16)

where t′ refers to a time post collision and α′ is the direction opposite to α.If one assumes that the solid surface is aligned with the lattice points, thentwo types of implementation can be made: on-grid and mid-grid [54].

The on-grid configuration implies that the solid boundary lies exactlyon the grid nodes, as depicted in Figure B.2. In this situation, the incom-ing directions of the distribution functions are reflected when encounter-ing the boundary nodes. By referring to Figure B.2, the distribution func-tions f0, f1, f2, f3, f5 and f6 are known from the streaming process, while theboundary conditions enforce f4 = f2, f7 = f5 and f8 = f6.

The mid-grid configuration is referred to the case where the solid bound-ary is aligned with the lattice points, but it is in between two grid lines, as in

137


Figure B.2: D2Q9 on-grid bounce-back

Figure B.3: D2Q9 mid-grid bounce-back

Figure B.3. In this case, it is necessary to consider some ghost nodes beyondthe solid wall and the reflection takes place at the fictitious nodes.

B.4.2 Higher-order boundary conditions

The bounce-back method is simple to implement; however, it does have sev-eral disadvantages. In particular, although the method is second order ac-curate when the solid boundary resides on the mid point of the link, this isnot the case for an arbitrary surface, for which the accuracy of the methoddegrades into a first order accurate velocity field. To improve the accuracyof the method, two different approaches have been proposed:

• Link-based methods

This method interpolates from fluid nodes adjacent to the fluid bound-ary nodes. A disadvantage to all interpolation methods is the lack ofmass conservation.

• Nodes-based methods

This second group of methods alters the fluid nodes adjacent to thesolid surface without creating links. These methods conserve globalmass in the simulation domain.

138

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145

Acknowledgements

I wish to express my sincere gratitude to my supervisor Prof. Paolo Zuninofor his constant support and guidance during these months. He has alwaysencouraged me towards higher goals in all the steps that led to this masterthesis.

I am grateful to Prof. Paolo Decuzzi from the Laboratory of Nanotechnol-ogy for Precision Medicine at IIT for his challenging suggestions. This workwould not have been so stimulating without the collaboration with his labo-ratory at IIT.

I would like to thank Dr. Silvia Lorenzani from the Department of Mathe-matics for her help and time with the deepening on the kinetic theory.

I would like to thank Dr. Alessandro Coclite from IIT for his precious con-tribution to this work with the simulations at the microscale as well as forhis suggestions.

I am very grateful to Federica Laurino, whom I worked with during thesemonths. We have shared each trouble and each goal of this work as a realteam.

I would like to thank Valeria Lusi from IIT for her contribution to this workwith the experimental data.

A special mention to my colleagues, Claudia, Cristina, Elisa, Manuela, MariaLaura, Roberta and Giuliano for their presence during these years at Politec-nico. Nobody can survive to PoliMi without friends.

A special gratitude to Andrea, who has always believed in me. Words arenot enough to express how much his constant presence has been precious tome.

I owe a debt of gratitude to my parents, who have taught me to always domy utmost and look ahead. I would never have reached this result withouttheir support. Thank you.

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