Failure of vascular tissue with applications tothe aneurysm wall, carotid plaque and

myocardial tissue

Caroline Forsell

Doctoral thesis no. 83, 2013KTH School of Engineering Sciences

Department of Solid MechanicsRoyal Institute of TechnologySE-100 44 Stockholm Sweden

TRITA HFL-0545

ISSN 1104-6813

ISRN KTH/HFL/R–13/13–SE

ISBN 978-91-7501-798-3

Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan i Stockholmframlagges till offentlig granskning for avlaggande av teknisk doktorsexamen fredag den 7Juni kl. 9.00 i D2, Kungliga Tekniska Hogskolan, Lindstedsvagen 5, Stockholm.

Somewhere, something incredible is waiting to be known.

Carl Sagan

i

Abstract

Cardiovascular disease is the leading cause of death in the modern world. Examples are tho-

racic aortic aneurysm (TAA), abdominal aortic aneurysm (AAA) and stroke due to plaque

rupture. Failure in soft tissues caused by medical devices is also a medical challenge. In

all these cardiovascular events a better prediction of failure of the tissue and a better under-

standing about the tissue properties will help in predicament and treatment. For example the

diameter-based indication for surgical repair of AAA and TAAs is not sufficient and refined

methods are needed. In this thesis failures of some soft vascular tissues, was studied. Ex-

periments have been combined with numerical modeling to understand the elastic and failure

properties of AAA, TAA and plaque tissue as well as the ventricular wall. Vascular tissue

is anisotropic, time-dependent, nonlinear and shows large deformations. Among others this

thesis showed the importance of viscoelasticity which motivates to develop a new continuum

mechanical framework. In addition a large part of this thesis dealt with anisotropy of vascu-

lar tissue. For the first time the collagen orientation distribution in the AAA wall has been

identified. Collagen and its distribution orientation is also an important feature of this tissue.

There was a correlation between the strength and stiffness of the AAA samples with the de-

creasing wall thickness. Increased stiffness was found in the aortic wall of patients with chronic

obstructive pulmonary disease (COPD) compared to patients that did not have COPD. As

well as difference in stiffness of TAA tissue, in patients with non-pathologic and pathologic

aortic valves. Some of the findings in this thesis could have a long-term consequence for

management of risk of rupture in AAA, TAA and plaque.

ii

Sammanfattning

Hjart- och karlsjukdomar ar den vanligaste dodsorsaken i det moderna samhallet. Ett par

exempel pa orsaker ar toracala aneurysm (TAA), buk-aneurysm (AAA) eller stroke pa grund

av plackruptur. Skador orsakade av medicintekniska produkter ar ocksa en medicinsk utman-

ing. Vid alla dessa kardiovaskulara sjukdomstillstand ar en battre forstaelse av skador pa

vavnaden och dess matrial egenskaper viktig for att hjalpa till vid diagnos och behandling.

Till exempel ar den numera diametern-baserade indikationen som anvans vid kirurgisks in-

grepp vid AAA och TAA inte tillracklig och battre metoder behovs. I denna avhandling

har skador pa nagra mjuka vavnader studerats. Experiment har kombinerats med numerisk

modellering for att forsta den elastiska och inelastiska responsen av AAA, TAA, plackvavnad

samt den ventrikulara hjartvaggen. Vaskular vavnad ar anisotrop, tidsberoende, icke-linjar

och karikteriseras av stora deformationer. I denna avhandling pavisades markant inverkan

av viskoelasticitet, som motiverade att en ny kontinuum mekanisk modell utvecklades. Aven

vikten av anisotropi undersoktes. For forsta gangen identifierades fordelningen av kollagenfi-

brer i AAA-vaggen. Vi fann att kollagenfibrerna och dess orientering ar en viktig egenskap

i denna vavnad. Det fanns en korrelation mellan styrkan och styvheten hos AAA-vavnaden,

med avseende pa minskad vaggtjocklek. Dessutom fann vi okad styvhet i AAA vaggen hos

patienter med kronisk obstruktiv lungsjukdom (KOL) jamfort med patienter som inte hade

KOL. Forutom detta fann vi en skillnaden i styvhet i TAA-vavnad, hos patienter med icke-

patologiska och patologiska aortaklaffar. Vissa resultat som pavisats i denna avhandling kan

fa en langsiktig relevans for hanteringen av risken for ruptur av AAA, TAA och plack.

iii

Preface

The work presented in this thesis has been carried out at the department of Solid Mechan-

ics, Royal Institute of Technology (KTH), Stockholm between 2008 and 2013. The work has

been financially supported by the Project Grants No. 2007− 4514 and 2010− 4446 from the

Swedish Research Council. The author is grateful for their financial support. First of all I

would like to express my sincere gratitude to my supervisor T. Christian Gasser for giving

me the opportunity to be a part of this project. I am thankful for all his support and encour-

agement and guidance.

During this project I have also have the opportunity to discuss and collaborate with sev-

eral inspiring people. Special thanks too Vincent M. Heiland, Xiao Xing, Joy Roy, Jesper

Swedenborg, Ulf Hedin, Sara Gallinetti, Per Eriksson, Anders Franco-Cereceda, Stanizlav

Polzer, Martin Oberg, Per Berg, Kurt Lindquist, Therese Olsson, Karin Lundstrommer and

Giampaolo Martufi.

I would also like to thank my colleagues and friends at KTH solid Mechanics for making this

a great place to work. In particular I would like to thank my room-mates Irene and Mats for

many interesting discussions. Last but not least I would like to thank my family, friends and

boyfriend Chris for their encouragement and love. Especially thanks to my mom Yvonne for

inspiring me to start with research.

Stockholm, May 2013

iv

List of appended papers

Paper A: Numerical simulation of the failure of ventricular tissue due to deep penetration:The impact of constitutive properties.C. Forsell and T.C. Gasser,Journal of Biomechanics 44 (2011) 45–51.

Paper B: The numerical implementation of invariant-based viscoelastic formulations at finitestrains. An anisotropic model for the passive myocardium.T.C Gasser and C. Forsell,Computer Methods in Applied Mechanics and Engineering 200 (2011) 3637–3645.

Paper C: Spatial orientation of collagen fibers in the Abdominal Aortic Aneurysm’s walland its relation to wall mechanics.T.C. Gasser, S. Gallinetti, X. Xing, C. Forsell, J. Swedenborg, and J. Roy,Acta Biomaterialia Issue 8 (2012) 3091–3103.

Paper D: The quasi-static failure properties of the Abdominal Aortic Aneurysm wall esti-mated by a mixed experimental-numerical approach.C. Forsell, J. Swedenborg, J. Roy and T.C Gasser,Annals of Biomedical Engineering, published online (2012).

Paper E: Identification of carotid plaque tissue properties using an experimental-numericalapproach.V.M. Heiland, C. Forsell, J. Roy, U. Hedin and T.C. Gasser,Journal of the Mechanical Behavior of Biomedical Materials, accepted for publication (2013)

Paper F: Failure properties for the thoracic aneurysm wall; Differences between BicuspidAortic Valve (BAV) and Tricuspid Aortic Valve (TAV) patients.C. Forsell, P. Eriksson, A. Franco-Cereceda and T.C. Gasser,Report 544, Department of Solid Mechanics, Royal Institute of Technology, Stockholm,To be submitted

v

In addition to the appended papers, the work has resulted in the following papers:

Automatic identification and validation of planar collagen organization in vascularwall from histological stains with an application to abdominal aortic aneurysmwall.S. Polzer, T. C. Gasser, C. Forsell, H. Druckmllerova, M. Tichy, R. Staffa, R. Vlachovsky,and J. Bursa,Submitted to Microscopy and Microanalysis.

In addition to the appended papers, the work has resulted in the following conference contri-butions:

Numerical Simulation of Deep Penetration of Ventricular Tissue.C. Forsell, T.C. Gasser, P. Gudmundson and G. Dohr,WCCM8 and ECCOMAS,Venice, Italy, June 30 - July 5, 2008.

Modeling of vascular failure with application to myocardial failure due to deeppenetration.C. Forsell and T.C. Gasser,10th US National Congress on Computational Mechanics, Columbus, Ohio, US, July 16-19,2009.

Modeling of myocardial splitting due to deep penetration.C. Forsell and T.C. Gasser,ECCMR 2009 - 6th European Conference on Constitutive Models for Rubber, Dresden, Ger-many, 7 - 10 September 2009.

Experimental and modeling of myocardial splitting.C. Forsell and T.C. Gasser,ECCM 2010 - IV European Conference on Computational Mechanics, Paris, France, May16-21, 2010.

The impact of constitutive properties on myocardial tissue perforation.C. Forsell and T.C. Gasser,WCB 2010 - 6th World Congress on Biomechanics, Singapore, Singapore, August 1-6, 2010.

Myocardial tissue deformation due to pacemaker lead contact-The impact of ma-terial anisotropy.C. Forsell and T.C. Gasser,CMBE2011-2nd International Conference on Mathematical Computational BiomechanicalEngineering, Washington D.C, USA, March 30-April 1, 2011.

Impact of material anisotropy on deformation of myocardial tissue due to pace-maker electrodes.C. Forsell and T.C. Gasser,SBC2011-Summer Bioengineering Conference, Farmington, Pennsylvania, USA, June 22-25,2011.

vi

A mixed experimental-numerical approach to identify the failure of the Abdom-inal Aortic Aneurysm wall.C. Forsell and T.C. Gasser,Euromech colloquium 534 Advanced experimental approaches and inverse problems in tissuebiomechanics, Saint- tienne, France May 29-31, 2012.

Mixed experimental numerical approach to identify properties of vascular tissue.C. Forsell and T.C. Gasser,WCCM 2012 - 10th Word Congress on Computational Mechanics, Sao Paulo, Brazil, July8-13, 2012.

vii

viii

Contents

Abstract ii

Sammanfattning iii

Preface iv

List of appended papers v

Introduction 1

Some cardiovascular challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Risk assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Aneurysm rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Plaque rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Pacemaker leads perforation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Histology of vascular tissue and pathophysiology . . . . . . . . . . . . . . . . . . . 5

Aorta and collagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Plaque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Ventricular wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Principles of vascular testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Constitutive models for vascular tissue . . . . . . . . . . . . . . . . . . . . . . . . . 10

Time-dependent properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Failure through penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Prediction of failure and damage models . . . . . . . . . . . . . . . . . . . . . . . . 11

Aim 13

Method 15

Experimental study design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

ix

Key Results 17

Discussion/Conclusion 21

Summary of appended papers 23

Bibliography 25

x

Introduction

Some cardiovascular challenges

Cardiovascular disease is the leading cause of death responsible for 17.3 million deaths world-

wide in 2008, of those around 7.3 million from coronary heart disease and 6.2 million died from

stroke [Mendis et al., 2011]. Specially, thoracic aortic aneurysm (TAA) and abdominal aortic

aneurysm (AAA) are diseases that require clinical attention. An aneurysm is a dilatation of

an artery and, due to the risk of rupture, it can become a life-threatening condition. Aortic

aneurysm rupture is the 16th leading cause of death for persons aged over 55 years and

ranked 19th for all age groups in US [CDC, 2010]. Another severe cardiovascular condition is

plaque rupture, especially if the rupture occurs in one of the major arteries [Richardson, 2002].

Carotid plaques (stenosis degree larger than 50%) are also, present in 29.2% of persons effected

by stroke [Kuroda et al., 2000].

Plaque consist of different components, where the proportion of these components influence

the properties of plaque, see the schematic picture Figure 1. Many plaques consist of a lipid-

rich necrotic core that is surrounded by an extracellular matrix (ECM) and is separated from

the blood stream by a fibrous cap. Plaque parts can travel in the arteries to the heart or the

brain leading to a stroke or a heart attack. Failure of soft tissues caused by medical devices

is another medical challenge. Specifically lead perforation is a rare but serious complication

of pulse generator implantation.

A pulse generator (defibrillator or pacemaker) is a device that is used to regulate cardiac

arrhythmias [Medical, 2006]. Figure 2 shows how the lead is placed in the heart. Prevalence

for lead perforation is about, 4.8% [Matsuura et al., 1990] for pacemaker leads and 5.2%

for [Molina, 1996] implantable defibrillator leads. Depending on the time from the operation

to the event, it can be acute, sub-acute, or delayed [Alla et al., 2010].

1

Extracellular Matrix

Fibrous capLipid core

Lumen

Artery

Figure 1: A cross section of an idealized plaque, illustrating the different (pseudo) tissues.

Figure 2: The heart consists of four chambers, and the figure illustrates how the pacemaker leads are placedin the heart [St.Jude, 2008].

2

Table 1: Abbreviations and expressions used in this thesis

AA Abdominal AortaAAA Abdominal Aortic Aneurysm

Anisotropy/ Orthotropic Property that depends on the direction in alldirections/some orthogonal directions

BAV Bicuspid Aortic ValveCEA Carotid Endarterectomy

Constitutive model Relation between stress and strainCOPD Chronic Obstructive Pulmonary DiseaseECM Extracellular Matrix

Hyperelastic The stress-strain relationship can be described witha strain energy function

Incompressible No volume change during deformationNon-linear material A material with non-linear stress-strain relationship

Stiffness The resistance offered by an elastic body to deformationStrain energy function The potential energy stored in a body due to a

deformation. It is equal to the work that must be performedto produce this deformation.

Strength Materials ability to withstand stress without failureTA Thoracic Aorta

TAA Thoracic Aortic AneurysmTAV Tricuspid aortic valve

Toughness Material ability to absorb energy and plastically deform withoutfracturing

UTS Ultimate Tensile StressWSS Wall Shear Stress

3

Risk assessment

Risk assessment is a critical step in the clinical decision making process, in determining the

patients treatment. Typically a clinical intervention is indicated if the risk from the clinical

condition exceeds those from the treatment.

Aneurysm rupture

Risk of aortic aneurysm rupture is related to its maximum diameter. Different diameter

thresholds are used depending where along the aorta the aneurysm is located, see Fig-

ure 3, for definitions of aortic segments. TAA repair is indicated if an aneurysm exceeds

5.5 cm for the ascending aorta, and 6.5 cm for the descending [Elefteriades, 2002]. For

AAA a large clinical study suggested that repair is indicated for a diameter that exceeds

5.5 cm [The UK Small Aneurysm Trial Participants, 1998]. Rupture risk is also related to the

growth rate, which depends on the diameter and other factors. Small aneurysms (less than 5.5

cm) growth in average by 4.4%, whereas the maximum growth reads 16% [Martufi et al., 2013].

Plaque rupture

Carotid endarterectomy (CEA) is a surgical procedure to prevent strokes. This procedure is

indicated for patients with carotid territory transient, due to rupture of vulnerable plaque,

ischemic attacks or a minor stroke with stenosis of 70 − 99%. For patients with 50 − 69%

stenosis a modest favoring for surgery has been found. For patents with less than 50% stenosis

anti-aggregants are recommended. [Biller and Thies, 2000]. The different compositions of

plaque affect the risk of rupture. For example a thin fibrous cap has been more commonly

observed in patients that died from acute myocardial infarction [Virmani et al., 2006].

Pacemaker leads perforation

Different factors have been reported to influence lead perforation [Carlson et al., 2008]. Con-

comitant therapies such as steroids and anticoagulants, implant techniques, and design char-

acteristics of the lead seem to have an influence [Carlson et al., 2008]. However currently it

is unclear why some leads perforate whereas others do not. To prevent lead perforation a

perforation-resistant lead design would be preferable, which in turn requires an understanding

of the mechanical failure mechanisms of the ventricular wall.

4

Figure 3: Aorta and its different parts [Isselbacher, 2005]

Histology of vascular tissue and pathophysiology

Aorta and collagen

Blood vessels consist of three layers, intima, media and adventitia. The intima contains en-

dothelial cells and its structural contribution in healthy subjects seems negligible. The media

is an active vascular layer that contains smooth muscle cells, collagen, elastin and ground

substance. From a mechanical perspective the adventitia mainly contains collagen fibers.

For the thoracic aorta collagen contents of 36.8% and 77.7% for media and adventitia, are

reported [Fung, 1981]. While the amount of collagen critically influences the mechanical be-

havior, the orientation of the collagen is also an important structural wall property. Previous

studies found that in large AAAs the collagen content increase by about 50%, while elastin

almost disappears when compared to the normal aorta [Rizzo et al., 2011]. Collagen is bire-

fringent, a property that can be used to measure the collagen orientation in the wall by

5

polarized light microscopy. The organization of collagen has been reported for intracranial

aneurysms [Canham et al., 1999] coronary arteries [Canham et al., 1989] and AAAs

[Gasser et al., 2012]. The AAA wall collagen fiber orientation was found to be orthotropic and

no difference between the medial and the adventitial layer was seen. A larger orientation vari-

ance has been found in the tangential plane than in the cross-sectional plane [Gasser et al., 2012].

In coronary arteries a main direction, of the fibers, have been found in the circumferential

direction [Canham et al., 1989]. Other methods like using small-angle light scattering have

also been suggested to study the collagen orientation in pericardial tissue [Sacks et al., 1997].

Unfortunately not many studies combine tensile testing and histology measurements. One

rare example is the study that used multi-photon microscope combined with tensile test-

ing [Hill et al., 2012].

Plaque

Plaque consist of different components, where the proportion of these components change in

different plaques [Kim et al., 2008]. Stiffness can vary significantly for the different (pseudo)

tissues. For fibrous cap tissue from human plaques a Young’s modulus of 500 kPa and for lipid

core of 5 kPa was reported [Ohayon et al., 2007]. In addition residual strain in the fibrous

cap seem to increase with increased size of the lipid core [Ohayon et al., 2007]. Another

approach reports cross-plaque properties [Loree et al., 1994] and divides it into hard, mixed

and soft plaques. Based on a Yeoh model [Yeoh, 1993], different material parameters have

been reported for these different groups of plaques [Lawlor et al., 2011]. Calcification can

occur in late stages of plaques, creating a more complex structure with altered mechanical

properties [Kim et al., 2008]. The effect of calcification on plaques instability is not yet clearly

understood [Virmani et al., 2006].

Ventricular wall

The ventricular wall consists of three layers; endocardium, myocardium and epicardium

[Bloom and Fawcett, 1970], from the inside to the outside. The thickest layer is the my-

ocardium that consist primary of fibroblasts and myocytes. The myocyte-fiber direction

changes across the thickness of the myocardial wall [Okamoto et al., 2000].

6

Principles of vascular testing

Several different test methods have been suggested to investigate mechanical properties of

vascular tissue in the literature. A brief summary of the tests methods is discussed below.

Common test methods are uniaxial tensile testing (Figure 4(b)) and uniaxial compression

testing (Figure 4(a)). To study shear deformation in biological soft tissue function, shear

testing can be performed (Figure 4(c)). To better mimic in-vitro loading, biaxial testing

(Figure 4(l)) is recommended. In-vivo vascular tissue is inflated such that inflation testing

(Figure 4(f), Figure 4(g)) is motivated. Splitting tests (Figure 4(h)) is also preformed where

a tear causing a cavity of fluid is formed in the tissue by injecting saline solution. In order

to study aortic dissection, tearing tests (Figure 4(d)) and pealing tests (Figure 4(e)) have

been used. In the literature how the tissue respond to penetration or bending has been

investigated by penetration testing (Figure 4(k)), probing tests (Figure 4(i)) and bending

tests (Figure 4(j)).

Experiments

Uniaxial tensile tests of aortic tissue have been performed in axial, circumferential

[Mohan and Melvin, 1982, Vorp et al., 1996, Vorp et al., 2003] and in radial

[MacLean et al., 1999, Sommer et al., 2008] directions. Results from these tests are repre-

sented for the human and porcine Thoracic Aorta (TA), TAA, Abdominal Aorta (AA) and

AAA. This data suggest a strength for aneurysm tissue (AAA and TAA) in axial direction

in the range of 0.65 − 1.21 MPa and circumferential direction 0.68 − 1.47 MPa. Compared

to normal aortic (TA and AA) with tissue strength of 1.47 − 1.71 MPa, 1.72 − 1.80 MPa,

0.061− 0.140 MPa in axial, circumferential and radial directions respectively. Stiffness have

been reported in a range of 3.51−4.48 MPa in axial direction and 4.76−5.81 MPa in circumfer-

ential direction, in aneurysm tissue (AAA and TAA). For non-aneurysmal tissue (TA and AA)

in the range of 2.61 − 5.69 MPa for longitudinal and circumferential direction have been re-

ported. Planar biaxial tensile testing has been used to determine parameters for a constitutive

relation of ventricular wall [Humphrey et al., 1990b, Humphrey et al., 1990a]. Shear testing

of the passive left ventricular myocardium suggest that the response to simple shear defor-

mation is dominated by the tensile material properties [Dokos et al., 2002]. Splitting tests of

the porcine tissue [Carson and Roach, 1990, Roach and Song, 1994, Roach et al., 1999] and

human tissue [Tiessen and Roach, 1993] is performed in thoracic and abdominal aorta respec-

tively. The splitting energy was found to be in a range of 1.88−15.9 mJ/cm2. Peeling testing

in AA [Sommer et al., 2008] found a dissection energy of 5.1 mJ/cm2. Tearing tests showed

that the fracture toughness of the aorta increases the further away the tissue was from the

heart [Purslow, 1983]. The mean breaking stress found in porcine TA was 2.19 − 3.64 MPa,

0.18 − 0.87 MPa in circumferential and axial directions, respectively. Finally inflation tests

7

done in TA [Mohan and Melvin, 1983, Groenink et al., 1999] showing a ultimate stress of

0.114− 2.7 MPa can be found in literature.

Uniaxial tensile testing of plaque tissue in circumferential direction

[Lawlor et al., 2011, Maher et al., 2009] and compression test in radial direction by

[Maher et al., 2009] can be found in the literature. The results of the Ultimate Tensile

Stress (UTS) varied between 0.131 − 0.779 MPa at a UTS strain of 0.299 − 0.588. Uni-

axial tensile testing has also been used to investigate viscous properties in human plaque

by relaxation testing [Salunke et al., 2001]. The small dimensions and irregularities in the

shape of plaque samples motivates probing at different sites to investigate compression re-

sponse [Tracqui et al., 2011]. Penetration tests have been performed in myocardial tissue

[Gasser et al., 2009]. The penetration pressure (penetration force/punch cross section area)

decrease slightly with larger punch diameter and has been found to be 1.76 MPa and 2.27 MPa

for diameters of 1.32 mm and 2.30 mm.

8

(a) (b) (c) (d)

(e)

P

(f)

P

(g)

P

(h) (i) (j)

(k) (l)

Figure 4: Testing principles to explore mechanical properties of vascular tissue: a)Uniaxial compression b)Uniaxial tensile c) Shear d) tearing e) Peeling f) Inflation g) Inflation h) splitting i) Penetration j)Bending k) Probing l) Biaxial

9

Constitutive models for vascular tissue

The literature on constitutive modeling of vascular tissue is rich [Fung, 1981]. Soft biological

tissue is non-linear elastic, anisotropic and incompressible material, which often is described

by hyperelastic theory. The incompressibility under many loading conditions is a consequence

of a large amount of non-mobile fluid, while macroscopic anisotropy is a consequence of the

organized structure of vascular tissue. Non-linearity arises from the engagement of fibrous

compounds under loading.

Originally, isotropic non-linear constitutive models have been developed for rubber mate-

rials. The simplest model is the NeoHookean [Mooney, 1940]. Others are the Mooney -

Rivling [Rivlin, 1948], Ogden, [Ogden, 1972], Yeoh [Yeoh, 1993] and others found in litera-

ture [Demiray, 1972]. However these models are not describing the anisotropy of the tissue

and hence have limited applicability in vascular mechanics. Anisotropic formulation for vas-

cular tissue are found as phenomenological [Humphrey, 1995] and histo-mechanical

[Gasser et al., 2006, Lanir, 1983, Holzapfel et al., 2000, Martufi and Gasser, 2011] approaches.

Time-dependent properties

Predicting the failure of soft biological tissue requires inelastic constitutive descriptions, where

specifically viscoelastic dissipative effects are important. Commonly used models to describe

viscoelasticity are the Voigt, Maxwell, and the standard linear solid models. For vascular

tissue there is a difference in stress for loading and unloading. It is however reasonable to

assume linear theory for oscillations of small amplitude about an equilibrium state. Nonlinear

generalizations of the standard linear solid model can be found in literature [Viidik, 1968].

A nonlinear differential equation can be used to describe the damper in the Maxwell model

or a quasi-linear solid can be considered where the non-linear viscoelasticity is modeled by

a series of nonlinear elements i.e. Maxwell-elements in parallel. This to capture a large

range of frequencies [Holzapfel et al., 2002, Gasser and Forsell, 2011, Puso and Weiss, 1998,

Idesman et al., 2001, Wu et al., 2006].

Failure through penetration

The application of an isotropic material assumption to tissue penetration found that a flat-

bottom punch leads to a mode-II ring while using the sharp tip formed a planar mode I

crack [Shergold and Fleck, 2004]. In contrast in-vitro experiments on heart tissue, which are

anisotropic, showed that even a flat bottom punch as well as a sharp tip punch created a

mode I failure [Gasser et al., 2009].

10

Prediction of failure and damage models

Failure in vascular tissues is defined by setting up a failure criteria [Volokh, 2011]. Examples

of failure criteria are maximum Von Mises stress, the maximum principal stress, the maximum

strain energy, etc... In the literature damage parameter are commonly used

[Hokanson and Yazdani, 1997, Rodrıguez et al., 2006, Balzani et al., 2006]. It can be imple-

mented on a mechanical level, a finite element level, or a continuum level.

11

12

Aim

This work aimed to advance our understanding in failure of vascular tissues, specifically the

AAA, TAA, the heart wall, and carotid plaques were studied. Constitutive material pa-

rameters were identified using numerical methods in combination with tailored experimental

testing.

Specific aims were:

• To estimate elastic and inelastic properties of the AAA wall.

• To investigate the difference in structural and mechanical properties of the TAA wall from

patients with normal and pathological aortic valves.

• To measure the three-dimensional structure of collagen in AAA and integrate it in a histo-

mechanical constitutive model.

• To identify tissue properties from carotid plaque, using bending testing.

• To study numerically pacemaker lead perforation of myocardium.

• To implement an improved viscoelastic formulation for a class of invariant-based constitu-

tive formulations.

13

14

Method

Experimental study design

Experimental design and results are detailed in the appended papers. In summary, uniaxial

failure tests have been performed with AAA and TAA tissue. The three-dimensional struc-

tural arrangement of collagen in the AAA and TAA wall, was measured from histological

slices using polarized light microscopy.

Dynamic bending testing with plaque samples was used to investigate the material properties

of fibrous tissue components. All tests where displacement controlled and performed in 37 ◦C

salt solution. The test specimens where directly collected from surgery at Karolinska Hospital

and testing was and preformed within one day.

Numerical modeling

The FE packages FEAP (University of California at Berkley) and ABAQUS (Dassault Sys-

tems) were used for modeling. The aneurysm wall was represented by the elasto-plastic

damage model, which allowed models to include the measured collagen orientation distribu-

tion. For the plaque tissue the thrombus and the lipid region were described by a NeoHooken

model with parameters from literature. For the fibrous plaque tissue, the isotropic version

of the GOH-model [Gasser et al., 2006] was used. Viscoelastic effects were captured by five

Maxwell devises. Finally, to study the stress field from contact with the pacemaker lead a

visco-elastic model for myocardial tissue was implemented in ABAQUS.

15

16

Key Results

One key finding was that rate-dependent effects of the bulk material strongly determine the

failure process, whereas dissipative effects directly related to failure zone did not have signif-

icant impact on the simulation results. Compare Figure 5a and Figure 5b, which illustrate

the limited influence of the fracture energy when contributed to changing viscosity-related

parameters.

−1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

Displacement [mm]

Fo

rce

[N

]

Fracture Set I

Fracture Set II

−1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Displacement [mm]

Fo

rce

[N

]

Prony Set I

Prony Set II

−1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Displacement [mm]

Fo

rce

[N

]

Elastic Set I

Elastic Set II

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

Relative Thickness

Fo

rce

[N

]

Simulation

Experiment Interventricular Septum

Elastic Set IProny Set I

Elastic Set IFracture Set II

(b)(a)

(c) (d)

Prony Set IIFracture Set II

Figure 5: Penetration force displacement response of myocardium against deep penetration. Impact of thefracture energy (a), the visco-elasticity (b) and the elastic properties (c) on the simulated results.(d) Data from experimental bovine myocardium penetration tests. [Forsell and Gasser, 2011]

17

Figure 6 illustrates the elastic response of a punch pushing down on a disk of porcine

myocardial tissue. The difference in the punch force, for isotropic and anisotropic formula-

tions, can clearly be seen. These results underline the importance of anisotropic constitutive

modeling in biomechanics. The softer cross fiber direction accumulates more strain then the

fiber direction. This deformation mechanism is not possible an the isotropic material, which

might be a reason for this observation.

1 1.05 1.1 1.15 1.2 1.250

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Stretch

Pun

ch F

orce

[N]

Anisotropic modelIsotropic model

Figure 6: The elastic response of a penetrator, pushing down on a disk of porcine myocardial tissue. Punchforce displacement response using the anisotropic (solid line) and isotropic (dashed line) constitutivedescriptions for myocardial tissue. [Gasser and Forsell, 2011]

The collagen dispersion was much larger in the tangential plane than in the cross-sectional

plane for the AAA wall. The adventitial and medial layers where found to have no significant

different in collagen orientation (see Figure 7) for polarized light microscopy images. This is

to the author´s best knowledge the first time that a constitutive model based on structural

properties predict the mechanical properties of AAA wall. So far only theoretical assumptions

are represented in the literature without experimental validation.

Another key result was that the strength and stiffness of the AAA wall increased with de-

creasing wall thickness, but no correlation between strength and aneurysm diameter was seen.

We also found that the AAA wall stiffness and strength increased in patients with chronic

obstructive pulmonary disease (COPD) compared to patients that did not have COPD. The

difference in strength for the two patient groups was statistical significant (p = 0.0290) and

can be seen in Figure 8. To our best knowledge this influence on strength on COPD has

never been reported. The collagen fibers of the COPD patients are thicker and that might be

one of the explanations for the difference among these patients. Alternatively an increase in

collagen content in the AAA tissue, for the patients with COPD, could explain this difference

in strength, in the AAA wall.

18

Figure 7: Polarized Light Microscopy (PLM) images taken from the media (left) and the adventitia (right) ofthree Abdominal Aortic Aneurysm (AAA) wall samples (a)-(c). The horizontal sides of the imagesdenote the circumferential direction. Picrosirius red was used as a birefringent enhancement stainand the images were taken at crossed polar on the microscope. Quantitative collagen orientationdata of the presented samples is given in Figure 4. (a) Typically observed collagen organizationsin the AAA wall, where the media and adventitia shows a very mixed bag of azimuthal alignment.(b) Rather circumferential alignment of collagen fibers that was seen in a few AAA wall samples.(c) Collagen organization that remembers on a normal aorta and seen in two AAA wall samples,i.e. where the adventitia shows two almost perpendicular families of collagen. [Gasser et al., 2012]

19

0.2

0.4

0.6

0.8

1.0

1.2

without CPOD Hn=10L with CPOD Hn=5L

Wa

llstr

en

gth

MP

aFigure 8: Strength of Abdominal Aortic Aneurysm (AAA) wall specimens, that were taken from patients

with and without COPD. The number of specimens is denoted by n, and the difference betweenboth groups is statistical significant (p=0.0290). The thick solid lines are the medians. The barsand boxes denote the standard deviations and the 50% quartile, respectively. [Forsell et al., 2012]

Similarly a difference between the stiffness and strength of TAA tissue, for patients with

normal and pathologic aortic valve was found. In addition to these key findings material

behavior was identified for the different vascular tissues and novel in-vitro mechanical testing

protocols was developed, see the appended papers for details.

20

Discussion/Conclusion

Computer simulations can help with a detailed patient specific analysis and help to predict

and diagnose diseases. Today we are close to a bioengineering knowledge when computational

simulations could be of big help for the clinicians. Due to variability in input parameters such

as loading conditions and constitutive properties models need further development. To get a

better understanding this diversity a mixed numerical approach was used to study the elastic

an inelastic properties of AAA, TAA, heart wall and carotid plaque tissue. This allowed

uncovering the mechanical properties of some vascular tissues. Some of these functions can

have long-term consequences for patient management.

Other approaches for the work in this thesis would be to use a more clinical statistical anal-

ysis and to follow up patients. The author thinks it is important to use new aproaches for

this problem to be able to get more knowledge in this important field. When doing me-

chanical testing in soft tissues it is preferable to mimic the in-vitro conditions. This is a

big challenge and we usually do not know the original stress state. To get an in-vivo load-

ing condition of the anisotropic vascular wall, and to characterize its inherent pseudo-elastic

properties [Humphrey, 2002] planar biaxial testing would be needed. Since we were primar-

ily interested in failure this would require a large number of specimens too get a reasonable

range of biaxial loading states. Consequently we used uniaxial tensile tests in this thesis. Me-

chanical testing has a large variability, and due to the limited number of samples, there are

statistical uncertainties in the conclusions of this thesis. One of our constitutive assumptions

was that collagen fibers are the main load carrier in TAA wall and AAA wall. This is found

to be reasonable for larger AAAs [Rizzo et al., 2011].

21

22

Summary of appended papers

Paper A: Numerical simulation of the failure of ventricular tissue due to deep penetration:

The impact of constitutive properties In this paper the myocardial failure due to deep pene-

tration was investigating. The impact of different constitutive parameters on the simulated

results was controlled. A phenomenological model described the failure with a traction sepa-

ration law and the finite element model considered the non-linear, isotropic and visco-elastic

properties of the myocardium. A non-linear FE modeling integrated the information from

tensile testing in cross-fibre with constitutive data from literature. The result showed the

importance of a viscoelastic formulation.

Paper B: On the numerical implementation of invariant-based viscoelastic formulations at

finite strains. A model for the passive myocardium. A framework for finite strain viscoelas-

ticity was developed and implemented. A superposition of an Elastic Body and Maxwell

Body where the reference configuration of the Maxwell Body moves in space and stores the

history of the deformation with a rate equation in strain space. The current configuration of

the continuum coincides with the reference configuration of the Maxwell body at the thermo

dynamical equilibrium. Two independent strain variables are describing the Helmholtz free

energy. An update equation of the over-stress allows variable time step to be used. The

model was implemented and the material behavior due to a rigid punch into myocardium

was studied. Specially the importance of anisotropy was looked at and the result shows the

importance of an anisotropic model.

Paper C: Spatial orientation of collagen fibers in the abdominal aortic aneurysm’s wall and

its relation to wall mechanics Collagen and its distribution orientation is important for the

mechanical properties of the wall. In this paper samples from AAA was collected at elective

AAA repair from Karolinska hospital. Tissue samples where fixated, embedded, sectioned,

stained and investigated by polarized light microscopy. The 3D orientation of the collagen

was measured and captured by a Bingham distribution function. This was performed to in-

tegrate the identified structural information in the AAA wall. No significant difference of the

orientation of collagen, between media and adventitia lager of the wall were found and the

dispersion was found lager in the tangential plane than in the cross-sectional plane.

23

Paper D: The quasi-static failure properties of the Abdominal Aortic Aneurysm wall esti-

mated by a mixed experimental-numerical approach. The present study estimated the elastic

and inelastic properties of the AAA wall. A mixed experimental-numerical approach was

used. Data from collagen orientation and results from failure tests of AAA wall was com-

bined into finite elements modeling. A histo-mechanical constitutive model was used were the

weakening of the fiber and remaining deformation were described by plastic fibril sliding. The

finite element models simulated the tensile tests, in longitudinal direction, for 16 specimens.

The stiffness and strength of the AAA tissue were found to be higher for patients with COPD

then for non COPD patients. No correlation with gender or, smoking were found to effect

the parameters related to irreversible deformation response.

Paper E: Identification of carotid plaque tissue properties using an experimental-numerical

approach The present study introduces an inverse parameter estimation approach to extract

tissue properties from fibrous cap of carotid plaque. Samples from Carotid Endarterectomy

(CEA) were used for in-vitro force displacement testing. Histology images were used to de-

velop Finite element- (FE-) models. An optimization procedure was then used to identify

material parameters and solve the inverse problem presented in the study.

Paper F: Failure properties for the thoracic aneurysm wall; Differences between Bicuspid

Aortic Valve (BAV) and Tricuspid Aortic Valve (TAV) patients. Aorta is predisposed for

hemodynamic events. Specially patients with aortic valve pathogeneses have an increased

risk of TAA. This study is investigating the mechanical properties of the TAA wall using

uniaxial failure testing and measurement of collagen orientation. The 3D orientation of col-

lagen is measured with help of polarized light microscopy. A FEM models integrate the 3D

collagen information with the tensile properties to identify material parameters for elasto-

plastic damage model. Especially the mechanical differences for the aneurysm tissue between

BAV and TAV patients were looked at. A statistical significant difference where found for

the TAA tissue in the ultimate tensile stress σult, between patients with TAV and BAV

(1.496SD0.845 MPa, 0.723SD0.336 MPa,p = 0.02). Also the collagen fiber stiffness-related

parameter (p = 0.01) had a statistical significant difference.

24

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