isogeometric analysis of cardiac electrophysiology ......the cardiac muscle is generally called...

POLITECNICO DI MILANO

M.Sc. IN COMPUTATIONAL SCIENCE AND ENGINEERING

MASTER THESIS

Isogeometric Analysis of CardiacElectrophysiology: Application to the

whole human heart

Advisor:Dr. Luca DedeCo-advisor:Professor Alfio Quarteroni

Student:Matteo Salvador

Matr. 872356

Academic Year 2017–2018

To Marilena, my grandmother

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Acknowledgements

Firstly I would like to show my appreciation to Alfio Quarteroni, for giving me the pos-sibility of working in the iHeart project within the Modelling and Scientific Computing(MOX) centre in Politecnico di Milano, and to Luca Dede, for being really supportive atany step of my master degree with valuable, constant advices, new ideas and prospec-tives on the numerous issues that we have dealt with together. I will never forget thehelp of Andrea Bartezzaghi, for giving me a better and faster comprehension of theisoglib codes.I would like to say a big thank you to all my colleagues from Mathematical Engineeringin Italy (and to Andrea and Federico in Switzerland by now), for the beautiful momentsthat we have spent together in these years, sharing also some headaches on several prob-lems that we have hopefully addressed properly, and intense group projects.I must also cite all the amazing people I have met during my Erasmus in Belgium, whichhas been full of incredible experiences that I will always remember in my life.A special thank you goes also to my entire family, who gave me all the economical andemotional support to go through my entire academic career. There are seriously no rightwords to express my complete gratitude for all the efforts that you have made for me inyour entire life.In the little town where I live, there are a bunch of strict friends, in the purest and morerealistic sense possible, that have always been there for me, even when I was completelyoverwhelmed by my duties and I was not able to give them back the time that theydeserve. A special thanks here goes to Fabio.Last but not least, I would like to thank Pamela, who makes me smiling every singleday and has been patiently on my side in these two years.

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Abstract

The aim of this thesis is to use Isogeometric Analysis for the spatial approximation of aset of PDEs in order to make several numerical simulations of the electrophysiology ofthe heart. IGA presents different kinds of advantages comparing to other techniques inNumerical Analysis for PDEs approximation: it often permits to obtain an exact repre-sentation of the geometry thanks to the usage of high order continuity basis functions,which also gives a very accurate numerical solution, and it has very limited numericaldispersion and dissipation errors. In this work we apply IGA to the numerical approx-imation of the so called Bidomain model, largely used in literature for the descriptionof the macroscopic behaviour of the electric activity of the heart. This model has beencoupled with two ionic models, which are able to capture the microscopic details of thecardiomyocytes: the Bueno-Orovio and the Luo-Rudy models. We start from simula-tions on the left ventricle, using both the mentioned microscopic models, and providing acomparison, in terms of differences and similarities, between them. Subsequently we seewhat are the significant changes when the elicoidal fibers are applied on the left ventri-cle. After that, we will put our attention on the biventricular case, trying to reproduceits normal behaviour firstly, and then addressing a situation of ventricular fibrillation,which consists of a disorganized contraction in the ventricles, leading to death if notfastly treated using a defibrillator. As a conclusion of this work, we simulate an elec-trophysiological coupling between the atria, modelled as surfaces, and the biventricle,modelled as a 3D structure, in order to reproduce the complete electric behaviour of anhealthy heart.

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Contents

1 Introduction 11

2 Heart anatomy and cardiac electrophysiology 13

2.1 The conduction system of the heart . . . . . . . . . . . . . . . . . . . . . 15

2.2 Categorization of cardiac cells . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 The development of the action potential . . . . . . . . . . . . . . . . . . 18

2.4 Coupling between contraction and excitation . . . . . . . . . . . . . . . 20

2.5 Wiggers diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 The electrocardiogram (ECG) . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Isogeometric Analysis 27

3.1 Introduction to IGA: NURBS and B-splines . . . . . . . . . . . . . . . . 27

3.2 Comparison between IGA and FEM . . . . . . . . . . . . . . . . . . . . 28

3.3 Knot vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 B-spline geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5.1 B-spline curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5.2 B-spline surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5.3 B-spline solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 hpk-refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.7 NURBS: basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.8 Construction of an idealized left ventricle . . . . . . . . . . . . . . . . . 35

3.9 Construction of the biventricle as single patch . . . . . . . . . . . . . . . 38

4 The microscopic models 41

4.1 Roger-McCulloch model . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Bueno-Orovio model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 The electrophysiology models 45

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Strong formulation of the Bidomain equations . . . . . . . . . . . . . . . 45

5.3 Mathematical derivation of the Bidomain equations . . . . . . . . . . . . 46

5.4 Well-posedness of the Bidomain equations . . . . . . . . . . . . . . . . . 49

5.5 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.6 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.7 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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10 CONTENTS

6 Numerical results 576.1 Fibers generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 Left ventricle: Bidomain and Bueno-Orovio models . . . . . . . . . . . . 596.3 Fibers effect: Bidomain and Bueno-Orovio models . . . . . . . . . . . . 656.4 Left ventricle: Bidomain and Luo-Rudy models . . . . . . . . . . . . . . 696.5 Biventricle: Bidomain and Bueno-Orovio models . . . . . . . . . . . . . 746.6 Electrophysiology of the full heart . . . . . . . . . . . . . . . . . . . . . 776.7 Ventricular fibrillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Conclusions and future developments 82

A The Luo-Rudy cellular model 84

Chapter 1

Introduction

Cardiovascular diseases are the most common cause of death globally as of 2008, ac-counting for 30% of them [49]. Several pathologies connected with several kinds of heartfailures are still not completely understood nowadays, or can be addressed in a bet-ter way from the medical viewpoint. In this context, Applied Mathematics can meetMedicine, in an attempt to use complex mathematical models to describe and simulatethe heart activity both in normal and pathological conditions.

The main role of the heart is to pump oxygenated blood, which contains all kinds ofnecessary nutrients, throughout the body, so that all the physiological activities can beperformed by the cells [24]. During the lifetime of an average human being, the heartbeats about 2.5 billion times and pumps in total 200.000 tons of blood into the circulatorysystem [43]. Each heart beat is characterized by a complex electrical excitation of thecardiac tissue, starting from the sinoatrial node and propagating to the other regionsof the organ [14, 21, 23]. This process can be accurately described by mathematicalequations: their numerical simulation presents difficulties that are due to the complexityof the models, which comes from the intrinsic multi-scale nature of the problem and thecharacteristic space-time behavior of the solution [35].

This thesis focuses on the simulation of the electric processes characterizing the heartby using Isogeometric Analysis for the spatial approximation of PDEs [7, 11, 22], whichaims at preserving the exactness of the geometric representation of the domain (fordifferent domain’s shapes) along the computational procedure. Moreover, if NURBSare used, high order continuity of the basis functions in the domain is guaranteed, thusyielding accurate and efficient numerical treatment of PDEs. A relevant reduction innumerical dissipation and dispersion is seen, which is an important requirement for thesolution of PDEs involving wave propagation (e.g. in Acoustic, Elastodynamics, CardiacElectrophysiology [33]) The main goal of this thesis is to address realistic simulationsinvolving ventricles, atria and complete models for the simulation of the cellular activityand the cardiac tissue. In particular, the Bidomain model [9, 10, 14, 40, 47] is used for thedescription of the macroscopic behavior. It is able to describe the electrical propertiesof cardiac muscle cells, taking into account the anisotropy of both the intracellularand extracellular spaces. We consider its coupling with two ionic models describingthe microscopic phenomena: the Bueno-Orovio model [5] and the model by Luo andRudy [27]. The first one is known to be a phenomenological model, i.e. it describesthe electrophysiological behaviour of an external observer, without accounting for the

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12 CHAPTER 1. INTRODUCTION

underlying physiology, whereas the Luo-Rudy model is a so called first generation model,as it provides more detailed and more precise description of the mechanisms of the cells[43]. We first focus on the simulation of the electrocardiology on the left ventricle, whichis modelled as a solid. Then, we perform a simulation involving both ventricles, which areagain seen as 3D structures. Finally, we discuss an application of the Bidomain model,namely the simulation of ventricular fibrillation, and we simulate the electrophysiologyof the complete heart in healthy condition, coupling together the propagation of theaction potential in the atria, developed as surfaces [33], and in the biventricle.

The thesis is structured as follows. Chapter 2 provides an overview of some conceptsof heart physiology, the focus being on cardiac electrophysiology. Chapter 3 is entirelycentered on the Isogeometric Analysis: it provides the reader with the main ideas thatform the basis of the method and the theory underlying the design of geometries. Wepresent, as an example of geometry generation, the same volume representation of theleft ventricle alone and both ventricles connected together, that will be both used inthe numerical simulations of Chapter 6. In Chapters 4 and 5 the microscopic modelsand the Bidomain equations are detailed both from the theoretical and the numericalviewpoints. We refer to chapters 1 to 5 as mostly review chapters. In Chapter 6, theBidomain equations coupled with the ionic model by Luo Rudy and Bueno Orovio aresolved on 3D representations of the ventricle/ventricles. We also discuss strategies forsimulating ventricular fibrillation by using the Bidomain model [3, 24, 40, 53], and thecoupling with the electric behaviour of the atria [33]. Finally, in Chapter 7 conclusionsfollow.

Chapter 2

Heart anatomy and cardiacelectrophysiology

The heart is a muscular organ, whose role is to pump blood through all the bloodvessels of the circulatory system. Blood provides the body with oxygen and all kinds ofnecessary nutrients for the body, as well as assists in the removal of metabolic wastes.In humans, the heart is located in the middle compartment of the chest, between thelungs [14, 21, 23].

The human heart is a double pump made by four chambers: two ventricles in thelower part, that are also called discharging chambers, divided by the interventricularseptum, and two atria in the upper part, separated by the interatrial septum. We referto the atria as receiving chambers [14].

Atria and ventricles are separated by the atrioventricular septum, which containsthe mitral valve in the left heart and the tricuspid valve in the right heart. The leftventricle is connected to the aorta through the aortic valve, whereas the right ventriclehas a connection with the pulmonary artery via the pulmonary valve [21].

The cardiac muscle is generally called myocardium. While the two atria can bemodelled as surfaces, the thickness of the ventricles is significant, so they must be treatedas 3D structures. Ventricles are differentiated into endocardium for the internal partand epicardium for the external one. The word myocardium is used for the lower partof the heart to indicate the central area. The left ventricular wall is known to be abouttwo/three times thicker than the right one [14]. There is also a sack that surrounds theheart, the so called pericardium.

The right atrium receives blood almost continuously from the body’s two majorveins, the superior and inferior venae cavae. A small amount of blood comes inside theright atrium also from the coronary circulation, through the coronary sinus, which isimmediately above and to the middle of the opening of the inferior vena cava. Thischamber has the role of driving blood towards the right ventricle, which synchronouslycontracts, in order to pump the blood itself through the pulmonary circulation, to thelungs and then back to the heart. The left atrium receives oxygenated blood backfrom the lungs via one of the four pulmonary veins, which is pushed after towards theleft ventricle. This last chamber exerts a huge force so as to pump oxygenated bloodtowards the aorta, and then to the entire systemic circulation around the body, untileven the smallest vessels are all reached by nutrients and oxygen. All the presented

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14 CHAPTER 2. HEART ANATOMY AND CARDIAC ELECTROPHYSIOLOGY

Figure 2.1: Schematic diagram of the anatomy of the heart, taken from [14]

Figure 2.2: Structure of the heart wall, taken from [51]

cardiac functions, mechanical and haemodynamical, are triggered and coordinated bythe electrical activity of the heart [23].

2.1. THE CONDUCTION SYSTEM OF THE HEART 15

Figure 2.3: Heart conduction system (taken from [50]). 1. Sinoatrial node. 2. Atrioventricularnode. 3. Bundle of His. 4. Left bundle branch. 5. Left posterior fascicle. 6. Left anteriorfascicle. 7. Left ventricle 8. Ventricular septum. 9. Right ventricle. 10. Right bundle branch

2.1 The conduction system of the heart

The contraction of cardiac cells, as in other muscle cells, is initiated by an electricalactivation due to an action potential, a depolarizing transitory membrane current thatraises the transmembrane potential of an excitable cell from its resting value rangingbetween −90 and −80 mV to slightly positive values, followed by a repolarizing currentthat returns the transmembrane potential to its resting value. Differently from skeletalmuscle cells, cardiac cells can activate themselves autonomously, independently of anervous stimulus. The electrical activity of the heart starts at the so called sinoatrial(SA) node, a group of cardiac pacemaker cells located on the top of the right atrium.The cells of the SA node have the fastest spontaneous depolarization speed of the cardiacconduction system and therefore they control the cardiac frequency in normal conditions.The autonomic nervous system and the endocrine system have direct influence overthe SAN cells, and they can modulate both beat-to-beat and long-term variations ofthe cardiac frequency. In normal conditions, these cells generate an action potentialthat propagates throughout the right atrium and, thanks to the Bachmann’s bundleand some other preferential lines of trasmission, to the left atrium, stimulating themyocardium of both atria so that they will contract in a coordinated and physiologicalway. The activation front reaches the atrioventricular (AV) node, located in the central


conjunction area between atria and ventricles, where, precisely, the interatrial septumand interventricular septum meet. The cells of the AV node have a relatively slowconduction velocity and they are responsible for the major part of normal conductiondelays between atrial and ventricular contractions. Such delays are properly timedin order to optimize the atrial pump activity and to protect the ventricles from tooearly stimulation. The AV node conducts the action potential through the nonexcitableatrioventricular septum, activating the specialized fibers of the bundle of His and thePurkinje network that spread as a tree-like left and right bundle branches ending on theendocardial surface of the ventricles. These Purkinje terminations transmit the actionpotentials to the ventricular walls and cardiac excitation then propagates throughoutthe ventricles. The electrical activation of the left ventricle, which is the largest cardiacchamber among the four that are present, normally starts at the interventricular septum.The electrical activation of the right ventricle triggers normally shortly after the leftventricle one [14, 41].

2.2 Categorization of cardiac cells

In the cardiac tissue, it is possible to differentiate the following categories of cells [14]:

Working cardiomyocytes: they define the atrial and ventricular tissues and are re-sponsible for force development. They present a tubular structure, with a diameterof about 10−20 µm and a length of about 50−150 µm. Myocytes volume and shapecan be complex and variable, according to the tissue region, developmental stageand disease processes. They are covered by a lipid membrane, the sarcolemma, andcontain one or more nuclei, mitochondria, myofibrils, the sarcoplasmic reticulum,sarcomeres, the cytoskeleton anchoring the different organelles and an aqueoussolution, the sarcoplasm, filling the intracellular space. The sarcolemma is a semi-permeable barrier and contains ion channels, pumps and exchangers that allowthe inward and outward currents involved in the action potential, as well as otherproteins involved in cell adhesion and signalling. Cardiomyocytes exhibit a peri-odic structure with crossed striations formed by alternating segments of thick andthin protein filaments. The sarcolemma envelops the so called T-tubules, whichallow quick penetration of the depolarization process to the interior of the cell andplay an important role in the excitation-contraction coupling.

Nodal cells: they are able to autonomously activate themselves thanks to thepresence of particular ionic channels. The SA and AV nodes are made by thiskind of cells. Compared with working myocytes, they are smaller, have limitedcontractile activity and lack T-tubules. They are also susceptible to autonomicinfluences thanks to the presence of specific innervation.

Conduction cells: they are also known as Purkinje cells and have a lot of aspectsin common with working myocytes, but with larger diameter, reduced contractileproteins and a few T-tubules.

Fibroblasts: they are mechano-electrical transducers (but also electrically non-excitable cells) located in areas between and surrounding cardiac myocytes. They

2.2. CATEGORIZATION OF CARDIAC CELLS 17

constitute the major non-myocyte cell population in the ventricles and are re-sponsible for the synthesis of extracellular matrix proteins such as different typesof collagen. Fibroblasts can become myofibroblasts and both types of cells areinvolved in the development of fibrosis in injured cardiac tissue.

Extracellular matrix and collagen: they furnish the passive mechanical propertiesof cardiac tissue. Particularly the extracellular matrix (ECM) occupies an impor-tant role in muscle development and maintainance of the functional integrity ofthe myocardium. The collagen network surrounds single myocytes instead.

Gap junctions: they are essentially ionic channels with a length of about 2−12 nmand a diameter of about 2 nm, used to interconnect myocytes mostly end-to-end(longitudinal gap junctions), but also laterally (transverse gap junctions).

Cardiac stem cells: they regulate myocytes turnover and act for myocardial re-covery after injury.


Figure 2.4: Internal structure of cardiomyocytes (copyright of [24])

2.3 The development of the action potential

We perform an analysis of the main phases of a ventricular action potential in normalconditions (as reported in Figure 2.5) [14]:

Phase 0 : myocytes undergo a rapid depolarization due to the opening of the fastNa+ channels. This causes a rapid increase in the membrane conductance, andthus a rapid influx of Na+ ions into the cell, the INa current, through sodiumchannels

2.3. THE DEVELOPMENT OF THE ACTION POTENTIAL 19

Figure 2.5: Schematic plot of a cardiac AP with its phases: 0 (depolarization or upstroke), 1(peak or notch), 2 (plateau), 3 (repolarization or recovery), 4 (resting). Main ions and currentsinvolved in each phase are also included. Taken from [14].

Phase 1 : inactivation of the fast Na+ channels. The transient net outward currentcausing the small downward deflection of the action potential is due to K+ andCl− ions carried by the Ito1 and Ito2 currents, respectively

Phase 2 : plateau created by a balance between the inward movement of Ca2+,the ICa current through L-type calcium channels, and the outward movement ofpotassium ions K+, contributing to the IKs current through the slow delayedrectifier potassium channels. The sodium-calcium exchanger current INa,Ca andthe sodium/potassium pump current INa,K also play minor roles during this phase

Phase 3 : rapid repolarization phase of the action potential, when the L-typeCa2+ channels close while the slow delayed rectifier (IKs) K

+ channels are stillopen. This ensures a net outward current, corresponding to negative change inmembrane potential, thus allowing more types of K+ channels to open. These areprimarily the rapid delayed rectifier K+ channels (IKr current) and the inwardlyrectifying K+ current IK1. This net outward, positive current causes the cell torepolarize. The delayed rectifier K+ channels close when the membrane potentialis restored to about −80 to −85 mV , while IK1 is maintained throughout phase4, contributing to set the resting membrane potential

Phase 4 : resting phase, where the transmembrane potential remains at the restingvalue of about−84mV until it is stimulated again by an external electrical activity.

We underline that the cardiac action potential is different when moving from oneregion of the heart to another one, as can be easily seen from Figure 2.6. It is possible tonotice that the shape, the duration of the different phases, and also the total duration ofthe action potential, change from the epicardial to the endocardial zone, and also fromatria to ventricles.


Figure 2.6: Heterogeneity of cardiac APs in different regions of the heart, taken from [14].

2.4 Coupling between contraction and excitation

As a conclusion of this chapter, we stress the importance of intracellular calcium concen-tration in the cardiac mechanisms. It influences the pacemaking at the level of the SAnode and the contraction of the cardiac tissue more generally [9]. Due to its role also inthe ventricles, we will treat the evolution of this variable in one of the numerical simula-tions in Chapter 6. Ca2+ increases during the first phases of the AP because of the fluxof ions coming from the extracellular space through the specialized long lasting (L-type)channels and because of the subsequent release calcium induced calcium release of cal-cium from the terminal cistarnae in the sarcoplasmic reticulum, which releases ions. Theinflux of Ca2+ is called trigger calcium [24]. Crossbridge mechanism, i.e. contraction ofthe cardiomyocytes, is initiated by this change of the intracellular calcium concentration(see [35]). At this point, Ca2+ ions bind to troponin-C in the thin filaments, which leadsto the detachment of troponin regulatory complex to the binding site in actin and tothe binding of the head of myosin to actin. Consequently, thin and thick filaments ofthe sarcomers start sliding past each other, the individual sarcomeres shorten, and thewhole cell contracts.

The contraction of the cell continues as long as the cytosolic calcium concentrationremains high enough. Ca2+ intracellular calcium concentration starts decreasing in thelast part of the AP, due to the ions leak through the calcium pump and the Na-Caexchanger and binding to cytosolic proteins (Ca2+ buffers) inside the sarcolemma. Inthis phase, some of the free ions are also trapped by the so called sarcoendoplasmicreticulum calcium ATPase, which is an ATP-dependent calcium pump. Calcium startsto dissociate from the binding sites and the troponin complexes bind again to the actin

2.5. WIGGERS DIAGRAM 21

Figure 2.7: Action of Ca2+ in the contraction process (image taken from [24])

fibers. The sarcomere returns then to its initial length and the cell relaxes.

2.5 Wiggers diagram

A Wiggers diagram is a standard plot used in cardiac physiology [29]. It presents timeon the X-axis and the following possible values on the Y-axis:

Blood pressure (specifically aortic pressure, ventricular pressure, atrial pressure)

Ventricular volume

Electrocardiogram (ECG), which is a recording of the electrical activity of theheart over a period of time using electrodes placed on the skin

Phonocardiogram, which is a plot that shows all the sounds made by the heart.It is recorded with a machine called phonocardiograph

The Wiggers diagram clearly illustrates the coordinated variation of these values asthe heart beats, assisting one in understanding the entire cardiac cycle. In particular,it permits to see clearly the connections and the coordination between the electrophys-iological behaviour, the mechanical one, and the characteristics in the fluid mechanicsfield, in terms of pressures in different significant points of the heart.

The cardiac cycle is divided into the following different phases:

Atrial contraction phase: the mitral valve is opened and the blood flows into theleft ventricle

Isovolumetric contraction phase, rapid ventricular ejection phase, slow ventricularejection phase: the aortic valve is opened whereas the mitral valve is closed. Theleft ventricle is contracting and pumping blood towards the aorta

Isovolumetric relaxation phase: both the mitral and the aortic valves are closedand the left ventricle is relaxing


Figure 2.8: Wiggers diagram (picture taken from [52])

Rapid inflow phase: the mitral valve opens again, the blood flows again into theleft ventricle and the cycle repeats itself

The cardiac cycle has been presented hereabove for the left part of the heart, but thesame phases hold also for the right part.

Looking at the aortic pressure plot, we can see that it has the value of almost80 mmHg until the rapid ventricular ejection phase occurs, when the blood is pushedtowards the aorta and it rises to the value of almost 120 mmHg and then comes back tothe original value in the relaxation phase. In the ventricular pressure diagram there isa huge variation during the isovolumetric contraction phase, because when the ventriclecontracts, the pressure inside it builds up and goes from 0 mmHg to 120 mmHg. Afterthat we observe a pressure drop when the ventricle stops contracting, i.e. during theisovolumetric relaxation phase. The next plot that we are going to analyze is the atrialpressure wave. We notice that it does not present huge variations during the cardiaccycle, and it rises a little only during the atrial contraction phase. The ventricularvolume diagram shows a huge drop in the volume during the rapid ventricular ejectionphase, because the blood is being pumped out of the left ventricle into the aorta. Thevolume starts to increase slowly when the mitral valve opens again, so that the leftventricle starts to accumulate blood once again coming from the left atrium.

Considering the electrocardiogram, we can distinguish different kind of waves and

2.5. WIGGERS DIAGRAM 23

Figure 2.9: Schematic view of a normal ECG waveform (picture taken from [14])

phases [14]:

P wave: it is associated with atrial depolarization. The completion of atrial re-polarization generates another wave, which is not visible in normal ECGs becausehidden by the QRS complex

QRS complex : it is usually the largest deflection in the ECG, associated withthe complex activation sequence and depolarization of the ventricles, resulting inupward and downward deflections and multiple peaks in most of the leads. TheQ wave corresponds to the activation of the interventricular septum, the R wavecorresponds to the septum activation completion and the apex activation, theS wave corresponds to the activation of the ventricular free walls and the basalregion. These three waves are not always detectable and can sometimes merge,yielding a quite variable morphology of the QRS complex.

ST segment : this is a period, which lasts a few hundreds milliseconds and followsthe QRS complex, where all cells are depolarized during the plateau of the actionpotential.

T wave: it is connected to ventricular repolarization, and its morphology dependson the repolarization sequence through the ventricles. In normal hearts, the startof the T wave is thought to correlate with the plateau ending of the epicardialcells, the T wave peak with the full repolarization of the epicardium, the end ofthe T wave with the repolarization of the midmyocardial M-cells.

U wave: it is a small diastolic deflection that occurs after the T wave, which isconnected to repolarization of the papillary muscles or Purkinje fibers.

The phonocardiogram contains the information about the heart sounds that occurduring the cardiac cycle. There are in particular only two strong sounds that should


be heard in an healthy heart: the first one is heard during the isovolumetric contrac-tion phase when the mitral valve closes, whereas the second one is originated from theisovolumetric relaxation phase, when the aortic valve closes.

When we talk about the cardiac cycle, we distinguish systolic phases, when theheart, or a part of it, is contracting, and diastolic ones, when the heart, or a part of it,is relaxing, as we can see from Figure 2.8.

2.6 The electrocardiogram (ECG)

The ECG is the recording of the extracellular potential on the human body surface, dueto the propagation of cardiac action potentials. There are several kinds of electrodes,usually placed at standard positions, in order to record this signal. We refer to them asEinthoven leads, augmented leads and precordial leads [14].

Einthoven leads: given the potentials at the left foot ΦF , at the right arm ΦR andat the left arm ΦL, Einthoven leads are defined by the subsequent potentials:

VI = ΦL − ΦR

VII = ΦF − ΦR

VIII = ΦF − ΦL

We have VI + VIII = VII by Kirchhoff’s voltage law. Einthoven assumed that the heartis located at the center of a homogeneous spherical conductor representing the torso,and that the three vectors associated with these three leads form an equilateral trianglein the vertical frontal plane of the body [14].

Augmented leads: these three additional leads, denoted by aVR, aVL, aVF and knownas augmented leads, are defined on the frontal plane in the directions that bisect thesectors associated with each pair of Einthoven leads [14]. We give here the example ofhow the aVF potential can be calculated:

VaVF = ΦF −ΦL + ΦR

2

In order to measure potentials close to the heart, six additional leads are defined on ahorizontal plane by using the Wilson central terminal:

ΦCT =ΦR + ΦL + ΦF

3

Precordial leads: The six precordial (or chest) leads, denoted by V1, ..., V6, are ob-tained by comparing the Wilson central terminal with unipolar electrode readings takenfrom six different locations on the chest. V1 and V2 are located at the fourth intercostalspace on the right and left side of the sternum, and the other four proceed around theleft chest just below the fourth rib, ending with V6 under the armpit [14]. The 12 leadsI, II, III, aVR, aVL, aVF , V1, ..., V6 form the most commonly used clinical ECG system.

The abovementioned explanation of how an electrocardiogram is recorded leads tothe fact that, in order to accurately reproduce this diagram from the numerical viewpoint, we need to consider a coupling between the heart and the torso, considering howthe propagation of the electric signal is induced in this bigger domain that contains theheart itself [43].

2.6. THE ELECTROCARDIOGRAM (ECG) 25

Figure 2.10: Schematic view of the Einthoven leads (picture taken from [14])

Figure 2.11: Schematic view of the augmented leads (picture taken from [37])


Figure 2.12: Schematic view of the six precordial leads V1, ..., V6 in a frontal view (left) andhorizontal cross-section (right) of the thorax (picture taken from [14])

Chapter 3

Isogeometric Analysis

Isogeometric Analysis is a method for the spatial approximation of PDEs which aims atpreserving the exactness of the geometric representation of the domain along the com-putational procedure; moreover, if NURBS are used, high order continuity of the basisfunctions in the domain is guaranteed, thus yielding accurate and efficient numericaltreatment of the PDEs [7, 11, 22].

3.1 Introduction to IGA: NURBS and B-splines

The idea of Isogeometric Analysis, often abbreviated as IGA, came to fill the existinggap between the world of computer-aided design (CAD) and finite element analysis.

In order to deal with real-life engineering problems in a proper way, complex geome-tries must be generated using the CAD approach, and these ones must be translated intoanalysis-suitable files, meshed and given as input to large-scale FEA codes. This taskusually takes over 80% of the total overall analysis time for a specific project. Moreover,when the Finite element treatment comes in, the exact CAD geometry boundary is ap-proximated in a piecewise way, giving an approximate representation of it. Due to thefact that Non Uniform Rational Basis-Splines (NURBS) are used in the design phase,the Isogeometric Analysis has the aim of using NURBS functions in both the design andmathematical treatment of the geometry, as well as in the numerical phase [11].

IGA improves the finite element spaces that give the numerical solution on the exactrepresentation of the geometry, based here on a different set of basis functions with thetypical characteristics of NURBS. This fact is in accordance with the isoparametric concept,which is common in classical finite elements, and consists of the definition of the samebasis functions for both the geometry and the numerical analysis and approximation.

There exist many stable and efficient algorithms to generate NURBS geometries[30]. They also have useful mathematical properties, such as the ability to be refinedthrough knot insertion, Cp−1 continuity for pth-order NURBS, the variation diminishingand convex hull properties. Nowadays there exist several generalizations of NURBStechnology, such as T-splines [2], that can be implemented in this context too. NURBS,also called rational B-splines, comes as an evolution of the older B-splines, which haveput the basis in this field.

We underline some important aspects that are typical of NURBS [11]. The basis

27

28 CHAPTER 3. ISOGEOMETRIC ANALYSIS

functions are usually not interpolatory. There exist two different meshes: the controlmesh and the physical mesh.

The control points define the control mesh, which do not necessarily lay in the actualgeometry, and the control mesh piecewise linearly interpolates the control points. Thecontrol variables are the degrees of freedom (DOFs) and they are located at the controlpoints. Control elements can collapse towards more primitive shapes, such as trianglesin 2D or tetrahedra in 3D. The control mesh may also be subjected to strong distortion,while at the same time, for sufficiently smooth NURBS, the physical geometry may stillremain valid.

The actual geometry is represented by the physical mesh. There are two notionsof elements in the physical mesh, the patch and the knot span. A patch can be seenas a subdomain, and when the geometry is divided in more patches we talk about amultipatch approach. Each patch has two representations, one in a parent domainand one in physical space. In two-dimensional topologies, a patch is a rectangle in theparent domain representation. In three dimensions it is a cuboid. Each patch can bedecomposed into knot spans. Knots are points, lines, and surfaces in 1D, 2D and 3D,respectively. Knot spans are bounded by knots. These define element domains wherebasis functions are arbitrarily smooth (i.e., C∞). Across the knots, basis functionswill be Cp−m where p is the degree of the polynomial and m is the multiplicity of theknot in question. The knots also have a representation in both a parent domain andphysical space. The index space of a patch is another important element, that permitsto discriminate between knots having multiplicity greater than one.

3.2 Comparison between IGA and FEM

Due to the importance and widespread diffusion of the Finite Element methodology inComputational Mechanics, the use of IGA is motivated by several advantages, some ofwhich can be already seen from Section 3.1. With Isogeometric Analysis it is potentiallypossible to:

Give an exact representation of the geometry under investigation, with the com-plete elimination of the so called geometry approximation error

Reduce in a relevant sense numerical dissipation and dispersion errors, which is astrict requirement for the solutions of all the problems involving wave and frontpropagations, as, e.g., in Acoustic, Elastodynamics, and of course Cardiac Elec-trophysiology [33]. This geometric advantage is given by high order continuity ofthe basis functions

IGA shows some drawbacks to be accounted for. Even if the space discretization isconstructed without the interaction with CAD geometry (reducing a lot in this way thetime which is needed to develop the domain itself) the mesh is made by rectangular (in2D) and parallelepiped (in 3D) structures generally speaking, which in practise shouldresemble squares and cubes respectively as much as possible. Moreover, if the mesh isnot fine enough, we can have several issues in terms of space and time propagation ofthe wave front, leading for instance to non-smooth fronts.

3.3. KNOT VECTORS 29

3.3 Knot vectors

A knot vector, expressed as Ξ = ξ1, ξ2, ..., ξn+p+1 where ξi ∈ R is the ith knot, i =1, 2, ..., n+p+1, p is the polynomial order of the basis functions, and n is the number ofbasis functions used to construct a B-spline curve, is a non-decreasing set of coordinatesin the parameter space. The knots divide the parameter space into elements.

A knot vector is supposed to be uniform in case of equal spacing in the parameterspace. Otherwise it is said to be non-uniform. A single knot value may be repeatedin the vector, and this fact changes the properties of the basis. We refer to an openknot vector if its first and last values appears p + 1 times. Open knot vectors are thestandard in CAD, and basis functions defined from them are interpolatory (as in the caseof ”nodes” in Finite Element Methods) at the ends of every parameter space interval[ξ1, ξn+p+1] defined in a suitable dimension.

In general, the boundary of a B-spline object with d parametric dimensions is itselfa B-spline object of dimension d − 1. For example, each edge of a B-spline surface isitself a B-spline curve.

3.4 Basis functions

After the introduction of a knot vector, the B-spline basis functions can be defined in arecursive way starting from a piecewise constant function (i.e p = 0):

Ni,0(ξ) = I[ξi,ξi+1)(s) 1 ≤ i ≤ n (3.1)

Using the so called Cox-de Boor recursion formula (Cox, 1971; de Boor, 1972), thefollowing development is considered for p = 1, 2, 3, ...:

Ni,p(ξ) =ξ − ξiξi+p − ξi

Ni,p−1(ξ) +ξi+p+1 − ξξi+p+1 − ξi+1

Ni+1,p−1(ξ) 1 ≤ i ≤ n (3.2)

This iterative procedure can be applied both for uniform and non-uniform knot vectors.In the second case, in order to use the previous formula properly, we simply have to addin this framework that 0

0 := 0.

For B-spline functions with p = 0 and p = 1, we have the same results as for standardpiecewise constant and linear finite element functions, respectively. From the quadraticone (p = 2), we can notice some differences. In each case the functions are exactly thesame, with a relative shift, whereas the shape of a quadratic finite element functiondepends on whether it corresponds to an internal node or an end node. This patterncan be observed for higher-order B-splines.

The basis constitutes a partition of unity, i.e:

n∑i=1

Ni,p(ξ) = 1 ∀ξ (3.3)

Moreover:

Ni,p(ξ) ≥ 0 ∀ξ (3.4)


Figure 3.1: Quadratic basis functions for open, non-uniform knot vector Ξ = 0, 0, 0, 1, 2, 3,3, 4, 5, 5, 5

Figure 3.2: Cubic basis functions for open, non-uniform knot vector Ξ = 0, 0, 0, 0, 1/3, 2/3,2/3, 1, 1, 1, 1

Another feature is that each pth order function has p − 1 continuous derivativesaccross the element boundaries (i.e, across the knots). Additionally, a B-spline functionof order p has always support on p+ 1 knot spans.

In Figure 3.2 a quadratic example with non-uniform knot vector is presented, usingΞ = ξ1, ξ2, ..., ξ11 = 0, 0, 0, 1, 2, 3, 3, 4, 5, 5, 5. It is possible to notice that the basisfunctions are interpolatory at the ends of the interval and also at ξ = 3, where repeatedknots are located. In this position, only C0-continuity is guaranteed. Elsewhere, thefunctions are C1-continuous. In general basis functions of order p have p−mi continuousderivatives across knot ξi, where the value mi is the corresponding multiplicity. When

3.5. B-SPLINE GEOMETRIES 31

the multiplicity of an internal knot value is exactly p, the basis is interpolatory atthat knot. If the multiplicity is p + 1, the basis becomes discontinuous, and the patchboundary is formed.

Derivatives of B-spline basis functions are efficiently represented in terms of B-splinelower order bases. It is certainly possible to give a formula to compute derivatives ofgeneric order k of the ith basis function, for a given polynomial order p and knot vectorΞ:

dk

dkξNi,p(ξ) =

p!

(p− k)!

k∑j=0

αk,jNi+j,p−k(ξ) (3.5)

with:

α0,0 = 1,

αk,0 =αk−1,0

ξi+p−k+1 − ξi,

αk,j =αk−1,j − αk−1,j−1

ξi+p+j−k+1 − ξi+jj = 1, ..., k − 1,

αk,k =−αk−1,j−1

ξi+p+1 − ξi+k.

The denominator of these coefficients can be zero in the presence of repeated knots.Whenever it happens, the coefficient is defined to be zero.

3.5 B-spline geometries

According to how knot vectors, univariate B-spline basis functions and control pointsare combined, we can construct three kinds of B-spline geometries: B-splines curves inthe 1D settings, B-splines surfaces in the 2D framework, and finally B-splines solids in3D.

3.5.1 B-spline curves

B-spline curves in Rd are constructed by taking a linear combination of B-spline basisfunctions, just as in classical Finite Elements. The vector-valued coefficients of the basisfunctions are referred to as control points. These are analogous to nodal coordinates infinite element analysis in that they are the coefficients of the basis functions, but thenon-interpolatory nature of the basis does not lead to a concrete interpretation of thecontrol point values.

Given n basis functions Ni,p i = 1, 2, ..., n, and corresponding control points Bi ∈Rd, i = 1, 2, ..., n, a piecewise-polynomial B-spline curve is obtained as follows:

C(ξ) =

n∑i=1

Ni,p(ξ)Bi (3.6)


Figure 3.3: Example of B-spline curve

Piecewise linear interpolation of the control points gives the so called control polygon.

An affine transformation of a B-spline curve is obtained by applying the transfor-mation directly to the control points. An affine transformation is mapping Φ : R3 → R3

such that ∀x ∈ R3:

Φ(x) = Ax + v (3.7)

for some matrix A ∈ R3x3 and vector v ∈ R3. The property of applying an affinetransformation to a curve by acting directly and only on the control points is calledaffine covariance.

Many properties of B-spline curves follow directly from the properties of their basisfunctions.

Additionally B-spline curves possess a variation diminishing property and are mono-tone.

3.5.2 B-spline surfaces

A B-spline surface is defined by:

S(ξ, η) =

n∑i=1

m∑j=1

Ni,p(ξ)Mj,q(η)Bi,j (3.8)

where Bi,j , i = 1, 2, ..., n, j = 1, 2, ...,m is the control net, p and q are the polynomialorders, Ξ = ξ1, ξ2, ..., ξn+p+1 and H = η1, η2, ..., ηm+q+1 are the knot vectors, andNi,p(ξ), Mj,q(η) are univariate B-spline basis functions.

Many of the properties of a B-spline surface are inherited from its tensor productnature.

3.5. B-SPLINE GEOMETRIES 33

Figure 3.4: Example of B-spline surface

Figure 3.5: Example of B-spline solid

3.5.3 B-spline solids

Tensor product B-spline solids are defined in the same manner of the B-spline surfaces.We introduce a control lattice Bi,j,k, i = 1, 2, ..., n, j = 1, 2, ...,m, k = 1, 2, ..., l, poly-nomial orders p, q, r, and knot vectors Ξ = ξ1, ξ2, ..., ξn+p+1, H = η1, η2, ..., ηm+q+1and Z = ζ1, ζ2, ..., ζl+r+1, a B-spline solid is defined by:

S(ξ, η, ζ) =n∑i=1

m∑j=1

l∑k=1

Ni,p(ξ)Mj,q(η)Lk,r(ζ)Bi,j,k (3.9)

The properties of B-spline solids are again trivariate generalizations of the B-splinesurfaces, and compatible with the one describes for the basis functions.


3.6 hpk-refinements

B-splines can be enriched in several ways, while leaving the underlying geometry andits parametrization exactly the same.

The first technique is called knot insertion where, given a knot vector Ξ = ξ1, ξ2, ..., ξn+p+1we introduce the notion of an extended knot vector Ξ = ξ1 = ξ1, ξ2, ..., ξn+m+p+1 =ξn+p+1, so that Ξ ⊂ Ξ. The new n + m basis functions can be computed using againthe Cox-de Boor recursion formula applied to the new knot vector Ξ, whereas the newcontrol points are formed as linear combinations of the original one.

This procedure can be assimilated to the h−refinement in Finite Element Analysis,due to the fact that elements are splitted into new ones.

The second mechanism is called order/degree elevation, which permits to raise thepolynomial order of the basis functions used to represent the geometry. Recalling thatthe basis has p − mi continuous derivatives across element boundaries, when p is in-creased, mi must be increased as well to preserve the discontinuities in the variousderivatives already existing in the original case. So, when order elevation is performed,the multiplicity of each knot value is increased by one, but no knot values are added.Also in this case, neither the geometry nor the parametrization are changed.

Order elevation is clearly similar to the classical p − refinement strategy in finiteelement analysis as it increases the polynomial order of the basis, but it preserves con-tinuity.

Lastly, there is a technique typical of Isogeometric Analysis only, that goes underthe name of k − refinement, where the order of the original, coarsest curve is elevatedto q, and then a knot value ξ is uniquely inserted. Of course the basis would have q− 1continuous derivatives at ξ. This higher-order method comes from a smart combinationof the two previous techniques.

3.7 NURBS: basis functions

Even though the B-spline basis defined by 3.2 allows to create a large variety of geome-tries, many very commonly used objects can not be designed using it. Just to give animportant example, circular shapes are not exactly described using a polynomial bases.In order to include also these cases we must introduce the concept of Non-UniformRational B-splines. NURBS entities are obtained through projection in Rd of B-splineobjects living in Rd+1. From the algebraic point of view, this means that we have tointroduce a new basis for NURBS as:

Rpi (s) =Ni,p(s)ωiW (s)

(3.10)

where

W (s) =

n∑i=1

Ni,p(s)ωi (3.11)

3.8. CONSTRUCTION OF AN IDEALIZED LEFT VENTRICLE 35

Figure 3.6: Definition of an arc with corresponding weights

is called weighting function and ωini=1 are called weights. The NURBS curve is thendefined as:

C(s) =n∑i=1

Rpi (s)Bi

It is now natural to extend this definition of basis also to manifolds of higher dimen-sions, thus finding:

Rp,qi,j (s, t) =Ni,p(s)Mj,q(t)ωi,j∑n

a=1

∑mb=1Na,p(s)Mb,q(t)ωa,b

,

Rp,q,ri,j,k (s, t, w) =Ni,p(s)Mj,q(t)Lk,r(w)ωi,j,k∑n

a=1

∑mb=1

∑lc=1Na,p(s)Mb,q(t)Lc,r(w)ωa,b,c

,

3.8 Construction of an idealized left ventricle

We present here the generation of an idealized left ventricle geometry, based on the socalled prolate ellipsoid, with the addition of an hole indicating the mitral valve. To do so,we have used the tools and functions available in the GeoPDEs MATLAB library [12].The easier way to generate this solid, which develops itself along the three parametricdirections s, t and k, is to start from the following knot vector:

Ξ = [0, 0, 0, 1/2, 1/2, 1, 1, 1]

which correspond to the definition of one parametric direction.From Figure 3.7 we see that the basic structure that is used to construct the left

ventricle is given by a combination of two arcs, constructed using the theoretical weights


Figure 3.7: Construction of a prolate ellipsoid with a hole, going from a curve to a surface toa volume with 9 elements

Figure 3.8: Knots refinement and top (refined) view of the left ventricle, with 3268 elements

3.8. CONSTRUCTION OF AN IDEALIZED LEFT VENTRICLE 37

Figure 3.9: Control points on the refined left ventricle (5670 in total)

shown in Figure 3.6. These two curves are then connected together in order to createthe green surface on the top right. After this union, performing a rotation of 180° ofthe surface, the final geometry is created. The idealized left ventricle is generated usinga single NURBS patch with P2/C0 basis functions. An order elevation is done at thispoint so as to have bases with regularity of 2 in the three parametric directions.

Once the domain is created, in order to obtain accurate results, also in terms ofpropagation of the action potential front and timing of the progation, more DOFS andelements must be added. Figure 3.8 shows an example of h-refinement. The top viewof the ventricle under investigation is also given. Figure 3.9 permits to see the controlpoints and the control mesh attached to the geometry. The manipulation of the controlpoints will be fundamental in the second example where, starting from the currentdomain, the biventricular geometry will be derived.


Figure 3.10: Control points on the initial left ventricle (75 in total)

3.9 Construction of the biventricle as single patch

We have taken the previous illustrated geometry and, by a combination of necessary knotinsertions and control points rearrangements, we have obtained two different cavities,representing the left and the right ventricles, with the corresponding valves with respectto the atria geometries (the mitral and tricuspid valves respectively).

Following Figure 3.11 we have firstly done an elongation of the left ventricle, in orderto create the placement for both cavities. The left ventricle will have more space becausethe wall is almost twice thicker than the one of the right ventricle, and the volume ofthe two chambers should be approximately the same. The second step is connected withthe closure of the interventricular septum. After this, the thickness of the right ventricleis reduced, so as to accomodate physiological values. In the end, a knot refine permitsto obtain more precision in the manipulation of the interventricular septum, so that theclosure can be guaranteed to be sharp and continuous.

Some conditions (known in the context of IGA as master-slave conditions [26]) are

3.9. CONSTRUCTION OF THE BIVENTRICLE AS SINGLE PATCH 39

Figure 3.11: Sequences of meshes obtained to construct both ventricles (from left-right, top-down)

applied so as to ensure that the solution in the conjunction DOFS is equal on both sides.Distorsion in the nearby of the interventricular septum can be seen. This is a well

known issue in IGA for single patch NURBS geometries, as discussed in [42], wheresome post-processing remedies are detailed. In our studies, we simply generate a veryfine mesh in that neighborhood, so as to reduce as much as possible this effect.


Figure 3.12: Lateral and top view of the two ventricles, with 6105 elements

Figure 3.13: Control points on the biventricle geometry (10283 in total)

Chapter 4

The microscopic models

We discuss the mathematical aspects of the electrophysiology of the heart by introducingthe models that have been used for simulating the individual ionic currents for the cellsin the ventricle. From a general point of view these models may be divided into threecategories: phenomenological models, first generation models, and second generationmodels. Phenomenological models describe the action potential from the perspectiveof an external observer, without accounting for the underlying physiology, whereas firstand second generation models attempt to include a description of the mechanisms ofthe cell [43].

In this paper we will discuss the Bueno-Orovio [5] and Roger-McCulloch [14, 18, 43]models, which are phenomenological models, as well as the Luo-Rudy model [27], whichis more involved and takes into account in a fairly more detailed manner the membranemechanisms that occur in the cardiomyocytes on the ventricles. The Luo-Rudy modelwill be fully addressed in appendix A, due to its huge mathematical background.

A general model in this context is made up of a certain number of gating variables,concentrations of ions that are effectively modelled, the transmembrane potential, andformulas essential to define some ionic currents or ionic current densities. The role of allthese quantities will be clarified in the next chapter, where the complete coupling withthe macroscopic model will be discussed.

4.1 Roger-McCulloch model

The equations that compose this very simple phenomenological reduced ionic model arethe following ones [14, 18]

dv

dt= Gv

(1− v

vth

)(1− v

vp

)+ η1vw t > 0 (4.1)

dw

dt= η2

(v

vp− η3w

)t > 0 (4.2)

v(0) = 100 (4.3)

w(0) = 0 (4.4)

where v is the transmembrane potential (expressed in mV ) and w is the unique adi-mensional gating variable of the model. The expression of the first right hand side can

41

42 CHAPTER 4. THE MICROSCOPIC MODELS

Roger-McCulloch parameters vth = 13 mV η1 = 4.4 ms−1

vp = 100 mV η2 = 0.012G = 1.5 ms−1 η3 = 1

Table 4.1: Values of the coefficients that appear inside Roger-McCulloch model. Taken from[18]

be used, giving suitable values to v and w, to perform an estimation of the total ioniccurrent, without distinction, in this case, of different fluxes accross the membrane.

Table 4.1 collects all the constants that are needed to set and calibrate the previoussystem from the numerical aspect:

Due to the fact that the Roger-McCulloch model is very simple and gives an un-physiological action potential range, the complete focus of this thesis is on the othertwo models, even if the Roger-McCulloch one constitutes a good starting point from thetheorical viewpoint.

4.2 Bueno-Orovio model

This phenomenological model is totally based on human heart experimental data [5]. Itis a minimal ventricular human model that is designed to reproduce important tissue-level characteristics of epicardial, endocardial and midmyocardial cells, including actionpotential (AP) amplitudes and morphologies, upstroke velocities, steady-state actionpotential duration (APD) and conduction velocity restitution curves, minimum APD,and minimum diastolic interval. The differential equations for the four variables thatare present in the model are as follows:

∂u

∂t= ∇ · (D∇u)− (Jfi + Jso + Jsi) t > 0 (4.5)

dv

dt=

(1−H(u− θv))(v∞ − v)

τ−v− H(u− θv)v

τ+v

t > 0 (4.6)

dw

dt=

(1−H(u− θw))(w∞ − w)

τ−w− H(u− θw)w

τ+w

t > 0 (4.7)

ds

dt=

1

τs

1 + tanh(ks(u− us))2− s

t > 0 (4.8)

u(0) = 0 (4.9)

v(0) = 1 (4.10)

w(0) = 1 (4.11)

s(0) = 0 (4.12)

where the three currents (fast inward current, slow outward current and slow inwardcurrent respectively) are given by the following equations:

Jfi =−vH(u− θv)(u− θv)(uu − u)

τfi(4.13)

4.2. BUENO-OROVIO MODEL 43

Jso =(u− u0)(1−H(u− θw))

τ0+H(u− θw)

τso(4.14)

Jsi =−H(u− θw)ws

τsi(4.15)

and H(x) is the standard Heaviside function. Equation (4.5) defines potentially thespace and time evolution along a tissue of the adimensional potential u ∈ [0, 1]. AMonodomain − like formulation is considered. The Monodomain model can be seenas a reduction of the Bidomain model, starting from the assumption that the intra andextracellular domains have equal anisotropy ratios. For the purposes of this project,this equation has been substituted by the Bidomain equations themselves. ODEs 4.6-4.8 define effectively the Bueno-Orovio cellular model, where the focus is only on thetime evolution.

In order to get the dimensional transmembrane potential the following formula mustbe applied:

VmV = 85.7u− 84 (4.16)

We refer to v, w, s as gating variables, and in particular the last one is knownto simulate approximately the behaviour of the Calcium concentration. Several timeconstants, function of the voltage variable u, are used in the previous formulation, andcan be introduced as follows:

τ−v = (1−H(u− θ−v ))τ−v1 +H(u− θ−v )τ−v2

τ−w = τ−w1 + (τ−w2 − τ−w1)(1 + tanh(k−w (u− u−w)))/2

τso = τso1 + (τso2 − τso1)(1 + tanh(kso(u− uso)))/2

τs = (1−H(u− θw))τs1 +H(u− θw)τs2

τo = (1−H(u− θo))τo1 +H(u− θo)τo2

Two infinity values are also defined:

v∞ =

1 u < θ−v0 u ≥ θ−v

w∞ = (1−H(u− θo))(1− u/τw∞) +H(u− θo)w∗∞

According to the considered area on the ventricle, i.e internal, intermediate or exter-nal part, along the thickness, the same set of model parameters can be calibrated in adifferent way. For all the simulations, for the sake of simplicity and in order to reduce thecomputational cost, an ”averaged” Bueno-Orovio model considering the midmyocardialvalues has been applied on the entire geometry. These parameter values are the onesreported in table 4.2: For further details about the model, the choice of internal andnumerical parameters, and the corresponding desired output, see [5].

44 CHAPTER 4. THE MICROSCOPIC MODELS

Symbol Value

uo 0

uu 1.61

θv 0.3

θw 0.13

θ−v 0.1

θo 0.005

τ−v1 80 s

τ−v2 1.4506 s

τ+v 1.4506 s

τ−w1 70 s

τ−w2 8 s

k−w 200

u−w 0.016

τ+w 280 s

τfi 0.078 s

τo1 410 s

τo2 7 s

τso1 91 s

τso2 0.8 s

kso 2.1

uso 0.6

τs1 2.7342 s

τs2 4 s

ks 2.0994

us 0.9087

τsi 3.3849 s

τw∞ 0.01 s

w∗∞ 0.5

Table 4.2: Parameters of the Bueno-Orovio model, taken from [5]

Chapter 5

The electrophysiology models

5.1 Introduction

The Bidomain model is a mathematical model able to describe the electrical propertiesof cardiac muscle cells (also called cardiomyocytes), taking into account the anisotropy ofboth the intracellular and extracellular spaces [9, 10, 14, 40, 47]. It is a continuum model,so it means that it is used to describe the average properties of many cardiomyocytes,and not the behaviour of single cells. This is why it is supposed to be a macroscopicmodel.

5.2 Strong formulation of the Bidomain equations

Let Ω ⊂ R3 be the domain, that could be the heart tissue or simply a part of it (as inthis case). Ω can be seen as the composition of two interpenetrating domains Ωi andΩe, representing the intracellular and extracellular space respectively. Considering eachpoint x ∈ R3, it is possible to associate the so called intra and extracellular potential, uiand ue respectively, that are averaged values over a suitable small subdomain includinga certain number of cells.

The Bidomain model can be now introduced in its so-called parabolic-parabolic (PP)formulation:

cm∂v

∂t−∇ · (Di∇ui) + χIion(v,w, z) = Isi in Ω × (0, T ) (5.1)

−cm∂v

∂t−∇ · (De∇ue)− χIion(v,w, z) = Ise in Ω × (0, T ) (5.2)

∂w

∂t− F(v,w) = 0 in Ω × (0, T ) (5.3)

∂z

∂t−G(v,w, z) = 0 in Ω × (0, T ) (5.4)

nTDi,e∇ui,e = 0 in ∂Ω × (0, T ) (5.5)

v = v0 in Ω × 0 (5.6)

w = w0 in Ω × 0 (5.7)

z = z0 in Ω × 0 (5.8)

45

46 CHAPTER 5. THE ELECTROPHYSIOLOGY MODELS

where the first two PDEs define effectively the Bidomain model, whereas the followingtwo ODEs are fundamental to make a coupling with the microscopic behaviour of asingle cell, already detailed Chapter 4. v = ui − ue is the transmembrane potential, cm= χCm is the specific capacitance, with Cm total membrane capacitance and χ beingthe area of cell membrane per tissue volume. The vector w = w1,w2, ...,wk expressesk recovery (or gating) variables. They play the role of probability density functions(whose values must be for this reason between 0 and 1), whose aim is to model thefraction of open ionic channels across the transmembrane of a single cell. The vectorz = z1, z2, ..., zk indicates m ionic concentration variables. The most important one inthis context is usually related to the calcium ion Ca2+.

Moreover Di = Di(x) and De = De(x) refer to the intra and extracellular conduc-tivity tensors, which are essential to describe the electric anisotropy of the heart tissue.This is due to the fact that muscle cardiac cells are oriented in different ways accord-ing to the part of the heart under consideration. Giving, for each point x ∈ R3, anorthonormal basis al(x) (parallel to fibers direction), at(x) (perpendicular to al(x) butlaying on the tangent plane of the muscular layer) and an(x) (normal to the previousdefined tangent plane), for the purposes of this work, Di(x) and De(x), following [14],have been modelled as follows:

Di,e(x) = σi,el (x)al(x)aTl (x) + σi,et (x)at(x)aTt (x) + σi,en (x)an(x)aTn(x) (5.9)

with σi,el (x), σi,et (x), σi,en (x) indicating the local intra and extracellular conductivitiesat each point.

The homogeneous Neumann boundary condition 5.5 comes from the assumption ofelectrically isolated domain. Three Dirichlet boundary conditions are also prescribedwith respect to v, w, z.

By summing the two PDEs (5.1) and (5.2), however, the system can be viewed underthe following parabolic-elliptic (PE) formulation:

cm∂v∂t −∇ · (Di∇v)−∇ · (Di∇ue) + χIion(v,w, z) = Isi in Ω × (0, T )

−∇ · (Di∇v)−∇ · ((Di +De)∇ue) = Ise + Isi in Ω × (0, T )∂w∂t − F(v,w) = 0 in Ω × (0, T )∂z∂t −G(v,w, z) = 0 in Ω × (0, T )

nTDi∇(v + ue) = 0 in ∂Ω × (0, T )

nT (Di +De)∇ue + nTDi∇v = 0 in ∂Ω × (0, T )

v = v0 in Ω × 0w = w0 in Ω × 0z = z0 in Ω × 0

5.3 Mathematical derivation of the Bidomain equations

In this part we derive the Bidomain equations under the assumption of no externalcurrent, based on [43].

Considering electromagnetism, the Maxwell-Faraday equation permits to link theelectric field E ∈ Rd and the magnetic one B ∈ Rd. In a generic medium, specified by aregion of space of dimensionality d, we have:

5.3. MATHEMATICAL DERIVATION OF THE BIDOMAIN EQUATIONS 47

∇×E +∂B

∂t= 0 (5.10)

We introduce the notion of potential u ∈ R:

u =

∫Γ

E · dl (5.11)

where Γ is a line connecting the zero potential point x0 to another point of interest x.Under the quasi static assumption for the magnetic field, equation (5.10) implies E tobe a conservative vector field. In this case, (5.11) becomes independent of the path andthe following relationship holds:

E = −∇u

The current density J ∈ Rd is the electric current per unit area passing through atransversal cross section. Ohms law implies that the drift contribution of this physicalquantity can be written as:

J = σE = −σ∇u (5.12)

where σ is the constant conductivity, under the hypothesis of isotropic medium.Equation (5.12) holds true for currents in the heart tissue, where the temporal vari-

ations of both the electric and the magnetic fields can be considered negligible, evenif the processes in the heart are quite fast. We also assume that at each point of thedomain, there is the coexistance of two different values of the potential, namely ue andui. As a consequence, in practise we can define two different current densities Je andJi:

Ji,e = −Di,e∇ui,e

where Di and De are the internal and external conductivity tensors respectively. Con-servation of charges inside an arbitrary volume V enclosing a portion of tissue can bewritten as follows:

∂

∂t

∫Vqi,edΩ =

∫V

∂qi,e∂t

dΩ = −∫∂V

Ji,e · ndl +

∫Vφi,edΩ = −

∫V∇ · Ji,edΩ +

∫Vφi,edΩ

(5.13)in this case, qi and qe represent the charge densities, φi and φe take into account thetransit of charges from the intra to the extracellular space and viceversa, and finally nis the outward directed unit vector normal to ∂V . φi = φe is given by conservation ofcharges. We remark that the first equality in (5.13) is valid under regularity assumptionsof qi and qe, whereas the last equality can be easily obtained using the divergencetheorem. Equalities given by (5.13) can be given pointwisely in V , due to its generality.Hence, we have:

∂qi,e∂t

= −∇ · Ji,e + φi,e (5.14)

The fluxes are strictly connected to the ionic membrane current Iion. Specifically, weassume that φi = −χIion and φe = χIion. The product by the geometric term χ isneeded because Iion is defined as the current per unit area of the membrane. Hence, in


order to find the value of the current per unit volume, Iion must be multiplied by theratio of membrane surface per unit volume. The signs of the ionic currents come fromthe fact that we consider as positive the currents going from intra to extracellular space.

We now observe that, since the membrane is thin, accumulated charges from one sideend up attracting charges from the other side, i.e. qi + qe = 0, which gives ∂

∂t(qi + qe) =0. Therefore, using (5.13), we find that:

∂qi∂t

+∂qe∂t

= −∇ · Ji −∇ · Je = 0

This is equivalent to:

∇ · (Di∇ui) +∇ · (De∇ue) = ∇ · (Di∇v) +∇ · ((De +Di)∇ue) = 0

which is exactly the second equation of the Bidomain model in the PE formulation. Thetransmembrane potential can be expressed as a function of membrane capacitance cmand the quantity of charges as follows:

v =1

2cm(qi − qe)

We derive at both sides and we use the relationship ∂qi∂t = ∂qe

∂t , obtaining:

cm∂v

∂t=∂qi∂t

By inserting this final relation into (5.14) referred to the intracellular space we get:

cm∂v∂t+∇ · Ji + χIion = cm∂v

∂t−∇ · (Diui) + χIion = cm

∂v

∂t−∇ · (Div)−∇ · (Diue)+

+χIion = 0

which we recognise to be the first equation of the Bidomain model in the PE formulation.

The assumption of electrically isolated domain leads to Neumann boundary condi-tions of the type nTJi = nTJe = 0. Hence:

0 = nT (Di∇ui) = nTDi(∇v +∇ue)

0 = nT (De∇ue) = nTDe∇ue + nTDi∇ui = nT (Di +De)∇ue + nTDi∇v

The ionic current Iion, as already said, is assumed to be dependent on the recoveryvariables w and concentration variables z. The ionic current expression and of theevolution laws F and G varies according to the membrane model that is chosen tocomplement the PDEs.

We summarize here below all the assumptions that are needed for the abovemen-tioned derivation of the Bidomain equations, in the order they are effectively encoun-tered:

5.4. WELL-POSEDNESS OF THE BIDOMAIN EQUATIONS 49

Quasi static assumption: the rate of change of electric and magnetic fields in theheart is negligible. For this reason, the current density and the electric field areproportional to the gradient of the potential.

Charge conservation: the rate of change of charge of the intra and extracellularspaces in a generic volume without sources is given by the current passing throughthat volume in one of the two possible spaces and the ionic current going betweenone space to the other.

No charge accumulation: attraction of charges of opposite signs from the twodifferent spaces, with final annichilation.

Ionic currents: the complete Bidomain model requires the choice of a specificmembrane model. Ionic currents are function of transmembrane potential, ionsconcentrations and recovery variables.

Isolated heart tissue: the heart is seen as a completely electrically isolated domain,so no currents can effectively go out from it.

5.4 Well-posedness of the Bidomain equations

Considering the following simplified form of the Bidomain model (with a single gatingvariable and without concentration variables), we mention some results of existence anduniqueness of the solution (in the R3 setting) [4]:

cm∂v∂t −∇ · (Di∇v)−∇ · (Di∇ue) + χIion(v, w) = Isi in Ω × (0, T )

−∇ · (Di∇v)−∇ · ((Di +De)∇ue) = Ise + Isi in Ω × (0, T )∂w∂t − F (v, w) = 0 in Ω × (0, T )

nTDi∇(v + ue) = 0 in ∂Ω × (0, T )

nT (Di +De)∇ue + nTDi∇v = 0 in ∂Ω × (0, T )

v = v0 in Ω × 0w = w0 in Ω × 0

Define the following Hilbert spaces:

Lp(Ω) = v : Ω→ R : ||v||Lp(Ω) <∞

Hp(Ω) = v ∈ L2(Ω) : Dαv ∈ L2(Ω) ∀α : |α| < p

with the norm:

||v||Lp(Ω) =

(∫Ω|v(s)|pdΩ

) 1p

α is a multi-index, with |α| = α1 + α2 + ...+ αn and:

Dαv(x) =∂|α|v(x)

∂x1α1x2

α2 ...xnαn


Take V = H1(Ω), H = L2(Ω), U = v ∈ V :∫

Ω v = 0. Additionally, let D(0, T ) be theset of C∞ functions on R with compact support on (0, T ).Definition 1 (Weak solution). Consider τ > 0 and the three functions v : t ∈ [0, τ)→ v(t) ∈ H, ue : t ∈ [0, τ) → ue(t) ∈ H, w : t ∈ [0, τ) → w(t) ∈ H. Given (v0, w0) ∈H × H, then (v, ue, w) is a weak solution to the Bidomain problem if and only if, forany T ∈ (0, τ):

v : [0, T] → H and w : [0, T] → H are continuous, with v(0)=v0 and w(0)=w0

in H

for a.e. t ∈ (0, τ), v(t) ∈ V, ue(t) ∈ U, and v ∈ L2(0, T ;V ) ∩ Lp(QT ), whereQT=(0, T) × Ω and (v, ue, w) verifies in D′(0, T ) :

cmd

dt(v(t), v) +

∫ΩDi∇(v(t) + ue(t)) · ∇vdΩ + χ

∫ΩIion(v(t), w(t))vdΩ =< Isi (t), v >

d

dt(w(t), w) +

∫ΩF (u(t), w(t))wdΩ = 0

∀v ∈ V , ∀w ∈ H, and∫ΩDi∇v(t) · ∇uedΩ +

∫Ω

(Di +De)∇ue(t) · ∇uedΩ =< Isi (t) + Ise (t), ue >

∀ue ∈ U .

Consider now the following hypotheses, essentials for the existence of the weak so-lution:

1. Ω has a Lipschitz boundary ∂Ω

2. Di,e ∈ L∞(Ω) and the two tensors are continuous elliptic, i.e there exist m ∈ Rand M ∈ R such that m|ξ|2 ≤ ξDi,eξ

T ≤M |ξ|2 ∀ξ ∈ R3

3. Isi,e : [0,∞]→ V ′ and∫

Ω[Isi (s, t)+Ise (s, t)]dΩ = 0 for almost any t > 0 (compatibilitycondition)

4. There exists some p ∈ [2, 6] such that the Sobolev embedding V = H1(Ω) ⊂ Lp(Ω)holds

5. Iion and F are affine with respect to w, i.e Iion(v, w) = I1ion(v) + I2

ion(v)w andF (v, w) = F 1(v) + F 2(v)w, where I1

ion, I2ion, F

1 : R → R are continuous functionsand F 2 ∈ R

6. There exist constants ci ≥ 0(i = 1, 2, ..., 6) such that, ∀v ∈ R:

|I1ion(v)| ≤ c1 + c2|v|p−1

|I2ion(v)| ≤ c3 + c4|v|p/2−1

|F 1(v)| ≤ c5 + c6|v|p/2

5.5. WEAK FORMULATION 51

7. There exist constants a, λ > 0, b, c ≥ 0 such that ∀v, w ∈ R:

λvIion(v, w) + wF (v, w) ≥ a|v|p − b(λ|v|2 + |w|2)− c

Theorem 1 (Global existence of a weak solution). Suppose that Ω and Di,e satisfyhypotheses 1. and 2., and Iion and F satisfy hypotheses 4. to 7. for some value of p≥ 2. Let be given v0, w0 ∈ H and let Isi,e ∈ L2(R+;V ′) satisfy hypothesis 3. Then thesimplified PE system has a weak solution (v, ue, w) in the sense of Definition 1 withτ=+∞.

Theorem 2 (Uniqueness of a weak solution). Consider all the hypotheses of the-orem 1. Moreover, ∀ v, w ∈ R, define the function G : R2 → R2 as G(v, w) =[µIion(v, w), F (v, w)]T for some µ > 0. Let z = (v, w) ∈ R2, Q(z) = [∇G(z)T+∇G(z)]/2be the symmetric part of ∇G. Let λ1(z) and λ2(z) be its eigenvalues, with λ1(z) ≤ λ2(z).If there exists c ∈ R such that, ∀ z = (v, w) ∈ R2, λ2(z) ≥ λ1(z) ≥ C, then the solutionobtained in Theorem 1 is unique.

These two theorems can be used to show well-posedness of the Bidomain equationscoupled with simple ionic models accounting for only one gating variable, such as theRoger McCulloch.

5.5 Weak formulation

We assume that w refers to the vector with both concentration and recovery variablesof the chosen ionic model. The total number of components of w is Nw. For this reasonthe general form of the ionic model can be represented as follows:

∂w

∂t− F(v,w) = 0 (5.15)

As we have done in the previous section, we put V = H1(Ω), H = L2(Ω) andU = v ∈ V :

∫Ω v = 0.

Consider the first equation of the PE version of the system. Take a test functionv ∈ V , multiply it both sides, integrate both members over Ω, and apply the Green′stheorem to the divergence term:

cm

(∂v

∂t, v

)+

∫ΩDi∇(ue+v)·∇vdΩ−

∫∂Ω

nT [Di∇(ue+v)]∇vdl+χ(Iion(v,w), v) = (Isi , v)

(5.16)where the boundary term that comes out is equal to zero due to the homogeneousNeumann condition.

The notation (·, ·) indicates the usual scalar product in L2(Ω). In order to ensurethe existence of all the forms in (5.16), we require that v ∈ V , ∂v∂t ∈ H and ue ∈ U . Thelast constraint permits to guarantee the uniqueness of the solution ue.

Introducing the bilinear form ai : V × V → R of the type ai(φ, ψ) =∫

Ω(Di∇φ) ·∇ψdΩ ∀φ, ψ ∈ V , equation (5.16) can be rewritten as:

cm

(∂v

∂t, v

)+ ai(v, v) + ai(ue, v) + χ(Iion(v,w), v) = (Isi , v)


Consider now the second equation of the PE formulation. Applying the same pro-cedure of the previous PDE we obtain as a result:∫

Ω(Di∇v+ (Di +De)∇ue) · ∇uedΩ−

∫∂Ω

nT [Di∇v+ (Di +De)∇ue]uedl = (Isi + Ise , ue)

(5.17)where the boundary term is again zero because of the effect of the homogeneous Neu-mann boundary condition.

A bilinear form a : V ×V → R defined as a(φ, ψ) =∫

Ω[(Di+De)∇φ]·∇ψdΩ ∀φ, ψ ∈V permits to rewrite the previous weak formulation as:

ai(v, ue) + a(ue, ue) = (Isi + Ise , ue)

The ODE connected to the microstructure requires a simpler treatment: it is neededto multiply both members by a test functions vector w ∈ [H]Nw and after to integrateover the domain Ω. These operations define:(

∂w

∂t, w

)= (F(v,w), w) (5.18)

where (·, ·) denotes the scalar product in [H]Nw . Requiring w ∈ [H]Nw and ∂w∂t ∈ [H]Nw

the existence of all the integrals in the previous equation.

A necessary condition to the well-posedness of the Bidomain equations is the com-patibility condition: ∫

Ω(Isi + Ise )dΩ = 0 (5.19)

that can be imposed from the numerical viewpoint using a Lagrange multiplier.

So, in order to sum up the results that have been just obtained, given v0 ∈ H,w0 ∈ [H]Nw , Isi,e ∈ L2(Ω × (0, T )) (satisfying the compatibility condition (5.19)), F ∈[L2(V, [H]Nw)]Nw , find v ∈ L2(0, T ;V ), ue ∈ L2(0, T ;U) and w ∈ [L2(0, T ; [H]Nw)]Nw

such that ∂v∂t ∈ L2(0, T ;V ) and ∂w

∂t ∈ [L2(0, T ; [H]Nw)]Nw , v(0) = v0 and w(0) = w0

and, ∀t ∈ (0, T ):

cmd

dt(v, v) + ai(v, v) + ai(ue, v) + χ(Iion(v,w), v) = (Isi , v) ∀v ∈ V

ai(v, ue) + a(ue, ue) = (Isi + Ise , ue) ∀ue ∈ Ud

dt(w, w) = (F(v,w), w) ∀w ∈ [H]Nw

(5.20)

5.6 Space discretization

Consider the first two variational equations in weak form:

cmd

dt(v, v) + ai(v, v) + ai(ue, v) + χ(Iion(v,w), v) = (Isi , v)

ai(v, ue) + a(ue, ue) = (Isi + Ise , ue)(5.21)

5.6. SPACE DISCRETIZATION 53

As already said in the chapter about IGA, a possible approach for building a finitedimensional space of the solution is to consider the space spanned by the basis func-tions used for the geometry generation. Denote with Vh ⊂ V the finite dimensionalapproximation of V built using the basis functions defining the geometry. ∀gh ∈ Vh:

gh =N∑i

Ni gi

where NiNi=1 are the basis functions and gi are the corresponding degrees of freedom.We introduce ∀t ∈ (0, T ) the finite approximations of v(t) and ue(t) in the space Vh, i.e:

vh(t) =N∑i

Ni vi(t)

uhe (t) =N∑i

Ni ui(t)

(5.22)

Let us now substitute v and ue in equations (5.21) with their approximation vh anduhe , and consider the expansion written in equation (5.22). This reasoning can be appliedalso to the test functions, substituting v and ue with vh and uhe respectively. Now it ispossible to notice that if equations (5.21) hold ∀vh ∈ Vh and ∀uhe ∈ Vh, equivalently, itholds ∀Ni with i = 1, ..., Nw. Hence for the first equation in (5.21) we get:

cm

N∑j=1

(Nj , Nk)d

dtvj +

N∑j=1

ai(Nj , Nk)vj +

N∑j=1

ai(Nj , Nk)uj + χ(Iion(v,w), Nk) =

= (Isi , Nk) ∀k = 1, ..., N

where vj = vj(t) and uj = uj(t). Collecting the DOFs in the vectors v(t) = (v1(t), ..., vN (t))T ,u(t) = (u1(t), ..., uN (t))T , and introducing the following matrices:

(M)lm = (Nm, Nl)

(Ki)lm = ai(Nm, Nl)

(K)lm = a(Nm, Nl)

we can rewrite the previous discretized weak formulation as follows:

cmMd

dtv +Kiv +Kiu + χIhion = Isi ∀t ∈ (0, T ) (5.23)

with v(0) = v0 = (v1,0, ..., vN,0)T , where (Ihion)j = (Iion(vh,w), Nj), (Isi )j = (Isi , Nj).The details in the computation of Ihion will be given later in this section. Following thesame steps, we can also perform a space discretization of the second equation in (5.21):

Kiv +Ku = Isi + Ise ∀t ∈ (0, T ) (5.24)


where (Ise)j = (Ise , Nj). The two discretizations just obtained can be now coupledtogether:

cm

[M 00 0

]︸︷︷︸M

d

dtz +

[Ki Ki

Ki K

]︸︷︷︸

K

z =

[Isi − χIhion

Isi + Ise

]︸︷︷︸

I

(5.25)

where the entries 0 stand for matrices composed only by zeros and with suitable dimen-sions and z(t) = [v(t),u(t)]T . Associating to the previous equation an initial conditionon z, we can give the complete semi-discrete formulation of the sole Bidomain equationsin PE form:

cmMd

dtz = I − Kz = G(z) t ∈ (0, T ) (5.26)

z(0) = z0 (5.27)

(5.28)

where z0 = (vT0 , 0)T .

Let us now address the discretization of the recovery and concentrations variablesw. A possible approach called nodal interpolation [32] would consist in making use ofthe same basis functions NiNi=1 to construct a suitable approximation of [H]Nw andto approximate the last equation in (5.20), as already done for the first two. We chooseto follow another approach called Gaussian integration [32]. We recall here the form ofthe ionic current term Ihion:

(Ihion)j =

∫ΩIion(vh,w)NjdΩ (5.29)

The standard strategy in finite element and IGA solvers to compute this kind of integralsis to exploit the special structure of basis functions which have compact and highlylocalized support and are piecewise polynomial, and utilize a gaussian quadrature ruleon each mesh element. This approach is convenient because, due to the locality of thebasis functions, only a very small number of elements contributes to the value of theintegrals. Let Ne be the number of elements making up the geometry Ω, Kj be the jth

element such that ∪Nej=1Kj = Ω, xjisi=1 be the quadrature points on the jth element

and ωisi=1 be the associated weights, with s number of points/weights on one singleelement. In the case of quadrature rules, integral (5.29) can be approximated as:

(Ihion)j =

∫ΩIion(vh,w)NjdΩ =

Ne∑j=1

∫Kj

Iion(vh,w)NjdΩ ≈

≈Ne∑j=1

s∑i=1

Iion(vh(xji ),w(xji ))Nj(xji )ωi

So we need only the value of w at the gaussian quadrature points, without consider-ing a discrete representation of it. We consider then the set of s×Ne vectors wj

i = w(xji )for i = 1, ..., s and j = 1, ..., Ne representing the values of recovery and concentrationvariables at each quadrature point. The evolution of wj

i can be computed through the

5.7. TIME DISCRETIZATION 55

ODE: d

dtwji = F(vjh,i,w

ji ) t ∈ (0, T ) (5.30)

wji (0) = w0(xji ) (5.31)

for i = 1, ..., s and j = 1, ..., Ne, where vjh,i = vh(xji ) is the interpolation of the approxi-

mate solution vh in the Gauss point xji .

5.7 Time discretization

In order to derive the fully discrete form of the Bidomain equations we consider adecoupled semi implicit scheme. We use Backward Differentiation Formulaes (BDFs)of order q for the time integration, which is suitable for the solution of stiff differentialequations. Increasing the order q of the scheme, the region of stability becomes smaller,even if we get a more accurate solution thanks to the fact that we use more temporalsteps to get the approximated value of the derivative in a specific instant of time [34].BDF scheme is a multistep method for this reason, and this characteristic permits toobtain a gain in accuracy of the estimations made over the time domain.

Consider now the general Cauchy problem, to see in the more general setting theapplication of this scheme:

dy

dt= f(t,y) t ∈ (t0, T ) (5.32)

y(t0) = y0 (5.33)

with y ∈ Rn, y0 ∈ Rn and f : R × Rn → Rn. Introducing a timestep ∆t such thattk = t0 + k∆t, the approximate solution at the (k + 1)th timestep yk+1 satisfies:

q∑i=0

αiyk+1−i = ∆t β f(t,yk+1) k ≥ q − 1 (5.34)

where y0 = y0 and αi ∈ R for i = 0, ..., q and β ∈ R (with α0 6= 0 and β 6= 0)are coefficients depending on the order of the method. Equation (5.34) can also beexpressed in the form:

αyk+1 − yk+1

BDF

∆t= f(t,yk+1) ≈ d

dty(tk+1) (5.35)

where α = α0β and

yk+1BDF = −

q∑i=1

αiα

yk+1−i k ≥ q − 1

The method obtained by applying (5.35) is implicit and requires the solution of apotentially non linear equation at each timestep. In order to decrease the computational


cost, we may introduce yk+1ext : this is an approximation of yk+1 obtained as extrapolation

using the values yj , with j = k, ..., k − (q − 1). We have then:

αyk+1 − yk+1

BDF

∆t= f(t, yk+1

ext )

which is an explicit method of order q.Let ∆t be the timestep of the time discretization, and tk = k∆t be the kth timestep.

We denote vk and uk the values of v and u respectively at the kth timestep, and withwj,ki the values of w at the time instance tk on the Gauss quadrature point xji . The

evaluation of the solution at time k+1 with the decoupled scheme, which is a first ordersplitting scheme, follows two steps:

1. Suppose to have vk+1ext and wj,k+1

ext,i expressing the extrapolated values of v and wji

at the time instance k + 1. Then the numerical solution of (5.30)-(5.31) can becomputed as:

αwj,k+1i −wj,k+1

BDF,i

∆t= F(vj,k+1

h,i ,wj,k+1ext,i )

∀i = 1, ..., s and j = 1, ..., Ne, where s is the number of quadrature nodes forelement, Ne is the number of elements, and vj,k+1

h,i is computed from vk+1ext by

evaluating the extrapolated solution in the quadrature points. As a consequence:

wj,k+1i = wj,k+1

BDF,i +∆t

αF(vj,k+1

h,i ,wj,k+1ext,i )

∀i = 1, ..., s and j = 1, ..., Ne.

2. Given wj,k+1i for i = 1, ..., s and j = 1, ..., Ne, the values vk+1 and uk+1 are

computed by solving:

cmMαvk+1 − vk+1

BDF

∆t+Ki(v

k+1 + uk+1) = Is,k+1i − χIk+1

ion

Kivk+1 +Kuk+1 = Is,k+1

i + Is,k+1e

where Ik+1ion is computed using vk+1

ext and wj,k+1i . We remark that, by using the

extrapolated transmembrane potential for the computation of the ionic current,we are linearizing the problem.

There are several other techniques to deal with the temporal discretization of this setof equations. We mention the Strang splitting scheme, which is a second order accuratenumerical method for solving differential equations that are decomposable into a sum ofdifferential operators [28], as in this case. It is used to solve multidimensional PDEs byreducing them to a sum of one-dimensional problems, and to speed up calculation forproblems which deal with very different time scales. Moreover we refer the reader to thework made by Spiteri et al. on high-order operator splitting for electrophysiology [8].

Chapter 6

Numerical results

We present in this chapter the numerical results, connected to the simulation of thecomplete action potential in the left ventricle (healthy and pathological conditions), inboth ventricles and in the complete heart: for this last part, we refer the reader to [33]for the simulations that have been performed on the atria. In all the treated cases,except for one which is explicitly detailed in a subsequent section, the simulations havebeen made neglecting the elicoidal fibers orientation, which is typical of ventricles. Asa result, the wave propagation will be more homogeneous and isotropic with respectto the one that occurs in the real human heart. Infact, thanks to fibers orientation,the wavefront presents more directionality and is distributed in a more anisotropic wayalong the entire myocardium: in particular, the conduction velocity in the longitudinaldirection is higher because of the high density of gap junctions.

In this work we study ventricular electrophysiology, AP and calcium development inthe different areas of the heart, positive aspects and limitations in the use of the Bido-main model and two important microscopic models in literature from the mathematicaland physical viewpoint.

All the simulations presented in this chapter are made with ∆t = 0.01 ms. Wewould like to stress that such as small timestep is effectively needed to ensure properconvergence of the numerical scheme in use, especially when the coupling with the Luo-Rudy ionic model is considered. Moreover, this choice is fundamental to capture ina proper way the most complicated phase of the AP from the numerical perspective,which is the action potential upstroke (given be the inactivation of fast Na+ channels),where there are strong gradient variations. A uniform mesh in the time domain, i.e. theconstant timestep just reported at the beginning of this paragraph, is considered for thesake of simplicity. Another reasonable decision would have been the one of non uniformtimestep, which is likely to be really small in problematic zones, and fairly higher inother areas where there are no significant gradient variations, such as the plateau. Wehave used BDF scheme of order q = 1 and q = 2 for the numerical integration ofw and v respectively. The order of the method for the recovery and concentrationvariables is chosen to increase its stability. Some parameters are fixed once for all thesimulations, such as cm = 1 mF cm−3 and ξ = 103 cm−1. Ise = −IsI , in order tosatisfy the compatibility condition 5.19. The initial conditions on v, gating variablesand concentrations are prescribed in the chapters dedicated to the two ionic modelsunder investigation.

57

58 CHAPTER 6. NUMERICAL RESULTS

All the numerical results of this thesis have been obtained using clusters and nodesof the parallel computing centre available at MOX, in the department of Mathematicsof Politecnico di Milano.

6.1 Fibers generation

Before addressing the simulations of the electrophysiology, we would like to present analgorithm taken from [38], that can be used to generate fibers and sheetlet directions onthe left ventricle.

The fibers in this framework are known to exhibit a rotation between about -60° and+60° from the epicardium to the endocardium. For this reason we initially have to setthe proper values of the angles θepi and θendo. Also the ventricular centerline vector k,pointing apex-to-base must be given as an input. Subsequently we have to solve thefollowing boundary value problem:

−∆φ = 0 (6.1)

φ|epi = 1 (6.2)

φ|endo = 0 (6.3)

∂φ

∂ν= 0, on Γbase (6.4)

Once φ is available, we compute the sheetlet direction as s0 = ∇φ||∇φ|| . After this, kp, i.e.

the projection of k on the plane orthogonal to s0, is given by kp = k − (k · s0)s0. An

initial fiber field is defined by f0 = s0 × kp

||kp|| . We then create a rotation matrix Rs0(φ)

which describes the rotation of the fiber field around the s0 axis. We end up with thefollowing formula for the fibers direction (that are already normalized between -1 an 1):

f0 = Rs0(φ)f0

Given the rotation angle θ = θ(φ) = (θepi − θendo)φ + θendo using a linear relationshipwith respect to the scalar potential φ, the rotation matrix Rs0(φ) is found through theRodrigues’ rotation formula:

Rs0(φ) = I + sin(θ)[s0] + 2 sin2

(θ

2

)(s0 ⊗ s0 − I)

where [s0] is the cross-product matrix defined as:

[s0] =

0 −s0,z s0,y

s0,z 0 −s0,x

−s0,y s0,x 0

The results of the application of the previous procedure on the generated geometry

for the left ventricle can be seen in Figure 6.1. In order to keep into account thefibers orientation for the left ventricle, a simplified version of the conductivity tensorformula (5.9) can be used. Choosing σil = 0.0023 Ω−1cm−1, σel = 0.0015 Ω−1cm−1,

6.2. LEFT VENTRICLE: BIDOMAIN AND BUENO-OROVIO MODELS 59

Figure 6.1: Distribution of the variable φ (left) and fibers (right) on the left ventricle

σit = 0.00024 Ω−1cm−1 and σet = 0.001 Ω−1cm−1:

Di,e(x) =

σi,et 0 0

0 σi,et 0

0 0 σi,et

+

(σi,el − σi,et )f0,1(x) 0 0

0 (σi,el − σi,et )f0,2(x) 0

0 0 (σi,el − σi,et )f0,3(x)

where f0 = [f0,1(x), f0,2(x), f0,3(x)]T are the fiber direction at the point x. There existalso algorithms and techniques that permit to apply the fibers in alternative ways withrespect to what has been presented in this section and to the biventricle [1, 19, 46].

6.2 Left ventricle: Bidomain and Bueno-Orovio models

We start from the simplest analysed framework, given by the simulation on the leftventricle (without the fibers effects) of the AP, as a result of the coupling betweenthe Bidomain equations and the Bueno-Orovio model. The simulation ran for T =600 ms, using 12 cores. This approach permits to distribute the efforts on differentpartitions of the original domain along different computational resources, which in theend communicates one with the other to share and connect the results in the correct way.The mesh contains 117608 elements, properly distributed along the three parametricdirections, so as to be as much similar to cubes as possible. This number of elementshas been verified to be sufficiently large to properly describe the development of the AP.


Indeed, another simulation has run in the meantime for a fairly lower final time T =100 ms, giving almost the same propagation pattern of the one reported in Figures 6.2and 6.3. The left ventricle has been stimulated at three points, in the middle positionbetween the apex and the base of the ventricle, which present an angle of 120° betweenone and the other, for 1 ms, considering a current equal to 1 s−1, due to the partialadimensional nature of the Bueno-Orovio model. This kind of stimulation replicateswhat effectively happens on the left ventricle of the human heart, keeping into accountthe stimuli coming from the bundle of His and the effect of the Purkinje network [25, 48].In Figure 6.2 and 6.3 we can see the space and time resolution of the transmembranepotential v and the gating variable s, which is known to have a behaviour similar to theconcentration of [Ca2+]i, even if the range of variation is not physiological. Infact wewould like to remind that the gating variables are probability density functions, so theyassume values between 0 and 1, whereas the order of magnitude of intracellular calcium,expressed in mM , is fairly different. Figure 6.4 shows the time evolution of the AP ands in the different points: point A at the epicardium, point B at the base, and point Cat the endocardium. All these three evaluated points are chosen at a distance sufficientto make the effects of the initial stimulation negligible. The plots have exactly the sameshape and present a shift in time connected to the different instant of time the actionpotential is triggered at that specific point. Results are in accordance with Figure 2.5.

As already discussed in Chapter 3, we know that the element size has dramaticconsequences on the wavefront velocity. For this reason we have considered a fine mesh,so as to obtain a trustworthy propagation of the action potential in the time domain,replicating the timing of the real AP of the human heart. This aspect is effectivelyfulfilled quite well in practise here. The non homogeneous size of the mesh elementshas in general consequences on the quality and smoothness of the wavefront in certainregions of the ventricle. As it can be seen from Figure 6.5, the wavefront is not completelysmooth already in the first ms of the simulation, where some oscillations can be easilyseen and are also enhanced in correspondance of the collision of different fronts comingfrom different initial stimuli, as clearly visible in Figure 6.6. One main issue that isintrinsically connected to the geometry is the sharp transition angle between the baseand the remaining part of the ventricle: when the front crosses this area, some numericaloscillations occur, as if both the mathematical models under study and IsogeometricAnalysis show some limitations in the propagation of wavefronts in presence of suchsudden changes in the domain.

Due to the fact that Figures 6.2 and 6.3 screenshots are taken in the same instant oftimes, we can conclude that the propagation of the gating variable s, which resemblesthe calcium behaviour, is slightly delayed comparing with the transmembrane potential,as it should be from the correct physiological viewpoint. Moreover, its behaviour seemsto be smoother than the one of the transmembrane potential, thanks to the fact that thenumerical treatment and/or resolution of the sole Bueno-Orovio model, once decoupledfrom the macroscopic model, permits to get more accurate data in the end.

Even if, as already said in the introduction of this chapter, the propagation withoutthe fibers should be in a certain sense quite isotropic, here there is an horizontal diffusionprevalence with respect to the vertical one, as shown looking at t = 4.5 ms and t = 15ms. This consequence relies again on all the issues already presented for IGA consideringmesh elements sizes and shapes.


(a) t = 4.5 ms (b) t = 12 ms

(c) t = 24 ms (d) t = 37.5 ms

(e) t = 150 ms (f) t = 255 ms

(g) t = 450 ms (h) t = 546 ms

Figure 6.2: Evolution of the dimensional transmembrane potential v in the left ventricle givenby the coupling of the Bidomain equations and the Bueno-Orovio ionic model


(a) t = 4.5 ms (b) t = 12 ms

(c) t = 24 ms (d) t = 37.5 ms

(e) t = 150 ms (f) t = 255 ms

(g) t = 450 ms (h) t = 546 ms

Figure 6.3: Evolution of the gating variable s in the left ventricle given by the coupling of theBidomain equations and the Bueno-Orovio ionic model


(a) Points of evaluation

(b) Action potential

(c) Gating variable s

Figure 6.4: Transmembrane potential v and gating variable s as a function of time in threespecific points


Figure 6.5: Zoom of the transmembrane potential at t = 4.5 ms. A lack of smoothness of thefront can be seen. This result is also given by the fact that the AP upstroke occurs exactly inthose positions, and it is the most difficult phase to be represented

Figure 6.6: Zoom of the transmembrane potential at t = 15 ms. The collision of the wavefrontsincreases the irregularity of the solution in that area, given by non homogeneous mesh elementsand a lack of resolution

6.3. FIBERS EFFECT: BIDOMAIN AND BUENO-OROVIO MODELS 65

6.3 Fibers effect: Bidomain and Bueno-Orovio models

In this simulation the only difference with respect to what has been presented in theprevious section, referring both to the geometrical and numerical settings, is given bythe additional presence of the fibers. This fact permits to define another propagationpattern for the AP, which is more in accordance with the reality, and keeps into accountthe elicoidal rotation of fibers between about -60 to +60 degrees between the epicardiumand the endocardium. The role of the fibers is mainly connected with the mechanicsof the heart, due to the fact that they define how the contraction and the relaxationof the myocardium occur [1]. In this case there is a kind of vertical stretch in thespacial development of the wavefront with respect to the one depicted without fibers.The propagation seems to be a little bit faster here. Considering t = 24 ms, numericaloscillations are strongly increased with the passage along the geometrical sharp edge.Similar considerations to those of the previous section hold also in the case, due to thefact that both the macroscopic and microscopic models involved in the computations areexactly the same. The gating variable s has been neglected here. However we can assertthat, due to the fact that the time and space behaviour of the transmembrane potentialare quite similar to the one of the previous simulation for the abovementioned reasons,we expect that the final outcome remains almost unvaried also for the gating variable s.The only difference that we can find in the time development of the action potential isconnected to the fact that, due to a different propagation path, the same shape presentsa time shift due to different triggering moments with respect to the previous simulation.


(a) t = 4.5 ms (b) t = 12 ms

(c) t = 24 ms (d) t = 37.5 ms

(e) t = 150 ms (f) t = 255 ms

(g) t = 450 ms (h) t = 510 ms

Figure 6.7: Evolution of the dimensionless transmembrane potential v in the left ventricle givenby the coupling of the Bidomain equations and the Bueno-Orovio ionic model, and keeping intoaccount the effect of the fibers

6.3. FIBERS EFFECT: BIDOMAIN AND BUENO-OROVIO MODELS 67



Figure 6.8: Transmembrane potential v as a function of time in three specific points




Figure 6.9: Comparison between the transmembrane potential v as a function of time computedwith the Bueno-Orovio model, with and without the effect of the fibers

6.4. LEFT VENTRICLE: BIDOMAIN AND LUO-RUDY MODELS 69

6.4 Left ventricle: Bidomain and Luo-Rudy models

In this case we have considered a more sophisticated ionic model, which is able to repro-duce in a more accurate way both the shape and the propagation of the action potential.The domain is given by the left ventricle without fibers, and the numerical configurationis exactly the same of the first simulation. We can easily notice that the peak of theAP, followed by the plateau, is very well captured, and the general propagation of thewave front along the ventricle is fairly faster compared to the one defined by the BuenoOrovio model: we have put the same screenshots of the space development of the AP inboth cases also to observe clearly this phenomenon. The AP depicted in blue at positionA has a fairly higher peak value with respect to the other two plots. This is probablyrelated to the fact the the propagation of the wavefronts in the nearby of the stimulationpoints is different with respect to the one far away from them. Again some numericalinstabilities raise already in correspondance of the first instants of time, when front col-lision occurs, and finally at the base (in this simulation the numerical oscillations aremore visible comparing to the ones obtained with the Bueno-Orovio model). Moreover,using the Luo-Rudy model, we can get an evaluation of the intracellular calcium concen-tration in the myocytes along all the tissue of the ventricle, eventually as a function oftime. This physical quantity is expressed in millimolar (mM). The calcium propagation,as reported in detail in section 1.4, has a direct impact in the way the contraction ofthe heart occurs, considering both the space and time mechanical excitation. From theprospective of the left ventricle, this kind of ion allows the ventricle itself to push bloodtowards the aorta, through the aortic valve. Due to stronger pressure and mechanicalmovements that are needed to transmit blood all over the body to give nutrients toall the cells, the concentration of intracellular calcium in the ventricle is one order ofmagnitude higher than in the atria (we refer to [33] to see this detail).


(a) t = 4.5 ms (b) t = 12 ms

(c) t = 24 ms (d) t = 37.5 ms

(e) t = 150 ms (f) t = 255 ms

(g) t = 450 ms (h) t = 510 ms

Figure 6.10: Evolution of the dimensional transmembrane potential v on the left ventricle


(a) t = 4.5 ms (b) t = 12 ms

(c) t = 24 ms (d) t = 37.5 ms

(e) t = 150 ms (f) t = 255 ms

(g) t = 450 ms (h) t = 510 ms

Figure 6.11: Evolution of the gating variable s on the left ventricle




(c) Cai

Figure 6.12: Transmembrane potential v and intracellular calcium concentration Cai as afunction of time in three specific points




Figure 6.13: Comparison between the transmembrane potential v as a function of time com-puted with the Luo-Rudy and the Bueno-Orovio models


6.5 Biventricle: Bidomain and Bueno-Orovio models

In this section we consider the biventricle geometry, applying all the machinery that wehave developed so far to the biventricular case. We have chosen not to include the fibers,because the algorithm proposed in Section 6.1 does not perform well in correspondanceof the interventricular septum in the single patch approach, that has been treated in thisthesis. The simulation ran for a lower time in this case (510 ms), because of the lack ofmass memory availability to store the different timesteps, and due to limited maximumtime per job on the cluster. Consequently, the repolarization has not been reached com-pletely and uniformly all over the ventricles. We have used a timestep ∆t = 0.01 s and16 cores, on a mesh that contains 170250 elements. The left ventricle has been againstimulated at three points, in the middle position between the apex and the base of theventricle itself, which present an angle of 120° between one and the other. The rightventricle has been stimulated in one point at a middle distance between the apex andthe base, and the same approach has been used for the interventricular septum. Theexcitation lasts for 1 ms and occurs simultaneously in the different zones, consideringagain a current equal to 1 s−1. This methodology is able to reproduce in an approximateway the behaviour of the Purkinje network. The evaluation of the action potential as afunction of time is considered here at three points, distributed over the right ventricle,the interventricular septum and the left ventricle. The shape remains almost unvaried,and only a shifting of the plot is effectively present, taking into account the differentinstants of time the AP is really triggered in the considered points. With reference toFigure 3.12, we have mesh distorsion all over the geometry, and the worst effects areconcentrated in the interventricular septum. This phenomenon could have been com-pletely avoided using the multipatch technique, unfortunately not available in the isoglibyet, instead of the single patch one [22]. According to this approach, the biventriculargeometry could have been generated as an union of the left and right ventricle, whichare developed in principle independently, and which are provided with two appropriatemeshes, that must be correctly connected in the junction points, increasing in this waythe size of the block matrices that are involved in these computations. Mesh distorsion,along with non homogeneous/non cubic mesh elements all over the domain and frontcollisions due to the variety of stimuli that are present, provide a numerical solutionwhich contains strong numerical oscillations and instabilities. In addition, the spacepropagation of the AP is faster with respect to the correct physiological one. Thesecomments show the limitations of the single patch technique in situations where the ge-ometry is extremely complex (as in this case) and how to interpret the results from IGAwhen several issues in the geometry/mesh development occur all together. A possibleremedy to all these issues is constructing the biventricle geometry using the multipatchtechnique: the left and the right ventricles are built separately and connected properlyafter. In this way two regular meshes without distorsion are generated on the two ven-tricles separately, and then the connections on some contact points are made, addingcompatibility conditions, in order to obtain the final geometry.

6.5. BIVENTRICLE: BIDOMAIN AND BUENO-OROVIO MODELS 75

(a) t = 6 ms (b) t = 12 ms

(c) t = 24 ms (d) t = 38 ms

(e) t = 150 ms (f) t = 256 ms

(g) t = 450 ms (h) t = 510 ms

Figure 6.14: Evolution of the dimensional transmembrane potential v on the ventricles




Figure 6.15: Transmembrane potential v as a function of time in three specific points

6.6. ELECTROPHYSIOLOGY OF THE FULL HEART 77

6.6 Electrophysiology of the full heart

We also simulate the electrophysiology of the complete heart. The action potential startsat the sinoatrial node on the right atrium, propagating towards the left atrium throughprefential lines, such as the Bachmann’s bundle, and towards the atrioventricular node.We have decided to trigger the action potential in the ventricles at t = 200 ms using thesame initial conditions presented in Section 6.5, in order to mimic the effect of the AVnode and Purkinje fibers, which are not represented here. A delay of almost 100 ms isapplied when the signal reaches the AV node, and the AP takes approximately 100 msto cover the distance between the sinoatrial node and the AV node. The two atria aredeveloped using the single patch technique, and the simulation for them has run usingthe Bidomain equations coupled with the Courtemanche Ramirez Nattel model, reallyprecise from the microscopical viewpoint for what concerns atria electrophysiology. Formore details about this model we refer to [33]. The simulation for the ventricles isthe one described in the previous section, where the coupling between the bidomainequations and the Bueno-Orovio model has been applied on the biventricular domain.Figure 6.16 and 6.17 depict the details of the action potential propagation in the twocases. Due to the geometrical issues already described on the ventricles, the propagationof the AP between the biventricle and the left atrium is not perfectly synchronized andcompletely physiological, because of an higher speed of the wavefront in the inferior partof the heart. Additionally, two scales with different ranges (in mV) are needed in thiscontext, due to the different nature, shape and values of the action potential in the atriaand the ventricles.


(a) t = 6 ms (b) t = 50 ms

(c) t = 100 ms (d) t = 150 ms

(e) t = 200 ms (f) t = 250 ms

(g) t = 300 ms (h) t = 350 ms

Figure 6.16: Evolution of the dimensional transmembrane potential v on the complete heart(part 1)

6.6. ELECTROPHYSIOLOGY OF THE FULL HEART 79

(a) t = 400 ms (b) t = 450 ms

(c) t = 500 ms (d) t = 550 ms

(e) t = 600 ms (f) t = 650 ms

(g) t = 700 ms (h) t = 750 ms

Figure 6.17: Evolution of the dimensional transmembrane potential v on the complete heart(part 2)


6.7 Ventricular fibrillation

We address here the problem of simulating an important pathological condition thatgoes under the name of ventricular fibrillation, which can leads to the death of a hu-man being if not treated immediately with the support of defibrillators [24, 53]. Itconsists in an irregular excitation and contraction of the two ventricles. This dangerousconsequence of a disorganized electrical activity is not completely understood, and theuncertainties over the physiological causes of the fibrillation makes the replication of thephenomenon difficult [40]. We propose here two possible alternatives to generate thiskind of behaviour:

Train of pulses: This approach can be followed on both single ventricle and biven-tricle geometries, and consists of giving multiple pulses in the same points afterevery n ms, where n can be arbitrarily fixed in the range in which the portionof heart tissue under consideration is in the refractory period, and such succes-sive pulses do not generate new wavefronts. This technique follows the so calledburst-pacing protocol [53].

Left ventricle blockage: This second methodology can be applied in the biventric-ular case only, and try to simulate the fact the stimuli do not even start on the leftventricle, and the entire propagation of the AP is entirely triggered for instanceon the right ventricle [3]. This condition is surely not physiological.

We have followed the first approach and pace the left ventricle in correspondence of onepoint in a mid position between the apex and the base at 20 Hz. In other words, wehave stimulated every 50 ms the ventricle with currents of 1 s−1 lasting for 1 ms. Thetotal duration of the experiment is T = 320 ms. The Bidomain equations have beencoupled with the Bueno-Orovio model for this simulation.

We observe that our model manages to reproduce the cellular refractory period,i.e. the period of time subsequent to the excitation of cardiac cells in which no otherexcitation is possible. Infact, the stimulations following the first one do not reflect in thegeneration of new wavefronts, as in Figure 6.18. Looking at Figure 6.19, showing thetime evolution of the transmembrane potential in the neighborhood of the stimulatedpoint, it seems that all the pulses after the initial one do not have any kind of effect onthe action potential. With reference to [33], if we would have waited until the stimulationpoint has reached the resting value, we would have observed the generation of a newwavefront. The new AP presents a shorter refractory period and a shorter total durationwith respect to the first one, meaning that the properties of the wavefront change heartbeat by heart beat.

Unfortunately, for the whole simulation we did not observe the insurgence of ventric-ular fibrillation. This could suggest that maybe the temporal scales we are consideringare not sufficient to simulate this type of arrhythmia and that it would be beneficial toextend the total duration of the experiment for several seconds [33].

6.7. VENTRICULAR FIBRILLATION 81

(a) t = 50 ms (b) t = 51 ms

(c) t = 53 ms (d) t = 55 ms

Figure 6.18: Stimulation at t = 50 ms of one point on the left ventricle. The colors refer tothe value of the transmembrane potential v. Since the tissue region surrounding the site is stillexcited and in the refractory period, such stimulus does not generate a new wavefront

Figure 6.19: Time evolution of the transmembrane potential in the neighborhood of the stim-ulated point

Chapter 7

Conclusions and futuredevelopments

This Master Thesis aims at the realistic simulation of the excitation front in cardiactissue in the two ventricles, with also atria coupling, in order to simulate the entire heartelectric behaviour by means of Isogeometric Analysis. Due to the fact that the expectedbehaviour of the solution of the Bidomain equations, its properties and convergenceof IGA in this context is verified in [33], our real first step has been to couple theBidomain model with the Bueno Orovio model on the left ventricle only, seeking alsofor the fibers effects, which permits the transmission in an anisotropic fashion, that isa consequence of a different transversal and longitudinal conduction velocity in cardiacfibers. After that, we have changed the microscopic model, moving towards the LuoRudy one, to see how the ionic model influences the final results, and to obtain higheraccuracy in the development of the action potential, thanks to the fact that the Luo Rudymodel falls into the category of first/second generation microscopic models, accordingto the version and set of equations that is effectively used. Then we have moved to thediscussion of both ventricles connected together using a single patch approach. Even ifwe could not recreate pathological conditions leading to the chaotic excitation of the leftventricle, connected to the formation of spiral waves, we have tried to give as an input atrain of pulses, which is not physiological, to see unusual effects on the action potentialpropagation, such as the shortening of the AP itself and refractory period. In the end,we have coupled together ventricles and atria, fixing properly the way, the positions andthe instants of time the different physiological stimuli are triggered, to simulate the APof an entire healthy human heart.

We hereby propose some possible improvements and/or future developments of thework that has been done in this project:

Multipatch technique on the ventricles: the final mesh detailed for both ventri-cles is single patch and subjected to mesh distorsion. In order to eliminate thisbad effect, two patches, one for the left and the other one for the right ventricle,should be created and connected properly, so that, in particular on the interven-tricular septum, the mesh becomes uniform and the geometry issues are stronglyattenuated.

Finer mesh on the ventricles: as an addition to what has been said in the previous

82

83

point, the consideration of smaller cubic elements increases a lot the regularity ofthe wavefront and improves the quality of its propagation also from the timeprospective.

Multiple heartbeats: the Bueno Orovio and Luo Rudy ionic models with standardinitial conditions do not start with a stable condition, namely multiple excitationsof the cells lead to a change in the resting values of the single variables. As a directconsequence, we should consider the AP development taking place after multipleexcitations of the tissue.

Improvement of the initial conditions on the ventricles: taking into account in aprecise way the effects of the 1D Purkinje network and bundle of His, along with abetter initial conditions, would be of great help to define a more realistic electricbehaviour for both ventricles.

Coupling of the electrical and structural models: the evolution in time and spaceof the calcium concentration provided by the Luo Rudy model could be used torealize the contraction of the heart and its deformation.

Simulation of ablations and spiral waves: one of the possible therapies for dealingwith ventricular fibrillation is the ablation of cardiac tissue, namely a change oftissue properties due to burning wounds made by surgeons. The effect of ablationscan be simulated by locally setting the value of the conductivity tensor to zero.This approach (which is not the only one) can be also used to create spiral waveson the cardiac tissue, that are normally generated in the presence of fibrillation.

Appendix A

The Luo-Rudy cellular model

With reference to [27], this set of equations is an attempt to model the ventricular cardiacaction potential, considering depolarization, repolarization and their interaction. Thismodel is based on the Hodgkin−Huxley philosophy in the calculation of ionic currentsand we refer to it as Luo-Rudy I model, due to the fact that more complex variant canbe taken into account, including more ionic currents and more gating variables.

In this context, a vector w(t) = [m,h, j, d, f, x]T of six recovery variables is con-sidered, in order to model several ionic currents that have a precise meaning from thephysiological prospective. The equations that describe the temporal evolution of thegating variables can be all seen under the following form:

dwidt

= αi(1− wi)− βiwi

wi(t = 0) = wi,0

where w(t) = [w1, ..., w6]T defines the vector of gating variables in a more abstractsetting. In addition, the only ion that is modelled is the Calcium. The parametersαi, βi are functions of the transmembrane potential value V only. A set of constants isinvolved in the development of the model. They are all written in the following table:The initial value of all the variables is here reported instead: Let us now consider indetails all the formulas and equations that permit effectively to build this model:

Transmembrane potential

dV

dt= −Iion + Iapp

C

Ionic current

Iion = INa + Isi + IK + IK1 + IKp + Ib

Fast Sodium current

ENa =RT

Fln(

NaoNai

)

INa = gNam3hj(V − ENa)

84

85

Symbol Definition Value

R Gas constant 8314 J K−1 kmol−1

T Temperature 310 K

F Faraday constant 96484.6 C mol−1

C Specific membrane capacitance 1 µF cm−2

gNa Maximal INa conductance 23 mS cm−2

Nao Extracellular Sodium concentration 140 mM

Nai Intracellular Sodium concentration 18 mM

PRNaK Na/K permeability ratio 0.01833

Ko Extracellular Potassium concentration 5.4 mM

Ki Intracellular Potassium concentration 145 mM

gKp Maximal IKp conductance 0.0183 mS cm−2

Eb Reversal background potential -59.87 mV

gb Maximal Ib conductance 0.03921 mS cm−2

Table A.1: Parameters and constants of the Luo-Rudy model

Symbol Initial value

V -84.3801107371 mV

[Ca2+]in 0.00017948816388306 mM

m 0.00171338077730188

h 0.982660523699656

j 0.989108212766685

d 0.00302126301779861

f 0.999967936476325

x 0.0417603108167287

Table A.2: Initial values of the variables of the Luo-Rudy model

86 APPENDIX A. THE LUO-RUDY CELLULAR MODEL

Fast Sodium current: m gating variable

αm =0.32(V + 47.13)

1− exp ((−0.1)(V + 47.13))

βm = 0.08 exp

(− V

11

)dm

dt= αm(1−m)− βmm

Fast Sodium current: h gating variable

αh =

0.135 exp

(80+V−6.8

)V < −40mV

0 otherwise

βh =

3.56 exp (0.079V ) + 310000 exp (0.35V ) V < −40mV1

0.13(1+exp (V +10.66−11.1 ))

otherwise

dh

dt= αh(1− h)− βhh

Fast Sodium current: j gating variable

αj =

(−127140) exp (0.2444V )−(0.00003474) exp ((−0.04391)V )(V+37.78)

1+exp (0.311(V+79.23)) V < −40mV

0 otherwise

βj =

0.1212 exp ((−0.01052)V )

1+exp ((−0.1378)(V+40.14)) V < −40mV0.3 exp ((−0.0000002535)V )

1+exp ((−0.1)(V+32)) otherwise

dj

dt= αj(1− j)− βjj

Slow Inward current

Esi = 7.7− 13.0287ln(Cai)

Isi = 0.09df(V − Esi)

Slow Inward Current: d gating variable

αd =0.095 exp ((−0.01)(V − 5))

1 + exp ((−0.072)(V − 5))

βd =0.07 exp ((−0.017)(V + 44))

1 + exp (0.05(V + 44))

dd

dt= αd(1− d)− βdd

87

Slow Inward Current: f gating variable

αf =0.012 exp ((−0.008)(V + 28))

1 + exp (0.15(V + 28))

βf =0.0065 exp ((−0.02)(V + 30))

1 + exp ((−0.2)(V + 30))

df

dt= αf (1− f)− βff

Time dependent Potassium current

gK = 0.282

√Ko

5.4

EK =RT

Fln(

Ko + PRNaKNaoKi + PRNaKNai

)

IK = gKxxi(V − EK)

Time dependent Potassium current: x gating variable

αx =0.0005 exp (0.083(V + 50))

1 + exp (0.057(V + 50))

βx =0.0013 exp ((−0.06)(V + 20))

1 + exp ((−0.04)(V + 20))

dx

dt= αx(1− x)− βxx

Time dependent Potassium current: xi gating variable

xi =

2.837(exp (0.04(V+77))−1)(V+77) exp (0.04(V+35)) V > −100mV

1 otherwise

Time independent Potassium current

gK1 = 0.6047

√Ko

5.4

EK1 =RT

Fln(

Ko

Ki)

IK1 = gK1K1∞(V − EK1)

88 APPENDIX A. THE LUO-RUDY CELLULAR MODEL

Time independent Potassium current: K1 gating variable

αK1 =1.02

1 + exp (0.2385(V − EK1 − 59.215))

βK1 =0.49124 exp (0.08032(V + 5.476− EK1)) + exp (0.06175(V − (EK1 + 594.31)))

1 + exp ((−0.5143)((V − EK1) + 4.753))

K1∞ =αK1

αK1 + βK1

Plateau Potassium current

EKp = EK1

Kp =1

1 + exp(

7.488−V5.98

)IKp = gKpKp(V − EKp)

Background current

Ib = gb(V − Eb)

Intracellular Calcium concentration

dCaidt

= (−0.0001)Isi + 0.07(0.0001− Cai)

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