Abdominal aortic aneurysm inception and 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-0605 ISSN 1654-1472
ISRN KTH/HLF/R–16/19–SE ISBN 978-91-7729-216-6
Akademisk avhandling som med tillst˚and av Kungliga Tekniska h¨ogskolan (KTH) i Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktorsexamen tisdagen den 20:e december 2016 kl. 10:00 i F3, Lindstedtsv¨agen 22, KTH, Stockholm.
Fakultetsopponent ¨ar Professor Frans N. van de Vosse, Technische Universiteit Eindhoven, 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) k¨annetecknas av en utbuktning hos aortav¨aggen i bu- ken. Tillv¨axt av en AAA ¨ar oftast asymtomatisk, men en s˚adan utbuktning kan pl¨otsligt brista, vilket har h¨og d¨odlighet. Tyv¨arr finns det inga mediciner som kan f¨orhindra AAA fr˚an att expandera eller brista. Patienter med uppt¨ackt AAA h˚alls d¨arf¨or under uppsikt tills operationskrav ¨ar uppn˚adda, s˚asom maximal AAA- diameter p˚a 55 mm eller expansionstakt p˚a 1 cm/˚ar. Modeller f¨or AAA-tillv¨axt kan bidra till att ¨oka f¨orst˚aelsen f¨or sjukdomsf¨orloppet och till att f¨orb¨attra be- slutsunderlaget p˚a en patientspecifik basis. AAA modeller f¨or tillv¨axt och struk- turf¨or¨andring (G&R) ¨ar ganska komplicerade och innan man tar sig an denna utmaning kr¨avs de god klinisk validering.
I Artikel A har en befintlig tjockv¨aggig modell f¨or tillv¨axt av ett skikt av en AAA-skiva ut¨okats till en tv˚a-skiktsmodell. Denna modell ˚aterspeglar b¨attre den skiktade strukturen hos k¨arlv¨aggen. Genom en parameterstudie unders¨oktes p˚averkan av mekaniska egenskaper och G&R-parametrar hos en s˚adan modell f¨or AAA-tillv¨axt.
I Artikel B utvidgades modellen fr˚an Artikel A till en organniv˚a-modell f¨or AAA-tillv¨axt. Vidare inkorporerades modellen i ett “Fluid–Solid–Growth”
(FSG) ramverk. En patientspecifik geometri hos bukaortan anv¨andes f¨or att il- lustrera m¨ojligheterna med modellen.
I Artikel C unders¨oktes utvecklingen av patientspecifika biomekaniska egenska- per hos AAA. Fyra patienter som skannats fem till ˚atta g˚anger med “Computed Tomography-Angiography” (CT-A) vid olika tillf¨allen analyserades. Flera icke tri- viala statistiska samband konstaterades mellan de analyserade parametrarna.
I Artikel D unders¨oktes effekten av olika tillv¨axt-kinematik f¨or AAA tillv¨axt.
En modell med transversellt-isotrop-i-tjockleken-tillv¨axt var den b¨ast l¨ampade f¨or AAA tillv¨axt, medans antagandet om fullt-isotrop-tillv¨axt och transversellt- isotrop-i-planet-tillv¨axt producerade orimliga resultat. Dessutom gav modellering av v¨avnadsvolymsf¨or¨andring ett f¨orb¨attrat v¨aggtjockleks resultat men en fortsatt
¨overskattning av v¨aggf¨ortunningen under AAA-expansionen.
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List of appended papers
Paper A: Influence of differing material properties in media and adventitia on 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 aortic aneurysm 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 aneurysm growth.
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-walled mixture 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 resulted in the following conference contributions:
A thick-walled fluid-solid-growth model of abdominal aortic aneurysm evolution.
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 aortic aneurysm 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 aneurysm evolution.
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. Contributed to the manuscript drafting. Schmid wrote the major part of the manuscript.
Paper B: Principal author. Further developed the model presented in Paper A and incorporated it into a Fluid-Solid-Growth framework. Designed the study and carried out all computations. Analyzed and interpreted the results. Drafted the manuscript. Holzapfel, Watton contributed with critical reviews.
Paper C: Designed the study together with Stevens, Gasser and Biasetti.
Developed the computational framework together with Stevens. Supervised Stevens and supported him throughout the computational efforts. Analyzed and interpreted results together with other co-authors. Contributed with critical review of the manuscript draft by Gasser.
Paper D: Principal author. Further developed the previously published model 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 with comments.
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Contents
Abstract . . . i
Sammanfattning . . . iii
List of appended papers . . . v
Contribution to the papers . . . vii
Introduction . . . 1
Structure of arterial wall . . . 2
Collagen . . . 4
Modeling the adaptation of vascular tissue . . . 6
Objective . . . 7
Methods . . . 8
Collagen growth and remodeling . . . 8
Elastin degradation . . . 9
Volume growth . . . 9
Fluid-solid-growth framework . . . 10
Key results . . . 11
Bilayer aneurysm model . . . 11
Fluid-solid-growth framework . . . 12
Patient-specific biomechanics . . . 13
Volume growth . . . 15
Conclusion . . . 16
Summary of appended papers . . . 17
Bibliography . . . 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 large aneurysm [7].
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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 2
Abdominal aortic aneurysm inception and evolution – A computational model
Figure 2: Idealization of arterial wall structure [14]. It consists of three layers: 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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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, 4
Abdominal aortic aneurysm inception and evolution – A computational model
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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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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Abdominal aortic aneurysm inception and evolution – A computational model
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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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 8
Abdominal aortic aneurysm inception and evolution – A computational model
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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Figure 3: Kinematics of growth. Deformation gradient F(τ, t) maps the reference configuration Ω0 into the current configuration Ωτ. On the other hand, the spatially homogeneous growth tensor Fg(τ ) (with det Fg = det F = ˆ
v) connects the reference configuration Ω0 to the intermediate stress-free configuration Ωg. This mapping relates to a time-scale τ of weeks. Then, the elastic deformation tensor Fe(t) connects Ωg to the current configuration Ωτ. Consequently, the total deformation gradient F(τ ) is split into volumetric growth 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.
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Abdominal aortic aneurysm inception and evolution – A computational model
(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 strain stressor
updated material
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 the growth 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 shear stress (WSS) is sent to (c). G&R model (c) adapts the material properties to the strain field and returns it to the solid model (a). Then, the solution starts 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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(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) Radius change over time, normalized by the radius in homeostasis. (b) Transmural plot of circumferential 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
Abdominal aortic aneurysm inception and evolution – A computational model
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| in the aneurysm domain and patient-specific up and downstream extensions is shown: 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
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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 s of the cardiac cycle, in four abdominal aortic aneurysm (AAA) patients.
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Abdominal aortic aneurysm inception and evolution – A computational model
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 the AAAs at twofold expansion. (a) thickness change relative to the thickness of aorta in homeostasis along AAA length. (b) Cauchy hoop stress σϕϕ across the wall. Dotted curves represent homeostatic values of the parameters.
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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.
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Abdominal aortic aneurysm inception and evolution – A computational model
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
Abdominal aortic aneurysm inception and evolution – A computational model
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.
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Bibliography
[1] G. R. Upchurch Jr and T.A. Schaub. Abdominal Aortic Aneurysm. Am Fam Physi- cian, 73:1198–1204, 2006.
[2] K. K. Hannawa, J. L. Eliason, and G. R. Upchurch, Jr. Gender differences in ab- dominal aortic aneurysms. Vascular, 17:S30–S39, 2009.
[3] S. Svensj¨o, M. Bj¨orck, and A. Wanhainen. Current prevalence of abdominal aortic aneurysm in 70-year-old women. Br J Surg, 100:367–372, 2013.
[4] A. Wanhainen, R. Hultgren, A. Linn´e, J. Holst, A. Gotts¨ater, M. Langenski¨old, K. Smidfelt, M. Bj¨orck, S. Svensj¨o, and Swedish Aneurysm Screening Study group (SASS). Outcome of the swedish nationwide abdominal aortic aneurysm screening program. Circulation, 134:1141–1148, 2016.
[5] K. C. Kent, R. M. Zwolak, N. N. Egorova, T. S. Riles, A. Manganaro, A. J. Moskowitz, A. C. Gelijns, and G. Greco. Analysis of risk factors for abdominal aortic aneurysm in a cohort of more than 3 million individuals. J Vasc Surg, 52:539–548, 2010.
[6] S. Shantikumar, R. Ajjan, K. E. Porter, and D. J. A. Scott. Diabetes and the abdominal aortic aneurysm. Eur J Vasc Endovasc Surg, 39:200–207, 2010.
[7] U.s. national library of medicine. https://medlineplus.gov/.
[8] R. M. Greenhalgh, L. C. Brown, J. T. Powell, S. G. Thompson, D. Epstein, M. J.
Sculpher, and United Kingdom EVAR Trial Investi. Endovascular versus open repair of abdominal aortic aneurysm. N Engl J Med, 362:1863–1871, 2010.
[9] A. R. Brady, S. G. Thompson, F. G. R. Fowkes, R. M. Greenhalgh, J. T. Powell, and UK Small Aneurysm Trial Participants. Abdominal aortic aneurysm expansion – Risk factors and time intervals for surveillance. Circulation, 110:16–21, 2004.
[10] F. L. Moll, J. T. Powell, G. Fraedrich, F. Verzini, S. Haulon, M. Waltham, J. A. van Herwaarden, P. J. E. Holt, J. W. van Keulen, B. Rantner, F. J. V. Schloesser, F. Se- tacci, and J-B Rica. Management of Abdominal Aortic Aneurysms Clinical Practice Guidelines of the European Society for Vascular Surgery. Eur J Vasc Endovasc Surg, 41:S1–S58, 2011.
[11] R. Scott, N. M. Wilson, H. A. Ashton, and D. N . Kay. Influence of screening on the incidence of ruptured abdominal aortic-aneurysm – 5-year results of a randomized controlled-study. Br J Surg, 82:1066–1070, 1995.
[12] L. C. Brown, J. T. Powell, and UK Small Aneurysm Trial Participants. Risk factors for aneurysm rupture in patients kept under ultrasound surveillance. Ann Surg, 230:289–296, 1999.
21
Andrii Grytsan
[13] E. Georgakarakos, C. V. Ioannou, Y. Papaharilaou, T. Kostas, and A. N. Kat- samouris. Computational Evaluation of Aortic Aneurysm Rupture Risk: What Have We Learned So Far? J Endovasc Ther, 18:214–225, 2011.
[14] T. C. Gasser, R. W. Ogden, and G. A. Holzapfel. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface, 3:15–35, 2006.
[15] S. Polzer, J. Bursa, T. C. Gasser, R. Staffa, and R. Vlachovsky. A numerical imple- mentation to predict residual strains from the homogeneous stress hypothesis with application to abdominal aortic aneurysms. Ann Biomed Eng, 41:1516–1527, 2013.
[16] M. Kazi, J. Thyberg, P. Religa, J. Roy, P. Eriksson, U. Hedin, and J. Swedenborg. In- fluence of intraluminal thrombus on structural and cellular composition of abdominal aortic aneurysm wall. J Vasc Surg, 38:1283–1292, 2003.
[17] S. M. Pradhan, D. Katti, and K. S. Katti. Steered molecular dynamics study of me- chanical response of full length and short collagen molecules. J Nanomech Micromech, 1:104–110, 2011.
[18] B. Depalle, Z. Qin, S. J. Shefelbine, and M. J. Buehler. Influence of cross-link struc- ture, density and mechanical properties in the mesoscale deformation mechanisms of collagen fibrils. J Mech Behav Biomed Mater, 52:1–13, 2015.
[19] S.J. Eppell, B.N. Smith, H. Kahn, and R. Ballarini. Nano measurements with micro- devices: mechanical properties of hydrated collagen fibrils. J R Soc Interface, 3:117–
121, 2006.
[20] J. A. J. van der Rijt, K. O. van der Werf, M. L. Bennink, P. J. Dijkstra, and J. Feijen.
Micromechanical testing of individual collagen fibrils. Macromol Biosci, 6:697–702, 2006.
[21] Z. L. Shen, M. R. Dodge, H. Kahn, R. Ballarini, and S. J. Eppell. Stress-strain experiments on individual collagen fibrils. Biophys J, 95:3956–3963, 2008.
[22] A. Gautieri, S. Vesentini, A. Redaelli, and M. J. Buehler. Hierarchical Structure and Nanomechanics of Collagen Microfibrils from the Atomistic Scale Up. Nano Lett, 11:757–766, 2011.
[23] E. Gentleman, A. N. Lay, D. A. Dickerson, E. A. Nauman, G. A. Livesay, and K. C.
Dee. Mechanical characterization of collagen fibers and scaffolds for tissue engineer- ing. Biomaterials, 24:3805–3813, 2003.
[24] A. G. Moon and R. T. Tranquillo. Fibroblast-populated collagen microsphere assay of cell traction force. 1. continuum model. AlChE J, 39:163–177, 1993.
[25] M. Chiquet, A. S. Renedo, F. Huber, and M. Fl¨uck. How do fibroblasts translate mechanical signals into changes in extracellular matrix production? Matrix Biology, 22:73–80, 2003.
[26] E. K. Rodriguez, A. Hoger, and A. D. McCulloch. Stress-dependent finite growth in soft elastic tissues. J Biomech, 27:455–467, 1994.
[27] L. A. Taber and D. W. Eggers. Theoretical study of stress-modulated growth in the aorta. J Theor Biol, 180:343–357, 1996.
[28] A. Rachev, N. Stergiopulos, and J. J. Meister. A model for geometric and mechanical adaptation of arteries to sustained hypertension. J Biomech Eng-T ASME, 120:9–17, 1998.
22
Abdominal aortic aneurysm inception and evolution – A computational model
[29] J. D. Humphrey. Remodeling of a collagenous tissue at fixed lengths. J Biomech Eng-T ASME, 121:591–597, 1999.
[30] J. D. Humphrey and K. R. Rajagopal. A constrained mixture model for growth and remodeling of soft tissues. Math Mod Meth Appl S, 12:407–430, 2002.
[31] W. Wan, L. Hansen, and R. L. Gleason, Jr. A 3-D constrained mixture model for mechanically mediated vascular growth and remodeling. Biomech Model Mechanobiol, 9:403–419, 2010.
[32] I. Karˇsaj and J. D. Humphrey. A multilayered wall model of arterial growth and remodeling. Mech Mater, 44:110–119, 2012.
[33] A. Valentin, J. D. Humphrey, and G. A. Holzapfel. A finite element-based constrained mixture implementation for arterial growth, remodeling, and adaptation: Theory and numerical verification. Int J Numer Method Biomed Eng, 29:822–849, 2013.
[34] P. N. Watton, N. A. Hill, and M. Heil. A mathematical model for the growth of the Abdominal Aortic Aneurysm. Biomech Model Mechanobiol, 3:98–113, 2004.
[35] P. N. Watton and N. A. Hill. Evolving mechanical properties of a model of Abdominal Aortic Aneurysm. Biomech Model Mechanobiol, 8:25–42, 2009.
[36] H. Schmid, P. N. Watton, M. M. Maurer, J. Wimmer, P. Winkler, Y. K. Wang, O. Roehrle, and M. Itskov. Impact of transmural heterogeneities on arterial adapta- tion. Biomech Model Mechanobiol, 9:295–315, 2010.
[37] G. Martufi and T. C. Gasser. Turnover of fibrillar collagen in soft biological tissue with application to the expansion of abdominal aortic aneurysms. J R Soc Interface, 9:3366–3377, 2012.
[38] P. Aparicio, M. S. Thompson, and P. N. Watton. A novel chemo-mechano-biological model of arterial tissue growth and remodelling. J Biomech, 49:2321–2330, 2016.
[39] H. Schmid, L. Pauli, A. Paulus, E. Kuhl, and M. Itskov. Consistent formulation of the growth process at the kinematic and constitutive level for soft tissues composed of multiple constituents. Comput Methods Biomech Biomed Engin, 15:547–561, 2012.
[40] T. S. E. Eriksson, P. N. Watton, X. Y. Luo, and Y. Ventikos. Modelling volumetric growth in a thick walled fibre reinforced artery. J Mech Phys Solids, 73:134–150, 2014.
[41] M. M. Dua and R. L. Dalman. Hemodynamic Influences on abdominal aortic aneurysm disease: Application of biomechanics to aneurysm pathophysiology. Vascul Pharmacol, 53:11–21, 2010.
[42] L. Virag, J. S. Wilson, J. D. Humphrey, and I. Karˇsaj. A computational model of biochemomechanical effects of intraluminal thrombus on the enlargement of abdom- inal aortic aneurysms. Ann Biomed Eng, 43:2852–2867, 2015.
[43] C. A. Figueroa, S. Baek, C. A. Taylor, and J. D. Humphrey. A computational framework for fluid-solid-growth modeling in cardiovascular simulations. Comput Method Appl M, 198:3583–3602, 2009.
[44] A. Sheidaei, S. C. Hunley, S. Zeinali-Davarani, L. G. Raguin, and S. Baek. Simulation of abdominal aortic aneurysm growth with updating hemodynamic loads using a realistic geometry. Med Eng Phys, 33:80–88, 2011.
23
24 BIBLIOGRAPHY
[45] P. Aparicio, A. Mandaltsi, J. Boamah, H. Chen, A. Selimovic, M. Bratby, R. Uberoi, Y. Ventikos, and P. N. Watton. Modelling the influence of endothelial heterogeneity on the progression of arterial disease: application to abdominal aortic aneurysm evolution. Int J Numer Method Biomed Eng, 30:563–586, 2014.
[46] S. Polzer, T. C. Gasser, B. Markert, J. Bursa, and P. Skacel. Impact of poroelasticity of intraluminal thrombus on wall stress of abdominal aortic aneurysms. Biomed Eng Online, 11:62, 2012.
[47] P. N. Watton, Y. Ventikos, and G. A. Holzapfel. Modelling the mechanical response of elastin for arterial tissue. J Biomech, 42:1320–1325, 2009.
[48] Cmiss: An interactive computer program for continuum mechanics, image analysis, signal processing and system identification. http://www.cmiss.org/.
[49] Feap: A finite element analysis program. http://www.ce.berkeley.edu/projects/feap/.
[50] A4clinics. http://www.vascops.com/en/vascops-a4clinics.html.
[51] J. S. Wilson, S. Baek, and J. D. Humphrey. Parametric study of effects of collagen turnover on the natural history of abdominal aortic aneurysms. Proc R Soc Lond A, 469(2150):20120556, 2013.
