Abdominal aortic aneurysm inception and evolution …kth.diva-portal.org/smash/get/diva2:1051142/FULLTEXT01.pdf · Abdominal aortic aneurysm inception and evolution { A computational

Abdominal aortic aneurysm inceptionand evolution – A computational model

Andrii Grytsan

Doctoral Thesis no. 98, 2016

Department of Solid Mechanics

School of Engineering Sciences

KTH Royal Institute of Technology

Stockholm, Sweden

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TRITA HLF-0605ISSN 1654-1472ISRN KTH/HLF/R–16/19–SEISBN 978-91-7729-216-6

Akademisk avhandling som med tillstand av Kungliga Tekniska hogskolan(KTH) i Stockholm framlagges till offentlig granskning for avlaggande avteknologie doktorsexamen tisdagen den 20:e december 2016 kl. 10:00 i F3,Lindstedtsvagen 22, KTH, Stockholm.Fakultetsopponent ar Professor Frans N. van de Vosse, Technische UniversiteitEindhoven, Netherlands.

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Abstract

Abdominal aortic aneurysm (AAA) is characterized by a bulge in the abdominal

aorta. AAA development is mostly asymptomatic, but a such bulge may suddenly

rupture, which is associated with a high mortality rate. Unfortunately, there is

no medication that can prevent AAA from expanding or rupturing. Therefore,

patients with detected AAA are monitored until treatment indication, such as

maximum AAA diameter of 55 mm or expansion rate of 1 cm/year. Models of

AAA development may help to understand the disease progression and to inform

decision-making on a patient-specific basis. AAA growth and remodeling (G&R)

models are rather complex, and before the challenge is undertaken, sound clinical

validation is required.

In Paper A, an existing thick-walled model of growth and remodeling of one

layer of a AAA slice has been extended to a two-layered model, which better

reflects the layered structure of the vessel wall. A parameter study was performed

to investigate the influence of mechanical properties and G&R parameters of such

a model on the aneurysm growth.

In Paper B, the model from Paper A was extended to an organ level model of

AAA growth. Furthermore, the model was incorporated into a Fluid-Solid-Growth

(FSG) framework. A patient-specific geometry of the abdominal aorta is used to

illustrate the model capabilities.

In Paper C, the evolution of the patient-specific biomechanical characteris-

tics of the AAA was investigated. Four patients with five to eight Computed

Tomography-Angiography (CT-A) scans at different time points were analyzed.

Several non-trivial statistical correlations were found between the analyzed pa-

rameters.

In Paper D, the effect of different growth kinematics on AAA growth was inves-

tigated. The transverse isotropic in-thickness growth was the most suitable AAA

growth assumption, while fully isotropic growth and transverse isotropic in-plane

growth produced unrealistic results. In addition, modeling of the tissue volume

change improved the wall thickness prediction, but still overestimated thinning of

the wall during aneurysm expansion.

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Sammanfattning

Bukaortaaneurysm (AAA) kannetecknas av en utbuktning hos aortavaggen i bu-

ken. Tillvaxt av en AAA ar oftast asymtomatisk, men en sadan utbuktning kan

plotsligt brista, vilket har hog dodlighet. Tyvarr finns det inga mediciner som

kan forhindra AAA fran att expandera eller brista. Patienter med upptackt AAA

halls darfor under uppsikt tills operationskrav ar uppnadda, sasom maximal AAA-

diameter pa 55 mm eller expansionstakt pa 1 cm/ar. Modeller for AAA-tillvaxt

kan bidra till att oka forstaelsen for sjukdomsforloppet och till att forbattra be-

slutsunderlaget pa en patientspecifik basis. AAA modeller for tillvaxt och struk-

turforandring (G&R) ar ganska komplicerade och innan man tar sig an denna

utmaning kravs de god klinisk validering.

I Artikel A har en befintlig tjockvaggig modell for tillvaxt av ett skikt av

en AAA-skiva utokats till en tva-skiktsmodell. Denna modell aterspeglar battre

den skiktade strukturen hos karlvaggen. Genom en parameterstudie undersoktes

paverkan av mekaniska egenskaper och G&R-parametrar hos en sadan modell for

AAA-tillvaxt.

I Artikel B utvidgades modellen fran Artikel A till en organniva-modell for

AAA-tillvaxt. Vidare inkorporerades modellen i ett “Fluid–Solid–Growth”

(FSG) ramverk. En patientspecifik geometri hos bukaortan anvandes for att il-

lustrera mojligheterna med modellen.

I Artikel C undersoktes utvecklingen av patientspecifika biomekaniska egenska-

per hos AAA. Fyra patienter som skannats fem till atta ganger med “Computed

Tomography-Angiography” (CT-A) vid olika tillfallen analyserades. Flera icke tri-

viala statistiska samband konstaterades mellan de analyserade parametrarna.

I Artikel D undersoktes effekten av olika tillvaxt-kinematik for AAA tillvaxt.

En modell med transversellt-isotrop-i-tjockleken-tillvaxt var den bast lampade

for AAA tillvaxt, medans antagandet om fullt-isotrop-tillvaxt och transversellt-

isotrop-i-planet-tillvaxt producerade orimliga resultat. Dessutom gav modellering

av vavnadsvolymsforandring ett forbattrat vaggtjockleks resultat men en fortsatt

overskattning av vaggfortunningen under AAA-expansionen.

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List of appended papers

Paper A: Influence of differing material properties in media and adventitiaon arterial adaptation – application to aneurysm formation and rupture.H. Schmid, A. Grytsan, E. Poshtan, P.N. Watton, M. Itskov.Computer Methods in Biomechanics and Biomedical Engineering, 2013, 16(1),33–53.

Paper B: A thick-walled fluid-solid-growth model of abdominal aorticaneurysm evolution: application to a patient-specific geometry.A. Grytsan, P.N. Watton, G.A. Holzapfel.Journal of Biomechanical Engineering, 2015, 137(3), 031008.

Paper C: Biomechanical changes during abdominal aortic aneurysmgrowth.R. Stevens, A. Grytsan, J. Biasetti, J. Roy, M. Lindquist Liljeqvist, T.C.Gasser.Report 603. Department of Solid Mechanics, KTH Royal Institute of Tech-nology. Submitted for publication.

Paper D: Growth description for vessel wall adaptation: a thick-walledmixture model of abdominal aortic aneurysm evolution.A. Grytsan, T.S.E. Eriksson, P.N. Watton, T.C. Gasser.Report 604. Department of Solid Mechanics, KTH Royal Institute of Tech-nology. To be submitted for publication.

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In addition to the appended papers, the work in this thesis has resultedin the following conference contributions:

A thick-walled fluid-solid-growth model of abdominal aortic aneurysmevolution.A. Grytsan, P. N. Watton, G. A. Holzapfel.Presented at 8th European Solid Mechanics Conference, Graz, Austria, 2012,(abstract and oral presentation).

Predictive model of AAA evolution: application to patient-specific geometry.A. Grytsan.Presented at Stockholm Aneurysm Research (STAR) group meeting, 2013.

Fluid-solid-volumetric-growth framework of abdominal aorticaneurysm evolution.A. Grytsan, P. N. Watton, T.S.E. Eriksson, T.C. Gasser.Presented at 7th World Congress of Biomechanics, Boston, USA, 2014, (ab-stract and oral presentation).

A novel fluid-solid-growth framework of abdominal aortic aneurysmevolution.A. Grytsan, P. N. Watton, T.C. Gasser.Presented at 11th World Congress on Computational Mechanics, Barcelona,Spain, 2014, (abstract and oral presentation).

A thick-walled fluid-solid-volumetric-growth model of abdomi-nal aortic aneurysm development.A. Grytsan, T.S.E. Eriksson, P.N. Watton, T.C. Gasser.Presented at 9th European Solid Mechanics Conference, Madrid, Spain, 2015,(abstract and oral presentation).

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Contribution to the papers

The author’s contribution to the appended papers is as follows:Paper A: Designed the study and carried out all computations. Analyzed

and interpreted the results together with Schmid and Poshtan. Contributedto the manuscript drafting. Schmid wrote the major part of the manuscript.

Paper B: Principal author. Further developed the model presented inPaper A and incorporated it into a Fluid-Solid-Growth framework. Designedthe study and carried out all computations. Analyzed and interpreted theresults. Drafted the manuscript. Holzapfel, Watton contributed with criticalreviews.

Paper C: Designed the study together with Stevens, Gasser and Biasetti.Developed the computational framework together with Stevens. SupervisedStevens and supported him throughout the computational efforts. Analyzedand interpreted results together with other co-authors. Contributed withcritical review of the manuscript draft by Gasser.

Paper D: Principal author. Further developed the previously publishedmodel for volumetric growth. Designed the study and carried out all com-putations. Analyzed and interpreted the results. Drafted the manuscript.Gasser contributed with critical review. Eriksson, Watton contributed withcomments.

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iSammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of appended papers . . . . . . . . . . . . . . . . . . . . . . . . vContribution to the papers . . . . . . . . . . . . . . . . . . . . . . . viiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Structure of arterial wall . . . . . . . . . . . . . . . . . . . . . 2Collagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Modeling the adaptation of vascular tissue . . . . . . . . . . . 6

Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Collagen growth and remodeling . . . . . . . . . . . . . . . . . 8Elastin degradation . . . . . . . . . . . . . . . . . . . . . . . . 9Volume growth . . . . . . . . . . . . . . . . . . . . . . . . . . 9Fluid-solid-growth framework . . . . . . . . . . . . . . . . . . 10

Key results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Bilayer aneurysm model . . . . . . . . . . . . . . . . . . . . . 11Fluid-solid-growth framework . . . . . . . . . . . . . . . . . . 12Patient-specific biomechanics . . . . . . . . . . . . . . . . . . . 13Volume growth . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Summary of appended papers . . . . . . . . . . . . . . . . . . . . . 17Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Paper A

Paper B

Paper C

Paper D

x CONTENTS

Abdominal aortic aneurysm inception and evolution – A computational model

Introduction

Aneurysm is a cardiovascular disease, that is characterized by an excessive

localized enlargement of a blood vessel. Aneurysms are most often found

in the abdominal and thoracic aorta, and within the circle of Willis in the

brain. Abdominal aortic aneurysm (AAA), hereon referred to simply as the

aneurysm, is diagnosed when the maximum diameter of the bulge is at least

50% larger than the diameter of the abdominal aorta, see Figure 1. Aneurysm

development is usually asymptomatic, and aneurysms are most often de-

tected incidentally or through screening programs. However, an aneurysm

can eventually rupture, which is a life-threatening event with an in-hospital

mortality of around 75% [1].

The aneurysms are rare in subjects under 55 years of age, and the risk

of developing an aneurysm increases significantly with age. Males tend to

develop aneurysm up to four times more often than females [2, 3, 4]. To-

bacco smoking increases the risk four times [5]. Other factors include high

cholesterol level in the blood, obesity, and hypertension. Diabetes mellitus

is a negative risk factor [6]. Genetic disorders such as Marfan syndrome and

Ehlers-Danlos syndrome correlate with the aneurysm development.

Figure 1: Schematic comparison of normal aorta versus aorta with largeaneurysm [7].

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Andrii Grytsan

Unfortunately, there is no medication available to prevent aneurysm growth

or rupture, and the common practice is to electively repair a non-ruptured

aneurysm by open surgery or endovascular repair (EVAR). Both the inter-

ventions are associated with a 30-day mortality of 2% (EVAR) and 4% (open

surgery). These interventions also have similar long-term outcomes [8] and

decrease the quality of life for the patients. Thus, the intervention risks

have to be carefully weighed against the risk of aneurysm rupture. To date,

population-based statistical criteria, such as the maximum AAA diameter of

55 mm [9, 10] or AAA growth of 1 cm per year [11], are used to indicate

the aneurysm repair. However, aneurysms below these thresholds can rup-

ture [12] while some larger aneurysms remain stable. Hence, patient-specific

criteria for the risk of aneurysm rupture would be potentially helpful in clin-

ical decision making. Estimation of such criteria has received considerable

attention in the literature [13]. However, such studies consider only the in-

formation from a single point in time and are unable to predict the evolution

of the risk of aneurysm rupture. A realistic model of the aneurysm evolution

can improve our understanding of the pathology of the disease and eventu-

ally inform clinicians about the changes to the risk of aneurysm rupture with

time.

Structure of arterial wall

Normal aorta The vascular wall is composed of three distinct layers,

namely the intima, the media and the adventitia, see Fig. 2. The intima

consists of a monolayer of vascular endothelial cells lining the luminal sur-

face, a thin basal membrane and a subendothelial layer. It is often assumed

that the intima does not contribute significantly to the load carrying ca-

pacity of the wall. However, sometimes normal, and often, the pathological

subendothelial layer has a significant thickness and stiffness. The media is

the middle arterial layer consisting of vascular smooth muscle cells (SMCs)

interwoven with elastin and bundles of collagen fibers and form a complex

three-dimensional structure. The media receives oxygen and nutrients from

the lumen. The media is separated from the intima and the adventitia by

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Figure 2: Idealization of arterial wall structure [14]. It consists of threelayers: intima (I), media (M) and adventitia (A).

the internal and external elastic laminae, respectively. The adventitia is the

outermost arterial layer that consists of mainly fibroblasts, thick bundles of

collagen fibers, and other ground substance constituents. It is surrounded

by loose connective tissue and the outer boundary is not clearly defined.

Small bloods vessels, vasa vasorum, supply the adventitia with oxygen and

nutrients from the outside.

The collagen fibers are very stiff and strong, serving as fiber reinforce-

ments of the tissue. They are wrapped around the lumen in double helical

pitches. Their orientations are distributed about a mean direction. However,

individual orientations deviate substantially from the mean. The mean di-

rection of fiber orientation depends on the arterial layer and the location of

the artery within the vascular system.

The artery in vivo is subjected to physiological loads, such as internal

pressure and axial pre-stretch. Furthermore, the arterial tissue has 3-D resid-

ual stresses, which are attributed to the arterial growth and adaptation [15].

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Andrii Grytsan

Abdominal aortic aneurysm The aneurysmal wall is usually less

structured, especially in the presence of a thick intra-luminal thrombus (ILT).

There is no clear boundary between the media and the intima, but to account

for the possible existence of thick subendothelial layer (or “neo-intima”), the

innermost layer of aneurysmatic or aged artery is sometimes referred to as

media-intima composite in biomechanical literature. Aneurysms have up to

90% less elastin than the normal aorta. In addition, elastin is fragmented in

the aneurysm wall. There is an increase in the collagen content, however the

collagen fibers are more dispersed and less densely interlinked. The media-

intima composite is often decellularized [16], especially in the presence of a

thick ILT layer. In this case, the adventitial capillaries can extend to media

in a process called neovascularization, which likely weakens the vessel wall.

There is often an increase in the collagen content in the adventitia, too.

Collagen

Collagen is one of the most important structural proteins in the human body.

Type I collagen is the most abundant, while as many as 28 types of collagen

have been identified so far. Collagen has a complex hierarchical structure,

and the mechanical properties vary widely at the different scales.

At the molecular scale, three procollagen chains form the collagen triple

helix of approximately 300 nm in length and 1.6 nm in diameter. When

pulled, collagen triple helix begins to unwind, which causes stretching of

the hydrogen bonds between the chains. Hence, the initial modulus of the

collagen molecule, which is about 4 GPa, is attributed mainly to the breaking

of hydrogen bonds and their reforming at new locations. At displacements

above approximately 18%, the molecular backbone begins to stretch. Tan-

gent modulus at large displacements is estimated to be as high as 75 GPa

[17].

At the fibrillar scale, collagen triple helices are arranged in a staggered

manner, forming collagen fibrils of 50 − 500 nm in diameter. The stretch-

ing mechanism of a collagen fibril is explained by the relative sliding of the

molecules [18]. The mechanical properties of the fibril depend on hydration,

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and crosslinking between the collagen molecules. The initial modulus of a

hydrated fibril has been experimentally determined to be in the range of 0.2

to 0.8 GPa [19, 20, 21]. An initial modulus of 0.3 GPa was also predicted by

atomistic modeling [22].

At the microscale, the bundles of collagen fibrils form collagen fibers. The

collagen fiber stretch mainly represents the interfibrillar sliding. An initial

modulus of the extruded, crosslinked collagen fiber has been reported to be

in the range of 260 to 560 MPa [23].

Finally, the arrangement of collagen fibers and fibrils at the macroscale

may lead to different structures with extreme variation in mechanical prop-

erties.

Collagen is produced by various types of cells, including fibroblasts, my-

ofibroblasts, and SMCs. Cells do not only secrete the components of collagen

molecules, they also play active role in the assembly of collagen fibrils and

fibers. In addition, cells are able to exert force on the collagen fibers and

therefore attach them to the matrix in a pre-stretched state [24]. Collagen

degradation is performed by matrix metalloproteinases (MMPs), which bind

to the collagen molecules and disassemble them into simpler proteins.

The process of constant production and degradation of collagen fibers is

referred to as the collagen growth and remodeling (G&R). Collagen-producing

cells act as strain gauges [25]. If they sense overstretch, they respond by

secreting more collagen. The opposite is also true, i.e., cells secrete less col-

lagen if they are stretched less. The stretch of the tissue also facilitates the

MMP-based collagen degradation. At larger stretches, more binding sites are

exposed for chemical reactions in a stretched fiber, provoking collagen degra-

dation. Similarly, a decreased stretch suppresses the collagen degradation.

In a certain, preferred state of stretch, the cell-based collagen production and

MMP-based collagen degradation are in a state of equilibrium and the col-

lagen mass is kept nearly constant. This state is referred to as homeostasis.

Departure from homeostasis may result in collagen net growth or resorption.

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Andrii Grytsan

Modeling the adaptation of vascular tissue

Soft tissues in the human body have a preferred state of stress and strain.

They can respond to alterations in stress and strain by growing and re-

modeling. This interaction is complex and mathematical modeling can be

helpful to improve our understanding of these processes. Early models of

vascular wall G&R focused on the tissue growth. They assumed that the

vessel wall adapts its geometry to maintain the wall stresses at a target level

[26, 27, 28]. Although such models were successful in predicting the tissue

growth, they provided limited insight into the underlying processes and mech-

anisms. Humphrey (1999) [29] has presented a constrained mixture model

of tissue remodeling based on continuous collagen turnover, i.e., deposition

and degradation, together with adapting its reference configuration. This

concept was further developed and applied to model arterial G&R in health

and disease [30, 31, 32, 33]. In contrast, Watton et al. (2004) [34] proposed

a G&R model for AAA, based on collagen net growth and adapting the ref-

erence configuration of the collagen. This concept was further developed for

membrane model [35] and the thick-walled model [36]. In addition, multiscale

model of collagen G&R was reported in the literature based on collagen fiber

stretch homeostasis [37]. Recently, collagen G&R has been also linked to a

biochemical pathway model [38]. Concurrently, soft tissue volume growth

has been described by structurally-motivated growth kinematics [39, 40].

Elastin degradation in AAA wall has been linked to the local blood flow

conditions [41] and to the exposure to the intra-luminal thrombus (ILT)

[16]. Therefore, the chemical reactions in the ILT have been coupled to

the AAA growth [42]. Similarly, modeling frameworks coupled the hemody-

namic simulations to G&R descriptions of the AAA wall [43, 44, 45]. These

efforts resulted in the coining of the term Fluid-Solid-Growth (FSG) model-

ing framework. Thus far, within such FSG models, the AAA wall was only

represented by membrane models.

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Objective

This study aimed at developing a thick-walled FSG framework of the AAA

evolution. The specific aims were

• to develop a thick-walled model of AAA growth that incorporates main

growth and remodeling mechanisms of the AAA wall;

• to develop the FSG framework, where the AAA growth model is one

way coupled to the patient-specific laminar blood flow;

• to investigate the effect of different growth kinematics and the key

model parameters on the predictions of AAA growth;

• to investigate how the biomechanical parameters change with time in

AAA patients, who are followed up by CT-A.

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Andrii Grytsan

Methods

We modeled the hypothetical normal aorta as a bilayer thick-walled cylindri-

cal structure. The two layers represent the media-intima composite and the

adventitia, and the arterial tissue was modeled as a nearly incompressible

material. Incompressibility is motivated by the high water content in the

tissue and low permeability of the arterial wall [46]. A structurally moti-

vated constrained mixture model captured the mechanical response of the

vascular tissue, thus allowing to model explicitly the response of each tis-

sue constituent. Constituents of mechanical relevance were collagen, elastin

and the ground matrix. The ground matrix included passive response of the

SMCs and all extra-cellular matrix except elastin and collagen. Elastin and

ground matrix were modeled with a neo-Hookean constitutive law [47], while

an exponential law based on the fiber stretch captured the collagen fibers.

Due to their waviness, collagen fibers have different reference configuration

as compared to the rest of the tissue. Recruitment stretch variable models

the collagen fiber engagement, i.e., the tissue stretch at which the collagen

fiber is stretched and starts to bear load. The SMC-based active response

and the presence of ILT are omitted in this work for the sake of simplicity.

The developed constitutive models were implemented in CMISS [48] in

Paper A and Paper B, and in FEAP [49] in Paper D. Solid mechanical

analysis in Paper C was performed using A4Clinics [50]. Finite element

formulation Q1P0 was used in all papers to avoid volumetric locking of the

elastically incompressible material.

Blood flow was modeled using ANSYS CFX in Paper B and Paper C.

In Paper B, blood was modeled as a Newtonian fluid and the steady state

flow was assumed. In Paper C, pulsatile flow was simulated. Carreau-

Yasuda viscosity model, which incorporates non-Newtonian effects of blood

flow, was used.

Collagen growth and remodeling

The collagen in the vessel wall is constantly degraded by MMPs and syn-

thesized by fibroblasts and SMCs. Vascular cells attach collagen fibers to

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the matrix in a pre-stretched state, often called deposition or attachment

stretch [35],[51]. Collagen deposition and degradation are balanced in a nor-

mal aorta in homeostasis. In this state, we assumed that the collagen fibers

are uniformly stretched at their attachment stretch. This is achieved by

remodeling the collagen recruitment variable until a homogeneous collagen

stretch is achieved in the entire vessel wall.

When the vascular tissue is overstretched, the balance between collagen

deposition and degradation is lost, leading to increased collagen production

and net growth. The opposite holds for decreased tissue stretch. Indepen-

dently, the collagen growth is modeled through the normalized mass change.

Consequently, two independent rate equations are used to describe evolution

of the recruitment stretch and the collagen mass, respectively.

Elastin degradation

The detailed cause of the onset of elastin degradation is not yet understood.

Elastin degradation may be due to the vascular wall injury or altered blood

flow conditions. In Paper A and Paper D, explicitly prescribing elastin

degradation [34] allowed to focus on the collagen-related G&R aspects. In

contrast, Paper B linked the elastin degradation to low wall shear stress

(WSS) levels to provide a more comprehensive investigation. In both cases,

elastin degradation is modeled through the normalized mass change.

Volume growth

Kinematics of the volume growth are illustrated in Figure 3. Isotropic, in-

plane and in-thickness volume growth models have been tested.

It is unclear, how the mass increments of vessel wall constituents are

turned into tissue volume, and the two scenarios of constant constituent vol-

ume (CCV) and constant constituent density (CCD) were considered. That

is, CCV assumed that the constituent’s mass change caused a change in the

density of the constituent. On the other hand, CCD leads to change in the

constituent’s volume in response to the constituent’s mass change.

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Andrii Grytsan

Figure 3: Kinematics of growth. Deformation gradient F(τ, t) maps thereference configuration Ω0 into the current configuration Ωτ . On the otherhand, the spatially homogeneous growth tensor Fg(τ) (with det Fg = det F =v) connects the reference configuration Ω0 to the intermediate stress-freeconfiguration Ωg. This mapping relates to a time-scale τ of weeks. Then, theelastic deformation tensor Fe(t) connects Ωg to the current configuration Ωτ .Consequently, the total deformation gradient F(τ) is split into volumetricgrowth Fg and elastic Fe parts, respectively.

Fluid-Solid-Growth framework

We assume that the arterial wall’s G&R description is coupled to the blood

flow sensed at the intimal layer. Due to clearly different time scales of the tis-

sue G&R (weeks) and the cardiac cycle (seconds), the two problems are only

loosely coupled, see Figure 4. Firstly, the momentum equations are solved

in the solid part of the aneurysm, i.e., for the AAA wall. Then, the blood

flow velocity is computed by solving the Navier-Stokes equations within the

updated aortic lumen. Up and downstream extensions are attached to obtain

a fully developed flow in the aneurysm region. Next, the aneurysm G&R de-

scription is informed of the WSS imposed by the blood flow. Consequently,

G&R model adapts the material properties accordingly. The new loop starts,

as the momentum equations are solved for the updated set of the material

parameters.

10


(c) G&R

COLLAGEN• growth• remodeling

TISSUE• vol. growth

ELASTIN• degradation

τ ,[yrs]

(a) SOLID

FEAP

t,[s]

(b) FLUID

ANSYS CFX

t,[s]

fluid iter?

geometry

false

true

WSS

strainor

stress

updatedmaterial

Figure 4: Flowchart of the Fluid–Solid–Growth (FSG) framework. Momen-tum equations are solved in the solid (a), and the strains are sent to thegrowth and remodeling (G&R) description (c). Every 0.2 years, the Navier-Stokes equations for the blood flow (b) are solved and the updated wall shearstress (WSS) is sent to (c). G&R model (c) adapts the material propertiesto the strain field and returns it to the solid model (a). Then, the solutionstarts over by solving momentum equations for the solid (a).

Key results

Bilayer aneurysm model

Influence of considering a second arterial layer, i.e., the adventitia, in the

aneurysm model has been investigated in Paper A. The relative stiffnesses

of the layers were set according to the literature. The specific case, where

the media is up to 10 times stiffer than the adventitia (MECH 1), was taken

as a reference. The material parameters for other three mechanical cases

(MECH 2 to 4) were fitted, such that the pressure-radius curve in homeostasis

closely resembled the MECH 1 case. Specifically, the two layers had the same

material parameters for the MECH 2 case, and the adventitia was up to 10

times stiffer than the media for MECH 3. Finally, for MECH 4, the media

was stiffer at lower strains and the adventitia “kicked in” at larger strains.

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Andrii Grytsan

(a) (b)

Figure 5: Effects of different sets of material parameters on aneurysm evolution:MECH 1 (blue), MECH 2 (green), MECH 3 (red), MECH 4 (black). (a) Radiuschange over time, normalized by the radius in homeostasis. (b) Transmural plot ofcircumferential first Piola-Kirchhoff stress Pθθ at the twofold aneurysm expansion,for internal pressure of 120 mmHg (solid curves) and 400 mmHg (dashed curves).

Figure 5 illustrates the effects of the different investigated cases. Specifi-

cally, the aneurysm expansion over time is shown in Figure 5(a), illustrating

that the MECH 1 and MECH 2 cases expand at a similar rate, while MECH 3

expands slower. Finally, MECH 4 is the fastest growing case.

Figure 5(b) shows the transmural stress plots at the time, when each

aneurysm has reached twice the original radius. Stresses are provided at

systolic blood pressure of 120 mmHg, as well as at hypertension of 400 mmHg.

It can be seen that adventitia has the highest stresses at systole in MECH 3

case. However, for MECH 4 and the hypertensive case, the adventitia “kicks

in”. This case illustrates the stress-protective function of the adventitia.

Interestingly, the adventitial stresses at hypertension in adventitia are not

significantly increased for MECH 1 and MECH 2 cases.

Fluid-solid-growth framework

The patient-specific geometry of the aneurysmatic abdominal aorta has been

obtained from computed tomography (CT) scans. Retrospectively, the aneurys-

matic region has been replaced by a conceptual model of a (hypothetical)

normal aorta. In addition, a small initial bulging was provoked by a local

12


elastin degradation in order to create a disturbance in blood flow, and hence,

in WSS. Since the proposed FSG framework links elastin degradation to low

levels of WSS, the evolution of the aneurysm is triggered, as shown in Fig-

ure 6. Specifically, the aneurysm propagates upstream and its evolution is

asymmetric.

Figure 6: The distribution of the wall shear stress (WSS) magnitude |τ | inthe aneurysm domain and patient-specific up and downstream extensions isshown: at time 0 yrs (left), after 5 yrs (center), after 10 yrs (right).

Patient-specific biomechanics

In Paper C, various geometrical and biomechanical parameters and their

evolution with time was extracted from the patient follow-up data. Four

patients were considered, with at least five CT scans at different points in

time. Geometrical parameters included maximum AAA diameter, AAA vol-

ume, ILT volume, and their changes with time. Biomechanical parameters

included stresses in the wall and the hemodynamic blood flow parameters

(velocity, shear rate, WSS). Figure 7 shows the patient-specific evolution of

the WSS for all patients. In Patients A and C, the ILT grows in regions

with low WSS, the lumen shrinks with time, and WSS values increase. Pa-

tients B and D remain relatively stable. In addition, several non-intuitive

13

Andrii Grytsan

correlations between parameters have been identified. These correlations

suggest, that the large ILT volume and its rapid increase with time may be

risk factors for AAA rupture.

Figure 7: Development over time of the wall shear stress (WSS) at t=0.25 sof the cardiac cycle, in four abdominal aortic aneurysm (AAA) patients.

14


Volume growth

Different volume growth models were tested in Paper D. In particular,

isotropic (IVG), in-plane (PVG) and in-thickness (TVG) growth models were

compared to predictions that suppress volume growth (NVG). TVG model

was found the most suitable for AAA modeling, i.e., TVG predictions were

the most plausible. Specifically, this model predicted slower AAA growth

in response to increased collagen net growth. On the other hand, IVG and

PVG showed unrealistic predictions, as the AAA grew faster in response to

increased collagen net growth.

Figure 8 compares the AAA wall thickness and transmural distribution of

Cauchy stresses between the growth models. Specifically, modeling volume

growth improved wall thickness and wall stress predictions, compared to the

predictions that suppress volume growth.

(a) (b)

Figure 8: Effect of the isotropic (IVG), in-plane (PVG), in-thickness (TVG)growth models and no volume growth model (NVG) on the properties of theAAAs at twofold expansion. (a) thickness change relative to the thickness ofaorta in homeostasis along AAA length. (b) Cauchy hoop stress σϕϕ across thewall. Dotted curves represent homeostatic values of the parameters.

15

Andrii Grytsan

Conclusion

The aims of the thesis were accomplished by developing a thick-walled model

of AAA G&R and integrating it into a stable FSG framework. Predictions

of the developed models were tested by varying different model parameters.

Specifically, the prescribed growth kinematics have severe influence on model

predictions. This aspect has not yet been thoroughly investigated. The

developed FSG model is a powerful tool for hypothesis testing and guiding

the experimental design, towards better understanding of the AAA evolution.

In addition, the evolution of the patient-specific AAA has been studied.

This exercise allowed to explore the evolution of the biomechanical param-

eters in real cases and provided valuable information for future modeling,

parameter calibration and model validation.

The developed FSG framework is subject to many limitations and can be

improved upon in the future. Modeling the ILT growth would probably be

the most crucial improvement of the existing framework. In addition, exper-

imental validation of the various parts of the model, such as collagen G&R,

elastin degradation, and volume growth should be performed. Finally, as for

any model with many parameters, parameter identification is challenging.

16


Summary of appended papers

Paper A: Influence of differing material properties in media and adventi-

tia on arterial adaptation – application to aneurysm formation and rupture.

Experimental and computational studies suggest a substantial variation in

the mechanical responses and collagen fiber orientations of the two struc-

turally important layers of the arterial wall. Some observe the adventitia

to be an order of magnitude stiffer than the media whilst others claim the

opposite. Furthermore, studies show that molecular metabolisms may dif-

fer substantially in each layer. Following a literature review that juxtaposes

the differing layer-specific results we created a range of hypothetical arteries

with different: (1) elastic responses, (2) fiber orientations, and (3) metabolic

activities during adaptation. We used a finite element model to investigate

the effects of those varying properties on: (1) the stress response in home-

ostasis; (2) the time course of arterial adaptation; and (3) an acute increase

in luminal pressure due to a stressful event and its influence on the likelihood

of aneurysm rupture. Interestingly, for all hypothetical cases considered, we

observed that the adventitia acted to protect the wall against rupture by

keeping stresses in the media and adventitia below experimentally observed

ultimate strength values. More significantly, this conclusion held true in

pathological conditions.

Paper B: A thick-walled fluid–solid–growth model of abdominal aortic

aneurysm evolution: application to a patient-specific geometry. We propose

a novel thick-walled fluid–solid–growth (FSG) computational framework for

modeling vascular disease evolution. The arterial wall was modeled as a

thick-walled non-linearly elastic bilayer cylindrical tube. The media-intima

and the adventitia were treated as a fiber-reinforced material with the fibers

corresponding to the collagenous component. Blood was modeled as a New-

tonian fluid with constant density and viscosity; no slip and no-flux condi-

tions were applied at the arterial wall. Disease progression was simulated

by growth and remodeling (G&R) of the load bearing constituents of the

wall. Adaptations of the natural reference configurations and mass densities

of constituents were driven by deviations of mechanical stimuli from home-

17

Andrii Grytsan

ostatic levels. We applied the novel framework to model abdominal aortic

aneurysm (AAA) evolution. Elastin degradation was initially prescribed to

create a perturbation to the geometry which resulted in a local decrease in

the wall shear stress (WSS). Subsequent degradation of elastin was driven by

low WSS and an aneurysm evolved as the elastin degraded and the collagen

adapted. The influence of transmural G&R of constituents on the aneurysm

development was analyzed. Both elastin and collagen strains evolved to be

transmurally heterogeneous and this may facilitate the development of tor-

tuosity. This multiphysics framework provides the basis for exploring the

influence of transmural metabolic activity on the progression of vascular dis-

ease.

Paper C: Biomechanical changes during abdominal aortic aneurysm growth.

The biomechanics-based Abdominal Aortic Aneurysm (AAA) rupture risk

assessment has gained considerable scientific and clinical momentum. How-

ever, such studies have mainly focused on information at a single time point,

and little is known about how AAA properties change over time. Conse-

quently, the present study explored how geometry-, wall stress- and blood

flow-related biomechanical properties change during AAA expansion. Four

patients with a total of 23 Computed Tomography-Angiography (CT-A)

scans at different points in time were analyzed. At each point in time,

patient-specific properties were extracted from (i) the reconstructed geome-

try, (ii) the computed wall stress at Mean Arterial Pressure (MAP), and (iii)

the computed blood flow velocity at standardized in and out flow conditions.

Testing correlations between these parameters revealed several non-intuitive

dependencies. Most interestingly, the Peak Wall Rupture Index (PWRI)

and the maximum Wall Shear Stress (WSS) independently predicted AAA

volume growth. Similarly, Intra- luminal Thrombus (ILT) volume growth

depended on both the maximum WSS and the ILT volume itself. In ad-

dition, ILT volume, ILT volume growth and maximum ILT layer thickness

correlated with PWRI as well as AAA volume growth. Consequently, a large

ILT volume as well as fast increase of ILT volume over time may be a risk

factor for AAA rupture. However, tailored clinical studies would be required

to test this hypothesis and to clarify whether monitoring ILT development

18


has any clinical benefit.

Paper D: Growth description for vessel wall adaptation: a thick-walled

mixture model of abdominal aortic aneurysm evolution. Modeling the soft

tissue volumetric growth has received considerable attention in the litera-

ture. However, due to the lack of experimental observations, the growth

kinematics are based on a number of assumptions. Present study tested the

plausibility of different growth descriptions when applied to the abdominal

aortic aneurysm (AAA) evolution model.

A structurally motivated material model and the multi-constituent tissue

growth descriptions were utilized. The mass increment of the individual con-

stituents preserved either the density or the volume of the constituent. Four

growth descriptions were tested, namely isotropic (IVG), in-plane (PVG),

in-thickness (TVG) growth and no volume growth (NVG) models.

Based on the model sensitivity to the increased collagen deposition, TVG

and NVG models were found to be more plausible scenarios, while the pre-

dictions of IVG and PVG models were found to be implausible. In addition,

TVG and NVG models were less sensitive to the initial volume fractions of

the constituents, than IVG and PVG models. In conclusion, the choice of

the growth kinematics is of crucial importance when modeling the soft tissue

growth and remodeling.

19

Andrii Grytsan

20

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