Repositorio Institucional de Documentos

Atheroma Plaque Vulnerability Based in a 3D Idealized Parametric Geometry. M´ aster Mec´ anica Aplicada Myriam Cilla Hern´andez Ingeniero Industrial    A todos los que me apoyan día a día… Atheroma Plaque Vulnerability Based in a 3D Idealized Parametric Geometry. RESUMEN Las enfermedades cardiovasculares constituyen la primera causa de mortalidad en los pa´ıses desarrollados, as´ı como en la pr´actica totalidad de los pa´ıses en desarrollo [47]. Dentro de las patolog´ıas cardiovasculares, una de las enfermedades que m´as muertes causan hoy en d´ıa es la aterosclerosis. Dicha enfermedad consiste en la degeneraci´on progresiva y cr´onica del engrosamiento y endurecimiento de la pared arterial como resultado de la acumulaci´on de dep´ositos de grasa, colesterol y otras sustancias en la pared interna del vaso. ´ Estas sustancias forman estructuras duras llamadas placas de ateroma y en ellas se puede diferenciar diferentes composiciones; calcificaci´on, l´ıpidos y tejido fibroso. Con el tiempo, si estas placas son vulnerables pueden romperse y provocar la formaci´on de co´agulos sangu´ıneos que bloquean el flujo sangu´ıneo desencadenando diferentes eventos tales como infartos, trombos o gangrena [1, 58]. La importancia de identificar la placa vulnerable antes de su rotura sigue siendo un reto para la medicina. Actualmente el diagn´ostico se puede realizar por m´etodos no invasivos, tales como es determinar la presencia de factores de riesgo, marcadores de vulnerabilidad de la placa, y m´etodos invasivos como es la angioscop´ıa. Aunque existen varios m´etodos, no existe uno que nos de toda la informaci´on morfol´ogica y de actividad de la placa necesaria. El objetivo principal de este trabajo es el estudio y modelado de vasos sangu´ıneos, concretamente de una coronaria, afectados por aterosclerosis. Para ello, se ha desarrollado un estudio param´etrico en tres dimensiones (3D) de los factores geom´etricos en la vulnerabilidad de la placa de ateroma y de la influencia de las tensiones residuales. Los modelos en 3D nos permiten incluir los efectos producidos por las tensiones residuales. Con dichos modelos se van a estudiar tres situaciones diferentes; sin tensiones residuales, considerando tensiones residuales en direcci´on longitudinal y por ´ultimo, considerando tensiones residuales tanto en direcci´on longitudinal como circunferencial. De este modo, se podr´a identificar el papel que juegan las tensiones residuales en la vulnerabilidad de la placa de ateroma y definir l´ımites de vulnerabilidad para cada uno de los par´ametros considerados. Los modelos se han simulado mediante elementos finitos con el software comercial ABAQUS. Los resultados obtenidos, nos permiten ir un paso m´as all´a en el diagn´ostico preventivo y en la planificaci´on preoperatoria en aplicaciones cardiovasculares. Con el fin de validar el modelo, se ha reconstruido mediante el software comercial MIMICS una geometr´ıa real a partir de un IVUS (ultrasonido intravascular) de un paciente adulto con placa de ateroma. Tras la reconstrucci´on se han medido los par´ametros estudiados en la geometr´ıa real y la hemos comparado con el correspondiente caso param´etrico de par´ametros similares. Contents 1Introduction 1 1.1 Cardiovascular diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Atherosclerosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Physiology and properties of the blood vessels . . . . . . . . . . . . . . . . . 3 1.2.1 Layers................................... 3 1.2.2 Residual stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Constitutive modelling of anisotropic material . . . . . . . . . . . . . . . . . 4 1.3.1 Hyperelastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Strain energy functions . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Objectives and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Contents...................................... 8 23D Parametric Study 9 2.1 Modeling of the atherosclerotic coronary artery . . . . . . . . . . . . . . . . 9 2.1.1 Idealized geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Parameters studied . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Mesh.................................... 11 2.1.4 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Results....................................... 15 2.2.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Trend analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 Vulnerability study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.4 Vulnerability factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Importance of 3D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3Real geometry versus idealized geometry 27 i CONTENTS ii 4Conclusions and Future work 30 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 1 Introduction 1.1 Cardiovascular diseases Cardiovascular diseases (CVD) are nowadays primary causes of mortality in the developed world and it has been calculated that they will become the first cause of death worldwide in 2020 [40]. Cardiovascular disease is divided into several categories of subclasses. The most important one, in terms of annual deaths, is Coronary Heart Disease (CHD), also referred to as Coronary Artery Disease (CAD), which refers to the disease of the blood vessels supplying the heart muscle [1, 44, 54] Due to the huge social and economical impact of CVDs, the study of the blood vessel and their associated pathologies have been one of the main research topics in medicine in the last decades. More recently, the mechanical factors influencing the vascular pathologies have been considered in Biomechanics [19]. Specifically, arteries have been focused most of the efforts due to their higher tendency to develop disease pathologies, whereas veins have not been so deeply studied. The studies developed until the date demonstrate that mechanics play a determinant role in the development and evolution of a variety of pathologies [31], which itself justifies the thorough investigation of their mechanical response both in healthy and diseased states. 1.1.1 Atherosclerosis The main dominant cardiovascular disease is arteriosclerosis, which is the process in which plaques - consisting of deposits of cholesterol and other lipids, calcium and large inflammatory cells called macrophages - are built up in the walls of the arteries causing narrowing (stenosis of the lumen), hardening of the arteries and loss of their elasticity, which leads to a reduction in the blood flow through the vessels. Nevertheless, the most serious damage occurs when the plaque becomes fragile and ruptures (vulnerable plaque). Plaque rupture 1 Chapter 1. Introduction 2 causes the formation of blood clots that can block blood flow or break off and travel to another part of the circular system thus producing heart attacks, strokes, difficulty in walking and eventually gangrene [25, 36, 62], see Figure 1.1. Figure 1.1: Coronary artery atherosclerosis. Until now, several methods have been used to evaluate the extent and location of atherosclerotic lesions; invasive methods such as IVUS (intravascular ultrasound) or X- ray angiography and non-invasive methods, which detect indicators of atherosclerosis such as classical risk factors [16, 33, 38]. Identifying vulnerable patients before plaque rupture occurs would help clinicians to provide early treatment as well as to take preventive measures. Many strategies have been proposed to achieve this goal, though available screening and diagnostic methods seem to be insufficient. The characteristics of vulnerable plaque have been well defined in several pathological studies [20, 55, 46, 64, 68] amongst others. Plaque rupture is believed to be related to plaque morphology, mechanical forces, vessel remodelling, blood conditions (levels of cholesterol, sugar, etc.), chemical environment and lumen surface conditions (inflammation) [63]. Regarding the mechanical forces, some authors [49, 67] consider the peak circumferential stress (PCS) as the most important biomechanical factor in the mechanisms leading to rupture of the atherosclerotic plaque, and have often used it as a predictor of atherosclerotic plaque rupture location. Previous works have shown that reduced fibrous cap thickness increases the maximal value of the PCS exponentially, and leads the cap stress to exceed the rupture threshold of 300 kPa [10, 39, 49] when the cap thickness becomes lower than 65 µm [17, 45, 66, 68]. The fibrous cap thickness has typically been identified as the key predictor of vulnerability and likelihood of rupture, but some clinical and biomechanical studies have shown that this single parameter is not a reliable predictor of plaque stability [35, 69], since Chapter 1. Introduction 3 plaque stability also depends on other intrinsic properties of the plaque, such as the size and the consistency of the soft atheroma core [17, 21], the cap and the core inflammation levels [4, 37] and the arterial remodelling index, which is defined as the external elastic membrane area at plaque divided by the external elastic membrane area at a nearest segment judged to be free of plaque. [48, 57, 65]. 1.2 Physiology and properties of the blood vessels 1.2.1 Layers The blood vessels are made of a layered structure which, when healthy, is usually composed of three different tunicas or layers. From the inner to the outer radius, they are the intima, the media, and the adventitia, whose main features are the following, (see Figure 1.2). •Intima: The intimal layer is the innermost layer of all blood vessels, being composed of the following structures: 1) a single layer of endothelial cells lining the vascular wall; 2) a thin, about 80 [nm] thick, elastic lamina; and 3) a subendothelial layer, composed of collagenous bundles, elastic fibrils, smooth muscle cells, and perhaps some fibroblasts. The subendothelial layer, however, is only presented in large elastic arteries such as the human aorta, whereas this sub-layer is missing in other smaller arteries. •Media: The tunica media, as the name indicates, is the middle layer of the vascular wall. It is made of smooth muscle cells, a variable number of elastic laminae, bundles of collagenous fibrils and a network of elastic fibrils. This layer is thicker in arteries than in veins, in which muscle cells may be absent in large veins like vena cava. In the aorta, the media layer may reach thicknesses of 500 [µm], whereas is about 20-50 [µm] in medium sized veins. •Adventitia: The tunica adventitia is the outermost layer of the vascular wall. Its thickness varies considerably depending on the type and location of the blood vessel. In all arteries, and most veins, the adventitia consists of dense fibroelastic tissue without smooth muscle cells, except in large veins, such as vena cava, where bundles of longitudinally arranged smooth muscle cells may be found. Also the nutrient vessels of the vascular wall that take nutrients to the muscle cells in the media, namely the vasa vasorum, are part of the adventitia, as well as nerves. The adventitia renders the vascular wall a fair amount of stability, and serves to connect blood vessels to the surrounding tissue. Chapter 2. 3D Parametric Study 10 2.1.2 Parameters studied The parametric model consists of a series of idealized plaque morphology models, mimicking different stages and variations of the atherosclerotic lesion growth. Previous analyses were first performed on the model in order to identify the most influential geometric parameters on the atheroma vulnerability risk. Geometric parameters such as the lipid core angle were rule out as one of the most influential parameters. Following the results of these previous test and several works of the literature [10, 17, 48, 72], the most influential geometric parameters considered were the fibrous cap thickness (fc), the stenosis ratio (sr) - which is obtained by dividing the lumen radius by the lumen radius of a normal artery (R= 1.5 mm), sr(%) = r(mm) R(mm)100 - the lipid core length (l) and the lipid core width (w). The lipid core width (w) was defined as the ratio between the percentage of the atheroma plaque width (w1)and the distance from the inner point of the lipid core to the outer point of the fibrotic plaque (w2),w(%) = w1(mm) w2(mm)100. The cross central section of the 3D model with the parameter studied marked is shown in Figure 2.1.b, where the lipid core length was measured along the longitudinal direction (Figure 2.1.a). Figure 2.1: (A) Idealized geometry of an atherosclerotic arterial model. Transversal section. (B) Geometrical parameters shown on the cross central section of the atherosclerotic vessel. Five values for each parameter were considered and combined which makes a total of 625 = 54idealized eccentric vessel models with atherosclerotic lesions. Realistic morphological data was investigated by varying the lipid core length (1 ≤l≤8, in mm), the stenosis ratio (46.7≤sr ≤66.7, in %), the fibrous cap thickness (0.025 ≤fc ≤0.25, in mm) and the lipid core width (30 ≤w≤90, in %) [18]. The values of the geometrical parameters used to define the idealized coronary plaque models are shown in Table 2.1. Chapter 2. 3D Parametric Study 11 Level l[mm] sr[%] fc[mm] w[%] 1 1 46.7 0.025 30 2 2 53.3 0.05 45 3 4 56.7 0.1 60 4 6 60 0.15 75 5 8 66.7 0.25 90 Table 2.1: Geometrical parameters used to generate the parametric 3D models. Residual stresses (RS) in longitudinal and circumferential directions were incorporated into the model to analyze the influence of the important mechanical factors in the vulnerability of the plaque. In order to assess the role of circumferential and axial residual stresses on the atheroma plaque ruptures, the set of 625 simulations described in the previous subsection (Subsection 2.1.2) were performed under three different hypothesis; •(i) without RS, •(ii) just considering axial RS •(iii) and taking account both circumferential and axial RS. In total, 1875 (625 x 3) analyses have been performed. 2.1.3 Mesh A fine mesh was created in the various regions of the model and the fibrous cap region was meshed with an adaptive mesh. Previous sensitivity analyses were performed on the mesh to choose the definitive one. Due to the symmetry of the problem only a quarter of the model with approximately 150.000 linear and hybrid tetrahedral elements and 30.000 nodes was considered. In general, the hybrid elements are used for incompressible materials to avoid volumetric locking. The mesh used and the number of nodes and elements vary from one simulation to another due to the geometry undergoes small alterations as the parameters are changed. Figure 2.2 show a transversal section of the mesh corresponding to one of the 625 models simulated. Chapter 2. 3D Parametric Study 12 Figure 2.2: Longitudinal section of the mesh used for one of the 625 models simulated. Detail of the lipid core mesh. C3D4 is the kind of elements used which correspond to a linear and hybrid tetrahedral elements. 2.1.4 Material properties All tissues were modelled as nonlinear, hyperelastic and incompressible materials [9, 28]. The lipid core and the atherosclerotic plaque were modelled as isotropic materials, while healthy wall was considered as an anisotropic material with two families of fibres, oriented at ±61.8oand ±28.35owith respect to the circumferential direction in the adventitia and the media layers, respectively. Both families of fibres were assumed to have the same mechanical properties [28] (see Figure 2.3). The behaviour of the tissue was modelled by using the Gasser, Ogden and Holzapfel (GOH) strain energy function (SEF) [22] Ψ = µ[I1−3] + k1 2k2X i=4,6 expk2κ[I1−3] + [1 −3κ][Ii−1]2−1,(2.1) where µ > 0 and k1>0 are stress-like parameters and k2>0 and 0 ≤κ≤1 3are dimensionless parameters (when κ=0 the fibres are perfectly aligned (no dispersion) and when κ=1 3the fibres are randomly distributed and the material becomes isotropic), I1is the first invariant of C=FTFwith Fthe deformation gradient tensor, I4=m0·Cm0 and I6=n0·Cn0are invariants which depend on the direction of the family of fibres at a material point Xthat is defined by the unit vectors field m0and n0[58]. Chapter 2. 3D Parametric Study 13 Figure 2.3: Fibres distribution in the adventitia and media layers. ±βis the orientation angle of the fibres with respect to the circumferential direction. m0and n0are the vectors which define the fibre orientations of each family of fibres. To obtain the material parameters for the constitutive law of the tissue, experimental data presented in previous works (the adventitia and the media properties from Holzapfel et al. [28] and the plaque and the lipid core properties from Versluis et al. [67]) were fitted using the Levenberg-Marquardt minimization algorithm [43]. Figure 2.4 shown the experimental stress-stretch model responses for each part of the model considered [28, 67]. Adventitia and media layers have different behaviour in the longitudinal and circumferential direction due to its anisotropy. Figure 2.4: Stress-stretch model responses of experimental data taken from the literature for each tissue of the model considered; adventitia and healthy media layers in both directions (circumferential and longitudinal), lipid core and atheroma plaque. Chapter 2. 3D Parametric Study 14 Table 2.2 shows the results of the parameter identification for each tissue fitted according the Gasser, Ogden and Holzapfel strain energy function [22]. Furthermore, the normalized mean square root error (ε) which is defined as ε=qχ2 n−q µ,(2.2) was used to check the goodness of the fit. Where qis the number of parameters of the (SEF), n is the number of data points, n−qis the number of degrees of freedom, and µ is the mean stress. µ[kP a]k1[kP a]k2[−]κ[−]ε[−] Adventitia 8.44 547.67 568.01 0.26 0.041 Healthy media 1.4 206.16 58.55 0.29 0.014 Atheroma plaque 9.58 17654.91 0.51 0.33 0.056 Lipid core 0.052 965.76 70 0.33 0.03 Table 2.2: Material parameters used in the finite element analysis for the adventitia, the healthy media, the atheroma plaque and the lipid core. 2.1.5 Boundary conditions Regarding the boundary conditions, the longitudinal displacements were constrained at the end of the vessel, whereas the radial displacement was allowed, to avoid the solid rigid behaviuor of the model. Symmetry conditions were imposed in the corresponding symmetry planes due to only a quarter of the model was simulated. Firstly, in order to introduce the circumferential residual stress, a cut with an opening angle of 23.5owas performed in the opposite side of plaque location according to the experimental data of severe atherosclerosis obtained by Jaroslav et al. [32]. The opened model was closed such that we finally obtain the configuration shown in Figure 2.5.a. Secondly, to introduce the longitudinal residual stress, the model was stretched a 4.4 % of the vessel length in the longitudinal direction, representing in-vivo conditions [28] as it shown in Figure 2.5.b. Thirdly, a constant internal pressure of 140 mmHg (18.7 kPa) was imposed in the inner surface of the lumen, simulating the blood flow pressure [48] (Figure 2.5.c). In the set of simulation without RS just the internal pressure is applied (Figure 2.5.c), in the set of simulation with the longitudinal RS included the second and third boundary condition are applied (Figure 2.5.b + Figure 2.5.c), and finally, in the set of simulation with longitudinal and circumferential RS included, the three boundary conditions are applied (Figure 2.5.a + Figure 2.5.b + Figure 2.5.c). Chapter 2. 3D Parametric Study 15 Figure 2.5: Boundary conditions applied for the different simulation set. A - Circumferential RS. B - Longitudinal RS and C - Internal Pressure. Maximal Principal Stresses (MPS) were considered as the mechanical factor for the purposes of comparison in this study. 2.2 Results It is important to remark that the maximum MPS was measured at the critical zones and always appeared in the circumferential direction. Some authors have shown that the maximum MPS sometimes appears at healthy areas, where rupture is unlikely [60, 61, 59]. Healthy areas where rupture is not probable, even if a local stress maximum occurred there, have been excluded from the analysis of the results. 2.2.1 Statistical Analysis To assess the influence of the geometrical parameters on the MPS, a statistical analysis was performed. The Lilliefors test (checking the normality of the distribution), the Student’s t-Tests and the analysis of variance (ANOVA) were used. The ANOVA test and the Student’s t-Test were performed at 1% and 5% significance level, respectively. Figure 2.6 shows the statistical analysis performed on the maximum MPS value in the critical region with respect to the distinct geometrical parameters for the three different studied cases, respectively. Each subfigure represents the results grouped for the different levels of each geometrical parameter, and the variation of this parameter becomes influential if the MPS is modified significantly as this parameter varies. In this Figure, the means of group nand the groups marked with ∗(n)are significantly different with the probability (p) indicated in the legend which is located in the top of each figure. Subfigures without any mark Chapter 2. 3D Parametric Study 16 mean performing two sample comparisons, the means are always significantly different. On each box, the central mark is the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers, and outliers are plotted individually. In the first row of Figure 2.6, the median for each variation of the lipid core length increases slightly with the lipid core length for all the cases, especially in the axial and circumferential RS case. For the cases with axial RS and without RS, considering all the two-sample comparisons (paired t-test), some significant differences are found, between the second group (l= 2 mm) and the groups marked with ∗(2) and between the fifth group (l= 8 mm) and the groups marked with ∗(5) (p < 0.01, see Figure 2.6). However, when the circumferential and the axial RS are considered, the lipid core length becomes more influential since the means are always significantly different for all the two-sample lipid core length comparisons considered (p < 10−5). The stenosis ratio statistical analysis is shown in the second row of the Figure 2.6. The medians and the dispersion are similar and very few significant differences between the means of the groups for the three cases considered are found, showing that the stenosis ratio do not play a significant role in vulnerability related with the MPS. The statistical analysis of the fibrous cap thickness is shown in the third row of Figure 2.6 and proves the influence of this parameter on the MPS for the three analyzed simulations set. A noteworthy remark is that the median and the dispersion of each variation of the fibrous cap thickness decreases dramatically as the fibrous cap thickness increases. Considering all the two-sample fibrous cap thickness combinations, the means are always significantly different (p < 10−5), reflecting the huge influence of this parameter on the MPS values, even if the RS are not considered. Finally, the lipid core width statistical analyses shown in the fourth row of Figure 2.6, shows that both medians and dispersions increase with the lipid core width. In a similar way than in the case of the fibrous cap thickness, for all the two-sample lipid core width considered comparisons, the means are always significantly different (p < 10−5) for the three studied cases, showing again a high dependence on this parameter. Chapter 2. 3D Parametric Study 17 Figure 2.6: Statistical analysis: Maximum MPS vs. the variation of each parameter for the three different cases; without RS, with the axial RS included and with the circumferential and the axial RS included. Chapter 2. 3D Parametric Study 18 2.2.2 Trend analysis In order to compare the variation of circumferential stress versus geometrical parameters, a normalized variation of each parameter has been defined. This parameter (ν) is obtained as ν=a−a0 a1−a0, where the variable “a” represents each of the four parameters (lipid core length (νl), lumen radius (νr), fibrous cap thickness (νfc) and lipid core width (νw)) in whatever position, and a0and a1are the lowest and highest values respectively of each parameter. Figure 2.7 represents the maximum MPS vs. normalized variation νof each parameter. There are 5 variations for each parameter, therefore 125 cases have been represented in each normalized variation of each parameter. A trend analysis has been performed in order to show the influence of the variation of each parameter in the maximum MPS. Therefore, a linear trendline (σ(kPa) = p1v+p2) has been added to the experimental data in each graph. The general trend observed is that the maximum MPS increases respect to the cases without RS if axial RS are considered, whereas the MPS decreases and the dispersion is reduced if circumferential and axial RS are included, as Figure 2.7 shown. In the first subfigure of Figure 2.7 the linear trendline has a positive gradient for the three cases studied (p1= 37.48kPa,p1= 45.3kPa,p1= 98.2kPa for the case without RS, with circumferential RS included and axial and circumferential RS included, respectively) and p1increases more than twice from the cases without RS to those with axial and circumferential RS included, reflecting an increased influence of this parameter on the MPS when the axial and circumferential RS are considered. Stenosis ratio evolution (second subfigure of Figure 2.7) has negative gradient for the cases without RS included (p1=−32.6kP a, so maximum MPS decreases as the lumen radius increases from 167.3 kPa to 134.7 kPa for sr = 46.7% and sr = 66.7 %, respectively. However, p1is positive for the cases with RS considered, so the general trend changes and maximum MPS increases as the lumen radius increases when RS are taken account. Furthermore, the gradient concerned to the stenosis ratio are the lowest, reflecting again a low influence of this parameter on the MPS. The gradients related to the fibrous cap thickness (third subfigure of Figure 2.7) are the steepest and they are negative (p1=−204.7kPa,p1=−293.6kPa,p1=−58.8kPa for the case without RS, with circumferential RS included and axial and circumferential RS included, respectively). Finally, the lipid core width analysis, (fourth subfigure of Figure 2.7), shows a pronounced slope with a positive value for the three set of cases studied (with axial RS, with circumferential and axial RS and without them). Eventually, p1decreases from the case without RS to the cases with RS considered. Chapter 2. 3D Parametric Study 19 Figure 2.7: Maximum MPS vs. the variation of each normalized parameter. Linear polynomial approximation is included (σ=p1v+p2). 2.2.3 Vulnerability study Regarding the vulnerability of the plaque, different threshold stress values have been proposed by different authors [39, 10, 42, 49]. In this study, a threshold value of 247 kPa has been used according to the set of experimental data obtained by Loree et al. (1994), assuming a normal distribution of the data. This threshold value indicates that the probability of having plaque rupture is 0.95 for the cases whose combination of parameters have a maximum MPS equal or higher than 247 kPa, according to the data by Loree et al. (1994). The maximum MPS for each combination of parameters is shown in Figure 2.8 and 2.9. In Figure 2.8, the two most influential parameter, the fibrous cap thickness and the lipid core width (fc and w), were chosen as the variable represented by the surfaces. In each subfigure, five surfaces are presented, one for each sr variation. The safety threshold plane at 247 kPa is presented. The results obtained for the three cases (with axial RS, with circumferential and axial RS and without them) have been compared both in Figure 2.8 and 2.9. Chapter 2. 3D Parametric Study 26 Figure 2.11: MPS (kPa). Comparison between 3D and plane strain plaque model. A - 3D model vessel. B - Plane strain model. Chapter 3 Real geometry versus idealized geometry In order to validate the assumption of an idealized geometry of the atherosclerotic artery, a patient-specific geometry of a coronary vessel has been segmented and modelled using the same material parameters, boundary conditions and dimensions. The performance of the 3D model used to carried out the 3D parametric analysis have been compared with a in-vivo reconstruction . A real geometry of a left coronary artery with atherosclerosis disease was obtained by in-vivo intravascular ultrasound (IVUS) images. The artery, which belong to an adult male patient of 50 years old, was imaged using an automatic pullback with a speed of 0.5mm/s, from the distality of the lesion to the tip of the guiding catheter, Figure 3.1.a. The MIMICS 10.0 commercial code was used to reconstruct the 20 cross-sections of the human coronary vulnerable plaque. The 3D plaque geometry of the patient is reconstructed by piling up and following the trajectory of the center of the catheter, Figure 3.1. Plaque components were characterized by their appearance on IVUS images, the contours delimiting lumen border, media, adventitia and plaque components (lipid core and atheroma plaque) were manually traced on each IVUS cross-sectional image. Figures 3.1.b and Figure 3.1.c show a cross and transversal section, respectively, of the 3D in-vivo atherosclerotic vessel obtained by IVUS. A noteworthy remark is that the cross section obtained by IVUS shown in Figure 3.1.a is the same cross section as that shown in 3.1.b. 27 Chapter 3. Real geometry versus idealized geometry 28 (a) One of the 20 crosssections of the human coronary obtained by IVUS (b) Cross section reconstructed from a real geometry (c) Transversal section reconstructed from a 3D real geometry Figure 3.1: 3D reconstruction of a vessel with atherosclerotic lesion obtained by IVUS. Notice that the cross section obtained by IVUS (a) is the same cross section as that shown in (b). Dimensions of lipid core and fibrous cap thickness were measured in the geometrical reconstruction from the IVUS for the purposes of comparison with a case with similar dimensions and properties in the 3D parametric study. The model was meshed with 252.216 linear tetrahedral elements of type C3D4H of similar size size to those of the 3D parametric mesh, and 52.312 nodes, Figures 3.1.b and 3.1.c. The dimensions measured on the real geometry obtained by IVUS images are shown in Table 3.1. l(mm) r(mm) fc(mm) w(%) 10 0.7 0.04 55 Table 3.1: Parameters measured in the 3D reconstruction of a vessel with atherosclerotic lesion obtained by IVUS Amongst all the 3D parametric models, the geometry with more similar parameters to the real geometry segmented from IVUS has been chosen, see Table 3.2. The lipid core length is 2 mm shorter in the idealized geometry than in the specific-patient model because 8 mm is the largest value studied for the lipid core length in the parametric study. However, Figure 2.9.a shows that the influence of lipid core length variation on the maximum MPS is not relevant for long lipid core lengths, so this can be considered as a good approximation. Furthermore, in order to simplify the finite element analysis, both models (idealized and patient-specific) have been simulated without RS. Chapter 3. Real geometry versus idealized geometry 29 l(mm) r(mm) fc(mm) w(%) 8 0.7 0.05 60 Table 3.2: Parameters selected to compare the idealized 3D with the real geometry. Note that the lipid core length is 2 mm shorter, however Figure 2.9.a shows that the influence of lipid core length variation is not relevant for long lipid core lengths. Both in the real reconstructed vessel and in the idealized model, the spatial distribution of the MPS is quite similar and the maxima are both located in the fibrous cap. The maximum MPS of the real and idealized geometries are 322 [kPa] (Figure 3.2.a) and 305 [kPa] (Figure 3.2.b), respectively, showing an error of 5%, and the MPS maps are similar in both cases, showing the validity of the idealized geometry, Figure 3.2. In both models the maximum MPS is greater than 247 kPa, so according to the defined threshold of 247 kPa defined in previous chapters, both cases could be considered as vulnerable plaque. Figure 3.2: MPS distribution (kPa). Comparison between the real geometry reconstructed from the IVUS and the idealized geometry of the 3D parametric study. A - Real geometry reconstructed from the IVUS. B - Idealized geometry. Chapter 4 Conclusions and Future work 4.1 Conclusions Quantifying the mechanical stress in the wall of an atherosclerotic vessel and, more specifically, in the fibrous cap, is a vital step in predicting the risk of plaque rupture based on biomechanical features, especially in 3D geometries [12]. For this reason, the mechanical behaviour of a 3D parametric atheroma plaque in a coronary vessel with atherosclerosis disease has been studied in this work by varying the four most influential geometrical parameters; lipid core length (l), stenosis ratio (sr), fibrous cap thickness (fc) and lipid core width (w), and including RS effects. Static finite element analyses were performed in order to study a group of idealized plaque morphologies and to try to predict the vulnerable plaque rupture. Furthermore, the performance of 2D plane strain plaque models versus 3D models have been analyzed, concluding that plane strain plaque models are not accurate enough to calculate MPS distribution. Plane strain hypothesis overestimates the maximum MPS, 510 kPa for plane strain versus 205 kPa for 3D analysis in the case presented in Figure 2.11, leading to similar results as those presented by Ohayon et al. [49, 47]. In the case presented here the plane strain analysis exceeds the threshold of 247 kPa, but the 3D analysis does not. This shows the limitation of the plane strain approach for such complex plaques. In addition, Figure 3.2 summarizes a first validation of the 3D idealized model, showing that the 3D model behaves in a very realistic way. Historically, fibrous cap thickness has been considered the most important and almost the exclusive factor determining plaque vulnerability [4, 8]. To date, very few 3D computational studies have been performed specifically to investigate the effect of the lipid core size on plaque stress distribution. Loree et al. (1992) and Tang et al. (2004) used a finite element model to study the influence of the lipid core width on plaque stress distribution in 30 Chapter 4. Conclusions and Future work 31 a small number of distinct models (n=6 and n=3, respectively). Imoto et al. (2005) came to the conclusion that the size of the lipid core had no influence on the peak circumferential stress, but as their models are particularized to concentric plaques, their conclusions are not generalizable to eccentric coronary lesions. Finet et al. (2004) performed a 2D parametric study which showed that a combination of measures including the arterial remodelling index, the cap thickness, and the necrotic core area or thickness are necessary for prediction. Ohayon et al. (2005) compared the in-vivo performance of 2D and 3D finite element models and concluded that 2D analysis tends to overestimate the amplitude of the maximum MPS, however residual stresses are not considered. The distribution of RS and its effects on the stress field in 3D parametric atherosclerotic coronary plaques have never been studied in detail. Owing to the difficulty of estimation stresses and strains in real geometries, the influence of residual stresses is usually ignored in structural analyses intended to predict plaque rupture location. Ohayon et al. [47] assessed RS and its impact on the in vivo stress distribution in human vulnerable coronary plaques, studying six real pathological epicardial coronary artery samples. Regarding the importance of RS, many authors have studied the role of circumferential RS, but mainly in non-stenotic arteries. Holzapfel et al. [28] performed statistical analysis to test for significant correlations between age and axial in situ stretch and there were significant negative correlations between both. This suggests that axial in situ stretches of the human LAD coronary artery decrease with age. Varnava et al. [65] simulated the effects of tissue aging on residual strain in the main right and left (ramus circumflexus) human coronary arteries, based on experimental data and they found that experimental opening angle scatters considerably with age. The factors affecting the opening angle are age, sex and the degree of atherosclerosis. Besides, their study showed the effect of including the circumferential RS in the final stress distribution. The vessel artery wall is under tension in the inner layers and under compressive stress in the outer layers, for positive opening angles. The difference between both layers increases as the opening angle increases. This fact tends to make the circumferential stress more uniform in the arterial wall under the constant internal pressure. The findings in the present work show a high dependency on some purely 3D parameters and factors on the MPS distribution, such as the lipid core length and the axial RS, affecting the vulnerability risk of the plaques. Figure 2.9 and 2.10 clearly show the influence of residual stresses since the vulnerable unsafe areas changes when residual stresses are considered. The predominant trend is that the incorporation of axial RS increases the maximum MPS. However, the incorporation of axial and circumferential RS reduces the maximum MPS. Therefore, 3D plaque models could produce more accurate predictions, and plane strain plaque models could not be enough to calculate a sufficiently accurate MPS distribution. Plane strain models not only overestimate the maximum MPS, as it is shown in the literature by Ohayon et al. [49, 35, 47], but they also miss RS effects and Chapter 4. Conclusions and Future work 32 other features as fibre orientation, which can not be consider in 2D models [28]. The general trend observed in this study is that the maximum MPS increases with the lipid core length, the lumen radius and the lipid core width, and also when the fibrous cap thickness decreases. Figure 2.8 and 2.9 and Table 2.3, 2.4 and 2.5 show that the most of the parameter combinations have MPS values lower than 247 kPa (VF<1), however, an important vulnerable plaque region, where the maximum MPS value is higher than the safety threshold (VF≥1) was found. This region is generally formed by low fibrous cap thickness values. Moreover, this unsafe region changes from the analysis in which axial RS are considered to the analysis in which RS effects are neglected (Figure 2.9) since when just axial RS are included (without internal pressure), the circumferential stresses are positive. Thus, if axial RS and internal pressure is imposed, the circumferential stresses are higher than without considering axial RS. To summarize, the vulnerable plaque region corresponds to a combination of the following parameters: w≥50%, fc ≤0.088 mm for any lumen radius and lipid core length. Table 4.1 shows the fc and wvulnerable limits for each land for all values of the lumen. With axial RS With circumferential and axial RS Without RS fc (mm) w (%) fc (mm) w (%) fc (mm) w (%) l=1 fc ≤0.088 w≥50% No vulnerable zone fc ≤0.077 w≥50% l=2 fc ≤0.085 No limit fc ≤0.025 w≥95% fc ≤0.050 w≥58% l=4 fc ≤0.080 No limit fc ≤0.052 w≥62% fc ≤0.079 w≥52% l=6 fc ≤0.081 No limit fc ≤0.055 w≥64% fc ≤0.079 w≥50% l=8 fc ≤0.082 No limit fc ≤0.080 w≥38% fc ≤0.080 w≥50% Table 4.1: Summary of vulnerable limits of fc and wfor each different lindependently of the sr parameter. The fibrous cap thickness and the lipid core width and length have been shown in this study to be critical geometric parameters to the overall plaque stability, whereas it has been shown that the lumen radius influence is lower, but non-negligible. The influence of these parameters on plaque rupture is shown in Figure 2.6. Similar results were previously obtained by other authors. Ohayon et al. (2008) obtained slight higher limits of these parameters, probably because they performed a 2D study. However, the global trends were similar. The remodelling index (a parameter equivalent to the stenosis ratio) and the lipid core width had a positive correlation with the maximum MPS, while the fibrous cap thickness had a negative correlation with the maximum MPS. Similar trends were found by Virmani et al. (2000) where they suggested that atherosclerotic lesions with a fibrous cap thickness of less than 65 µare most likely to rupture. Several studies in the literature shown that plaques containing a highly thrombogenic lipid-rich core are more at risk for rupture if the size of the lipid core is large and is less consistent. Several investigators have reported on the relation of the amount of extracellular gruel and plaque fissuring [23, 14, 51, 15]. Davies et al. [14] estimated that Chapter 4. Conclusions and Future work 33 when at least 40% of the plaque consists of lipid, an atheroma is at risk for rupture. The 3D parametric study presented on this work can be considered as an additional step towards the development of a tool to assist clinicians in the identification of vulnerable atheroma plaques. The large-scale computational analysis aids the clinical staff to identify the critical morphological parameters that indicate plaque vulnerability and the likelihood of rupture. 4.2 Limitations Some limitations of this study should be mentioned. First, an idealized straight geometry has been used to perform the parametric analysis. Second, the material model was assumed to be isotropic and incompressible for the fibrous plaque and the lipid core and anisotropic and incompressible for the wall vessel. However, these assumptions have been widely accepted as allowable for the assessment of the biomechanical properties of atherosclerotic lesions [10, 41]. Third, the material properties and residual stresses have been taken from experimental data in the literature [32, 28, 67]. There is no data of axial RS for isolated plaques in the literature, so the axial RS used corresponds to a non-stenotic artery. The opening angle was assumed to be constant for all of the geometries though it is known that it depends on the plaque geometry. Fourth, viscoelastic effects were not considered [2, 3, 52]. Fifth, the analysis does not reproduce the pulsatile nature of physiological blood pressure. Also, the fluid-structure interaction effects resulting from such cyclic loading were not considered [34]. It was assumed that there were no shear stresses, torques, time-varying forces or flow-related forces. Only static blood pressure was considered to be acting on the lesion in the models. Nevertheless, it has been documented that the effect of fluid shear stress is insignificant when compared to the effect of tensile wall stresses as a direct component in plaque fracture dynamics [30, 26], although it is considered essential in plaque formation and growth. The estimation of stresses induced by the static pressure load has been proved to be valid to identify stress concentrations in atherosclerotic lesions [10] since the location of stress concentration does not significantly differ between models including static pressure and models with complex dynamic pressure profiles. Sixth, although similar studies in the literature include other parameters such as the lipid core angle or the remodelling index which is related to the lumen radius, the present study takes into account four of the most influential parameters [67, 48]. Several preliminary tests were performed to exclude the lipid core angle as an influential parameter, see Supplementary data. Seventh, calcifications were not considered in order to simplify the study [7]. Different properties of the atheroma plaque were not taken into account in this parametric study. Properties of calcified, cellular and hypocellular plaques have been identified by other authors [42]. Finally, the 3D parametric study only could be validated qualitatively. It is not possible to measure the stress concentration in real Chapter 4. Conclusions and Future work 34 atheroma plaques and to correlate with the main geometrical risk factors in vivo conditions and later verify the plaque rupture. Actually, in the literature, the only way to extract stress information is by performing computational simulations reproducing geometry and in vivo conditions, and validate the model qualitatively, measuring the geometrical risk factors, but not measuring directly the stresses. 4.3 Future work Following the investigations described in this master work, a number of projects could be taken up, mainly focused on improving the model and its goal of helping to the clinicians to make decisions on atheroma plaque vulnerability. More careful examination of stress distributions in plaques reveals that it may be caused by the local stress behaviors caused by cap thinning, inflammation, macroscopic heterogeneity, and recently, the presence of microcalcifications, at critical sites. However, the role of microcalcifications is not yet fully understood and most finite element models of blood vessels with atheroma plaque do not take into account the heterogeneity of the plaque constituents at the micro-scale. Some studies indicate beneficial effects in stabilizing the plaque [10, 30], whereas others suggest that the microcalcifications increase the plaque vulnerability and shift the maximal principal stress (MPS) to the region of the microcalcification [7, 71]. Further parametric studies could include microcalcifications and also take into consideration the position and size of the microcalcification as influential parameters on the MPS. Figure 4.1 shows the central cross section of a possible further parametric study including microcalcifications and considering the microcalcification position angle (α), the microcalcification diameter (d), the microcalcification eccentricity measured by dand again the fibrous cap thickness (fc) since it is the most influential parameter on the atheroma vulnerability. Figure 4.1: Further parametric study. Possible new geometrical parameters related to the microcalcification shown on the cross-section coronary model. Chapter 4. Conclusions and Future work 35 In addition, a 3D model with a eccentric atheroma plaque and negative remodelling have been used in this work, however, other configurations could be analyzed, varying the atheroma plaque distribution (concentric or eccentric) or taking account the remodelling defined by Glagov et al. [24]. Their study demonstrated that coronary arteries can enlarge in response to the development of atherosclerotic plaques. Coronary arteries may respond to plaque growth by either outward expansion of the vessel wall (positive) or vessel shrinkage (negative). The compensatory remodelling process can maintain luminal dimensions during early atherosclerosis. These plaques grow further and the plaque does not generally begin to encroach on the lumen until it occupies 40% of the cross-sectional area. Figure 4.2 summarize other possible configurations of the atheroma plaque. Figure 4.2: (a) Positive arterial growth. (b) Negative arterial growth with an eccentric atheroma plaque. (c) Negative arterial growth with a concentric atheroma plaque. Finally, it would be of interest to develop a quantitative method for cumulative risk assessment of vulnerable patients based on atheroma plaque morphology which could replace the time consuming biomechanical simulations used in cardiovascular mechanics. A parametric tool based on machine learning techniques (such as Artificial Neural Networks (ANNs) and Support Vector Machines (SVMs) and used to predict the atheroma plaque rupture could be implemented from the results of this 3D parametric study (input). Procedures to detect plaque prone to rupture and to predict rupture location are very valuable for clinical diagnosis. Nowadays clinical procedures for detection of these vulnerable plaques are only performed by image analysis. The use of FEM computations presents the disadvantage of very high computational cost, usually hours or even days, when an immediate response is required. However, FEM analysis are used in pre-operatory surgical planning when clinical staff have enough time to perform the computational model and analyze the results. The main objective of this tool would be to search for alternatives to direct FEM simulations in a specific clinical field, detection of vulnerable plaques, when an instantaneous response is needed. Summing up, the procedure proposed would be carried as follows: for a specific patient, clinical staff should measure just four parameters in standard IVUS images and then, by using the ANN or SVM techniques, they would have an immediate response on the atheroma plaque vulnerability. The ANN or SVM BIBLIOGRAPHY 42 [32] Jaroslav, V., Jiri, S., Dita, V., Karel, V., 1999. Residual strain in human atherosclerotic coronary arteries and age related geometrical changes. Bio- Medical Materials and Engineering 9, 311–317. [33] Kips, J. G., Segers, P., Van Bortel, L. M., 2008. Identifying the vulnerable plaque: A review of invasive and non-invasive imaging modalities. Artery Research 2, 21–34. [34] Kock, S. A., Nygaard, J. V., Eldrup, N., Fr¨und, E.-T., Klærke, A., Paaske, W. P., Falk, E., Kim, W. Y., 2008. Mechanical stresses in carotid plaques using mri-based fluid-structure interaction models. Journal of Biomechanics 41, 1651 – 1658. [35] Krishna Kumar, R., Balakrishnan, K. R., 2005. Influence of lumen shape and vessel geometry on plaque stresses: possible role in the increased vulnerability of a remodelled vessel and the shoulder of a plaque. Heart 91, 1459–1465. [36] Kyriacou, S., Schwab, C., Humphrey, J., 1996. Finite element analysis of non-linear orthotropic hyperelastic membranes. Computational Mechanics 18, 269–278. [37] Lee, R., 2000. Atherosclerotic lesion mechanisc versus biology. Zeitschrift fur Kardiologie 89, 80–84. [38] Lee, R., Loree, H., Cheng, G., Lieberman, E., Jaramillo, N., Schoen, F., 1993. Computational structural analysis based on intravascular ultrasound imaging before in vitro angioplasty: Prediction of plaque fracture locations. Journal of the American College of Cardiology 21, 777–782. BIBLIOGRAPHY 43 [39] Lendon, C. L., Davies, M., Born, G., Richardson, P., 1991. Atherosclerotic plaque caps are locally weakened when macrophages density is increased. Atherosclerosis 87, 87–90. [40] Lloyd-Jones, D., Adams, R., Carnethon, M., De Simone, G., Ferguson, T. B., Flegal, K., Ford, E., Furie, K., Go, A., Greenlund, K., Haase, N., Hailpern, S., Ho, M., Howard, V., Kissela, B., Kittner, S., Lackland, D., Lisabeth, L., Marelli, A., McDermott, M., Meigs, J., Mozaffarian, D., Nichol, G., O’Donnell, C., Roger, V., Rosamond, W., Sacco, R., Sorlie, P., Stafford, R., Steinberger, J., Thom, T., Wasserthiel-Smoller, S., Wong, N., Wylie-Rosett, J., Hong, Y., for the American Heart Association Statistics Committee, Subcommittee, S. S., 2009. Heart disease and stroke statistics–2009 update: A report from the american heart association statistics committee and stroke statistics subcommittee. Circulation 119, 21–181. [41] Loree, H., Kamm, R., Stringfellow, R., Lee, R., 1992. Effects of fibrous cap thickness on peak circumferential stress in model atherosclerotic vessels. Circulation Research 71, 850–858. [42] Loree, H. M., Grodzinsky, A. J., Park, S. Y., Gibson, L. J., Lee, R. T., 1994. Static circumferential tangential modulus of human atherosclerotic tissue. Journal of Biomechanics 27, 195–204. [43] Marquardt, D. W., 1963. An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 11, 431–441. BIBLIOGRAPHY 44 [44] Mathers, C., Murray, C., 2006. Global burden of disease and risk factors, chapter the burden of disease and mortality by condition: Data, methods, and results for 2001. Oxford University Press , 45–240. [45] Moreno, P. R., Purushothaman, K. R., Fuster, V., O’Connor, W. N., 2002. Intimomedial interface damage and adventitial inflammation is increased beneath disrupted atherosclerosis in the aorta: Implications for plaque vulnerability. Circulation 105, 2504–2511. [46] Naghavi, M., Libby, P., Falk, E., Casscells, S. W., Litovsky, S., Rumberger, J., Badimon, J. J., Stefanadis, C., Moreno, P., Pasterkamp, G., Fayad, Z., Stone, P. H., Waxman, S., Raggi, P., Madjid, M., Zarrabi, A., Burke, A., Yuan, C., Fitzgerald, P. J., Siscovick, D. S., de Korte, C. L., Aikawa, M., Juhani Airaksinen, K., Assmann, G., Becker, C. R., Chesebro, J. H., Farb, A., Galis, Z. S., Jackson, C., Jang, I.-K., Koenig, W., Lodder, R. A., March, K., Demirovic, J., Navab, M., Priori, S. G., Rekhter, M. D., Bahr, R., Grundy, S. M., Mehran, R., Colombo, A., Boerwinkle, E., Ballantyne, C., Insull, William, J., Schwartz, R. S., Vogel, R., Serruys, P. W., Hansson, G. K., Faxon, D. P., Kaul, S., Drexler, H., Greenland, P., Muller, J. E., Virmani, R., Ridker, P. M., Zipes, D. P., Shah, P. K., Willerson, J. T., 2003. From vulnerable plaque to vulnerable patient: A call for new definitions and risk assessment strategies: Part I. Circulation 108, 1664–1672. [47] Ohayon, J., Dubreuil, O., Tracqui, P., Le Floc’h, S., Rioufol, G., Chalabreysse, L., Thivolet, F., Pettigrew, R. I., Finet, G., 2007. Influence of residual stress/strain BIBLIOGRAPHY 45 on the biomechanical stability of vulnerable coronary plaques: potential impact for evaluating the risk of plaque rupture. American Journal of Physiology - Heart 293, H1987–H1996. [48] Ohayon, J., Finet, G., Gharib, A. M., Herzka, D. A., Tracqui, P., Heroux, J., Rioufol, G., Kotys, M. S., Elagha, A., Pettigrew, R. I., 2008. Necrotic core thickness and positive arterial remodeling index: emergent biomechanical factors for evaluating the risk of plaque rupture. American Journal of Physiology - Heart 295, H717–H727. [49] Ohayon, J., Finet, G., Treyve, F., Rioufol, G., Dubreuil., O., 2005. A three dimensional finite element analysis of stress distribution in a coronary atherosclerotic plaque: In-vivo prediction of plaque rupture location. Biomechanics Applied to Computer Assisted Surgery 17, 225–241. [50] Ohayon, J., Teppaz, P., Finet, G., Rioufol, G., 2001. In-vivo prediction of human coronary plaque rupture location using intravascular and finite element method. Coronary Artery Disease 12, 655–663. [51] Pasterkamp, G., de Kleijn, D. P., Borst, C., 2000. Arterial remodeling in atherosclerosis, restenosis and after alteration of blood flow: potential mechanisms and clinical implications. Cardiovascular Researcher 45, 843–852. [52] Pe˜na, E., Alastru´e, V., Laborda, A., Mart´ınez, M., Doblar´e, M., 2010. A constitutive formulation of vascular tissue mechanics including viscoelasticity and softening behaviour. Journal of Biomechanics 43, 984–989. BIBLIOGRAPHY 46 [53] Pe˜na, E., del Palomar, A. P., Calvo, B., Mart´ınez, M. A., Doblar´e, M., 2007. Computational modelling of diarthrodial joints. Physiological, pathological and pos-surgery simulations. Archives of Computational Methods in Engineering 14(1), 47–91. [54] Roger, V. L., Go, A. S., Lloyd-Jones, D. M., Adams, R. J., Berry, J. D., Brown, T. M., Carnethon, M. R., Dai, S., de Simone, G., Ford, E. S., Fox, C. S., Fullerton, H. J., Gillespie, C., Greenlund, K. J., Hailpern, S. M., Heit, J. A., Ho, P. M., Howard, V. J., Kissela, B. M., Kittner, S. J., Lackland, D. T., Lichtman, J. H., Lisabeth, L. D., Makuc, D. M., Marcus, G. M., Marelli, A., Matchar, D. B., McDermott, M. M., Meigs, J. B., Moy, C. S., Mozaffarian, D., Mussolino, M. E., Nichol, G., Paynter, N. P., Rosamond, W. D., Sorlie, P. D., Stafford, R. S., Turan, T. N., Turner, M. B., Wong, N. D., Wylie-Rosett, J., 2011. Heart disease and stroke statistics - 2011 update. Circulation 123 (4), e18–e209. [55] Salunke, N. V., Topoleski, L. D. T., Humphrey, J. D., Mergner, W. J., 2001. Compressive stress-relaxation of human atherosclerotic plaque. Journal of Biomedical Materials Research Part A 55, 236–241. [56] Simo, J. C., 1991. Algorithms for Static and Dynamic Multiplicative Plasticity that Preserve the Classical Return Mapping Schemes of the Infinitesimal Theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112. [57] Smedby, O., 1998. Geometrical risk factors for atherosclerosis in the femoral artery: A longitudinal angiographic study. Annals of Biomedical Engineering 26, 391–397. BIBLIOGRAPHY 47 [58] Spencer, A. J. M., 1971. Theory of invariants. In: Continuum Physics. Academic Press, New York, pp. 239–253. [59] Tang, D., Teng, Z., Canton, G., Yang, C., Ferguson, M., Huang, X., Zheng, J., Woodard, P. K., Yuan, C., 2009. Sites of rupture in human atherosclerotic carotid plaques are associated with high structural stresses: An in vivo MRI-based 3D fluidstructure interaction study. Stroke 40, 3258–3263. [60] Tang, D., Yang, C., Zheng, J., Woodard, P., Saffitz, J., Petruccelli, J., Sicard, G., Yuan, C., 2005. Local maximal stress hypothesis and computational plaque vulnerability index for atherosclerotic plaque assessment. Annals of Biomedical Engineering 33, 1789–1801. [61] Tang, T., Howarth, S., Li, Z., Miller, S., Graves, M., U-King-Im, J., Trivedi, R., Walsh, S., Brown, A., Kirkpatrick, P., Gaunt, M., Gillard, J., 2008. Correlation of carotid atheromatous plaque inflammation with biomechanical stress: Utility of USPIO enhanced MR imaging and finite element analysis. Atherosclerosis 196, 879 – 887. [62] Thubrikar, M., 2007. Vascular mechanics and pathology. Spinger Science. [63] Van der Wal, A. C., Becker, A. E., 1999. Atherosclerotic plaque rupture - pathologic basis of plaque stability and instability. Cardiovascular Research 41, 334–344. [64] VanEpps, J. S., Vorp, D. A., 2007. Mechanopathobiology of atherogenesis: A review. Journal of Surgical Research 142, 202–217. BIBLIOGRAPHY 48 [65] Varnava, A. M., Mills, P. G., Davies, M. J., 2002. Relationship between coronary artery remodeling and plaque vulnerability. Circulation 105, 939–943. [66] Vengrenyuk, Y., Carlier, S., Xanthos, S., Cardoso, L., Ganatos, P., Virmani, R., Einav, S., Gilchrist, L., Weinbaum, S., 2006. A hypothesis for vulnerable plaque rupture due to stress-induced debonding around cellular microcalcifications in thin fibrous caps. Proceedings of the National Academy of Sciences 103, 14678–14683. [67] Versluis, A., Bank, A. J., Douglas, W. H., 2006. Fatigue and plaque rupture in myocardial infarction. Journal of Biomechanics 39, 339–347. [68] Virmani, R., Burke, A. P., Farb, A., Kolodgie, F. D., 2006. Pathology of the vulnerable plaque. Journal of the American College of Cardiology 47, C13–18. [69] Virmani, R., Kolodgie, F. D., Burke, A. P., Farb, A., Schwartz, S. M., 2000. Lessons from sudden coronary death: A comprehensive morphological classification scheme for atherosclerotic lesions. Arteriosclerosis, Thrombosis, and Vascular Biology 20, 1262–1275. [70] Weiss, J. A., 1994. A constitutive model and finite element representation for transversely isotropic soft tissues. Ph.D. thesis, The University of Utah. [71] Wenk, J. F., 2011. Numerical modeling of stress in stenotic arteries with microcalcifications: A parameter sensitivity study. ASME Journal of Biomechanical Engineering 133, 014503. BIBLIOGRAPHY 49 [72] Williamson, S. D., Lam, Y., Younis, H. F., Huang, H., Patel, S., Kaazempur-Mofrad, M. R., Kamm, R. D., 2003. On the sensitivity of wall stresses in diseased arteries to variable material properties. ASME Journal of Biomechanical Engineering 125, 147–155.