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2013 56 Pablo Sáez Viñas Theoretical and computational study of the mechano-biology in hypertension disease Departamento Director/es Ingeniería Mecánica Peña Baquedano, Estefanía Martínez Barca, Miguel Ángel Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Pablo Sáez Viñas THEORETICAL AND COMPUTATIONAL STUDY OF THE MECHANO-BIOLOGY IN HYPERTENSION DISEASE Director/es Ingeniería Mecánica Peña Baquedano, Estefanía Martínez Barca, Miguel Ángel Tesis Doctoral Autor 2013 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Theoretical and computational study of the mechano-biology in hypertension disease Thesis by Pablo Saez For the Degree of Doctor in Philosophy University of Zaragoza Zaragoza, Spain Theoretical and computational study of the mechano-biology in hypertension disease Thesis by Pablo Sáez For the Degree of Doctor in Philosophy Faculty advisors: Prof. Miguel Ángel Martínez Barca Dr. Estefanía Peña Baquedano University of Zaragoza Zaragoza, Spain Acknoweldge The work of this PhD thesis was carried out at the department of Mechanical Engineering of the University of Zaragoza, first at the GEMM and later in the AMB lab, both group being part of the I3A, a broader cluster of research institutes. We were quite bunch of people back on these days so this first acknowledge go to all of those how are or were part of them and that we share time together. I would like to thanks my supervisors, Dr. Estefania Peña and Prof. Miguel Ángel Martínez for the opportunity I had to be part of this great group. I will also like to thanks them for the freedom that they gave me and their confidence in my research during my thesis and all the time they spent on my work. I would also like to acknowledge to Prof. Manuel Doblaré which was the PI of my graduate grant, BES-2009-028593, which made me able to accomplish this thesis. I want to thanks also some professors that have been and will be a reference and that I had the pleasure to work with. Prof. Andreas Menzel with who I had the opportunity to work in its group at TU Dormund. To Prof. Ellen Kuhl from Stanford University. She has been a fundamental part of this work and I will always be in debt for the six months I spent at her lab. I would also like to thank Prof. John M.Tarbell, from the City College of New York, for the opportunity to work in his lab. Finally, I always will be in debt with Prof. Micheal Ortiz, who accept me to spend three months in his group. In was astonishing to spend that time in one of the most amazing computational solid mechanics group in the world. And above all, I would like to remember firstly to all my class colleges: Myriam Cilla, Belen Hernandez, Carlos Borau, Fares El Halabi and Sergio Gabarre. It was amazing to share all this time with you guys and I will miss you all!! Hopefully we will end up somewhere close enough to hang out together again! I would also like to thank Mauro. You were a great friend here and a great travel mate. Also to JF that helps, not only me but everyone else, in the lab to anything we ask him for. I hope your new life in the boot-shaped country goes 6 great. Sarita for her positiveness and happiness and for being a great person. Marina and Raquel!!...They both are making me to wish never leave the lab, at least during they stay in. I was really great all the useless time we spend taking coffee, getting a nice tone or just hanging out!! And many others: Noelia, Mar, Jorge, Jamal, Clara, Carlos, Enrique, Berto, V. Alastrue., V. Acosta, Galavis, Olfa, Libardo, Horacio, Mena, Tobias, Claudia, Sonja, Ronny, Alex, Manuel, Jon, Adrian, Nick... I sure I will miss you guys wherever I end up! List of Figures 1.1 A widespread idea of basic science (Tropea, 2011). . . . . . . . . . 23 1.2 Representation of the different circulatory system. . . . . . . . . . 26 1.3 Workflow in current biomechanics research field. . . . . . . . . . . 35 1.4 Computational model of remodeling of fibers in a tendon-like structure (Kuhl and Holzapfel, 2007). . . . . . . . . . . . . . . . . . . . 38 1.5 A computational model of density growth for bone (Waffenschmidt etal.,2012). .............................. 39 1.6 Computational tensegrity-type model of a cell (Kardas et al., 2012). 40 1.7 Experimental results of vascular smooth muscle cell cyclically loaded (Hayakawa et al., 2001) . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Microstrutural representation of rat aorta (O’Connell et al., 2008). 63 3.2 Microstrutural components of arterial tissue. . . . . . . . . . . . . 66 3.3 Representation of the longitudinal (l) and circumferential (c) strips and the angle of the fibers. . . . . . . . . . . . . . . . . . . . . . . 71 13 14 LIST OF FIGURES 3.4 Bingham fitted functions for the micro structural organization of collagen by Garcia (2012). It represented the probability of finding a collagen fibrils in a given direction of space. . . . . . . . . . . . 74 3.5 Numerical fitting with and without considering residual stresses using data from Sommer and Holzapfel (2012). . . . . . . . . . . . 77 3.6 Fitting results of the mechanical behavior of individual collagen fibrils from data presented in the literature. . . . . . . . . . . . . . 79 3.7 Schematic representation of the structural organization in the media of the carotid artery, being nthe preferential direction. . . . . 80 3.8 Uniaxial test in longitudinal and circumferential direction in a pig aortaartery................................ 81 3.9 Experimental and fitted curves of the uniaxial test of the proximal pigCCA. ................................ 84 3.10 Experimental and fitted curves of the uniaxial test of the distal pigCCA. ................................ 85 3.11 Experimental data presented by Sommer and Holzapfel (2012) and fitted curves using the model presented in Eq. 3.10. . . . . . . . . 89 3.12 Reconstruction of the carotid artery from human carotid CT image (Alastrue et al., 2010a). . . . . . . . . . . . . . . . . . . . . . . . . 91 3.13 Finite element mesh of a human carotid artery. . . . . . . . . . . . 92 3.14 Spatial distribution of the mechanical material parameters along thecarotidartery. ........................... 93 3.15 Some details of the fiber distribution defined along the carotid artery. 94 3.16 Representative slice of the CCA, ICA and ECA from the carotid model................................... 95 3.17 Maximum principal stress at different cuts in the whole CA for the homeostaticloads. ........................... 96 3.18 Maximum principal stress at different cuts in the whole CA for the hypertensiveloads............................ 97 3.19 Maximum principal stretch at homeostatic and hypertensive and ratio at hypertensive and homeostatic state from left to right. . . . 98 LIST OF FIGURES 15 4.1 (a) represents a sketch of the SMCs growth via hypertrophy and (b) a pulmonary artery of a hypertensive rat (Jeffery and Wanstall, 2001). ..................................103 4.2 Data collected in Feihl et al. (2008) showing the variability of the diameter (D), ration media thickness to internal diameter (M/D) and cross-sectional area (CSA). Bars indicate control group (C), essential hypertension (EH), essential hypertension and diabetes (EH+D) and renovascular hypertension (RVH). . . . . . . . . . . . 105 4.3 Kinematics of growth. Composition of a elastic deformation gradient Feand a growth tensor Fg. ..................108 4.4 Representation of a SMC based on experimental data from (Thakar etal.,2009). ..............................109 4.5 Growth variable and stress evolution. . . . . . . . . . . . . . . . . 115 4.6 Slice of the CCA presented in chapter 3. . . . . . . . . . . . . . . 118 4.7 Growth evolution in CCA section at different time steps. (b-e) shows the evolution of the the CCA carotid section at four time steps. ..................................118 4.8 Evolution of the SMC in the middle section of the media layer at differenttimesteps. ..........................118 4.9 Comparison of the evolution of growth for experimental findings (Fridez et al., 2002) and numerical study. . . . . . . . . . . . . . . 119 4.10 Growth variable and stress evolution. . . . . . . . . . . . . . . . . 119 4.11 Growth for the carotid geometry at t= 100 days for different longitudinal and transversal cuts . . . . . . . . . . . . . . . . . . . . . 121 4.12 Evolution of the growth (d-g) and maximal principal stresses in the adventitia (h-k) media (l-o)layer at different times steps in the sliceoftheCCA.............................122 4.13 Evolution of the growth (d-g) and maximal principal stresses in the adventitia (h-k) media (l-o)layer at different times steps in the sliceoftheICA. ............................123 16 LIST OF FIGURES 4.14 Evolution of the growth (d-g) and maximal principal stresses in the adventitia (h-k) media (l-o)layer at different times steps in the sliceoftheECA ............................124 5.1 Molecule representation (PDBe, 2013). . . . . . . . . . . . . . . . 129 5.2 Sensitivity of TGF −βcontent, ρTGF−β, with respect to sensitivity parameter γTGF−β, evolution exponent mTGF−β, initial value ρ∗ TGF−β, and saturation value λ∗ TGF-β. ................134 5.3 Evolution of TGF-β, TIMP, and MMP concentrations involved in progressive collagen turnover. . . . . . . . . . . . . . . . . . . . . 135 5.4 Evolution of the mass production of TGF-βdue to SMC activity in different depth cuts. . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.5 Evolution of the mass production of TGF-βdue to SMC activity in different transversal cuts. . . . . . . . . . . . . . . . . . . . . . . 138 5.6 Evolution of the mass production of TIMP due to SMC activity in different depth cuts. . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.7 Evolution of the mass production of TIMP due to SMC activity in different transversal cuts. . . . . . . . . . . . . . . . . . . . . . . 140 5.8 Evolution of the mass production of MMP due to SMC activity in differentdepthcuts. ..........................141 5.9 Evolution of the mass production of MMP due to SMC activity in different transversal cuts. . . . . . . . . . . . . . . . . . . . . . . . 142 5.10 Evolution of the mass production of (a) TGF, (b) MMP and (c) TIMP at six featured points due to SMC activity. . . . . . . . . . . 143 5.11 Sketch of the layers and boundary conditions in the mass transport problem. ................................148 5.12 Evolution of the TGF-βcontent in the adventitia layer due to mass transport in the carotid wall in different depth cuts. . . . . . . . . 150 5.13 Evolution of the TGF-βcontent in the adventitia layer due to mass transport in the carotid wall in different transversal cuts. . . . . . 151 5.14 Evolution of the TIMP content in the adventitia layer due to mass transport in the carotid wall in different depth cuts. . . . . . . . . 152 LIST OF FIGURES 17 5.15 Evolution of the TIMP content in the adventitia layer due to mass transport in the carotid wall in different transversal cuts. . . . . . 153 5.16 Evolution of the MMP content in the adventitia layer due to mass transport in the carotid wall in different depth cuts. . . . . . . . . 154 5.17 Evolution of the MMP content in the adventitia layer due to mass transport in the carotid wall in different transversal cuts. . . . . . 155 5.18 Evolution of the mass transport in the adventitia of TGF (a), MMP (b) and TIMP (c) at six featured points due to SMC activity.156 5.19 Evolution of the collagen content in hypertensive patients driven by an increase in TGF-βand a decrease in MMP. . . . . . . . . . 158 5.20 Evolution of the collagen content in ρcol/ρ∗ col due to fibroblast realease and MMP degradation in the media layer. . . . . . . . . . 161 5.21 Evolution of the collagen content in ρcol/ρ∗ col due to fibroblast realease and MMP degradation in the adventitia layer. . . . . . . . 162 5.22 Evolution of the collagen content in the media and adventitia stress at the six nodes depicted in Fig. 5.18. . . . . . . . . . . . . . . . . 163 6.1 Results presented by Hoyakawa and co-works in a cyclically stretched cells test over 90 minutes. Looking at the cells we observe two different processes, a reorientation of the mean direction of the cell and a morphological change of the cells due to adaptation of the internal cell elements, such as microtubules and stress fibers. A more pointed distribution of the cell is obtained at the end of the experiment while the reorientation of the mean direction is gathered at the beginning of the test. . . . . . . . . . . . . . . . . . . 171 6.2 Shapes of the Bingham ODF and probability density values for different values of κ1,2,3and Q=ex⊗ex+ey⊗ey+ez⊗ez. (a) and (b) represent the same distribution shape but rotated 90o, depending on where the non zero values are placed. (c) provides a planar-type distribution and (d) presents a von Mises distribution, that can be considered as a particularization of the Bingham ODF. 178 18 LIST OF FIGURES 6.3 Representation of the fibrils within the fibered structure for the Bingham ODF represented in Fig 6.2. . . . . . . . . . . . . . . . . 179 6.4 Cauchy stresses along stretching directions ex,eyand ezfor the concentration parameters given in Fig. 6.2b, c and d. . . . . . . . 180 6.5 Evolution of each integration direction. . . . . . . . . . . . . . . . . 184 6.6 Evolution of Qdriven by C(o),S(∗)and M(+) and detail fora specifictime. ..............................191 6.7 Evolution of stress, anisotropy and dissipation. . . . . . . . . . . . 192 6.8 Evolution of anisotropy for a 1D fiber. . . . . . . . . . . . . . . . . 193 6.9 Evolution of Qdriven by different quantities. Blue crosses indicate the results for ζΞ ∗= 2 and red circles those for ζΞ ∗= 10. . . . . . . 194 6.10 Anisotropy measure δfor the different driving quantities and ζΞ ∗= 2and ζΞ ∗= 10. .............................195 6.11 Dissipative evolution for the different driving quantities and ζΞ ∗= 2and ζΞ ∗= 10. .............................196 6.12 Evolution of principal direction of the ODF for different steps. (a) the distribution at step 1, (b) for step 33 ,(c) for step 66 and finally (d) at the end of the analysis. . . . . . . . . . . . . . . . . . . . . 197 6.13 Evolution of the microstructure by means of the diagonal components of ρ.................................199 6.14 Evolution of the fibrils for different steps. (a) the initial distribution with κ1,2,3= 0,8,10, (b) for step 33 leading to κ1,2,3= 0,28,38,κ1,2,3= 0,43,75 for step 66 in (c) the end of the analysis with κ1,2,3= 0,48,112 in(d). ....................200 6.15 Evolution of the distribution for different steps. (a) the distribution at step 1 with κ1,2,3= 0,8,10, (b) for step 33 leading to κ1,2,3= 0,28,38,κ1,2,3= 0,43,75 for step 66 in Fig. (c), the end of the analysis with κ1,2,3= 0,48,112 in (d). For t7→ ∞ the Bingham distribution leads to very concentrated von Mises distributions. . 201 6.16 Dissipation of the model for the different driven quantities and material parameters ¯ ζΞ ∗= 4,20. ...................202 LIST OF FIGURES 19 6.17 Description of the initial values of the ODF distribution for the three marked points and boundary condition of the finite element model with micro-structural information. . . . . . . . . . . . . . . 205 6.18 Stress field in X direction for the static simulation. The stress field shows a highly non-uniform distribution of the stresses due to the random distribution of the micro-structure. . . . . . . . . . . . . . 206 6.19 Evolution of the displacement and reaction forces for different boundary conditions problems. . . . . . . . . . . . . . . . . . . . . 207 6.20 Evolution of stress for different boundary value problems, Dirichlet (a-b) and Neumann (c-d) for remodeling driven by strain. . . . . . 208 6.21 Evolution of stress for different boundary value problems, Dirichlet (a-b) and Neumann (c-d) for remodeling driven by stress. . . . . . 209 6.22 Evolution of anisotropy for different boundary value problems, Dirichlet (a-b) and Neumann (c-d) for remodeling driven by strain. 210 6.23 Evolution of anisotropy for different boundary value problems, Dirichlet (a-b) and Neumann (c-d) for remodeling driven by stress. 211 6.24 Anisotropy evolution for the three points of interest for strain (C and CF) and stress (S and SF) driven problems and different boundary conditions, Dirichlet (C and S) and Neumann (CF andSF)..................................213 6.25 Cell morphology in a no-flow situation and athero-prone and protective flow condition reported by Dai et al. (2004). . . . . . . . . 214 6.26 Fluid simulation variables in the model of the human carotid. . . . 215 6.27 Structure of the cell from real images (a-b) and the proposed model (c-d)....................................216 6.28 Evolution of the strain driven problem with Dirichlet condition at differenttimesteps. ..........................217 6.29 Evolution of the cell structure for different combination of WSS and OSI values at different time steps. . . . . . . . . . . . . . . . . 218 6.30 Evolution of the SI for different values WSS values for experimental data (a) and the model presented here (b). . . . . . . . . . . . . . 219 A.1 Representación de la microestructura de una aorta de cerdo a través de Scanning Electron Microscopy (SEM) (O’Connell et al., 2008). ..................................231 A.2 Geometría reconstruida de una carótida humana para su análisis por el método de los elementos finitos. . . . . . . . . . . . . . . . 233 List of Tables 3.1 Material constants obtained for the proximal curves obtained from Garciaetal.(2011). .......................... 71 3.2 Material constants obtained for distal curves obtained from Garcia etal.(2011). .............................. 72 3.3 Composition of pig carotid artery of elastin, collagen and SMC measured by Garcia (2012). . . . . . . . . . . . . . . . . . . . . . 73 3.4 Results of material parameters fitted for inflation test for the CCA and ICA for the adventitia and media layers provided by Sommer andHolzapfel(2012). ......................... 76 3.5 Percentage of the material constituents, elastin, collagen and SMC given by Sommer and Holzapfel (2012). . . . . . . . . . . . . . . . 76 3.6 Weight percentage of each component respect to fry weight in human carotid arteries obtained by Humphrey and Rajagopal (2003). 78 3.7 Results of the OP1, solving for the Neo-Hookean parameter C10, for pig carotid specimens (Garcia et al., 2011). First and second line are the I-VI and VII-XIII specimens of the distal and proximal part respectively while third and fourth are the I-VI and VII-XIII specimens of the distal and proximal samples respectively. . . . . 82 21 3.8 Results of the OP2 for proximal pig carotid specimens. . . . . . . 86 3.9 results of the OP2 for distal pig carotid specimens. . . . . . . . . 86 3.10 Comparison of the fitting results presented in Garcia et al. (2011) and the fitting proposed in previous sections. . . . . . . . . . . . . 87 3.11 Material parameters of the OP2 procedure for human carotid specimens in Sommer and Holzapfel (2012). . . . . . . . . . . . . . . . 88 4.1 Algorithm for an implicit Euler scheme of volumetric growth . . . 112 5.1 Algorithm to compute the local main substances content using an implicit Euler backward scheme. . . . . . . . . . . . . . . . . . . . 133 5.2 Collagen turnover model. Material parameters for hypertensive casestudy. ...............................135 5.3 Collagen turnover model. Material parameters for hypertensive casestudy. ...............................157 5.4 Collagen turnover model. Relevant material parameters, their physical interpretations and units. . . . . . . . . . . . . . . . . . . 159 5.5 Variation on the lumen diameters at different positions due to the increase in collagen content . . . . . . . . . . . . . . . . . . . . . . 164 6.1 Time step in which equilibrium is achieved for different degrees of stiffness and different driving quantities. . . . . . . . . . . . . . . 192 6.2 Time step in which equilibrium is achieved for different driving quantities and material parameters ζΞ ∗= 2 and ζΞ ∗= 10. . . . . . 194 6.3 Energy dissipated for each type of driving quantity. . . . . . . . . 197 6.4 Energy dissipated per each type of driving quantity and for ¯ ζΞ ∗= 4,20. ..................................199 6.5 Algorithm to compute remodeling of cell-like structures. . . . . . . 204 The cardiovascular system 29 as age increases. In men, this increase levels off around age 45 to 50 years. In women, the increase continues sharply until age 60 to 65 years (Jousilahti et al., 1999). One of the proposed explanations for the gender difference in cardiovascular disease is hormonal difference. Among women, estrogen is the predominant sex hormone. Estrogen may have protective effects through glucose metabolism and hemostatic system, and it may have a direct effect on improving endothelial cell function. The production of estrogen decreases after menopause, and may change the female lipid metabolism toward a more atherogenic form by decreasing the HDL cholesterol level and by increasing LDL and total cholesterol levels. Aging (Jani and Rajkumar, 2006) is also associated with changes in the mechanical and structural properties of the vascular wall, which leads to the loss of arterial elasticity and reduced arterial compliance and may subsequently lead to coronary artery disease. In this thesis, we are going to focus in hypertension. In the next section we discuss briefly the main aspects of hypertension, from causes to consequences. 1.1.2 Hypertension Hypertension is a chronic medical condition in which the blood pressure is elevated. This makes the heart to work harder than normal to pump out blood through the blood vessels. Blood pressure involves two measurements, systolic and diastolic, which depend on whether the heart muscle is contracting (systole) or relaxed between beats (diastole). JNC7 (the Seventh Report of the Joint National Committee on Prevention of Detection, Evaluation and Treatment of High Blood Pressure) (Chobanian et al., 2003) classification set normal pressures below 140 mmHg systolic and 90 mmHg diastolic. High blood pressure is considered when it is at or above 140/90 mmHg for long time based on JNC7 classification. Recent international hypertension guidelines have range values to indicate a continuum of risk with higher blood pressures in the normal range. JNC7 uses the term prehypertension for blood pressure in the range 120-139 mmHg systolic and/or 80-89 mmHg diastolic. There is also a classification for different hypertension stages. Isolated systolic hypertension refers to elevated systolic pressure 30 Motivation with normal diastolic pressure and is common in the elderly. A first stages refers to blood pressure between 140 and 159 or diastolic pressure between 90 and 99. A second stage for people with systolic blood pressure exceeding 160 mmHg systolic or a diastolic pressure over 100 mmHg. Hypertension is classified as either primary (essential) hypertension or secondary hypertension; about 90 −95 of cases are categorized as "primary hypertension" which means that blood pressure increases with no obvious underlying medical cause (Carretero and Oparil, 2000). The remaining 5−10 of cases (secondary hypertension) are caused by other conditions that affect the kidneys, arteries, heart or endocrine system. Primary hypertension is the most common form of hypertension, accounting for 90 - 95 of all cases of hypertension (Carretero and Oparil, 2000). Blood pressure rises with aging and the risk of becoming hypertensive in later life is considerable (Narayan et al., 2003). Hypertension results from a complex interaction of genes and environmental factors. Numerous common genes with small effects on blood pressure have been identified as well as some rare genes with large effects on blood pressure but the genetic basis of hypertension is still poorly understood (Lifton et al., 2001). Several environmental factors influence blood pressure. The possible role of other factors such as stress, caffeine consumption, and vitamin D deficiency are less clear cut (Chobanian et al., 2003). Many mechanisms have been proposed to account for the rise in peripheral resistance in hypertension. Most evidences implicates either disturbances in renal salt and water handling, particularly abnormalities in the intrarenal reninangiotensin system and/or abnormalities of the sympathetic nervous system (Anderson et al., 1989). These mechanisms are not mutually exclusive and it is likely that both contribute to some extent in most cases of essential hypertension. It has also been suggested that endothelial dysfunction and vascular inflammation may also contribute to increase peripheral resistance and vascular damage in hypertension (Brunner et al., 2005) In most people with established essential (primary) hypertension, the increased resistance to blood flow (total peripheral resistance) trigger a high pres- The cardiovascular system 31 sure output while cardiac outflow remains normal (Conway, 1984). There is evidence that some younger people with prehypertension or ‘borderline hypertension’ have high cardiac output, an elevated heart rate and normal peripheral resistance, termed hyperkinetic borderline hypertension. These individuals develop the typical features of established essential hypertension in later life as their cardiac output falls and peripheral resistance rises with age. The increased peripheral resistance in established hypertension is mainly attributable to structural narrowing of small arteries and arterioles, although a reduction in the number or density of capillaries may also contribute (Folkow, 1982). Hypertension is also associated with decreased peripheral venous compliance which may increase venous return, increase cardiac preload and, ultimately, cause diastolic dysfunction (Safar and London, 1987). Whether increased active vasoconstriction plays a role in established essential hypertension is unclear (Schiffrin et al., 2000). Secondary hypertension results from an identifiable cause. Renal disease is the most common secondary cause of hypertension (O’Brien et al., 2007). Hypertension can also be caused by endocrine conditions, such as Cushing’s syndrome, hyperthyroidism, hypothyroidism, acromegaly, Conn’s syndrome or hyperaldosteronism, hyperparathyroidism and pheochromocytoma. Other causes of secondary hypertension include obesity, sleep apnea, pregnancy, coarctation of the aorta, excessive liquorice consumption and certain prescription medicines, herbal remedies and illegal drugs. The World Health Organization has identified hypertension, or high blood pressure, as the leading cause of cardiovascular mortality. Last data available shows that around one billion people all over the world, close to the 26%of the total adult population in the world suffer of hypertension. The World Hypertension League (WHL) recognized that over the 50%of the hypertensive population worldwide do not know their condition. Hypertension does not represent a danger by itself, but is a major risk factor for stroke, myocardial infarction, heart failure, aneurysms, peripheral arterial disease among many others and it is closely related to a shorten of time life. Contrarily, hypertension is the most important 32 Motivation preventable disease for early death worldwide. USA leads many of the survays and studies about hypertension as they do for suffering of hypertension and it is the most common chronic medical problem, prompting visits to primary health care providers in USA (Association, 2012). The American Heart Association estimated the direct and indirect costs of high blood pressure in 2010 as 76.6 billion. 34%of the US population (over the world mean) and African American adults have among the highest rates of hypertension in the world, being lower in whites and mexican americans. What change in US, as in other developed countries, is that 80%of people are aware of their condition while just 71%take some medication or are adequately controlled. Inadequate management of hypertension could be boosted by inadequacies in the diagnosis, treatment, and/or control of high blood pressure. Hypertension is rarely accompanied by any symptoms, and its identification is usually through screening, or when seeking healthcare for an unrelated problem. A proportion of people with high blood pressure reports headaches, as well as lightheadedness, vertigo, tinnitus (buzzing or hissing in the ears), altered vision or fainting episodes. A "hypertensive emergency" is diagnosed when there is evidence of direct damage to one or more organs as a result of the severely elevated blood pressure. This may include hypertensive encephalopathy, caused by brain swelling and dysfunction, and characterized by headaches and an altered level of consciousness (confusion or drowsiness). Breathlessness, cough, and the expectoration of blood-stained sputum are characteristic signs of pulmonary edema, the swelling of lung tissue due to left ventricular failure, an inability of the left ventricle of the heart to adequately pump blood from the lungs into the arterial system. The first line of treatment for hypertension is identical to the recommended preventative lifestyle changes and includes: dietary changes, physical exercise, and weight loss (Chobanian et al., 2003). These have all been shown to significantly reduce blood pressure in people with hypertension. If hypertension is high enough to justify immediate use of medications, lifestyle changes are still recommended in conjunction with medication. Different programs aimed to reduce psychological stress such as biofeedback, relaxation or meditation are advertised Passive behavior of the vessel wall 33 to reduce hypertension. However, in general claims of efficacy are not supported by scientific studies, which have been in general of low quality. With the aim of studying the hypertension disease in the arterial tissue within a computational mechanics framework we have to look at some aspects of the arterial wall to characterize the biological process. We assume two behaviors of this tissue, a passive and an active one. 1.2 Passive behavior of the vessel wall Among the most important features of the cardiovascular tissue we find a highly non-linear and anisotropic material with a nearly incompressible response due to the high water content. They area made up of different components as collagen, elastin, smooth muscle cells and fibroblast, turning them into heterogeneous composite-like materials. We devote Chapter 3 to describe the actual passive behavior of the arterial tissue and the carotid artery in particular. 1.3 Active behavior of the vessel wall In contrast with other materials, traditionally studied in the field of continuum mechanics, biological tissue is a very active matter. It does not only bear mechanical load. They exhibit a very optimized and advanced auto-regulatory mechanisms under mechanical and chemical stimuli. The cardiovascular system in particular undergoes a constant evolution over mammalian life. Moreover, these changes can be dramatically speed up in some diseases such a hypertension, atheroesclerosis, etc. The experimental and computational study of such processes are a very important research field. In the following section we will address those changes observed in hypertension disease. 1.3.1 Role of smooth muscle cells Smooth Muscle Cells (SMC) play an important role in both the passive and active behavior of the arterial tissue. However, it is in the active and evolution 34 Motivation aspects where SMC play a differentiate role. After a mechanical stimulus has occurred, up-regulating or down-regulating the homeostatic state, we can give a time-line description of its behavior. Focusing the problem on the hypertensive disease, an increase in blood pressure, and changes in blood flow the arterial wall experience an increase in the stretches and therefore stresses. The first active response to such changes is the contraction of the SMC in order to reestablish both normal stresses over the arterial thickness and normal values of the wall shear stress on the endothelial layer. The change of the basal tone is described usually as a short-term response and depends on the type and health of the artery. While important in all arteries, the active behavior of the SMC are known to have a primarily role on peripheral or resistance arteries. They are the ones that control the cardiovascular resistance to fulfill the requirements of the cardiovascular network. However, large and medium size arteries have lower contraction capabilities. SMC also boost the release of many different substances which have the main goal of acting on other cells and on the Extracellular Matrix (ECM) to adapt the internal structure to the new mechanical environment. Among the most important substances are the Transforming Growth Factors (TGF), such as TGF-β, Metalloproteinases enzymes (MMP) and Tissue Inhibitors of Metalloproteinases (TIMP). In the long-term response SMC experience a change in volume which have been well documented in terms of thickening of the arterial wall. Chronically, SMC start to growth, via hypertrophy, hyperplasia or both (Owens et al., 1981; Owens, 1989), to restore the homeostatic stress state. This chronic adaptation leads to the well-documented thickening of the vessel wall. These changes are also more pronounced in small or resistance vessels (Folkow et al., 1958; Mulvany and Aalkjaer, 1990). Bishop and Lindahl (1999) showed that the increase in extracellular matrix (ECM) deposition can be caused by hyperplasia or by the increase in cell synthesis, while Owens et al. (1981) identified hypertrophy to be the main origin of SMC growth. Active behavior of the vessel wall 35 1.3.2 Role of endothelial cells Endothelial cells (EC) are cells of the selectin family (E-Selectin) which conform the inner layer of blood vessels. They are distributed along the lumen in a monocell layer and are in charge, jointly with the SMC and fibroblast, of the regulatory system of the peripheral cardiovascular system. While SMC sense and react to changes of the blood vessel, EC do to variations of the shear stress, transforming mechanical stimulus into intracellular signals leading to functional, mechanical and chemical variations of the cell. Concerning the mechano-sensing receptors, several hypothesis have been investigated and established as sensors of shear stress. Ion channels represent one of the most important receptors, in particular by increasing K+permeability, increasing Ca2+ influx and activation of NonSelective Cation channels (NSC) and Cl−channels. Integrins are known to play an important role in mechanosensing of EC. They are placed on the cell membrane and represent an interface between the ECM, fibronectin and collagen, and the cell. They and their glycoproteins content are sensitive to displacement due to strain or shear stress. Platelet endothelial cell adhesion molecule-1 (PECAM-1) bind adjacent endothelial cells and have been also observed to increase due to shear strees. They play a role in abgeogenesis and cell adhesions. G-ProteinsCoupled receptors (GPCRs) and G-Proteins also have been reported to activate due to shear stress. Glycocalyx covers the surface of some cells, like in the EC, and plays a fundamental role in the mechonosensing and transduction of the shear stress, see Weinbaum et al. (2007); Tarbell and Shi (2012) for more details. All these receptors activate other huge cascade of signaling intracellular pathways. Among the most identified molecules involved we can point out Protein Kinase C (PKC), Rho family GTPases, PI3K and MAPKs. All this machinery effects and leads to changes on several aspects of EC. Shear stress have shown increase of gene expression in of Platelet-Derived Growth Factor (PDGF)-A and -B, Transforming Growth Factors (TGF −β), Intercellular Adhesion Molecule-1 (ICAM-1) and vasodilators NO and decreasing of Vascular Cell Adhesion Molecule-1 (VCAM-1) and the expression of the vasoconstrictor ET-1. 36 Motivation 1.4 Computational models The mathematical modeling of natural processes is a nice path to a better knowledge of what is happening in them. In particular the modeling of biomechanics and mechanobiology is a enterprising area of research (Humphrey, 2003; Holzapfel, 2004). Moreover, the inclusion of such models in a computational framework, the finite element method for example, allows researchers to study real patient specific geometries. Note that one of the main goals of computational and theoretical studies of biomechanics and mechonobiology related problems is the application of these studies to improve current diagnostic tool and drugs as well as a new research field in building synthetic organs (see an interesting New York Times article by Fountain (2012)). These two issues brought together are a wonderful tools to study many different topics within the biological and computational mechanics community. Among many of the topics to study in biomechanics, researchers have been focusing more strongly on the modeling of different kinds of tissue as the heart (McCulloch et al., 1998), bones (Rice et al., 1988; Weinbaum et al., 1994), vessels (Holzapfel et al., 2000), tendoms (Peña et al., 2006), eye (Pandolfi and Manganiello, 2006; Alastrué et al., 2006), etc. Many efforts have been also invested to model the electrical behavior of the heart (Heidenreich et al., 2010; Wong et al., 2011) and arteries (Murtada et al., 2010). There are also a high amount of works about the evolution of biological tissue over time like remodeling and growth, see e.g. the review in Humphrey (2009) and Ambrosi et al. (2011). Molecular dynamics is other useful tool for the simulation of movement of atoms and molecules (Rapaport, 2002). In terms of the mechanical behavior, a large number of constitutive laws have been recently proposed for modeling the mechanical behavior of soft biological tissues and blood vessel in particular. In general, this kind of tissues are made up of an extra-cellular matrix, composed of an isotropic ground substance with a high water content in which a network of fiber composed of elastin, different kinds of collagen and proteoglycans are embedded. The combined contribution of these constituents determines the mechanical response of the tissue, which turns out to be highly non-lineal. Those fibers are also responsible for the anisotropic Computational models 37 Figure 1.3: Workflow in current biomechanics research field. response of the tissue due to the existence of clear preferential orientations of the fibre bundles (Rhodin, 1980; Landuyt, 2006). They are commonly incorporated to the constitutive models as discrete anisotropy directions by means of, e.g., additional structural tensors (Gasser et al., 2006). Nevertheless, histological studies have shown that it does exist some fiber dispersion around the preferential orientation, which varies as a function of many variables as the vessel layer, the vessel type or even the position along the vessel length (O’Connell et al., 2008; Gasser et al., 2012). Thus, the incorporation of anisotropy considering fiber dispersion is one of the challenges for modeling the mechanical behavior of soft tissues. There exist also many models considering damage, viscoelastic and plastic effects (see e.g. the monographs of Fung (1990) and Humphrey (2002)). Firstly developed anisotropic constitutive models for blood vessels were purely phenomenological. Fung et al. (1979) proposed an exponential law of the deformations in the principal cylindrical directions in order to account for the anisotropic behaviour of the tissue. Later, directional fibre dispersion was introduced in the 38 Motivation models by means of different methodologies. In Holzapfel et al. (2005), a phenomenological parameter served to account for dispersion, whereas Gasser et al. (2006) used the von Mises Orientation Density Function (ODF) (Fisher, 1953) to determine a structural tensor representing the fibre distribution. More recently, models including fiber dispersion from a micro-structurally-based approach have been proposed (Alastrué et al., 2009). Hemodynamics is also an important topic in computational mechanics. FSI has been applied to cardiovascular problems. Coupling medical images with computing resources made possible the study of the blood flow in human arteries through Computational Fuid Dynamics (CFD) simulations and, in more recent years, through Fluid Structure Interaction (FSI) simulations with the aim to evaluate the influence of vessel properties (e.g., geometrical properties and wall compliance) on arterial pathologies. Steinman et al. (2002) proposed a novel approach for non-invasively reconstructing artery wall thickness and local hemodynamics at the human carotid bifurcation, and reported the first direct comparison of hemodynamic variables and wall thickness. On FSI analysis, Perktold and Rappitsch (1995) investigated the effect of a distensible artery wall on the local flow field and determined the mechanical stresses in the artery wall where incrementally linearly elastic behavior was assumed. Tang et al. (2001, 2003, 2004, 2005, 2008) conducted extensive research on stress analysis in plaque MRIbased models. Gao et al. (2009) compared the differences in stress distribution on plaque locations, between different diseased carotid bifurcations and analyzed the impacts of the specific combination of fibrous cap thickness and lipid core volume to the stress distribution (Gao and Long, 2008). Besides the fluid dynamics involved in the blood through the vessel there is an important fluid contribution within the arterial wall. Plasma, the fluid component of blood, comprises around 55%of the blood volume and it is part of the fluid that goes through the endothelial layer. However what is really important to be considered is the mass transfer happening in the arterial wall which makes many biochemical substances to react over time. They are usually taken into account by diffusion and convection equations. Mathematical models and computational schemes have been widely used to account for it. Given such Objectives and Thesis outline 45 and its genetics expression. We address a model for reorganization of the endothelial cytoskeleton based on its mechanical environment. The model is exploited in a computational way and the results of such simulation are presented. The model is also applied to the real carotid geometry. We also present a simple model of the synthesis and diffusion of the NO through the vessel wall. •Chapter 6 discusses the most important conclusions as a whole and outlines future lines of research in the different topics studied in this thesis. 2 Continuum mechanics framework The chapter outlines the basic notions of continuum mechanics that our models are based on. It describe the mapping configurations, kinematics, master balance, balance of mass, balance of momentum„ balance of mechanical energy and enthropy inequality. At the end we will provide some particularization used in the constitutive modeling of arterial tissue. The goal of this chapter is to provide a minimum framework in the context of solid mechanics that allow us to exploit later developments. This is not a complete review of solid mechanics and many important topics are omitted. Readers with a background in solid mechanics will see it very basic. This chapter would be more focus on give to the more biological oriented reader a basic notion of solid mechanics. Marsden and Hughes (1994); Ogden (1996); Bonet and Wood (1997) and Truesdell and Noll (2004), among many others, are monographs that the interested readers should not avoid for more comprehensive view of continuum mechanics. 2.1 Configuration and associated manifolds Let’s star by considering a body of interest MP ⊂ R3and let’s consider a point P. We need to provide him with a reference or a configuration. Let’s consider a configuration for MP called B. A configuration, B, is a smooth, so differentiable, 47 48 Continuum mechanics framework manifold, such that B:MP → R3. In other words, any particular point of MP in the configurationB, e.g. X∈R3, is a funtion that associated points of our body in euclidean reference system, that is B0:MP → B0(MP)⊂R3B0(P) = X.(2.1) Let’s take the a point Pcontained in a chart. A chart is a portion of the manifold from where we can define a reference coordinates for the manifold. We can define the tangent space T B0of MP at P. A tangent space is a vector space R3 containing all the possible tangent vector from P. We have given MP, and therefore to P, a place and a reference system. Let’s now consider the configuration at time t Bsuch that B:MP → B(MP)⊂R3B(P) = x.(2.2) At this point we have set not only the initial configuration of our body but also a the final configuration, and the reference system associated to that configuration. We adopt in the entire work a Euclidean reference system. This issue have important implication in the description of continuum mechanics and differential geometry in general. The material and spatial metrics called in literature Gand grenders the second order identity tensor δij. As consequence Christtoffel symbols vanish so the covariant derivative correspond to the derivative of a quantity in the direction of {◦} (vector or tensor) for any scalar, vector or tensor. ∇{◦}{•} =∇{•}[{◦}].(2.3) And finally, we need to give to our body of a path or a way to move from the initial to the final configuration. We allow to that motion to be reversible, that is invertible at any time. We do that by the so called, material and spatial motion respectively from now on, as ϕ:B0(MP)→ B(MP), φ(x) = X(x, t), φ:B(MP)→ B0(MP), ϕ(X, t) = x(X, t).(2.4) Configuration and associated manifolds 49 It is usual to write, given a composition of function as ϕ◦B0(MP) = B(MP) φ◦B(MP) = B0(MP)(2.5) Remark 1 Time derivatives The derivation of a given quantitity with respect to time t have some parcularirties dependending on the features of the configuration frame we are working on. Let’s consider a quantity {•} as a function of material points Xand time t. Let’s also define other quatity {◦} in terms of spatial point §and time t. We define the following called material derivatives that representent the time derivatives of material and spatial quantities respectively as Dt{•} =∂t{•}|Xwe call it material local derivative Dt{◦} =∂t{◦}|X=∂t{◦}|x+∇x{◦}·vwe call it material derivative of a spatial quantity (2.6) Let’s consider that we fixed the spatial position §and derivate with respect to time, called spatial derivatives. dt{◦} =∂t{◦}|xwe call it material local spatial derivative dt{•} =∂t{•}|x=∂t{•}|X+∇x{•}·vfor points in the spatial configuration (2.7) Let’s define two important time derivatives, the velocities of the motion and the inverse of the motion. We call them material and spatial velocity of the motion and are defined (see Remark 1) as V(X) = Dtϕ(X, t)and v(x, t) = V(X)◦φ(x, t).(2.8) The convective velocity can be obtained by the pull-back of the spatial velocity as V=F−1v. The ϕ∗(•)define the pull-back and ϕ∗(•)the push-forward operator. 50 Continuum mechanics framework And finally, the acceleration in the different configuration can be expressed as A(X) = DtV(X, t), a(x) = dtv(X, t) = ∇xv·vand A(x) = dtv(X, t) = ∇xv, (2.9) . 2.2 Kinematics We want to start by recalling some basic notions of the kinematic assumption. The deformation gradient is the tangent of the motion and represents a two-point linear map over the reference configuration. Based on the motion ϕwe defined the deformation gradient F(X, t) : TB0→ T B F(X, t) = ∇Xϕ(X, t)(2.10) between the tangent spaces T B0and T B of B0and Brespectively. The Jacobian of the deformation gradient will be denoted as J=det(F)>0. Assuming X,Y∈ B0being neighbouring points the material length element dX=Y−X=d=kY−Xkn0,(2.11) where the n0is associated to the material configuration B0fulfilling kn0k= 1, describing the direction of a material line element at the point X. The deformation gradient Fcan be used to express the material line element in its spatial counterpart as dx= (y−x)as dx=F·dX=F·[Y−X].(2.12) Kinematics 51 And similarly the vector n0can be transformed into n(X, t) = F(X, t)·n0.(2.13) λ=knkrepresents the stretch in the direction of n. We also introduce the right and left Cauchy-Green strain tensors C=Ft·Fand b=F·Ft,(2.14) which are symmetric and positive-definite tensors. It is also interesting to represent the polar decomposition of the deformation gradient tensor as F=R·U=v·R,(2.15) Rsatisfies the relation Rt=R−1, and the stretch tensors Uand ,v are unique positive definite symmetric tensors defined in the material and spatial configurations respectively. The material Land spatial velocity gradient lthen takes the following multiplicative representation l=˙ F·F−1with l=∇xv,(2.16) and L=˙ Fwith L=∇XV,(2.17) The symmetric part of the spatial velocity gradient defines the spatial rate of deformation tensor d=lsym. Among other quantities, we can also express the material deformation rate tensor as ˙ E=FtdF . To conclude, the time rate of the Jacobian of the deformation gradient is given by ˙ J=JF−t:˙ f=JdivV.(2.18) In connection with the modeling of biological tissue the mechanical response 52 Continuum mechanics framework of nearly incompressible material can be multiplicative split as F=J1 3I·¯ F.(2.19) where J1 3Iis associated to the volumetric part and ¯ Fwith the isochoric contribution, so that det( ¯ F)=1. And the right and left Cauchy-Green strain tensors transform into ¯ C=¯ Ft·¯ F=J−2 3C,¯ b=¯ F·¯ Ft=J−2 3b.(2.20) and ¯ n=¯ F·n0=J−1 3n,and ¯ λ=k¯ nk=J−1 3knk.(2.21) Remark 2 Objectivity. One of the most important principles of mechanics is the notion of objectivity. This principle comes up from the idea that any quantity describing the behavior of a body remains unchanged when an observer attached to and rotating with the body remain unchanged. Although the nature of such quantity remain unchanged, its description can be altered. This time dependent rotation Q(t)∈SO(3) is a proper orthogonal transformation in the proper orthogonal group. For example, the superimposition of a rigid body motion of two any points preserve distances, so the motion is said to be rigid or isometric. However, there are some tensorial forms that do not transform in this same objectively manner. The velocity gradient tensor l=˙ F·F−1is one of this non-objective tensor. Stress tensors in spatial configuration are also important tensorial forms that do not transform objectively. This particular quantity will be reviewed later as well as the principle of material frame invariance for hyperelastic description. 2.3 Stress Given some forces acting both on the surface of the body and we assumed that this body is cut by a plane passing though a spatial point x∈ Bt. A differential surface element dsin the cutting plane around x, characterized by its outward oriented normal n. The resultant of the internal forces are df=tds, where Stress 53 trepresents the Cauchy traction vector exerted on dswith outward normal n. The counterpart surface element dS(associated to the X∈ B0point), and first Piola-Kirchhoff traction vector T, defined in the reference configuration as df=tds=TdS, (2.22) t=t(x, t, n)and T=T(X, t, N),(2.23) where tand Tare usually defined as surface traction. Cauchy’s theorem assumed the existence of unique second-order tensor fields so that t(x, t, n) = σ(x, t)·nand T(X, t, N) = P(X, t)·N,(2.24) where σdenotes the symmetric spatial Cauchy stress tensor, while Pcharacterizes the two-point first Piola-Kirchhoff stress tensor. Given that force does not depend on the body geometry, a relation between the Cauchy stress and the first Piola-Kirchhoff stress tensor must exist, we write σ(x, t)·nds=P(X, t)·NdS, (2.25) Using the Nanson’s formula ds=JF−tdV, which relates elements in the reference and deformed configurations by making use of the volume ratio, then Pand σcan be related by P=Jσ·F−t.(2.26) In addition to these two measures of stress, other definitions of stress tensors can be described. The spatial Kirchhoff stress tensor, e.g., can be expressed as τ=Jσ, the second Piola-Kirchhoff tensor, convinient for the definition of constitutive theories is given by S=F−1·τ·F−t. We give in the following sections a brief description of the balance equations. Nevertheless, we can move between configuration with push-backs and pull-forward operation in a fashion way. 54 Continuum mechanics framework 2.4 Balance of mass Although biological tissue undergo a huge amount of mass production and transport it is quite usual to treat them as closed system. A closed system it is that one that do not allow mass to cross over the boundary ∂B0. A closed system is, therefore, characterized by a fixed mass in the system. We call system a collection of matter in space. Let’s define a mass density function ρ(x, t), such that dm =ρ(x, t)dv. For the material description the desnsity is given by ρ0(X, t) = ρ(x, t)◦ϕand the convective as %(X, t) = ρ(X, t). The mass balance laws give, in its spatial, material and convective description respectively, ∂tρ+div(ρv)=0 (2.27) Jρ0=ρ(2.28) ∂t%+Div(%V)=0 (2.29) 2.5 Balance of linear momentum To obtain the balance of momentum we identify our three classical terms. The variation with respect to time of the lineal momentum have to be balanced by the external volume specific momentun sources ρband the spatial momentum fluxes σin the spatial configuration. And as usual with standard transformation we can get the spatial, material and convected terms as ρa=div(σ) + ρb,(2.30) ρ0A=Div(P) + ρ0Band (2.31) %A=DIV(J−1S) + %B.(2.32) Constitutive equations 61 split again into volumetrixc-isochoric terms C= 2∂CS= 2∂CSvol + 2∂CSich =Cvol +Cich.(2.58) The volumetric contribution to the elastic tensor end up as Cvol = 2J[p+J∂Jp]C−1⊗C−1−2JpC−1C−1,(2.59) . And the isochoric contribution does as Cich =P:C:Pt−2/3Tr(J−2/3Sich)˜ P−2/3[S⊗C−1+C−1⊗S],(2.60) where ˜ P=C−1C−1−1/3C−1⊗C−1and ¯ C= 2 J−4/3∂¯ C¯ S. Remark 6 Identity tensors and diadic products. For the use in the entire work we summarize some notations. We use the fourth order identity tensor and fourth order symmetric tensor as Iijkl =δikδjl and Isym ijkl =1 2[δikδjl +δilδkj ].(2.61) In connection with this expression we define the standard and two non-standard diadic product as ({•}⊗{◦})ijkl ={•}ij{•}kl, ({•}¯ ⊗{◦})ijkl ={•}ik{•}jl and ({•}⊗{◦})ijkl ={•}il{•}jk and ({•}{•})ijkl = 1/2[{•}ik{•}jl +{•}il{•}jk]. (2.62) Remark 7 Objectives stress rates To properly define the constitutive equation in the updated lagrangian approach that it is followed in this work a proper definition of the stress rate is needed. In contrast with the objective nature of the stress tensor, its material derivative is not objective. In the spirit of getting the hyperelastic rate constitutive equation the variation of the Kirchhoff stress tensor 62 Continuum mechanics framework have to be derived properly. The Lie derivative of a spatial quantity is defined as the push-forward of the material time derivative of its pull-back, provide the needed tool, expressed as L(•) = ϕ∗(Dtϕ∗(•)).(2.63) The known as Jaumann rate, given by the Lie derivative of the Kirchhoff stress tensor, neglecting the stretch components of F, can be expressed Lv(τ)|l=w=◦ τ=Dtτ+τ·w+w·τ,(2.64) with w=˙ RRT. 3 Structure and Passive Behavior of the arterial tissue The artery wall has been object of research since a long time ago. The vessel wall is the conduit responsible for driving the blood flow from the heart all over the body. It is, together with the heart, key factors in terms of feeling and supplying the extra pressure needed to transport nutrients to the organs and tissue. Arteries can suffer from different diseases as we pointed out in the Introduction chapter. Any of these diseases can modify or cancel the normal work of the organism. This is the reason why the study of the arteries, whether from a biological or a mechanical point of view, has gained so much attention. What we could call normal behavior of materials, as a purely mechanical response to stimulus, is only a small part of the biological tissue behavior. They do underlie damage, plastic and viscoelastic behavior, as many conventional materials (rubbers, steel, aluminum etc.). But they also present a very active component where chemical reactions, cell synthesis and apoptosis, electrical signaling etc., are involved in its response. In this Chapter we focus on the mechanical response of the arterial wall, where no other chemical, active or any other kind of response is assumed. The vessel wall is a very complex, heterogeneous material. Arteries show a bunch of characteristic mechanical properties. They present a high level of incompressibility due to a high 63 64 Structure and Passive Behavior of the arterial tissue water content (≈70%), a high non-linear response primarily due to the collagen fibers that made up this tissue and they also show a complex microstructure where different layers and structural organization of its components can be found. We focus on these features over the next sections as we review some experimental data found in literature and provide a constitutive model to characterize them. Finally we reconstruct a patient specific carotid artery geometry in a finite element model, which is used over the entire work to check the performance of our models in a real artery geometry. 3.1 Structural organization Arteries are made up of three main components and have a clear composite-like structure (Fung, 1990). The arterial wall has three differentiated layers called intima, media and adventitia (Clark and Glagov, 1985). The intima layer is a thin sheet of endothelial cells (Fig. 3.2(a)) and is the one in contact with the blood flow and taking care of the mechanotransduction due to fluid dynamic changes (Runanyi et al., 1990). We study some aspects of this phenomenon in Chapter 6. The media, the next layer in radial direction, is made of sheets of elastin fibers randomly packed. In most of the arteries SMC are placed in a preferential circumferential direction and bundles of collagen fibers are placed along the SMC. Arteries are also classified by muscular and elastic arteries depending on the percentage of each constituent. Elastic arteries have a high content of elastin and collagen and they are the vessels in a closer distance from the heart and usually called large and medium size arteries. Muscular arteries have a higher percentage of SMC and they are further away form the heart constituting medium and small size arteries. The adventitia is made of collagen in a more random fashion. Fibroblast are also an important component of adventitia and are mainly the responsible of the synthesis of collagen (Bell et al., 1979). O’Connell et al. (2008) shows a nice reconstruction of a pig aorta artery (see Fig. 3.1) where we can observed all these properties. In terms of its components, elastic fibers, collagen and smooth muscle cells (SMC) are the responsible to accomplish structural purposes. Structural organization 65 Elastic fibers (Fig. 3.2(b)) are bundles of elastin proteins, secreted mainly by fibroblast and SMC, and fibrillin-microfibrils, secreted by fibroblast and which are essential for the correct formation of the elastic fibers (Rosenbloom et al., 1993). Fibrillin is secreted into the ECM forming microfibrils which have been seen to provide a scaffold-like structure for elastin deposition and play a key role in deposition of tropoelastin (Kielty et al., 2002). *[Confocal laser scanning microscopy reconstruction. In (a) the elastin (green), SMC (blue) and collagen fibers (red). (b) represent the elastin, (c) SMC and (d) the collagen bundles.] *[Scanning electron microscopy reconstruction.] Figure 3.1: Microstrutural representation of rat aorta (O’Connell et al., 2008). These components have different elastic properties. Sherratt et al. (2003) investigate the stiffness of fibrillin micro-fibrils. They reported Young’s modulus values of 78-96 [MPa]. They suggested that these elements behave as reinforcement of the elastin fibers. Lillie et al. (1998) also showed that removal of fibrillin 66 Structure and Passive Behavior of the arterial tissue within the arterial tissue decreased the Young’s modulus at low strain while slightly increasing at higher strains also suggesting the stiffening role of fibrillin micro-fibrils. However, how they interact to increase the overall stiffness of the tissue is not yet well understood. Mechanical characterization of elastic fibers have been reported in the literature for bovine ligament in the order of 0.4-1.2 [MPa] (Aaron and Gosline, 1981), in equine ligament (Koenders et al., 2009) from to 0.3-1.5 [kPa] and 0.56-0.74 [MPa] for the fibrillin-microfibrils. These authors stated that microfibrils are not essential in the mechanical behavior of elastic fibers in vertebrates. Sherebrin (1983) reported values for the elastic fibers in dog and sheep aorta in the rage of 0.13-0.65 [MPa], which are in accordance with other experimental findings. Elastin has important flexibility and extensibility features for blood vessels and endows the ECM with resilience during the loading of the cardiac cycle. The elastic fibers are randomly packed forming sheets. During pathological conditions (Kielty, 2006), e.g. cardiovascular diseases, there exist remodeling and turnover of elastin which lead to the modification of the mechanical properties of the arterial wall. Structurally, collagen (Fig. 3.2(c)) is the most important protein found in many connective and fibrous tissues. The extracellular collagen molecule, procollagen, is made up of three left-handed helix polypeptides, the so-called α-chains, coiled-up in a right-handed helical structure, about 300 nm long and 1.5 nm in diameter (Bella et al., 1994; Orgel et al., 2001; Bhattacharjee and Bansal, 2005). Collagen molecules assemble along a given direction through covalent bonds to form collagen microfibrils, constituting the basic building block of collagen fibrils (Baselt et al., 1993; Hulmes et al., 1995; Orgel et al., 2006, 2011). Collagen fibrils are gathered in the extracellular matrix to form bundles of collagen fibers. Fig.3.2(c) shows a representation of such structure. Collagen bears the major part of the load transmitted through the tissue, and a lot of research has been devoted to understand its hierarchical microstructure at and across the different scales (see Fratzl (2008) for a review). Its highly non-linear behavior makes the arterial wall to "block" under too large deformation preventing the tissue to damage. Strain of collagen fibers is achieved by stretching Structural organization 67 of collagen molecules and fibril sliding. Mechanical properties of the different scales of collagen, in molecules (Bozec and Horton, 2005), fibrils (Wenger et al., 2007) and fibers (Silver et al., 2003) have been well studied. We encourage the reader with the thesis works of van der Rijt (2004) and Yang (2008) who did an extensive experimental work characterizing collagen fibrils. Although the mechanical properties of the different collagen hierarchical constituents is well understood, the mechanisms by which all the levels interplays represents a more complicated process and the connections from different scales is more poorly known. A complete understanding of these mechanisms would allow to predict the macroscopic behavior of collagen fibers by means of the behavior of its smaller structures. It is known, however, that in a first strain regime, molecular stretching predominates and beyond that point slipping of fibrils take place (Sasaki and Odajima, 1996). Sasaki and Odajima (1996) show how as we move from smaller (molecule) to higher scale stiffness of the structure decreases due to the different cross-liking mechanism between scales. Moreover, the waving at which collagen fibrils ensemble into the ECM to form collagen bundles also modifies the behavior of the tissue. As in elastin, collagen also experiences a very important turnover both in healthy and different pathologies. We address some of these issues in Chapter 5. Smooth Muscle Cells (SMC) (Fig. 3.2(d)) are a very important part of the puzzle in terms of its active response. SMC accomplish the so called myogenic tone. SMC react mechanically to external stimuli by contraction or dilating the vessel. SMC also have the important role of synthesizing different kind of biological substances which make the arterial tissue to remodel. Finally, we would like to point out an important feature of arterial tissue related to the SMC, the myogenic tone. As we barely described above, myogenic tone is an active force exterted by SMC during the cardiac circle. This force is used to accomodate the arterial tissue to the pressure load to maintain normal values of stress and strain in the arterial wall. Normal myogenic tone can be changed, and exert larges forces, when blood pressure increases, e.g. in hypertensive disease. Myogenic tone is also an key factor in small arteries and arterioles which, due to neurological mechanisms, contract and increase the blood 68 Structure and Passive Behavior of the arterial tissue (a) Endothelial layer (SCIENCEphotoLIBRARY, 2013). (b) Elastic fibers (Koenders et al., 2009). (c) Collagen fibrils packed in fibers (SCIENCEphotoLIBRARY, 2013). (d) Smooth muscle cells (SCIENCEphotoLIBRARY, 2013). Figure 3.2: Microstrutural components of arterial tissue. pressure gradient between the heart and capillaries. This process is one the main mechanisms by which essential hypertension occurs. There are come attemps in literature to model this behavior, see e.g. (Zulliger et al., 2004; Murtada et al., 2010). In the first one the authors propose a SEDF to incorporate the active contraction and relaxation of the SMC while the second relies on a kinematic model of the cells based on Ca+and K concentrations. All these material features are important in order to properly characterize the arterial wall. During the next Sections we particularize to different models to capture the carotid artery behavior. The carotid artery: Previous studies and models 69 3.2 The carotid artery: Previous studies and models The carotid artery have been widely studied in literature because of it is prone to develop atherosclerosis, stenosis, remodeling, etc. Experimental results in carotid artery are wide in literature but they used to be focused on a specific issue of the tissue, using different species on human, pig, dog, etc. Some of them rely on the mechanical properties of the wall using uniaxial or inflation test. Others also present values of the pre-stretch of the carotid or micro-structural features of the material. However there are few of them where a complete characterization of the tissue is performed for a specific study, therefore the researcher have to rely on different studies from different samples, species, etc. Here, we review two studies (Garcia et al., 2011; Sommer and Holzapfel, 2012) that, for our understanding, present the most completed set of experimental tests. First, it is convenient to present the constitutive modeling, which is the same for both studies. They were interested in the mechanical properties of the mechanical behavior of the tissue. These authors used a hyperelastic framework (see Chapter 2) to characterize the tissue and used the same constitutive model use to fit the experimental findings. The elastin, which is usually described with a isotropic behavior, was described with a Neo-Hookean model as Ψelas(I1) = C10[I1−3] (3.1) where µis a stress-like parameter and I1=tr(C)is the first invariant of the isochoric part of the deformation. The isochoric Piola-Kirchhoff stress reads as in Eq. 3.2 while the isochoric elastic tensor ∂2 CΨelas(I1)nulls in this case. Siso =∂CΨelas(I1) = ∂I1Ψelas(I1)∂CI1=C10I1I(3.2) For the description of the collagen fibers, they adopted the exponential-type 70 Structure and Passive Behavior of the arterial tissue model presented by Holzapfel et al. (2005), which is given as follows. Ψcoll(I1, I4) =      0,if λ < 1 k1 2k2 [exp(k2[(1 −ρ)[I1−3]2+ρ[I4−1]2)−1] if λ≥1(3.3) where k1∈R+represents a stress-like parameter, k2∈R+measures the exponential behavior of the response, ρ∈R, ρ ⊂[0,1] represents a measure of the dispersion of collagen. I4=C:Nis the isochoric anisotropic invariant with N=n⊗n, being nthe direction of the fibers. This model represents a symmetric helicoidal distribution of the fiber within the vessel thickness. For more information of this model, as well as a with other constitutive models, we address to Holzapfel et al. (2005). Other important issue when studying arterial tissue is the residual stresses within the arterial wall. Residual stresses were shown for the first time in Fung and Liu (1989) to be an important feature of arterial tissue. It has been hypothesized that they play a role to maintain physiological stresses in the arterial wall. The modeling of pre-stress have been described from a kinematic point of view. The usual way to characterized residual stresses is to cut the artery up in rings and then longitudinal to release those stresses in the circumferential direction. The final geometry of the ring once cut is taken to describe the stress-free state of the tissue. Further discussion is addressed in following section. Remark 8 Pre-stress in biological tissue. It has been also pointed out the importance of pres-stress in the behavior of arterial tissue. It is not clear if this feature is associated to the elastin, collagen or SMC. Moreover, if this is associated to collagen or elastin fibers is neither clear if these elements achieve this residual stress by generating an active force by themselves, which seem awkward, or if it comes from the state at which collagen and elastin is deposited. It could be suggested that not incorporating the residual stresses do not reflect the actual behavior of the artery. At some extend we could agree with that statement. However, given the phenomenological models of arterial tissue this argument seems also arguable since the fitting experimental results in samples with residual stresses The carotid artery: Previous studies and models 77 to have a very different behavior. And although some residual stresses information was collected, it was not incorpareted later in the constitutive relation. 3.2.2 Human carotid artery Sommer and Holzapfel (2012) provided data from human carotid arteries tested by inflation test. A general strategy to fit these material parameters is to consider an analytical thick wall problem under pression and axial stretch. We refer to Holzapfel et al. (2000) for the complete developed equations. They presented a bunch of experimental results of pressurized vessels and stretch imposed in axial direction. In a series of papers they provide both residual stresses and mechanical behavior of human carotid arteries. At the end, the problem relies on finding a set of material parameters that best fit a pressure-stretch test for the particular expression of σtt,σrr and σzz see, e.g., Holzapfel et al. (2000). The material parameters are obtained by means of a non-linear least-square optimization by minimizing the following objective function. E2= n X j=1 hpj−pψ ji2+hFj−Fψ ji2(3.5) with p=ˆr0 ri (σrr −σtt dr r)and (3.6) Fz=ˆr0 ri (2σz−σrr −σtt)rdr (3.7) The results achieved in this work for the common carotid artery (CCA) and the internal carotid artery (ICA), where pres-stress was included, is summarized in Table 3.4. For more specific data we refer to Sommer and Holzapfel (2012) where some data for the different specimens are provided. Remark 12 Age-related data. Looking at the results and the specimens used by Sommer and Holzapfel (2012) one could argue that the sample represents a old sample field. As it is explained in that work, the carotid arteries were extracted from a mean 74 years old humans. It is well known that arterial wall stiffens 78 Structure and Passive Behavior of the arterial tissue C10[kP a]k1[kP a]k2[−]θ[o]ρ[−] CCA adv Mean 59.6 180.9 109.8 30.1 0.8 CCA adv SD 33.8 316.6 104.4 10.6 0.1 CCA media Mean 122.3 24.7 16.5 6.9 0.8 CCA media SD 12.2 10.7 9.3 2.4 0.2 ICA adv Mean 28.3 112.1 100.6 31.8 0.9 ICA adv SD 59.5 179.1 101.1 11.2 0.1 ICA media Mean 17.6 21.3 17.3 9.8 0.8 ICA media SD 17.8 10.3 9.5 2.9 0.2 Table 3.4: Results of material parameters fitted for inflation test for the CCA and ICA for the adventitia and media layers provided by Sommer and Holzapfel (2012). during the lifetime. In this study they also reported material content and gave the results shown in Table 3.5. As it can be seen the value of elastin, and mainly of collagen seems too high compared to previous values and other data (see Table 3.6) in literature (Humphrey and Rajagopal, 2003). This could be related to the old age of samples and therefore to the stiff response of those samples. φcol φelas φSMC 59 ±8% 31 ±6% 10 ±3% Table 3.5: Percentage of the material constituents, elastin, collagen and SMC given by Sommer and Holzapfel (2012). In connection with Remark 8 we also performed the material fitting including the residual stresses as an example of the residual stresses role on phenomenological models. We have fixed the angle of the fibers to reduce variability. In the following figure we present the fitting curves by Sommer and Holzapfel (2012) with and without pretension. The pre-stress is introduced by kinematical consideration as is described in, e.g., Holzapfel et al. (2000). Results of the fitting procedure were almost identical for the C10 and k1and the angle of the fibers, only k2increase ≈10%. This results indicate that pre-stress provide a higher stiffness at higher strain values. However, it can not be concluded further details of the actual pre-stress disposition of the fibers and how this phenomenon occurs by means of this kind of data. However the biggest problem we deal with in this fitting problem is the non- A new approach toward microstructural fitting 79 1 1.05 1.1 1.15 0 10 20 30 40 50 σ [KPa] λ [−] To fit long Fitted long To fit circ Fitted circ (a) Fitting without considering pre-stress. 1 1.05 1.1 1.15 0 5 10 15 20 25 30 35 40 45 σ [KPa] λ [−] To fit long Fitted long To fit circ Fitted circ (b) Fitting considering pre-stress. Figure 3.5: Numerical fitting with and without considering residual stresses using data from Sommer and Holzapfel (2012). uniqueness of solution. In fact, we can gather a good fitting with different set of parameters. In the next section we will try to reduce these drawbacks. 3.3 A new approach toward microstructural fitting In this section we want to include different aspects of the arterial tissue to fit again those tests shown before to, our understanding, better describe the microstructural behavior described by, e.g., Garcia (2012) and O’Connell et al. (2008), and reduce the ill-possenes of the optimization problem. Beginning from the biggest to the smallest scale in our model, our first attempt is to obtain values of the material content of SMC, elastin and collagen. The SEDF is used to be described in terms of energy per mass unit or, equivalently, in terms of the Helmholtz free energy per volume unit, which give us stress units for those quantities. This is usually done to avoid scale problems in the definition of the constitutive models. When splitting the SEDF into different parts one should take into account the percentage of substance for each one of the splitting quantities. Usually these measures are into the mechanical parameters, 80 Structure and Passive Behavior of the arterial tissue φcol φelas φSMC 50% 20% 30% Table 3.6: Weight percentage of each component respect to fry weight in human carotid arteries obtained by Humphrey and Rajagopal (2003). but it should be more correct to state them explicitly. A Helmholtz free energy function U =ρΨis introduced with Ψthe SEDF, reorganizing to have them in its volume and mass functions we get U/m= Ψ/V. Therefore we can write the splitting of both forms of the energy as U m=Ψ V=φwΨvol(J) + φelasΨiso(I1) + φcolΨani(I4,) + φSMCΨSMC(I4).(3.8) where φw,φelas,φcol and φSMC represent the relative amount of water, elastin, collagen and SMC respectively. Note that we are not considering the mechanical contribution of the SMC from now on since collagen and elastin are supossed to bear the major part of the mechanical load. Also note that Pi≥1φi= 1 with φithe percentage in volume of each constituent (Humphrey and Rajagopal, 2003). Water content is usually measured to be a 70%of the material, while the remaining correspond to the dry part, that can be related to the isochoric part of the SEDF. Out the 30%remaining, the percentages are split based on the elastin, collagen and SMC contect described in previous sections. This approach is the base of the Mixture-theory. We refer to Humphrey and Rajagopal (2003) for a complete development of this theory with application to arterial tissue. Values of material content are provided by Humphrey and Rajagopal (2003) in human carotids. Regarding the collagen behavior, as we pointed out in the introduction of this section, collagen fibers are made up of collagen fibrils which gain into the extracellular matrix achieving some degree of pretension and are organized following a preferential direction. The mechanical behavior of the different components in the hierarchical structure are quite different. However the mechanical properties of the collagen molecules and the collagen fibrils maintain a similar behavior be- A new approach toward microstructural fitting 81 tween different tissues and different species. What change from one to another is the degree of link and the deposition state of the fibrils within the ECM, which give also the pre-stress behavior. In Fig. 3.6 we fit the mechanical properties of a collagen fibril from a tensile test by Yang (2008). We use a SEDF initially proposed by Holzapfel et al. (2000), Ψfib(I4) = k1 2k2 [exp(k2[I4−1]2)−1] (3.9) 1 1.01 1.02 1.03 1.04 1.05 1.06 0 5 10 15 20 25 σ [KPa] λ [−] To fit long Fitted long (a) Experimental test by Yang (2008) 1 1.01 1.02 1.03 1.04 1.05 1.06 0 5 10 15 20 25 30 σ [KPa] λ [−] To fit long Fitted long (b) Experimental test by van der Rijt et al. (2006) Figure 3.6: Fitting results of the mechanical behavior of individual collagen fibrils from data presented in the literature. However, this behavior only takes into account the collagen fibril. The behavior of a collagen fiber is softer than the fibrils due to the cross-links so an upper bound can be stated in the mechanical values of the fibrils. This model does not account for high strain states where breaks of cross-linking between fibril are important. Once we know the individual mechanical behavior of the collagen fibrils and material parameters are fitted we want to describe the structural organization of the arterial layers. O’Connell et al. (2008) have shown that arterial fibers distribution follows a helicoidal distribution from the outer to the inner layer. As they stated, collagen fibers are bundled around the SMC with some collagen fibrils linking the main fiber in a predominant perpendicular direction. Adventitia layer is known to have a random distribution of collagen with almost any SMC and 82 Structure and Passive Behavior of the arterial tissue some fibroblast. In the media layer there are several sublayers, every one with a preferential direction of the SMC and therefore of collagen fibers, and elastin sheets separating these sublayers. In the case of the media layer of carotid arteries previous studies report that SMC keep a very circumferential direction (Garcia, 2012). Having in mind these works we modified the SEDF in Eq. 3.3 to recover these features, sketchy drawn in Fig. 3.7. Figure 3.7: Schematic representation of the structural organization in the media of the carotid artery, being nthe preferential direction. Following Sommer and Holzapfel (2012), Garcia et al. (2011) and O’Connell et al. (2008) observations we can consider that the preferential direction of the fibers are in circumferential direction. The collagen fibers are linked each other by other collagen fibers, cross-linking fibrils, that we consider isotropic and perpendicular to the main one, representing the amount of cross-linking between main fibers. This contribution is multiplied by a factor αto measure the amount of these cross-links. Given that the cross-linking are isotropically connecting to the main fibers we propose an isotropic contribution of it, taking out the contribution of the main fibers. We can write this as Ψcoll(I1, I4) = N X i=1 [1 −α]Ψi fib +αk1 2k2 [exp(k2[I1−3]2)−1] −Ψi fib = N X i=1 [1 −2α]Ψi fib +αk1 2k2 [exp(k2[I1−3]2)−1] (3.10) A new approach toward microstructural fitting 83 where αrepresents the amount of cross-links being α≤0.5.α= 0 represents no cross-links and α= 0.5that the degree of links is high enough to consider an isotropic distribution of the fibers. Our approach consists on a preferential direction of the main fiber, which, in the present case is in the circumferential direction for the carotid artery. So we set N= 1 as we consider a unique family of fibers in the circumferential direction. For shake of scouting possible values of αwe use some experimental tests performed in a pig aorta artery in our laboratory. Fig. 3.8 shows the fitting of such uniaxial tests. As we could predict, values for the adventitia layer are higher than values in the media layer since orientation of the collagen fibers is much randomly packed in adventitia layer than in the media one. 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 300 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (a) Adventitia fitting results. C10 = 11.2kPa, k1= 18.72 kPa, k2= 1.84 and α=0.41. 1 1.2 1.4 1.6 1.8 0 50 100 150 σ [kPa] λ [−] To fit circ To fit long Fitted circ Fitted long (b) Media fitting results. C10 = 13.59 kPa, k1= 15.93 kPa, k2= 1.20 and α=0.13 Figure 3.8: Uniaxial test in longitudinal and circumferential direction in a pig aorta artery. 3.3.1 Results of pig carotid The proposed workflow is initially applied to the experimental result in Garcia et al. (2011). There are some evidences of non-isotropic behavior of the elastin in arterial walls, however we will ignore the anisotropic contribution in this work. As pointed out by Zou and Zhang (2009), the behavior of the first part of the stress-strain curve of the arterial wall is due to the compliance of the elastin. Therefore, we initiate our optimization problem (Eq. 3.11) with the minimization 84 Structure and Passive Behavior of the arterial tissue function Fto fit the material parameter corresponding with the first part of test up to 3−4% of strain. (OP1) For pgiven minimize F(p, C10) = NOP 1 X j=1 hσlj−σlψ ji2+hσcl−σcψ ji2,(3.11) where NOP 1∈N+is the number of points in the strain interval [1-1.04]. σljand σcjrepresents the longitudinal and circumferential stress at the sample point j. Applying OP1, fitting only the elastin part, we obtain the results in Table 3.7 φelastC10[kP a]Mean SD Distal 24.43 21.65 14.72 21.81 33.39 19.81 Specimen I-XIV 19.66 20.58 18.24 22.68 21.34 22.94 21.81 4.25 Proximal 22.84 24.42 22.23 19.21 21.79 26.77 Specimen I-XIV 20.98 19.02 21.96 18.72 29.78 27.06 22.89 3.49 Table 3.7: Results of the OP1, solving for the Neo-Hookean parameter C10, for pig carotid specimens (Garcia et al., 2011). First and second line are the I-VI and VII-XIII specimens of the distal and proximal part respectively while third and fourth are the I-VI and VII-XIII specimens of the distal and proximal samples respectively. Now we focus on each artery constituent to get as much information as possible. Concerning the elastin behavior of the material which, as we said before, is usually modeled by means of a Neo-hooken law. The Young modulus of elastin fibrils has been measured to 560-740 kPa (Koenders et al., 2009) although, as we discussed previously, the value of the elastic modulus for the elastic fibrils depend on the fibrillin and cross-linking degree. In a low strain regime C10 =E/6is usually approached, so we get a C10 ≈100kP a. Interestingly, if we apply the content values of elastin in carotid arteries of an rough averaged of 20 −30% we get the C10 values for our fitting of ≈20 −30 kPa. Note that this is an upper bound, since there will be some collagen working in the strain regime. Note also that the values presented by Koenders et al. (2009) are for individual elastin fibers, the elastin lamelae we are dealing with have other cross-linking that probably stiffen A new approach toward microstructural fitting 85 the material at some level (Lillie et al., 1998; Sherratt et al., 2003). As we can see this approach gives a very similar values for all the specimens, as we highlight in the introduction of this chapter. Now we apply OP2 to extract the rest of the material parameters. (OP2) For pgiven minimize F(P, Mat) = NOP 2 X j=1 hσlj−σlψ ji2+hσcl−σcψ ji2.(3.12) where Mat=[k1k2α] subject to α≤0.5. The experimental and fitted curves are presented in Fig. 3.9 and 3.10 for the proximal and distal place respectively. In Table 3.8 and 3.9 we resume the material parameters for such fittings. Values of C10 are allowed to change with respect to those obtained previously for a better fitting result. 86 Structure and Passive Behavior of the arterial tissue 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [kPa] λ [−] To fit circ To fit long Fitted circ Fitted long (a) Specimen I 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (b) Specimen II 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (c) Specimen III 1 1.2 1.4 1.6 1.8 2 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (d) Specimen IV 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (e) Specimen VI 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (f) Specimen VII 1 1.2 1.4 1.6 1.8 2 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (g) Specimen VIII 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (h) Specimen IX 1 1.2 1.4 1.6 1.8 2 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (i) Specimen X 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (j) Specimen XI 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (k) Specimen XII 1 1.2 1.4 1.6 1.8 0 50 100 150 200 250 σ [KPa] λ [−] To fit circ To fit long Fitted circ Fitted long (l) Specimen XIII Figure 3.9: Experimental and fitted curves of the uniaxial test of the proximal pig CCA. Patient specific human carotid artery modeling 93 Figure 3.12: Reconstruction of the carotid artery from human carotid CT image (Alastrue et al., 2010a). experimental data in Sommer et al. (2010) to describe the thickness for the different layers of the arterial wall. They provide thickness values of the media and adventitia layer in the CCA and ICA. We apply a linear transition of the thickness over the arterial wall from the CCA and the ICA. Therefore, the finite element mesh has two different and important groups. The first group consists of 5 elements in radial direction and a total of 104,005 elements, representing the media layer of the carotid artery. The other group, representing the adventitia layer, contains three element in thickness and a total of 62.403 elements. Fig. 3.13 shows some details of the final finite element mesh. 94 Structure and Passive Behavior of the arterial tissue (a) Complete mesh of the CA. (b) Detail of the bifurcation in the CA. (c) Detail of the lumen in the CA. (d) Detail of the front side of the CA. Figure 3.13: Finite element mesh of a human carotid artery. Unlike usually in computational model of blood vessels, besides the linear variable distribution of the thickness, we adopt a variable distribution of the mechanical parameters. In a very same fashion, we linearly interpolate the values of the mechanical properties obtained in Table 3.11 at both parts of the artery to describe in a more realistic way the mechanical properties along the artery length. The results of the three mechanical constants are plotted in Fig. 3.14. An important issue in this kind of finite element models is the imposition of boundary conditions reproducing the actual in-situ conditions of the vessel. These arteries, and carotid artery in particular, are surrounded by connective tissue, therefore we have included a set of spring elements all over the surface of the carotid model to simulate then and avoid solid rigid motions. These springs block rigid solid motion but allow deformation of the arterial wall. The mechanical properties of these springs have been chosen so that this condition is fulfilled. Patient specific human carotid artery modeling 95 Figure 3.14: Spatial distribution of the mechanical material parameters along the carotid artery. The differences of the element face have to be also taken into account to avoid mesh-related issues. To close-up the mechanical and structural description of the model at hand, we left to describe the structural organization of the constituents. As we discuss before, we assume an isotropic distribution of the elastin component. The collagen and SMC follow the same organization and they are preferentially oriented in circumferential direction. The differences in collagen distribution between the media and adventitia layer are considered by means of the cross-link-related parameter αbut the preferential direction stays circumferential. Fig.3.15 shows details of the fiber distribution in some representative places. The orientation of the fibers is gathered by means of the principal stresses directions of the carotid modeled with an isotropic model, under internal pressure and the displacement of the external face constrained (Alastrue et al., 2010a). Maximal principal stress appears in the circumferential direction from where we can get the unitary vectors nwhich can be associated with the orientation of the fibers. This slice is 96 Structure and Passive Behavior of the arterial tissue used in the following for fitting and visualization purposes. Figure 3.15: Some details of the fiber distribution defined along the carotid artery. 3.4.3 Results in carotid artery In this section we present the finite element results of the carotid artery geometry shown in the previous section at different slices (Fig. 3.16). In Fig. 3.17 we shown the maximum principal stresses for the whole artery and the three slices of the CCA, ICA and external carotid artery (ECA) for the normotensive at 13.3 [kPa] and Fig. 3.18 does for the hypertensive sate at 16 [kPa]. As we can see, due to the mechanical properties of the carotid, stresses in the ICA and ECA are higher than those in the CCA. We can also appreciate the increase of stress due to the overpressure in hypertensive case and the transition of the media and adventitia layer. Conclusions 97 Figure 3.16: Representative slice of the CCA, ICA and ECA from the carotid model. In Fig. 3.17 and 3.18 we show results in the homeostatic and hypertensive cases respectively and for the whole artery and in different longitudinal cuts. Besides the stresses difference in the CCA and ICA and ECA is also appreciable similar stress concentration in a ring just before the bifurcation and a ring at the bifurcation part of the ICA. To close up the results section we present in Fig. 3.19 the maximum principal stretch at normotensive and hypertensive conditions and the ratio between hypertensive and normotensive. This variable will be the mechanical stimulus that will trigger different adaptation processes developed in the following chapters. 3.5 Conclusions The study of arterial tissue from its structural organization and its mechanical behavior have been widely researched over the last decades. The most important features of the material structure from a load bearing point of view are that they are made up of collagen, elastin, SMC and water, that collagen and SMCs are distributed mainly in a preferential circumferential direction and that collagen has highly non-linear behavior. Based on these observations, researches have investigated different phenomenological models to capture the behavior of arterial tissue (see e.g. Fung et al. (1979); Demiray et al. (1988); Holzapfel et al. (2000), among others). Usually, elastin is modeled as a Neo-Hookean material while collagen has been object of further investigation, where it is common the use of exponential-type expressions, given the uniaxial response of collagen fibers. Although these phenomenological models reproduce very nicely some type of 98 Structure and Passive Behavior of the arterial tissue Figure 3.17: Maximum principal stress at different cuts in the whole CA for the homeostatic loads. Conclusions 99 Figure 3.18: Maximum principal stress at different cuts in the whole CA for the hypertensive loads. 100 Structure and Passive Behavior of the arterial tissue Figure 3.19: Maximum principal stretch at homeostatic and hypertensive and ratio at hypertensive and homeostatic state from left to right. tissues or arteries others do not fit so well. In other situations, although they fit, the fitted parameters do not have a physical meaning or they do not match with some experimental observations. This is usually because of the lack of a particular feature of the tissue in the model or because of the ill-possess of the fitting problem. The aims in constitutive modeling of soft tissue should go towards the micro-structural characterization of its components, both its organization and its mechanical behavior. In any case, there are much efford to be done in the modeling of soft tissue, e.g., from more micro-structural oriented models to a multiscale model, which for some reason there have not been introduced in mechanics of soft tissue. In this chapter we have looked into several structural data of the arterial wall and the mechanical behavior of the elastin (Lillie et al., 1998; Koenders et al., 2009) and the collagen fibers (van der Rijt et al., 2006; Yang, 2008), which are known to be the most important constituents in terms of mechanical behavior. In this direction, we have modified a well known and widely used SEDF to describe some features that this previous model did not include. For example, Garcia (2012) and O’Connell et al. (2008) showed that some arteries have a preferentially Conclusions 101 circumferential direction. Based on this previous model, the experimental finding could not be fitted, in our opinion, because they do not consider small amount of collagen fibrils that link the main fibers, conferring a non negligible stiffness in the transversal direction of the fibers. To this end, we have introduced a cross-linked-related parameter. We have also looked into the particular behavior of elastin and collagen fibers in order to reduce the variability of the results in the fitting problem. We have been able to overcome the limitations that we found with previous models and our results in different carotid arteries were improved with respect to previous models. Our final goal in this work was to apply this mechanical model, and others that we will show in the following chapters, to real patient specific carotid arteries to be able to simulate it computationally in a finite element framework. We obtained a geometrical model from CT images of a human carotid artery. We also obtained a circumferential distribution of the fibers over the arterial thickness which correlated well with experimental data (Garcia et al., 2011; Garcia, 2012; O’Connell et al., 2008). Latter, we gather information in literature to create a variable distribution of the mechanical properties along the carotid artery. Our results showed a good agreement with experimental findings of uniaxial and inflation tests. In terms of other FE simulations, our results showed similar (Alastrue et al., 2010a) or slightly lower stress values than other models (Delfino et al., 1997). Usually the external face of the artery is a free face which does not reflect the physiological conditions of the artery, surrounded by other connective tissue. The direct inclusion of this tissue or spring elements in the external face, as in our case, describe the physiology of the arterial tissue, in our opinion, in a more realistic way. There are also important limitations in the current model. Firstly, we did not consider the pre-stress of the arterial wall and, although we show that it can be recovered by means of constitutive models that do not include implicitly the pre-stress, it can not be omitted if an accurate model of arterial tissue wants to be performed. Pre-stress have been extensively studied in analytical models (Holzapfel et al., 2000; Rachev and Greenwald, 2003) and simple FE models (Alastrué et al., 2007) but an approach for general geometries is more singular 102 Structure and Passive Behavior of the arterial tissue task. There are also a lack of pre-stress models where the background meaning of these phenomena is described (what do create it, how is the mechanism?) since there used to be purely phenomenological models. This is, by far, the most important limitation of our model. They would be also interesting to consider the myogenic tone of SMC (Zulliger et al., 2004; Murtada et al., 2010; Stalhand et al., 2011) since this active force can modify completely the stress field over the arterial wall. In summary, we have extended previous SEDF by means of a cross-link-related parameter that allow to gather mechanical features that previous models did not. With this approach we have been able to fit a set of mechanical tests of carotid arteries that produced better results than with previous models. This information, together with a geometrical reconstruction of a human carotid artery, leads to a finite element model of this geometry, with variable properties over the arterial length and a micro-structural description of the wall components based on experimental data. Kinematics of finite growth 109 4.2 Kinematics of finite growth Within the framework of finite growth, the key kinematic assumption is the multiplicative decomposition of the deformation gradient Finto an elastic part Feand a growth part Fg(Rodriguez et al., 1994), F=Fe·Fgwith F=∇Xϕ,(4.1) adapting a concept first proposed in the context of finite elasto-plasticity (Lee, 1969). During this Chapter, we consider the growth part of the deformation gradient to be, at least, function of a growth tensor ϑ. From now on we consider this variable as a scalar quantity, ϑ. The Jacobians of the deformation gradient and its elastic and growth counterparts will be denoted as J=det(F),Je= det(Fe), and Jg=det(Fg), respectively, such that J=JeJg. We can then introduce the right Cauchy Green tensor C, and, in complete analogy, the elastic right Cauchy Green tensor Ce. C=Ft·F Ce=Ft e·Fe=F−t g·C·F−1 g.(4.2) Recall that the pull back of the spatial velocity gradient to the intermediate configuration F−1 e·l·Fe=Le+Lg(4.3) results in its additive decomposition into the elastic velocity gradient Leand the growth velocity gradient Lg Le=F−1 e·˙ Fe=˙ Fe·F−1 eand Lg=˙ Fg·F−1 g,(4.4) such that the rate of deformation tensor of the intermediate configuration can be expressed as dg=lsym gwith lg=Fe·Lg·F−1 e. Figure 4.3 illustrates the kinematics of finite growth, the deformation tensors Cand Ce, and the mappings F=Fe·Fgand F−t=F−t e·F−t gbetween tangent spaces TB,TBgand TB0 in the material configuration, the intermediate configuration, and the spatial configuration. 110 Computational modeling of hypertension-induced anisotropic growth. Figure 4.3: Kinematics of growth. Composition of a elastic deformation gradient Feand a growth tensor Fg. Remark 13 (Growth of arterial tissue) For the particular problem of growth in arterial tissue, we adopt the formulation proposed by Himpel et al. (2005) and Goektepe et al. (2010). To account for experimental observations of SMC growth (Owens et al., 1981; Owens and Schwartz, 1983; Owens, 1989), we define the growth tensor as Fg=I+ [ ϑ−1 ] nr⊗nr where ϑis the scalar-valued growth multiplier that defines the level of growth and nrcharacterizes the radial direction (Rausch et al., 2011). This particular format of the growth tensor characterizes smooth muscle cell thickening in the radial direction nrwhile the ellipsoidal muscle cells maintain the same length, see Figure 4.4. Acutely, SMC maintain their original length by contracting. Chronically SMC grow in the radial direction to reduce elevated wall stresses Some experiments indicate that SMC may also contract in the axial direction, to decrease the lumen diameter. However, for simplicity, here we assume that the axial dimension remains constant. Hyperelastic model of growth 111 (a) Nuclei and actin staining of a SMC (Thakar et al., 2009) (b) Sketch representation of a SMC. Figure 4.4: Representation of a SMC based on experimental data from (Thakar et al., 2009). 6 e2 H Hj e3  e1 4.3 Hyperelastic model of growth We consider a hyperelastic framework for soft tissue, motivated by their highly nonlinear behavior and the finite strains they are subjected to in-vivo. Given a Helmmholtz strain energy density function (SEDF) in its material description Ψ(C,Fg), and based on the dissipation inequality from classical thermodynamics we can write, neglecting heat sources and thermal effects given the constant temperature in living organs, −˙ Ψ = −∂CΨ : ˙ C−∂FgΨ : ˙ Fg,(4.5) S= 2∂CΨ = 2∂CeΨ : ∂CCe=F−1 g·2∂CeΨFt g=F−1 g·Se·F−t g.(4.6) To obtain the tangent moduli, essential for a consistent finite element implementation, we evaluate the total derivative of the Swith respect to C. C= 2dCS= 2∂CSFg+ 2∂CSF= 2∂CSFg+ 2 ∂FgS:∂ϑFg⊗∂CϑF(4.7) The first term reads the classical tangent modulus and which is no more that a pull-back of the Cewith Fg. We will call Cethe elastic tangent modulus in the intermediate configuration, ˆ Cethe same tensor in the initial, stress-free 112 Computational modeling of hypertension-induced anisotropic growth. configuration and cethe elastic modulus in the actual configuration. ˆ Ce=F−1 g·F−1 g·2∂2 CeΨ·F−t g·F−t g=F−1 g·F−1 g·Ce·F−t g·F−t g.(4.8) The second sum term is related with the linearization of the stresses for the growth model. ∂ϑFgand ∂Fϑare specific of the Fgchosen and is provided in following sections. Finally, ∂FgSreads ∂FgS=−F−1 g⊗S+S⊗F−1 g−F−1 g⊗F−1 g:ˆ Ce 2:F−t g⊗Ce+Ce⊗F−t g(4.9) with ({•}¯ ⊗{◦})ijkl ={•}ik{•}jl and ({•}⊗{◦})ijkl ={•}il{•}jk. Last, we need to define the evolution of the growth multiplier ϑ. We adopt the assumption of stress-driven growth described in Goektepe et al. (2010) ˙ ϑ= κ(ϑ)φ(Ξ)where Ξrepresents a given stimuli. κ(ϑ)is a function ensuring that the tissue does not grow over a threshold limit and φ(Ξ)represents the growth criteria. In the next two sections, we discuss two different approaches to the description of the growth evolution. The first one is a Mandel-based driven evolution and the second one is based on a strain measure. 4.3.1 Numerical implementation For the numerical implementation of the growth evolution law, we adopt an implicit Euler backward scheme ˙ ϑ= [ϑn+1 −ϑn]/∆t(4.10) We introduce the discrete residual R=ϑn+1 −ϑn−κ(ϑ)φ(Ξ)∆t. (4.11) Hyperelastic model of growth 113 To solve the non-linear equation we expand the residual form up to the first order term. This allows us to solve the problem within a Newton iterative scheme as R(ϑ)n+∂ϑR(ϑ)nϑn[ϑ−ϑn] = 0.(4.12) The tangent of the residual form reads then K=dϑR= 1 −[κ∂ϑφ+ϑ∂ϑκ] ∆t, (4.13) and the local update of the growth variable can be updated as ϑn−R/K → ϑn+1 to evaluate the growth parameter determined by the current stress state and the loading history. Table 4.1 shows the algorithm used to compute the growth model. It remains to establish the function κ(ϑ)and the growth criteria φ(Ξ). In the following we use κ(ϑ) = 1 τϑmax −ϑ ϑmax −1γ ∂ϑκ(ϑ) = −γ ϑmax −ϑκ(ϑ),(4.17) ϑmax sets the threshold value of growth. τand γrepresent parameters that control the speed and the non-linearity of the growth respectively. The particular expression for φ(Ξ)is established in the next two sections. 4.3.2 Stress-type anisotropic growth Last, we need to define the evolution of the growth multiplier ϑ. We adopt the assumption of stress-driven growth described in Goektepe et al. (2010) ˙ ϑ= κ(ϑ)φ(Me)with φ(Me) = tr(Me)−Mcrit e.(4.18) Note that Me=Ce·Seand Mcrit erepresent the growth criteria value above which growth occurs. The derivative of the growth criteria with respect to the growth variable reads ∂ϑφ=−∂ϑCe:Se+Ce:∂ϑSe,with (4.19) 114 Computational modeling of hypertension-induced anisotropic growth. Input:Ft+1, ϑt 1. Compute different kinematic variables: Ct+1,Ft+1 e=Ft+1 ·Ft gand consitutive variables St+1,St+1 e. 2. Check state of tissues IF (λSMC > λcrit)THEN Apply local newton iteration ELSE GO TO 4 3. Local newton iteration WHILE R> Tol DO R=ϑn+1 −ϑn−κ(ϑ)φ(Ξ)∆t, (4.14) K= 1 −[κ∂ϑφ+ϑ∂ϑκ] ∆t, (4.15) ϑn+1 ←ϑn−K−1·R.(4.16) 4. Compute Cauchy stresses σt+1. 5. Compute tangent operator related to the Jaumann rate that ABAQUS uses as ∇ ct+1 = [ct+1 + 1/2[δτt+1 +τt+1δ+δτ t+1 +τt+1δ]]/J . Output:σt+1,∇ c, ϑn+1 Table 4.1: Algorithm for an implicit Euler scheme of volumetric growth A growth model problem in hypertension 115 ∂ϑCe=−F−t g·∂ϑFt g·Ce−Ce·∂ϑFg·F−1 gand ∂ϑSe=1 2Ce:∂ϑCe.(4.20) Finally, to completely define Eq. 4.7, the derivative of the growth variable with respect to the Cauchy-Green strain tensor lead to ∂Cϑ=∂Ceϑ:∂CCe=kg∆t K1 2Ce:Ce+Se.(4.21) 4.3.3 Strain-driven anisotropic growth And the evolution of the growth multiplier ϑfor the strain-driven growth based on the description given in Goektepe et al. (2010) as φ(λ) = λ−λcrit and ∂φ ∂ϑ =−λ/λ2 g.(4.22) Note that λSMC can be expressed as λ=Fe: [nz⊗nz], this is the elastic stretches in the longitudinal direction of the SMC. It remains to state the sensitivities of the scaling function κ, the growth law φ, and the SMC stress with respect to the growth multiplier. 4.4 A growth model problem in hypertension In this section we present some results of the two approaches described above and we discuss about its applicability to modeling volumetric growth in arterial tissue due to hypertensive conditions. 4.4.1 Preliminary results The aim of this first subsection is making a first approach to the volumetric growth in the arterial wall to analyze some growth features, e.g. what driven quantity is more suitable for hypertension-induced growth. The procedure is as follow: •Apply internal pressure up to normotensive state, this is an internal pressure of 13.3 kPa. At this point, the critic stretch or stress values are calculated 116 Computational modeling of hypertension-induced anisotropic growth. and saved to trigger the growth evolution. •Apply elevated pressure to simulate a hypertensive pressure at 16.0 kPa. •Allow the tissue to growth to compensate the extra deformation of the SMC. First, we apply this methodology to an idealized artery slice, made up of media and adventitia layer. Mechanical parameters are taken from the values of the CCA which were described in chapter 3. The growth parameters were chosen as ϑmax = 2,γ= 2 and τ= 1. A growth model problem in hypertension 117 (a) Maximun principal strain at normotensive conditions. (b) Maximun principal strain at hypertensive conditions. (c) Maximun principal Piola-Kircchoff stress at normotensive conditions. (d) Maximun principal Piola-Kircchoff stress at hypertensive conditions. (f) Growth for the straindriven problem. (g) Growth for the stressdriven problem. (h) Maximun principal Piola-Kircchoff stress for the strain-driven problem. (i) Maximun principal Piola-Kircchoff stress for the stress-driven problem. Figure 4.5: Growth variable and stress evolution. 118 Computational modeling of hypertension-induced anisotropic growth. In Fig. 4.5(a-b) we picture the maximum strain field in the slice of the CCA for the normotensive and hypertensive load state. In Fig. 4.5(f-g) we show the evolution the growth variable at two different time steps for the two driven cases while 4.5(h-i) shows the maximum principal stress. Result are quite different. In the Mandel case the volumetric growth goes outward to inwards and in a very marked way. As we discuss in previous section, mechanical properties of the adventitia layer are much stiffer than properties of the media layer, these higher stresses in the adventitia layer makes the material of this layer to growth much faster. Moreover, given the higher rate of growth in adventitia, the expansion of the material in the perpendicular direction toward the center of the slice makes the internal material, corresponding to the media layer, to decrease its circumferential direction. In this simulation we have adopted a growth model that applies all over the thickness. Himpel et al. (2005); Rodriguez et al. (2007) also relied on computational models that do not consider the separation of the media and adventitia layers. Our results show that taking into account different layers, growth occurs in the adventitia layer when imposing a Mandel-driven stimulus. Experimental findings, as we discuss in the introduction, show that thickening occurs meanly in the media layer due to the hypertrophy and hyperplasia of the SMC, which are located in the media layer. In the case of the strain driven problem (Fig. 4.5(f-g)), the growth shows a more uniform distribution over the thickness. Since the circumferential stretch is more uniform than the stress distribution growth occurs in a more spread way with higher growth in the outer than in the inner layer. There is an open debate about what is the actual quantity that better describes different processes in cells, as differentiation, migration, etc., and in this direction the choice of a strain or stress based approach to describe the actual volumetric growth due to hypertension is also not clear. From these results we can conclude that mechanical stimuli for growth in arterial tissue is an important issue. In the particular case of growth in arteries is also known that growth occurs basically in the media layer since there is where the SMC are placed. In order to better describe the actual behavior of the arterial wall we focus on a model where growth is allowed to occur only in the media layer. A growth model problem in hypertension 125 (a) Homeostatic state (b) Hypertensive state (d) Time t=5 days (e) Time t=15 days (f) Time t=40 days (g) Time t=100 days (h) Time t=5 days (i) Time t=15 days (j) Time t=40 days (k) Time t=100 days (l) Time t=5 days (m) Time t=15 days (n) Time t=40 days (o) Time t=100 days Figure 4.13: Evolution of the growth (d-g) and maximal principal stresses in the adventitia (h-k) media (l-o)layer at different times steps in the slice of the ICA. 126 Computational modeling of hypertension-induced anisotropic growth. (a) Homeostatic state (b) Hypertensive state (d) Time t=5 days (e) Time t=15 days (f) Time t=40 days (g) Time t=100 days (h) Time t=5 days (i) Time t=15 days (j) Time t=40 days (k) Time t=100 days (l) Time t=5 days (m) Time t=15 days (n) Time t=40 days (o) Time t=100 days Figure 4.14: Evolution of the growth (d-g) and maximal principal stresses in the adventitia (h-k) media (l-o)layer at different times steps in the slice of the ECA Conclusions 127 4.5 Conclusions The study of remodeling and growth of biological tissue have been an extensive research field over the last decades (Ambrosi et al., 2011). Researchers deal with different kinematics formulations, approaches to balance equations, define appropiate evolution equations and identification of suitable stimuli to drive the growth process. In this Chapter we have focused on a classical kinematics decomposition of the deformation gradient (Lee, 1969; Rodriguez et al., 1994) into an elastic and a growth part. Moreover, this formulation was consistently linealized and it was embedded into a finite element framework. Our model looks into the volumetric growth process of the arterial tissue. This growth, or arterial thickening, is due to the hypertrophy of the SMC. We studied how the growth of a micro-structural component, the SMCs, undergo a macroscopically thickening of the arterial wall. Our approach establishes that stretch of the SMC is the stimulus that triggers the SMC growth. We also consider that SMC thicken only along its radial direction. Our results show a good agreement in terms of thickening and growth rates in comparison with experimental findings. We also showed the FE simulation of a real patient-specific carotid geometry. Our numerical results showed a very homogeneous growth over the medial layer all over the carotid length and only small portions of the carotid bifurcation showed slight variations. Our simulations also raise an interesting aspect of the stress distribution. Stresses in the media layer decrease due to the effect of the SMC cell. This is a classical assumption in adaptation of biological tissue, it reacts to compensate the over stimuli and goes back to a physiological situation. Cells are responsible for this sensing task and SMCs, in particular, are for sensing strains in the arterial wall. However, stresses in the adventitia layer rise up in a 20-30% (note the highly non-linear behavior pointed out in chapter 3). SMC growth in the media layer induces strains in the adventitia and these, in turn, make the stresses to increase. These results also meet the well accepted idea that the adventitia layer acts as a protection layer. There are also not few limitations. Our model is based on purely passive underlying elasticity. SMC have a very important active component, the myogenic 128 Computational modeling of hypertension-induced anisotropic growth. tone, which makes the arterial wall to contract or expand to maintain a similar lumen radio. The arterial model could improved by the inclusion of this feature, althought, up to date, only a few computational models of myogenic tone are available in literature. The growth stimuli could be better characterized based on this basal tone. Note, however, that the myogenic tone is a response to the over stretch of the SMC, which are the stimuli that we are considering and results should not differ markedly from our present results. It is also important to note that the evolution equation for growth pre-imposes both the growth level and its rate. Rate-related parameters can be experimentally fitted to describe specific arteries and specimens. This leads other two important limitations. Experimental results are usually studied at normotensive and at the final stage of hypertension, which do not allow to fit rate-dependent parameters. Although some of them do, none of them make and extensive study of different spots of an artery, which neither allow to stablish more accurate growth models. Moreover, experimental results are amazingly different from arteries, specimens and even some arteries of the same species. There are, therefore, drawbacks in the experimental part of the study that turn to induce limitations of the computational model. In summary, we have adopted a well established computational model for volumetric growth and we have adapted it to describe the thickening of a human carotid artery. Our model takes a generic approach of growth and bases the actual thickening of the arterial wall on the microstructural growth of the SMC. This and the consideration of variables related to the micro-structure, nicely establish a multiscale view of the problem. In this way we can simulate arterial thickening of any kind of artery in hypertension. This thickening is a very common complication in hypertensive patients because it leads to a decrease in the blood flow and other related complications. Computational models like the one presented here can, and do, help to understand the underlying mechano-chemical processes and to provide a frame work for biologists and clinicians to develop drugs and devices to prevent and deal with this kind of complications. 5 Computational model of collagen density-growth. The goal of this chapter is the study of the turnover of collagen content in arterial tissue in hypertensive patients. This process can be divided in three different stages. The flowchart begins studying the SMC synthesis of different biological substances. As we described in chapter 3, SMC are located mainly in the media layer. The trigger stimulus of this mass production is assumed to be the over-stretching of the SMC due to the increase of pressure reported in hypertensive patients. The next step is the mass-transport simulation through the arterial wall up to the adventitia layer of the substances previously released into the extracelullar matrix. The third, and last step, is to compute the turnover of collagen based on the amount of substances in the arterial wall, which interact each other and also trigger other cells, like fibroblast, to modify the turnover rate of collagen. The overall change in the collagen density can be attributed to the perturbation of the deposition/degradation rate of collagen in the ECM. 129 130 Computational model of collagen density-growth. 5.1 Mass production of biological substances 5.1.1 Introduction In this section we study the mass production of different substances of interest within the context of collagen turnover in arterial tissue. Classical close-systems in continuum mechanics keep mass changes constant; however, this is not the case of biological tissues. For this reason we consider the thermodynamic theory of open systems in this work. This assumes a non-vanishing term, called source term R, that fulfill mass balance equation. Next we will discuss some particular substances and processes of interest in hypertensive disease. In terms of collagen deposition, which is our final goal, fibrogenic cytokine proteins such as the transforming growth factor TGF-βare the most important regulators of collagen synthesis (see e.g. Border and Noble (1994); Wrana et al. (1994)) by vascular fibroblasts (Burke and Ross, 1979; Roberts et al., 1986). TGF-βis a protein that influences cellular functions as proliferation or differentiation and plays a key role in numerous cardiovascular diseases (Massague et al., 2000) and cancer (Massague, 2008). The activation pathway (Massague, 2000) of the TGF-βfamily, which is part of a superfamily of proteins, is still not completely understood. They also play a prominent role in SMC proliferation (Owens, 1995; Raines, 2004). Butt et al. (1995) reported that SMC release TGF-βgrowth factor associated with an increase of procollagen deposition and a decrease of collagen degradation. They also showed that both SMC and endothelial cells are the source of platelet-derived growth factor PDGF, another important growth factor. The molecular structure of TGF-βis shown in Fig. 5.1(a), presenting a 25 [kDa] molecular weight and an equivalent external radius of around 3.8·10−9[m]. In terms of degradation, Metalloproteinases Enzymes (MMP) (Nagase and Woessner, 1999) and Tissue Inhibitors of Metalloproteinase (TIMP) (WojtowiczPraga et al., 1997), and in particular their aspect ratio MMP/TIMP (Visse and Nagase, 2003), might be the most important metric to quantify the evolution of collagen degradation (Galis and Khatri, 2002). MMP (Galis et al., 1994), in addition to other regulatory mechanisms such as differentiation and apoptosis of cells, Mass production of biological substances 131 are responsible for extracellular matrix degradation in general, and for collagen degradation in particular. TIMPs are a type of inhibitor of metalloproteinase. In hypertension, TIMP have been reported to increase, decreasing the total MMP, which, in turn, decreases the rate of collagen degradation. O’Callaghan and Williams (2000) have shown that the amount of collagen turnover increases with the magnitude of the strain imposed to SMC in in-vitro experiments. They also reported the production of MMP-2, a gelatinase-degrading enzyme and TGF-β (see also Sarzani et al. (1989); Hamet et al. (1991)), which could be stimulated by cyclic stretching. It acts as an important regulator of ECM production, mainly by inhibition of MMP-1 and by increasing the activity of MMP-2. The molecular structure of MMP-1 and TIMP-1 is shown in Fig. 5.1(b) and (c), presenting a 52 [kDa] and 28 [kDa] molecular weight and an equivalent external radius of around 4·10−9[m] and 7·10−9[m] respectively. MMP-1 is known as one of the most important collagenase and will be the form used here as well as its inhibitor TIMP-1. (a) TGF −βmolecule (b) MMP-1 molecule (c) TIMP-1 molecule Figure 5.1: Molecule representation (PDBe, 2013). 5.1.2 Model In this section a model for synthesis and degradation of TGF-β, MMP and TIMP is presented. We allow the material density to evolve in time according to the balance of mass for open system thermodynamics, and adopt a source term, R, similar to the one described by Harrigan and Hamilton (1992); Kuhl and 132 Computational model of collagen density-growth. Steinmann (2003), as ˙ρ=Rwith R=ρ ρ∗−m λ−λ∗,(5.1) where the exponent mtypically vary between two and three, λ∗is the stretch of the homeostatic equilibrium state and λ= [r·C·r]1/2the stretch of a fiber with orientation r.ρis the density of the substance at hand, ρ∗is the initial density and ˙ρthe material time derivative of ρ. Depending on the stretch of the SMC λsmc, which we understand to be the main driving force for these processes, the substance density will increase or decrease. In particular, we hypothesize that an increase in the stretch will increase growth factors such as TGF-β, responsible for collagen deposition. RTGF−β= ˙ρTGF−β=γTGF−β "ρTGF−β ρ∗ TGF−β#−mTGF−β λsmc −λ∗ TGF−β (5.2) Both deposition and degradation can occur at the intracellular (collagen molecules) or extracellular (tropocollagen) levels. There have been several in-vitro and invivo experiments studying these processes in detail, as we discussed in the introduction. There is a strong agreement that the TGF −βplays a fundamental role in the turnover of collagen. This implies that TGF-βwill be upregulated, RTGF−β>0, for blood pressures above a characteristic threshold level, hρTGF−β/ρ∗ TGF−βi−mTGF−βλsmc > λ∗ TGF−β, downregulated, RTGF−β<0, for blood pressures below, hρTGF−β/ρ∗ TGF−βi−mTGF−βλsmc < λ∗ TGF−β, and otherwise remain constant, RTGF−β= 0. We further hypothesize that an increase in blood pressure will increase tissue inhibitors of metalloproteinase, TIMP, causing a decrease in metalloproteinase, responsible for collagen degradation. The mass production of TIMP is also expressed as RTIMP = ˙ρTIMP =γTIMP "ρTIMP ρ∗ TIMP −mTIMP λsmc −λ∗ TIMP#(5.3) Mass production of biological substances 133 This implies that TIMP will be upregulated, RTIMP >0, for blood pressures above a characteristic threshold, [ρTIMP/ρ∗ TIMP]−mTIMP λsmc > λ∗ TIMP, downregulated RTIMP <0, for blood pressures below, [ρTIMP/ρ∗ TIMP]−mTIMP λsmc > λ∗ TIMP, and otherwise remain constant, RTIMP = 0. Basically, we consider the changes of TGF-βand TIMP as the primary variables, assuming that the upand downregulation of TGF-βand TIMP is driven by the local SMC state. The material parameters mTGF−β, mTIMP ∈R+govern the evolution of TGF-βand TIMP respectively, while γTGF−β, γTIMP ∈R+set the sensibility of these substances to changes of the SMC stretch. Finally, we define the source term of the MMP, which directly change the rate of absorption of collagen. For the sake of simplicity, we hypothesize that it directly correlated to the source term of TIMP as RMMP =γMMP RTIMP (5.4) where γMMP ∈R−defines the sensitivity of MMP to changes in TIMP. 5.1.3 Computational treatment To solve the non-linear differential equations of collagen deposition and absorption ( Eqs. 5.2 and 5.3) we adopt a standard Euler backward scheme, ˙ρ0= [ ρj+1 0−ρj 0]/∆t(5.5) for given initial conditions ρ0|t0=0 =ρ∗ 0. The temporal discretization is given by dividing the time interval Tinto sdiscrete subintervals, T=Ss−1 j=0[tj, tj+1], with a time increment ∆t=tj+1 −tj≥0. We transform the evolution equations Eqns. 5.2 and 5.3 for TGF-βand TIMP into their residual formats, RTGF−β=ρj+1 TGF−β−ρj TGF−β− RTGF−β∆t= 0 RTIMP =ρj+1 TIMP −ρj TIMP − RTIMP ∆t= 0.(5.6) We solve these equations by applying a local Newton-Raphson iteration, based on a Taylor expansion at ρk TGF−βand ρk TIMP up to terms of first order, see e.g 134 Computational model of collagen density-growth. Kuhl et al. (2003a); Kuhl and Steinmann (2003) for more details. To this end, we calculate the derivative of the residuals with respect to the current TGF-β and TIMP concentrations. ∂RTGF−β ∂ρTGF−β −1 ρk TGF−β = 1 −∂˙ρk TGF−β ∂ρk TGF−β ∆t= 1 + mTGF−β ρk TGF−β"ρk TGF−β ρ∗ TGF−β#−mTGF−β λsmc ∆t ∂RTIMP ∂ρTIMP  −1 ρk TIMP = 1 −∂˙ρk TIMP ∂ρk TIMP ∆t= 1 + mTIMP ρk TIMP ρk TIMP ρ∗ TIMP −mTIMP λsmc ∆t (5.7) We can then calculate the discrete changes in the TGF-βand TIMP concentrations, ∆ρTGF−β=−∂RTGF-β ∂ρTGF−β −1 ρk TGF−β RTGF−β(ρk TGF−β) ∆ρTIMP =−∂RTIMP ∂ρTIMP  −1 ρk TIMP RTIMP(ρk TIMP) (5.8) and update the current concentration values. ρk+1 TGF-β=ρk TGF-β+ ∆ρTGF−β ρk+1 TIMP =ρk TIMP + ∆ρTIMP (5.9) Once the local Newton iteration is converged, we can calculate the MMP concentration ρj+1 MMP =γMMP ρj+1 TIMP (5.10) Table 5.1 summarizes the algorithm to compute the update of the local collagen content. 5.1.4 Sensitivity anaylisis In Fig. 5.2 we present a sensitivity analysis of the material parameters involved in the evolution of the TGF−βcontent for baseline values of m= 1,ρ∗ TGF−β= 1.0, γTGF−β= 1.0and λ∗ TGF−β= 1.1. We impose a λSMC = 1.2in this case. We study the problem for s= 100 discrete time steps of ∆t= 1. We illustrate in Fig. 5.2(a) the sensitivity of the TGF−βconcentration when we vary the γTGF-β Diffusion through the vessel wall 141 (a) t=0 (b) t=2 (c) t=5 (d) t=19 Figure 5.6: Evolution of the mass production of TIMP due to SMC activity in different depth cuts. 142 Computational model of collagen density-growth. (a) t=0 (b) t=2 (c) t=5 (d) t=19 Figure 5.7: Evolution of the mass production of TIMP due to SMC activity in different transversal cuts. Diffusion through the vessel wall 143 (a) t=0 (b) t=2 (c) t=5 (d) t=19 (e) t=0 Figure 5.8: Evolution of the mass production of MMP due to SMC activity in different depth cuts. 144 Computational model of collagen density-growth. (a) t=0 (b) t=2 (c) t=5 (d) t=19 (e) t=0 Figure 5.9: Evolution of the mass production of MMP due to SMC activity in different transversal cuts. Diffusion through the vessel wall 145 (a) Featured points in the media layer. 0 20 40 60 80 100 50 100 150 200 time [−] TGF [ng/mL] Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 (b) Evolution of TGF-β. 0 20 40 60 80 100 400 500 600 700 800 900 1000 time [−] TIMP [ng/mL] Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 (c) Evolution of MMP 0 20 40 60 80 100 26 28 30 32 34 36 time [−] MMP [ng/mL] Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 (d) Evolution TIMP Figure 5.10: Evolution of the mass production of (a) TGF, (b) MMP and (c) TIMP at six featured points due to SMC activity. phenomena (see, e.g. Philibert (2006) for a review), about Brownian movement by Einstein (1905) as well as the works of related atomistic and macroscopic transport by Maxwell (1871) and Huang (1987) have boosted a great amount of works and efforts in the study of mass transport. In the field of biology and biomechanics mass transport toward and within the arterial wall is a fundamental process to understand not only different vascular diseases but also the normal evolution of arteries. Mass transport in arterial tissue of different kind of molecules occurs due to concentrations and pressure gradients across the thickness by the well known diffusion and convective phenomena (Tarbell, 2003a). Low-density proteins (LDL) are probably the most studied molecular transport since they are the main initiator of atheroesclerotic plaque (Lusis, 2000) and many works have showed up during the last years, see e.g. Curmi et al. (1990); Glagov et al. (1988); Cancel et al. (2007) and Kim and Tarbell (1994); Huang and Tarbell (1997); Tada and Tarbell (2002) for experimental and numerical works respectively. There are however other important molecules involved in hypertension that gain less attention but also play an im- 146 Computational model of collagen density-growth. portant role in the collagen turnover. As we discussed in the introduction, we are looking at three substances: TGF-β, TIMP and MMP. In the next section we study how these three molecules move through the arterial wall based on their specific features. 5.2.2 Preliminary study of transport phenomena Diffusion and convection of molecules is an important issue to be considered in mass transport within the arterial wall. In order to discern if these two phenomena are important in mass transport within the arterial wall is usual to calculate the Peclet number (Pe) (Eq. 5.11). For Pe >> 1convection have to be studied while for Pe << 1diffusion dominates the problem. Pe =vl D,(5.11) where v [m/s] is the fluid velocity through the thickness, l [m] is the thickness and D [m2/s] the diffusion coefficient. Diffusion coefficients are related to the molecular size and the size of the porous media where the molecule goes through. For a free media the diffusion coefficient, for the three substances considered in this work, is given by the Stokes-Einstein Equation (Einstein, 1905) as D∞[m2/s] = kBT 6πµa =       1.9227 ·10−12 TGF-β, a = 3.8·10−9m 3.8455 ·10−12 TIMP, a = 7 ·10−9m 2.1974 ·10−12 MMP, a = 4 ·10−9m, (5.12) where kB= 1.38·10−23[J/K]is the Boltzmann constant, T = 293[K] the absolute temperature which remain constant, µ= 4 ·10−3[N.s/m2] is the viscosity of the fluid and ais the particle radius obtained by superposing an sphere that contains the particle (see Fig. 5.1). Diffusivity in a fiber-matrix media was derived by Ogston et al. (1973) and used later on by Kim and Tarbell (1994), among others. The effective diffusion Diffusion through the vessel wall 147 coefficient can be expressed as Dm[m2/s] = D∞exp −[(1 −)0.5(1 + a/rf)]=       8.081 ·10−13 for TGF-β 1.613 ·10−12 for TIMP 9.229 ·10−13 for MMP, (5.13) where rfis the fiber radius which has an average value of 1µm and = 1 −Vf the void fraction and Vfthe fiber volume fraction, which, based on data given by Huang and Tarbell (1997) and O’Connell et al. (2008), is approximately equal to 25%. Moreover, the contribution of the presence of SMC can be recovered (Huang and Tarbell, 1997) by Deff [m2/s] = 1 1−SMC 1 f(SMC)D=       4.444 ·10−13 for TGF-β 8.887 ·10−13 for TIMP 5.075 ·10−13 for MMP, (5.14) where fSMC = 2.3899 given a SMC volume fraction of SMC = 0.45% (Huang and Tarbell, 1997). Going back to Eq. 5.11 we can obtain the Pecklet number for this problem as Pe =vl Deff ≈102,(5.15) where v take a mean value of 0.05 [µm/s] (Levick, 1987; Wang and Tarbell, 1995), l 1-0.6 [mm] thickness and the effective diffusive parameters given in Eq. 5.14, gives a Peclet number in the order of hundreds, which points out the necessity of taking into account the convective phenomena. The coefficient related to the convective term, the lag coefficient of solute in the interstitial phase with fiber matrix, was showed by Curry (1984) to be 148 Computational model of collagen density-growth. expressed as Kcf [−] = 2 −Φ =        1.006 for TGF-β 1.011 for TIMP 1.006 for MMP, (5.16) where Φis the partition coefficient defined by Ogston et al. (1973) as Φ[−] = exp −[(1 −)(2a/rf+a2/r2 f)]=       0.994 for TGF-β 0.989 for TIMP 0.994 for MMP. (5.17) As we did in previous coefficient, we need to weight the lag coefficient to include the SMC cell contribution as Kcfeff [−] = 1 1−SMC Kcf =       1.348 for TGF-β 1.341 for TIMP 1.348 for MMP. (5.18) 5.2.3 Convection-Diffusion problem in the arterial wall Considering changes in mass as its basic characteristic, see e.g. the monographs by Welty et al. (2008) and Deen (2011), the balance of mass (Eq. 5.19) has to fulfill that the flux of mass and the mass source be in equilibrium with the change of material density Dtρ0=DivJ+R0in Ω,t∈(0, T),(5.19) where ρ0is the material density, Jthe mass flux and R0the source of mass. The domain of interest, as previously introduced, will be denoted by Ω. As it was demonstrated in the previous Section, the mass transport in arterial tissue is a diffusion-convection problem. To recover these issues the flux term is given by J=−D∇ρ0−ρ0Dtu,(5.20) Diffusion through the vessel wall 149 where the first term represents the diffusive contribution and is driven by density gradients. The second term represents the convective contribution and it is assumed to be controlled by pressure gradients with uthe displacement vector. We can obtain the velocity through the solid by means of Darcy’s law (Eq. 5.21). Note therefore, that Navier-Stokes equations are not used and velocity is not obtained independently but based on pressure gradients. Some authors (Tada and Tarbell, 2002) have also used Brinkman equation (Brinkman, 1947) for model the transmural flow. ∇p=−µ/KpDtu,(5.21) where p is the pressure and µthe viscosity of the fluid. The parameter need to study the convection related term in blood vessels, according to the fiber-matrix theory, is the hydraulic permeability Kpwhich can be calculated with the Kozeny-Carman equation as discussed by Curry (1984). Kp[m] = r2 f2 4ck(1 −)2= 1.389e−13,(5.22) where ckis the Kozeny constant with a value equal to 5 (Huang and Tarbell, 1997). And again the permeability have to be recalculated to take into account SCM distribution. In a similar way Kpeff [m] = Kp 1−SMC −0.3058SMC 1−SMC + 0.3058SMC = 7.9721e−14.(5.23) 5.2.4 Computational aspects The partial differential equation presented in Eq. 5.19 is managed numerically to be solved within a Finite Element scheme. The basic procedure is here described for sake of clarity. The problem is solved in ABAQUS with standard elements and formulation therefore it does not give any new insight on the issue. By evaluating Eq. 5.19 in the domain Ω0and applying Divergence Theorem we get ˆΩ0 Dtρ0dV =ˆδΩ0 J·ndV +ˆΩ0R0dV.(5.24) 150 Computational model of collagen density-growth. layer endo ext media adv. Figure 5.11: Sketch of the layers and boundary conditions in the mass transport problem. The boundary of the problem (Eq. 5.25) δΩ0is conditioned by Dirichlet boundary conditions over δΩd 0and Neumann boundary conditions over δΩn 0. ρ0=ρ∗ 0on δΩd 0, J·n=Ron δΩn 0. (5.25) In particular we consider the following set of boundary conditions in the boundary problem in Fig. 5.11 J·n= 0 on δΩn endo (5.26) Jlayer+·nlayer+=Jlayer−·nlayer−on δΩn layer (5.27) Jext ·next = 0 on δΩd ext.(5.28) 5.2.5 Finite element results In this section we present the results of the mass transport from the media layer through the arterial wall. The results are presented in two longitudinal cuts and only for the adventitia layer, for sake of clarity. Note that fibroblast are located mainly in the adventitia layer and they are responsible for the synthesis of collagen molecules. Therefore, we focus on the substances turnover only in the adventitia layer. In Figs. 5.12 and 5.13 we show the evolution of the TGF-β content in the adventitia layer for different times and both cuts. 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