Modelling Evolution of Vascular Disease: Aneurysms

Dr Paul Watton

Department of Computer Science & INSIGNEO Institute of in silico Modelling, University of Sheffield, UK

Department of Engineering Science, University of Oxford, UKSchool of Mathematics & Statistics, University of Glasgow, UK

Cerebral aneurysm evolution

Abdominal aortic aneurysm evolution

ROADMAP AHEADInterdisciplinary collaborative research

APPLICATION: Modelling other cardiovascular disease evolutionatherosclerosis, hypertension, in-stent restenosis, heart disease, tissue engineering

Basic ScienceVascular biologists

TranslationClinicians, engineers

Modelling aneurysm evolution

Illustrative example: Modelling framework for cerebral aneurysm evolution

Modelling Aneurysm Evolution

provide insight into (patho)physiology of disease

predictive models which have clinical application:

• AAA - predict growth rates and guide frequency of monitoring

• IA: identify stable aneurysms

Framework for Modelling Aneurysm Evolution

Mechanical environment

of vascular cells

Structural analysis

synthesis degradation

CFD

• CLINICAL CASE

• GROW ANEURYSM

• TEST HYPOTHESES FOR G&R

Aparicio et al (2014), IJNMBE

DOI: 10.1002/cnm.2620

Modelling developments:

endothelial heterogeneity

adventitial adaption (conceptual mathematical model)

Signalling pathways related to collagen regulation.

Influence of SMC apoptosis on aneurysm evolution.

Cerebral Aneurysms

OCCURRENCE: 3-5% of population.

DETECTION: increasingly diagnosed (improved imaging).

RUPTURE: LOW risk <1% per year.

RUPTURE OUTCOME: 30%-40% fatality.

What is the best treatment ?

1. Clip 2. Coil

3. Coil & Stent

4. Blood Flow divertors

Intervention: risky (1-7% morbidity)

expensive

Low rupture risk (0.1%). Do Nothing ?

6 months

MODELLING ANEURYSM EVOLUTION (& more generally vascular disease):

Model healthy arterial wall (Holzapfel et al 2000)

theoretical framework to describe growth and remodelling of tissue.

Aneurysm evolution: enlargement

Abrupt destruction of medial layer:

fragmentation/loss of elastin fibres

adaption of collagen fabric

Aneurysmal artery(cerebral)

Layers of the Arterial Wall

intima (endothelial cells)

media (elastin, collagen, smooth muscle cells)

adventitia (collagen, fibroblasts)

Healthy artery

Natural reference configurations for constituents

CHALLENGES:

For Biological tissues, the unloaded configurations of constituents/cells:

Modelling Evolution of Vascular Disease:

Mathematical framework needed to keep track of the unloaded geometrical reference configurations of tissue constituents.

Our skin grows with us!

may evolve

may be distinct

wavy collagen fibres

Shu Chien (2007)

Key Concept: Mechanotransduction

Modelling Evolution of Vascular Disease:

• Quantify mechanical environment of vascular cells.

• Understand/Model the influence of the mechanical stimuli on cell functionality.

Mechanical forces stimulate vascular cells…

- leading to modulations gene/protein expression and cellular functions

- influencing growth and remodelling (G&R) of tissue.

Mechanobiology

Structural analysis

(Quasi-static: Systolic/ diastolic deformations)

Mechanical environment

of vascular cells

synthesis degradation

Computational

fluid dynamics

ANSYS CFX

Perl wrapping script; Loosely coupled Fluid-Solid-Growth Framework

Aparicio et al (2014), IJNMBE DOI: 10.1002/cnm.2620

Solid Mechanics: Nonlinear elastic membrane

Principal of Virtual Displacements governs equilibrium displacement field.

0int =Π−Π extδδ

Watton et al (2004) A Mathematical Model for the Growth of the Abdominal Aortic Aneurysm, Biomechanics and Modelling in Mechanobiology, 3:98-113

Functional forms for strain energy density functions of arterial tissue required such that growth and remodelling can be simulated.

Solves steady (large) deformations: as tissue adapts (growth and remodelling)

DevelopmentThick-walled model: Schmid, Watton et al (2010)

Thick-walled FSG framework: Grytsan, Watton & Holzapfel (2013)

Volumetric Growth: Eriksson, Watton et al (submitted)

Thick-walled FSVG: Grytsan, Eriksson, Watton, Gasser (in progress)

+ simple conceptual mathematical models (still many open questions!)

Unloaded configuration

recruitment stretch:

onset of load bearing of collagen fibre

Recruitment Stretch (in 1D):

0Ω

Rλ

LRλ

L

Collagen fibers: - crimped in unloaded artery

- recruited to load bearing at physiological pressures

R

C

λ

λλ =

Elastinous and collagenous constituents have different natural reference configurations at which they begin to bear load.

Theoretical formulation (extended to 3D) enables:

- mechanical response of constituents to be defined relative to configuration they begin to bear load

- remodelling scheme for collagen to be implemented.

1

1

R

C

γ

γγ

λ

λλ =

zλ

λ

zλ

recλ1=C

γλ

γ

Rγλ

1=γλ

Cγλ

γλ

UNLOADED

CONFIGURATION

RECRUITMENT

CONFIGURATION

UNLOADED

RECRUITMENT

CONFIGURATION

LOADED

FIELDS OF RECRUITMENT STRETCHES DEFINED THROUGHOUT MEDIA AND ADVENTITIAL LAYERS

),( tRR

Xγγ λλ =

0Ω

R

γΩ

tΩ

C

γF

R

γ

C

γFFF =

R

γF

F

γa

recruitment stretch fields

normalised mass density elastin

( ) ( )( )0,

,,

=≡

t

ttm

E

EE

X

XX

ρ

ρ

( ) ( )( )γγ

γγ

γ

γγ

λ

λλ

aaC

aaC

×

×=

=

:

:2

2

R

R

C

)3( 1 −=Ψ Ikm E

EE

SEF for Elastin (defined relative to unloaded configuration)

( )( )0,

,

==

t

tm

C

C

C

X

X

γ

γγ

ρ

ρ

SEF for Collagenous constituents (defined relative to fibre stretch)

Strain Energy Density Function of Aneurysmal Tissue

COLLAGEN G&R

( )tR ,Xγλ

normalised mass density collagen

ELASTIN DEGRADATION

(Neo-Hookean SEF)

∑Ψ+Ψ=Ψγ

)()( C

γFF

CER

γ

C

γFFF =

)(trtr1 FFCT

I ==

R

T

RR FFC =

(1-2) IMPLY

collagen remodels to maintain its max stretch to

ASSUMPTIONS(1) Fibroblasts configure the collagen to achieve a maximum stretch during the

cardiac cycle – denoted the attachment stretch:

(2) Attachment stretch is constant during aneurysm evolution.

Cells: secrete collagen molecules and collagenase:

Continual turnover of collagen. New fibres attach in a state of stretch (Alberts et al)

C

ATλ

Collagen Remodelling

C

ATλ

−=

∂

∂C

AT

C

ATMAX

CR t

t

t

λ

λλα

λ γγ),(),(

0

XXevolve collagen reference configurations to restore maximum stretch of collagen fibres in the physiological configuration to the attachment stretch. (RATE-BASED APPROACH)

MAX

Ct),(Xγλ Denotes maximum stretch of collagen during cardiac cycle

C

ATλ

synthesis removal

Growth/Atrophy Collagen

ECM synthesis

Collagenase

INCREASE

STRETCH

ECM synthesis

Collagenase

Collagen mass increases

Collagen mass decreases

Fibroblasts reconfigure integrin attachments to matrix

DECREASE

STRETCH

synthesis removal

HomeostasisHomeostasis

ASSUMPTIONS

• Concentration of fibroblasts α concentration of collagen:

Ref. configuration of fibroblasts = Ref. configuration of collagen fabric: CF

γγ λλ =

[ ] [ ] [ ]

<∂

∂⇒<

>∂

∂⇒>

=∂

∂⇒= 0&0&0

t

m

t

m

t

mC

F

ATMAX

F

C

F

ATMAX

F

C

F

ATMAX

F γγ

γγ

γγ λλλλλλ

−=

∂

∂⇒

C

AT

C

ATMAX

C

C

C

E

EEm

t

m γ

γγ β0

Requirements for functional form:

CFmm γγ β0=

HOMEOSTASISINCREASE COLLAGEN

DECREASE COLLAGEN

)( FF

C

hmt

mγγ

γ λ=∂

∂

2/)1( 2 −= λE

Application of Aneurysm Evolution: Clinical Case

Mechanical environment

of vascular cells

Structural analysis

synthesis degradation

CFD

• CLINICAL CASE

• RECONSTRUCT HEALTHY ARTERY

• GROW ANEURYSM

• TEST HYPOTHESES FOR G&R

Alisa Selimovic, Patient-specific modelling of cerebral aneurysm evolution, PhD Thesis, Dept. of Eng.

Science, University of Oxford, Oct 2009-2012.

Clinical case: courtesy of James Byrne, John Radcliffe Hospital, Oxford

Linking Growth and Remodelling to the

Haemodynamic Environment: WSS

Prescribe aneurysm inception:

degrade elastin in localised region to create perturbation to the geometry

Creates altered haemodynamic environment.

Watton et al (2009) ASME Journal of Biomechanical Engineering

Linking elastin degradation to Haemodynamics

)(τDf

τ

( )

≥

<≤−

<−

=

n)degradatio no(2:0

25.0:

n)degradatio of rate maximum(5.0:

τ

ττ

τE

DD

E

DE

maf

ma

dt

dm

75.0;1)(0:

)(

=≤≤

−=

DD

E

DD

E

af

mafdt

dm

τ

τ

Watton et al (2009) Coupling the Haemodynamic Environment to the Evolution of Cerebral

Aneurysms: Computational Framework and NumericalExamples, JBiomech Eng, 131:101003.

WSS== ττ

WSS Elastin Concentration

Elastin Strain Collagen Strain

MODEL

Patient Aneurysm

MODEL: Elastin degradation linked to low wall shear stress.

Collagen remodels to achieve maximum stretch during cardiac cycle.

Collagen growth linked to magnitude of cyclic deformation of cells during cardiac cycle.

Results consistent with clinical observations: tentative support for modelling hypotheses

≥

<

−−

=

h

h

h

h

hD t

tt

f

ττ

τττ

ττττ

0

),(

),(),(

),(

2

X

XX

)0,(),( == tth XX ττ dttT

t

t

TtL

h

L

∫−

= ),(1

),( XX ττ

Heterogeneous Homeostatic

WSS (non-adaptive)

Heterogeneous Homeostatic

WSS (adaptive)

Spatially Heterogeneous and Temporally Adaptive Homeostatic WSS

WSS distribution is spatially heterogeneous in the arterial tree.

Endothelial Cells are in continual state of turnover.

Are endothelial cells preprogrammed with a homeostatic WSS or is this calibrated to local

haemodynamic environment ?

Assumptions: Elastin degradation driven by deviations of WSS from homeostiatic values.

(i) homeostatic WSS is spatially hetergeneous.

(ii) Is temporally adaptive

E

DhD

E

mafdt

dm),( ττ−=

Time Period = 10 years

Definition of Endothelial cell homeostasis has implications for disease evolution.

Guidance is needed from vascular biology.

Heterogeneous Homeostatic

WSS (non-adaptive)

Heterogeneous Homeostatic WSS

(adaptive)

Aparicio P, Mandaltsi A, Boamah J, Chen H, Selimovic A, Bratby M, Uberoi R, Ventikos Y,

Watton PN (2014) Modelling the Influence of Endothelial Heterogeneity on Progression of

Arterial Disease: Application to Abdominal Aortic Aneurysm Evolution, International

Journal for Numerical Methods in Biomedical Engineering’ DOI: 10.1002/cnm.2620

The cyclic deformation is important too!• Cummins et al (2007) Cyclic strain-mediated matrix

metalloproteinase regulation within the vascular endothelium: a

force to be reckoned with. AJPHCPUniaxial cyclic stretch

Healthy artery 1.1

Equi-biaxial cyclic stretch

Collagenous artery 1.02

Cyclic stretch affects functionality and alignment of vascular cells:

• Need to quantify the cyclic deformation mechanical environment

• Link G&R to cyclic deformation stimuli

BIOLOGY

• Fibroblasts sensitive to magnitude of cyclic deformation.

• Greater magnitude of cyclic deformation greater rate of synthesis

QUANTIFY CYCLIC DEFORMATION:

CYCLIC AREAL STRETCH CYCLIC STRETCH

diastole21

systole21

aa

aa

×

×=CS

A

DIAS

C

SYS

C

CS

γ

γ

γλ

λλ =

( )C

ATMAX

CCCSCS

C

EEmAt

m−=

∂

∂γγγγ

γ λξ ),(

( )

−

−=

==

0,1maxexp0,1maxexp,0;0

0 CS

t

CS

CS

CS

t

CSCS

A

CSCS

A

AA

γ

γγγγ

λ

λξξξλξ

Linking Collagen Growth to Cyclic Deformation

Growth linked to cyclic areal stretch

If G&R is not linked to cyclic deformation – Neck region can develops large cyclic deformation.

MODEL: Linking G&R to cyclic deformation contracts neck region and helps to promote aneurysms with a well-defined necks

Cyclic stretch of aneurysm 2-4% - consistent with clinical studies

Hypothesis: Collagen growth linked to cyclic deformation – a mechanism for neck formation ?

Cyclic Deformations

0=BSIχ

Biaxial Stretch Index (BSI)

MOTIVATION: Cell alignment and functionality is influenced by cyclic deformation.

cylindrical artery – cyclic deformation is 1-dimensional

idealised aneurysm (sphere) – cyclic deformation is equi-biaxial.

How does the cyclic deformation environment evolve and how can this be characterised ?

Watton et al (2011) PROPOSE NEW INDEX: Biaxial Stretch Index (BSI)

Watton et al (2011) Modelling evolution and the evolving mechanical environment of

saccular cerebral aneurysms, Biomechanics and Modeling in Mechanobiology, 10:109-132.

1=BSIχ

cyclic deformation is 1-dimensional

cyclic deformation is equi-biaxial.

( )( )CSCS

CSCSSIB

21

21

,max

,min

εε

εεχ =

1−= CSCS

αα λε magnitude of cyclic (linearised) strain during cardiac cycle

(defined in direction of principle stretches)

Cyclic Areal Stretch Biaxial Stretch Index (Watton, 2011)

cyclic deformation

is 1-dimensional

cyclic deformation

is equi-biaxial.

Watton et al (2011) Modelling evolution and the evolving mechanical environment of

saccular cerebral aneurysms, Biomechanics and Modeling in Mechanobiology, 10:109-132.

Adaption of the functional role of the adventitia.

[4]

[3]

Abrupt destruction of medial layer:

fragmentation/loss of elastin fibres

adaption of collagen fabric

Aneurysmal artery(cerebral)

Healthy young artery: adventitia acts as protective sheath to prevent overdistension

Cerebral aneurysm: Adventitia may become primary load bearer – How ?

Change in definition of homeostasis for collagen fabric

Haoyu Chen(Oxford 2010-14)

Simple mathematical models can guide understanding...

REALITY: distributions of attachment and recruitment stretches

Observation of collagen waviness distribution in the unloaded configuration

Schrauwen, J. T. C.,

Stergiopulos, N., Vosse, F. N.

van de, Bovendeerd, P. H. M.,

Rezakhaniha, R., & Vilanova,

A. (2012). A method for the

quantification of the

pressure dependent 3D

collagen configuration in the

arterial adventitia. Journal of

Structural Biology.

Distribution of collagen attachment stretches in the physiological configuration

Distribution of collagen recruitment stretches

Need for a novel mathematical model…

Distribution of (3 parameters)

Distribution of medial collagen fibre attachment and recruitment stretches

Assumption: medial collagen bears load in physiological configuration.

EQUIVALENTLY: Distribution of

Physiological stretch

∙

∙

∙

stretched crimped

! 1

MODELLING: Use triangular distribution function: 2 additional parameters

Distribution of #

Application: adventitial collagen fibres distributions $ %&'& at ( )

Assumption: adventitial collagen bears no load at physiological stage (* 0). # , 1

Conceptual Model of aneurysm evolution

Watton, P. N., Ventikos, Y., &

Holzapfel, G. A. (2009). Modelling

the growth and stabilization of

cerebral aneurysms. Mathematical

medicine and biology : a journal of

the IMA, 26(2), 133-164.

(i) 1D two-layer non-linear cylindrical membrane subjected to blood pressure and axial stretch

(ii) Media: elastin + collagen, Adventitia: collagen

(iii) Adventitia layer starts load bearing at systolic configuration

0 -./

.0

* -12 12

3 451-

66 7897: 9 4

1-67:#6

Model overview

Mechanical governing equation

6786 ;8<8 1 = 1

1>?

Modelling elastinous constituents (elastin + passive SMC) as Neo-hookean solid

Need to integrate NOVEL collagen model!

Collagen strain energy function (SEF)

7:@∗ :@ <B@2 :@ = 1 >

Distribution of @

Where: D=@, E=@ and F @.

G @ H@

@

Recruited Crimped

G @

0 1 I @2 @ = DF = D E = D D , @ , E2 F = @F = D F = E E I @ , F

0 @ ! F7:@ ;:@ J 7:@∗

@ G @ H@

K

L

Mechanical response of a single fibre

Hill, M. R., Duan, X., Gibson, G. A., Watkins, S., & Robertson, A. M.

(2012). A theoretical and non-destructive experimental approach for

direct inclusion of measured collagen orientation and recruitment

into mechanical models of the artery wall. Journal of Biomechanics.

SEF of a fibre

SEF of all fibre

Stress-stretch response of the collagen constituents

1st P-K stress-stretch response

67:@6 ;:@

H M 7:@∗ @ G @ H@K

LH

A nonlinear mechanical response of collagen can be obtained by assuming a linear response

for each collagen fibre and gradual recruitment of the fibres. (Piecewise analytic function)

67:@6

MODEL: Destruction of Media and growth of adventitia

ASSUMPTION

Prescribe degradation of medial elastinous constituents (elastin/SMC) and medial collagen

;8 ;: NOP0

Medial layer degraded during aneurysm evolution.

Watton, P. N., Ventikos, Y., & Holzapfel, G. A. (2009). Modelling the growth and stabilization of

cerebral aneurysms. Mathematical medicine and biology : a journal of the IMA, 26(2), 133-164.

6;:#6* ;:#

:# = ##

Adventitial collagen growth: simple stretch-based evolution law

Collagen remodelling

GQ@RNew formulation: evolve collagen recruitment stretch distribution GQ@R to restore the

collagen stretch distribution in LOADED configuration towards attachment stretch distribution.

H@H* S ∙ :@

= @Q*R@Q*R

H@H* S ∙ :@

= @Q*R@Q*R

MEDIAL COLLAGEN: Assume attachment stretch distribution fixed during evolution

Q*)=Q*=0) Q*)= Q*=0)

@ * T * , @Q*R U * , @Q*R V *

#Q*R= #Q* 0R

Q*RMODEL: Attachment stretch distribution can adapt but

functional form (triangular distribution) does not change.

MODEL: Evolve recruitment stretch distributions, e.g

Collagen remodelling: adaption of adventitial collagen attachment distribution $Q%&'WRPhysiological motivation

- Change of functional role of the adventitia from a protective sheath in normal

physiological condition to the main load bearer in the aneurysmal condition

H#H* S ∙ :#

= #Q*R#Q*R

H#H* S ∙ :#

= #Q*R#Q*R

Prescribe evolution of attach

stretch distribution

ADVENTITIAL COLLAGEN: Assume attachment stretch distribution adapts during

evolution.

# * T * , #Q*R U * ,

#Q*R V *(NEED TO LINK TO BIOLOGY)

Results illustration: mass density & stretches

Adventitial collagen growthMedial fibre degradation

:# #Media degradation prescribed, adventitial collagen evolves, system expands by a factor of

about 2.5, :# stabilises at the values of #.

Expansion of the arterial radius Evolving maximum and minimum adventitial

collagen stretch

Results illustration: remodelled thickness & thickness/radius ratio

Thinning of the arterial

wall during cerebral

aneurysm evolution

Model with adaption of attachment stretch distributions shows more realistic:

collagen growth, volumetric growth, evolution thickness/radius ratio.

Thickness/radius ratio (new vs old model)

The old model assumes single and

constant values of .

How the adventitia adapts may influence whether an aneurysm stabilises or ruptures:

Guidance from experiments to improve modelling.

Summary and Outlook

Integrated Fluid-Structure-Growth (FSG) computational framework for vasculature adaption:– Physiological geometries

– Model of the arterial wall

– (membrane (F77), thick-wall (FEAP/CMISS), volumetric growth (FEAP))

– G&R linked to cyclic deformation and steady haemodynamics.

Development (ongoing):– Improved representation of fibroblast mechanobiology

– Coupling framework with agent-based models of vascular cells.

– Improved models for adaption of collagen fabric.– (fibril distribution function and fibre reorientation/dispersion remodelling)

Development (outlook/collaborations):– Application to animal models (guidance/validation)

– Experimental guidance of how mechanical stimuli influence functionality

of vascular cells

– Modelling framework a basis for modelling range of vascular diseases.

– Application to vascular construct design.

– Translation (clinicially useful predictive models of aneurysm evolution)

IA: Stable/enlarging; AAA: rate of enlargement (frequency of monitoring)

Theoretical Mechanobiology Group

Hamna

Afaq

www.themebio.org

Chen, Mandaltsi, Boamah, Hornsby, Aparicio, Dickinson, Chan, Afaq

Acknowledgements:

Nick Hill (Glasgow), Yiannis Ventikos (UCL),

Gerhard Holzapfel (Graz), Anne Robertson (Pittsburgh),

www.insigneo.org

INSIGNEO Institute of in silico Medicine

Over 100 academics from engineering, medical physics , computer

science and hospital

Largest institute of its type Europe, Established May 2013

Open questions: How to couple with signalling

pathway models…

Modelling of Signalling Pathways

in Vascular Mechanobiology

[4]

[3]

Pedro AparicioOct 2013-2016

Need for collaborations with vascular biologists and experimental arterial

mechanobiology

Flowchart

update

arterial

layer

radii

update

constitutive

model

update

inner

radius

update

thrombus

radius

P

MECHANICAL EQUILIBRIUM OF MEMBRANE

λzL

λR X Y Z[\]]^_`, Z\b(^_`

L: Media, Adventitia

J: Elastin, Collagen, SMCs

OXYGEN MASS TRANSPORT

O2 diffusion throughL = T, M, A

TROMBUS

LUMEN

WALL

cW Y dW, eWfeWf Y dW, cW

GROWTH AND REMODELLING

ghiWg( Y jiW ,hiW , %iW , %iW&'

g%iWkg( Y liW , %iW , %iW&'

THROMBUS PROPAGATION

WALL

Shear regulated

thrombus

propagation

m' nof

update

O2 regulated

growth rates

jiW YQcWR

( )( ) 073.02/12

=−= C

AT

C

ATE λ

073.0

30,30 −+=Mβ

60,60 −+=Aβ

Adventitial collagen strain

Medial collagen strain

Normalised time

Pressure (Pa)

073.0

Azimuthal elastin strain

Healthy Artery Model

Material, geometric & remodelling

parameters

( )

( )

+

−+=

=

∑≥±== 0:;,

2

22

42

cosexp11

1

)(

C

pJp

ppp

EpAMJR

J

J

C

J

CC

J

CC

JJ

z

E

AA

E

MM

z

EaEakhkHkH

R

pp

λ

γ

λλλ

λ

40 expCollagen

4/ (A)Collagen

KPa2 (M)Collagen

10/ (A)Elastin

KPa135 (M)Elastin

=

=

=

=

=

C

C

M

C

A

C

M

E

M

E

A

E

M

a

kk

k

kk

k

3

years1.0

years 8.0

1-

0

-1

=

=

=

CS

Aβ

β

α

oo

oo

60,60, (A) angles Fibre

30,30, (M) angles Fibre

3/ thickness(A)Adventitia

3/2 thicknessMedia(M)

5/ thickness wallUnloaded

1 radius Unloaded

25.1 (systole)stretch ntialCircumfere

3.1 stretch Axial

0

−=

−=

=

=

=

=

=

=

−+

−+

AA

MM

A

M

z

HH

HH

RH

mmR

γγ

γγ

λ

λ

Pressure-stretch relationship for cylindrical membrane model of artery (derived from variational equation).

GEOMETRIC

MATERIAL

Remodelling

Radius (mm)

Pressure (kPa)

http://www.scsitaly.com/products/AIMA

@neurist projectSemi-automated simulation pipeline: Segmentation

to CFD

Boundary conditions: 1D circulation model

included in AIMA

by

AIMA: a software features a medical imaging pipeline to create computational models of AIMA: a software features a medical imaging pipeline to create computational models of

the haemodynamics of cerebral aneurysms. Integrated with ANSYS ICEM and CFX

The importance of aneurysm research….

2014: Current biomechanical model (UK) used by clinicians for predicting risk of abdominal aortic aneurysm rupture

Need: Improved biomechanical models to aid clinical decisions!!!

economic savings and health benefits

Intervention may be recommended

No intevention

Decision on whether to operate based on aneurysm diameter

Inference of arterial wall thickness during aneurysm evolution

Remodelled wall thickness evolution

Remodelled wall thickness/evolving radius ratio

p8 , p: dryweightmassfractions of fibres in the media

Vtuv * w *w * 0

xyz|~y~∙

4

* ~yzy

zzy|z~

* =ℎtuv *

*=ℎtuv *

- *=

w *

w5 0 + w 0

4

-

1

* >

w *

w * = 0=

p8;8* + p:;:

* w5(0) + ;:#w 0

w5 0 + w 0

Assumptions

1. Constant mass fraction of water in the arterial wall.

2. Volumetric growth of the wall is proportional to the mass growth of the constituents.

Wall thickness evolution is approximated by assuming that the volumetric growth is a

function of the mass growth /atrophy of the constituents.

Is the attachment stretch constant ?

Lzλ

R0λ

Lzλ

Rt)(λ

t

0Ω tΩ

Rr 0λ= Rtr )(λ=

tΩt

0Ω

[ ] [ ] ???0 ⇒=∧→ C

AT

CEm λλ

( ) ( )( )CCCEE

z

mmH

Rp λσλσ

λλλˆˆ

1+

=

( ) ( )( )CCCEE mmH

Rp λσλσ

λλˆˆ

22

+

=

Given functional forms for stress of elastin and collagen.

Assume elastin degrades and collagen adapts to achieve

new homeostasis.

What are the implications of:

(i) Constant attachment stretch

(ii) Collagen stress-strain function independent of time (fibril undulation distribution constant))

Watton PN, Ventikos Y, Holzafpel GA (2009) Modelling the growth and stabilisation of cerebral aneurysms, Mathematical Medicine and Biology, 26:133-164.

( ) ( )( ) )1(ˆˆ1 CCCEE

z

mmH

Rp λσλσ

λλλ+

=

( )( ) ( )C

AT

CE

ECEP

λσλσ

λσ

ˆˆ

ˆ

0

0:

0+

=

( )Z

CCEE Hthhmmth

λλϕϕ =+≈ )(:)(

~

constant1

~

0

0

:

=

−=

r

h

Pr

h

CE

Cϕ

fractions mass dry weight, =CE ϕϕ

Material equilibrium [ ]C

AT

Ct λλ =)(

Volumetric growth (assuming mass fraction water is constant)

( ) )0()( =+= tVmmtV CCEE ϕϕ

Mechanical equilibrium (cylinder)

Wall thickness evolution (loaded configuration)

[ ] [ ] [ ]

−

→⇒=∧→∧

CE

CC

AT

CE

Pmtm

:

0

2

0 1

1)(0(1)

λ

λλλ

Proportion of load borne by elastinous constituents at t=0.

Analytic expression for ratio of remodelled thickness to radius (loaded configuration)

Theoretical prediction of (remodelled wall thickness/radius) ratio

as aneurysm enlarges:(Independent of G&R approach and functional form of stress functions)

LzλRt)(λ

( ) ( )( )CCCEE

z

mmH

Rp λσλσ

λλλˆˆ

1+

=

( )

−=

42

2 11ˆ

λλλλσ

z

E

Ek ( ) ( )

5

2ˆ

=

C

AT

C

C

CCC

E

Ekλλσ

5/01.0)( tEtm =

8.0:

0 =CEP 5.05.0 == CE ϕϕ

KPa165/1/1.1/3.1

1.13.13.1

0

0

===

===

pRHR

C

ATz

λ

λλλ

( )C

AT

CCC

EEmdt

dm−= β

( )C

AT

CR

EEdt

d−= α

λ

0

0

:123.0

~

r

h

Pr

h

CE

C

−=→

ϕ

G&R

Parameters

Verification of theory with

numerical model

constant attachment

stretch implies unrealistic

increases in collagen mass

Conclusion: attachment

stretch not constant.

- Assume:

(i) Distribution of attachment stretches

(ii) Distribution of attachment stretches can adapt

Novel Model for microstructural remodelling of collagen

- Explore consequences using conceptual model of cerebral aneurysm evolution.

Inception: Growth and Remodelling

RadiusElastin ConcentrationCollagen strain

(media, postively wound)

WSS Elastin E22 Collagen concentration

Growth and Remodelling (G&R)

Growth/Atrophy

Remodelling

Changes in mass of the constituents

Reorganization of the constituents (no change in mass), e.g. Constituent reference configurations, orientations.

synthesis degradation

Cells continuously produce ECM and enzymes to degrade the ECM.

Homeostasis

G&R approaches: Integral based (Humphrey and co-workers) and rate based

(Humphrey).