MECHANICAL PROPERTIES OF ARTERIAL WALL by JORGE OCTAVIO VIRUES DELGADILLO Bachelor in Chem. Eng., Universidad Veracruzana, 2001 Master in Engineering (Chem. Eng.), Universidad Nacional Autonoma de Mexico, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Chemical and Biological Engineering) THE UNIVERSITY OF BRITISH COLUMBIA February 2008 © Jorge Octavio Virues Delgadillo, 2008 11 ABSTRACT The incidence of restenosis has been shown to be correlated with the overstretching of the arterial wall during an angioplasty procedure. It has been proposed that slow balloon inflation results in lower intramural stresses, therefore minimizing vascular injury and restenosis rate. The analysis of the biomechanics of the arterial tissue might contribute to understand which factors trigger restenosis. However, few mechanical data are available on human arteries because of the difficulty of testing artery samples often obtained from autopsy while arteries are still considered "fresh". Various solutions mimicking the physiological environment have been used to preserve artery samples from harvesting to testing. In vitro mechanical testing is usually preferred since it is difficult to test arteries in vivo. Uniaxial and biaxial testing has been used to characterize anisotropic materials such as arteries, although methodological aspects are still debated. Several objectives were formulated and analyzed during the making of this thesis. In one study, the effect of deformation rate on the mechanical behavior of arterial tissue was investigated. The effect of several preservation methods, including cryopreservation, on the mechanical properties of porcine thoracic aortas was also analyzed. Finally, the differences in the mechanical behavior between three different types of sample geometry and boundary conditions were compared under uniaxial and equi-biaxial testing. Thoracic aortas were harvested within the day of death of pigs from a local slaughterhouse. Upon arrival, connective tissue was removed from the external wall of the artery. Then the artery was cut open along its length and cut out in rectangular samples for uniaxial testing, and square and cruciform samples for biaxial testing. Samples belonging to the freezing effect study were preserved for two months at -20°C and -80°C in isotonic saline solution, Krebs-Henseleit solution with 1.8 M dimethylsulfoxide, and dipped in liquid nitrogen. Samples belonging to the deformation rate effect study were tested uniaxially and equi-biaxially at deformation rates from 10 to 200 %/s. The uniaxial and biaxial experiments were simulated with the help of an inverse finite element software. The use of inverse modeling to fit the material properties by taking into account the non-uniform stress distribution was demonstrated. A rate-dependent isotropic hyperelastic constitutive equation, derived from the Mooney-Rivlin model, was fitted to the experimental results (i.e. deformation rate study). In the proposed model, one 111 of the material parameters is a linear function of the deformation rate. Overall, inverse finite element simulations using the proposed constitutive relation accurately predict the mechanical properties of the arterial wall. In this thesis, it was found that easier attachment of samples (rectangular and cruciform) is accomplished using clamps rather than hooks. It was also found that the elastic behavior of arteries is nonlinear and non-isotropic when subjected to large deformations. Characterization of the arterial behavior at large deformations over a higher deformation range was achieved using cruciform samples. The mechanical properties of arteries did not significantly change after preservation of arteries for two months. Under uniaxial and biaxial testing, loading forces were reduced up to 20% when the deformation rate was increased from 10 to 200 %/s, which is the opposite to the behaviour seen in other biological tissues. The differences observed in the mechanical behavior of fresh and thawed samples were not significant, independently of the storing medium or freezing temperature used. The lack of significant differences observed in the freezing study was likely due to the small number of samples tested per storing group. Further studies are required to clarify the impact of cryopreservation on extracellular matrix architecture to help tailor an optimized approach to preserve the mechanical properties of arteries. From the results obtained in the deformation rate study, it is concluded that the stiffness of arteries decreases with an increase in the deformation rate. In addition, the effect of deformation rate was observed to be higher than the effect of anisotropy. The inverse relationship between stiffness and deformation rate raises doubts on the hypothesized relationship between intramural stress, arterial injury, and restenosis. iv TABLE OF CONTENTS Abstract^ ii Table of Contents^ iv List of Tables vi List of Figures vii Acknowledgements ^ xi Co-Authorship Statement^ xii Dedication^ xiii 1. Chapter 1: Introduction & Literature Review^ 1 1.1. Introduction 1 1.2. Literature Review^ .2 1.2.1. Physiology of Arteries^ 2 1.2.1.1. Introduction 2 1.2.1.2. Classification of Arteries^ 4 1.2.1.3. Collagen, Elastin and Smooth Muscle Cells^ 5 1.2.2. Mechanical Behavior of Arteries 6 1.2.2.1. Non-linear Elasticity^ 6 1.2.2.2. Anisotropy^ .9 1.2.3. Experimental Testing 9 1.2.3.1. Basic Experiments .9 1.2.4. Modelling the Arterial Wall^ 15 1.2.4.1. Strain energy density function^ 15 1.2.4.2. Neo-Hookean and Mooney models 17 1.2.4.3. Models developed to obtain the behavior of arteries^ 18 1.2.4.4. The Holzapfel Model^ 21 1.2.4.5. The Guccione Model .22 1.2.5. Finite Element Modeling 23 1.3. References^ 27 2. Chapter 2: Scope of Work^ 34 2.1. Introduction ^ 34 2.2. Objectives^ .35 2.3. Thesis Organization^ . 37 2.4. References 39 3. Chapter 3: Mechanical Characterization of Arterial Wall: Should Cruciform or Square Sample Be Used in Biaxial Testing?^ 41 3.1. Introduction^ .41 3.2. Materials and Methods^ .42 3.2.1. Experimental Setup 42 3.2.2. Constitutive Equations 45 3.2.3. Inverse Modeling^ 45 3.3. Results^ .49 3.3.1. Experimental Results 49 3.3.2. Inverse Modeling Results^ 50 3.3.2.1. Uniaxial Fit 53 3.3.2.2. Biaxial Cruciform Fit ^ 54 V3.3.2.3. Biaxial Square Fit^ 56 3.4. Discussion^ 56 3.5. Summary 59 3.6. References 60 4. Chapter 4: Effect of Freezing on the Biaxial Mechanical Properties of Arterial Samples^ 62 4.1. Introduction^ 62 4.2. Materials and Methods .63 4.2.1. Experimental Setup^ 63 4.2.2. Statistical Analysis .66 4.3. Results^ .66 4.3.1. Experimental Results^ . 66 4.4. Discussion 72 4.5. Summary^ . 75 4.6. References 76 5. Chapter 5: Effect of Deformation Rate on the Mechanical Properties of Arterial Samples^ 81 5.1. Introduction^ .81 5.2. Materials and Methods .82 5.2.1. Experimental Setup^ 82 5.2.2. Statistical Analysis .83 5.2.3. Inverse Modeling 84 5.3. Results^ .86 5.3.1. Experimental Results^ 86 5.3.2. Constitutive Modeling 93 5.3.3. Inverse Modeling Results 93 5.3.3.1. Uniaxial & Biaxial Cruciform Fit (Simultaneous)^ 93 5.3.3.2. Uniaxial Fit^ 99 5.4. Discussion^ 102 5.5. Summary 104 5.6. References 106 6. Chapter 6: Conclusions, Recommendations and Contribution to the Knowledge ^ 111 6.1. Introduction^ ..111 6.2. Conclusions 111 6.3. Contribution to knowledge^ 113 6.4. Recommendations for future work 114 6.5. References^ 116 Appendix A: Uniaxial-Biaxial Simulations using Ogden & Guccione Constitutive Equations 117 Appendix B: Inflation of Cylindrical Arteries^ 119 Appendix C: Relaxation Test of Thoracic Aorta 124 Appendix D: Active Response of Thoracic Aorta .. 125 Appendix E: Second Piola-Kirchhoff Stress Tensor Derived Equations^ 127 Appendix F: Modulus D Matrix Implemented in the Finite Element Code 133 LIST OF TABLES 1.1. Neo-Hookean Model applied to Uniaxial and Equibiaxial Extensions^ 24 3.1. Shape, attachment method, applied condition and predicted values for each experimental test performed^ 47 3.2. Mooney-Rivlin fitted parameters 50 4.1. Number of samples for each storage condition^ 65 5.1. Number of force-stretch experimental data averaged per deformation rate 84 5.2. Effect of deformation rate: Wilcoxon test p values 92 5.3. Effect of Anisotropy (Circumferential vs axial forces, n = 4 ): Wilcoxon test p values^ 93 5.4. First simulation results (material parameters) of uniaxial and biaxial cruciform tests^ ..94 5.5. Second simulation results (material parameters) of uniaxial and biaxial cruciform tests .94 5.6. Third simulation results (material parameters) of uniaxial and biaxial cruciform tests^ ..95 5.7. Final simulation results (material parameters) of uniaxial and biaxial cruciform tests ..95 5.8. Linear regression adjusted parameters^ . 96 5.9. Mooney-Rivlin fitted parameters in uniaxial extension^ 100 A.1. Ogden model fitted parameters 117 A.2. Guccione model fitted parameters^ 118 B.1. Mooney Rivlin model fitted in Cylindrical Arteries^ 121 vi vii LIST OF FIGURES 1.1. The circulatory system: The blood (rich in oxygen) leaves the heart through the arteries, and then it crosses the smaller arteries called 'arterioles' and finally arrives to the different organs through the capillaries. Now, the blood (rich in carbon dioxide) passes to the veins through the capillaries and venules and comes back to the heart. (Pugsley and Tabrizchi, 2000) 3 1.2. Arterial wall layers: Tunica Intima, Tunica Media and Tunica Adventitia (Holzapfel, et al., 2000) 5 1.3. (A) Collagen and (B) Elastin fibres (Alberts, et al., 2003) 6 1.4. Mechanical response of an artery to inflation, showing the non-linear wall tension, T, versus internal radius, R (Shadwick, 1999) 7 1.5. Inflation-deflation cycle 8 1.6. Schematic representation of (A) uniaxial, (B) biaxial, and (C) Planar extension. Solid lines show the direction of the applied displacement and dash lines show the response of the material in the other principal directions. In (C), the material is prevented to deform in the direction perpendicular to the applied displacement 10 1.7. Different geometries and gripping methods used for biaxial tests: (a) suture attachment or (b) clamps on a square sample, and (c) clamps on a cruciform sample (Sun, et al., 2003) 11 1.8. Biaxial square specimen (Prendergast, et al., 2003) 12 1.9. Stress-Stretch curves for pig aorta: (a) Uniaxial tests, (b) Biaxial tests (Prendergast, et al., 2003) 12 1.10. Schematic representation of the arterial dimensions: Undeformed and deformed configurations in inflation test (van Andel, et al., 2003)^ 13 1.11. Cyclic extension-inflation tests of a preconditioned adventitia specimen. Each of the curves was associated with an initial axial stretch (referred to the difference between the load-free gauge length and the in situ gauge length at 13.3 kPa), which is specified by the labels "50%," "75%," "100%," and "150%" (Schulze- Bauer, et al., 20 2) 3 1.12. Circumferential stress-strain curves for (A) fetal, (B) lamb, and (C) adult thoracic sheep aortas (Wells, et al., 1998). Dashed reference lines show mean wall stresses corresponding to blood pressure conditions obtained in vivo (control, 30% reduced, and 70% increased arterial blood pressure) 14 1.13. Comparison of stress-strain curves obtain for the polynomial strain energy function given by equation 1.18 with experimental data from Fung, et al. (1979) 19 1.14. Comparison of stress-strain curves obtain for the exponential strain energy function given by equation 1.19 with experimental data from Fung, et al. (197 ) 20 1.15. Plots of the Cauchy stresses in the circumferential, (700^ ^axial, G„ and radial, an directions through the deformed media and adventitia layers in the physiological state with p, = 13.33 [kPa] and X, = 1.7. Holzapfel, et al. (2000) 22 viii ^3.1.^Sample dimension in millimetres for (a) uniaxial testing: rectangular sample clamped with grips; and (b, c) biaxial testing: (b) cruciform sample clamped with grips, (c) Square sample attached with hooks^ 42 ^3.2.^(a) Biaxial test bench: (b) cruciform setup and (c) square setup^ 43 ^3.3.^Schematic representation of the inverse modeling technique used to obtain the material properties for the biaxial experiments with a cruciform sample^ 47 ^3.4.^Undeformed meshes and boundary conditions for (a) uniaxial extension tests, (b) biaxial tests with cruciform samples, and (c) biaxial tests with square samples 48 3.5. Force — stretch ratio curves for (a) uniaxial extension, (b) biaxial testing with cruciform samples, and (c) biaxial testing with square samples. Circumferential data are represented by dashed lines and hollow symbols. Axial data are represented by solid lines and solid symbols 49 3.6. Mooney-Rivlin experimental and computed reaction forces versus stretch ratios: (a) circumferential and (b) axial directions for uniaxial tests, (c) biaxial tests with cruciform samples and (d) biaxial tests with square samples 50 3.7. Optimization of the Mooney-Rivlin parameters for a biaxial test with a cruciform sample. (a) Objective function, and (b) a 30 , (c) a10 , (d) a01 , (e) an , (f) a70 parameter values as a function of iteration number, for three different initial guesses for material parameters: Case_A (all parameters = 0.1 MPa), Case_B (all 52parameters = 1.0 MPa), and Case_C (all parameters = 0.0001 MPa)^ ^3.8.^Deformed configurations at 1.80 nominal stretch ratio for rectangular mesh in uniaxial tension: distributions of (a) o ^stress and (b) stretch ratio in the principal direction 53 3.9. Simulated uniaxial and biaxial tests using Mooney-Rivlin parameters fitted to uniaxial experimental data: (a) circumferential and (b) axial force-stretch curves 53 3.10. Deformed configurations at 1.60 nominal stretch ratio for cruciform mesh in biaxial stretching: distribution of (a) csxx stress and (b) stretch ratio in the principal direction 54 3.11. Simulated uniaxial and biaxial tests using Mooney-Rivlin parameters fitted to biaxial experimental data with cruciform samples: (a) circumferential and (b) axial force-stretch curves 55 3.12. Deformed configurations at 1.64 nominal stretch ratio (at the puncture sites) for square mesh in biaxial stretching: distribution of (a) 6,, stress and (b) stretch ratio in the principal direction 56 3.13. Simulated uniaxial and biaxial tests using Mooney-Rivlin parameters fitted to biaxial experimental data with square samples: (a) circumferential and (b) axial force-stretch curves 57 4.1. Sample dimension in millimetres for biaxial testing and (b) biaxial test bench used for cruciform sample clamped with grips 64 4.2. Medians of the force-stretch behavior of fresh cruciform samples tested at 110 %/s (n =18). Open triangles and closed circles represent axial and circumferential mean forces, respectively. The percentile of the data points is also shown 66 ix 4.3. Medians of cruciform samples tested at a deformation rate of 110 %/s, and stored in saline solution (n = 3 ): Arterial wall behavior in (a) circumferential and (b) axial directions 67 4.4. Medians of cruciform samples tested at a deformation rate of 110 %/s, and stored in Krebs solution with dimethyl sulfoxide, DMSO (n = 3 ): Arterial wall behavior in (a) circumferential and (b) axial directions 68 4.5. Medians of cruciform samples tested at a deformation rate of 110 %/s, dipped in liquid nitrogen and stored in air (n =3): Arterial wall behavior in (a) circumferential and (b) axial directions 69 4.6. Comparison of Thawed/Fresh force ratios per storage group at the maximum stretch ratio applied (2 =1.55): (a, b) circumferential and (c, d) axial direction F Thawed 1 F Fresh ratios of samples stored at -20 and -80 °C 70 5.1. Mesh and boundary conditions used in simultaneous simulation of uniaxial and biaxial testing^ 85 ^5.2.^Loading-unloading circumferential force-stretch cycles of a typical biaxial cruciform sample: Five steady-state cycles at a deformation rate of 100%/s ^ 86 5.3. Sample to sample variability of thoracic aorta. Open diamonds and squares represent the mean force-stretch behavior at 10 and 200 %/s, respectively (n = 4 ): (a) circumferential, and (b) axial directions. Standard error bars are also shown 87 ^5.4.^Effect of deformation rate on the force vs. stretch ratio curve of arteries: circumferential force vs. stretch ratio curves in uniaxial and biaxial testing^ 88 ^5.5.^Effect of deformation rate on the force vs. stretch ratio curve of arteries: axial force vs. stretch ratio curves in uniaxial and biaxial testing^ 89 ^5.6.^Deformation rate effect on (a, b) uniaxial and (c, d) biaxial forces at maximum stretch ratio ( 2=1.5): (a, c) circumferential and (b, d) axial direction^ 90 ^5.7.^Comparison of the effect of deformation rate versus the effect of anisotropy. Typical sample forces at maximum stretch ratio (1.5)^ 92 5.8. Parameter a30 dependency on deformation rate for both, uniaxial and biaxial extensions^ 96 5.9. Mooney-Rivlin computed reaction forces versus stretch ratios in uniaxial and biaxial testing at 10 and 200 %/s. An increase in the deformation rate decreases the reaction forces at a particular stretch ratio 97 5.10. Optimization of the Mooney-Rivlin parameters for both uniaxial and biaxial tests at a deformation rate of 100%. (a) Objective function, and (b) parameter a 30 values as a function of iteration number, for three different initial guesses for material parameter a 30 : Case_A ( a30 , = 0.1 MPa), Case_B ( a 30 , = 1.0 MPa), and Case_C (a30 , = 0.0001 MPa)^ 98 5.11. Deformed configurations at 1.50 nominal stretch ratio for uniaxial and cruciform mesh: Distribution of (a) a' ^and (b) stretch ratio in the principal direction. The stress and stretch distributions at 10, 100, and 200 %/s are shown inside the tables^ 5.12. Parameter a30 dependency on deformation rate for uniaxial extension: (a) Circumferential and (b) axial directions 98 100 x5.13. Mooney-Rivlin computed reaction forces versus stretch ratios in uniaxial testing at a deformation rate of 100 %/s^ 101 5.14. Deformed configurations at 1.50 nominal stretch ratio for uniaxial mesh using the material parameters of Table 6.3 (Circumferential direction): distribution of (a) o-xx stress and (b) stretch ratio in the principal direction. The stress and stretch distributions at 10, 100, and 200 %/s are shown inside the tables^ 101 A.1. Experimental and Ogden model computed reaction forces versus stretch ratios: (a) circumferential and (b) axial directions for uniaxial tests. Experimental data is represented by the dash figures and modeling data by the solid line. (c) biaxial cruciform and (d) biaxial square. Cruciform experimental data: circumferential (plus) and axial (cross) direction. Square experimental data: circumferential (closed squares) and axial (open squares) direction. Modeling data are represented by solid (circumferential fit) or dash (axial fit) lines^ 117 A.2. Experimental and Guccione model computed reaction forces versus stretch ratios: (a) circumferential and (b) axial directions for uniaxial tests. Experimental data is represented by the dash figures and modeling data by the solid line. (c) biaxial cruciform and (d) biaxial square. Cruciform experimental data: circumferential (plus) and axial (cross) direction. Square experimental data: circumferential (closed squares) and axial (open squares) direction. Modeling data are represented by solid (circumferential fit) or dash (axial fit) lines^ 118 B.1. Inflation test machine^ 119 B.2. IVUS Image: Cross sectional area of a left coronary artery^ 120 B.3. Inflation test: (a) Left coronary artery surrounded by cardiac muscle, and (b) carotid artery attached to deployment tester; (c) sample dimension in millimetres, (d) undeformed mesh^ 121 B.4. Pressure-Stretch experimental curves of left coronary (diamonds) and carotid (open squares) arteries. Solid (coronary) and dash line (carotid) are the computed results using Mooney fitted parameters^ 122 B.S. Deformed configurations at the maximum applied pressure for left coronary artery: distribution of (a) stress and (b) stretch ratio in the principal direction ^ 122 B.6. Deformed configurations at the maximum applied pressure for carotid artery: distribution of (a) stress and (b) stretch ratio in the principal direction^ 123 C.1. Relaxation Experiment of Rectangular Porcine Thoracic Aorta Samples (Uniaxial Testing)^ 124 D.1. Active Test Setup 125 D.2. Rabbit Aorta Passive and Active Behaviour^ 126 ACKNOWLEDGEMENTS The interdisciplinary research done between biomedical disciplines and applied sciences, such as engineering, was an amazing experience, including the difficult and rewarding moments I found writing this thesis, which significantly contributed to my personal and professional development. I express my sincere gratitude and appreciation to my supervisor Prof. Savvas G. Hatzikiriakos for his assistance and encouragement throughout the making of this thesis. I would also want to thank my co-supervisor Dr. Sebastien Delorme for sharing his wide knowledge and enthusiasm on the biomechanical field. His critical approach and ideas were essential in the achievement of this work. Thanks to the Mexican Council of Science and Technology (Consejo Nacional de Ciencia y Tecnologia), CONACYT for the financial support. Thanks to Marc-Andre Rainville for his guidance and assistance in sample preparation and mechanical testing. Special thanks to Professor Robert DiRaddo for giving me the opportunity of performing my research in the installations of the Industrial Material Institute (IMI, CNRC — NRC). I thank all the scientists who took time from their busy schedules to engage in the resolution of subjects presented in this thesis. They include: Rouwayda El_Ayoubi, Denis Laroche, Vincent Mora and Francis Thibault. On a more personal level, I would like to thank my parents and all my family in Mexico for their continued support and encouragement during last three years. Thanks to all my friends, who gave me their unconditional affection and sincerity; and with whom I shared wonderful memories. Especially, I wish to thank Leon Demetrio Vasquez de la Cruz, Omar Delgadillo Velasquez and Iman Brouwer for been my best friends. xi xii CO-AUTHORSHIP STATEMENT I did mostly all the literature review, manuscript preparation, and the experimental and modeling analysis performed during the making of this thesis. However, the following researchers contributed in one way or another to the resolution of some of the problems I went through while making the research presented in this thesis: • Prof. Savvas G. Hatzikiriakos • Dr. Sebastien Delorme • Dr. Robert DiRaddo • Dr. Rouwayda El_Ayoubi • Dr. Vincent Mora • Dr. Francis Thibault DEDICATION To my dad, mom, brother, grandmothers, uncles, aunts, cousins & friends 1CHAPTER 1 1. INTRODUCTION & LITERATURE REVIEW 1.1. INTRODUCTION The development of research techniques for use in the cardiovascular system requires an understanding of the basic physiological and mechanical properties of arterial wall and other biological tissues associated with the cardiovascular system. Soft tissues have complex material properties that are difficult to characterize. The mechanical properties of arteries greatly influence cardiovascular function, since they determine the relationships between blood pressure, blood flow, and arterial wall dimensions (Coulson, et al., 2004). These properties depend on the relative proportion and arrangement of the arterial wall constituents. The most precise and compact way to express the mechanical properties is to use appropriate constitutive equations that define how the materials deform upon the application of forces. Constitutive models for arterial walls must be able to represent the mechanical behavior of healthy and diseased tissue (Ottensmeyer, et al., 2003). In particular, a constitutive model is required to reflect the actual experimental data over a wide range of deformations as accurately as possible and to be consistent with both mechanical and mathematical requirements. Additionally, the constitutive model selected and/or developed should involve material parameters that can be related to the structure of the material. In this work, the mechanical properties (i.e. fitted material parameters) of porcine thoracic aortas subjected to uniaxial and biaxial extensions are studied experimentally and numerically using an inverse modeling technique. The mechanical properties of biological tissues are necessary for accurate surgical simulation and diagnostic purposes. For instance, the parameters of an angioplasty and stenting procedure, such as inflation pressure, can be optimized with the help of finite element simulation (Dumoulin and Cochelin, 2000; Etave, et al., 2001; Gasser and Holzapfel, 2007; Gourisankaran and Sharma, 2000; Holzapfel, et al., 2002; Kiousis, et al., 2007; Lally, et al., 2005; Liang, et al., 2005; Migliavacca, et al., 2002; Oh, et al., 1994; Rogers, et al., 1999; Yang, et al., 2003; Wu, et al., 2007). Overstretch of the arterial wall during an angioplasty or stenting procedure has been shown to be correlated to the incidence of restenosis (in-growing tissue re-blocking the artery lumen). It has been found that 30% to 40% of the patients treated with an angioplasty 2procedure develop restenosis after surgery (Holmes, et al., 2000). Restenosis occurs when the balloon produces vessel injury and intimal rupture (Ellis and Muller, 1992; Schwartz, et al., 1992). Arterial injury leads to platelet deposition and thrombus formation followed by elastic recoil, intimal hyperplasia and smooth muscle cell proliferation (Ellis and Muller, 1992). Based on the hypothesis that lower deformation rate results in lower intramural stresses, slow balloon inflation has been proposed as a means to minimize vascular injury and reduce restenosis incidence (Tenaglia, et al., 1992). Early studies did not conclude there was any difference in restenosis rates between conventional and slow balloon inflation (Miketic, et al., 1998; Tenaglia, et al., 1992; Timmis, et al., 1999) while some observed better immediate results (Eltchaninoff, et al., 1996; Ohman, et. al., 1994). In more recent studies, significantly lower restenosis rates were clinically observed with slow balloon inflation (Umeda, et al., 2004; Weiss, et al., 2007). Slow stent deployment has also been proposed as a means to minimize arterial injury (Theriault, et al., 2006). The analysis of the biomechanics of the arterial tissue might contribute to understand which factors trigger restenosis. For instance, other mechanisms may play a role in leading to lower restenosis rates, such as endothelium denudation or arterial injury being dependent on stretch ratio rather than on intramural stress. Therefore, the mechanical properties gathered here might be useful in the development of simulation software that, in addition to further studies (i.e. active response effects), can be helpful to understand the mechanical changes that occur within the arterial wall as a consequence of an angioplasty procedure; and therefore, the finding of the approach that minimizes intramural stresses and arterial injury and the reduction in the rate of restenosis. The research made in this work might also be used as a part of a reliable artery mechanics database used in angioplasty surgical simulation for training medical students (Dayal, et al., 2004; Neequaye, et al., 2007). 1.2. LITERATURE REVIEW 1.2.1. PHYSIOLOGY OF ARTERIES 1.2.1.1. Introduction The circulatory system of the human body and almost all the vertebrates is constituted of large number of vessels that allow blood flow from the heart to the different Blood Flow —57.12,74',51Z0/ Blood Flow /// Endothelium Tunica intima Tunica media Tunica adventitia Ir ► ► 8 ► I ► 3 organs and its return to the heart. Figure 1.1 shows the different constituents of the circulatory system (Pugsley and Tabrizchi, 2000). It is only through this network of arteries, arterioles, capillaries, venules and veins that cellular growth and development is achieved. Artery ^ Capillary ^ Vein Fig. 1.1. The circulatory system: The blood (rich in oxygen) leaves the heart through the arteries, and then it crosses the smaller arteries called 'arterioles' and finally arrives to the different organs through the capillaries. Now, the blood (rich in carbon dioxide) passes to the veins through the capillaries and venules and comes back to the heart. (Pugsley and Tabrizchi, 2000). The arteries are subjected to higher physiological pressures than the veins, which makes them more likely to develop several diseases such as atherosclerosis (reduction of the internal radius of the artery due to the accumulation of cholesterol in their internal wall). The incidence of arterial disease can be as high as 30% and 48% among individuals greater than 65 and 84 years old (Stewart, et al., 1997; Otto, et al., 1999). Factors such as male gender, smoking, hypertension and diabetes mellitus increase this incidence (Aronow, et al., 1987; Lindroos, et al., 1993; Ouriel, 2001). Studying the mechanical properties of arteries is essential because it is considered that such mechanical factors may be important in triggering the onset of atherosclerosis (Holzapfel, et al., 2000). 41.2.1.2. Classification of Arteries Classification of arteries is usually done by their structure. In general, arteries are categorized into two broad types; namely elastic and muscular (Keshaw, 2001). The larger arteries leading from the heart and the first few branches are known as the elastic arteries, because of the large amount of elastin in them. These include the aorta, main pulmonary artery, common carotids and common iliacs. As the distance from the heart increases, the amount of muscle cells increases and thus changes the structure of the arterial wall. The smooth muscle cells lie in rings around the lumen (space enclosed by the walls of a tube) of the artery. These types of arteries, i.e. with higher content of smooth muscle cells and lower proportion of elastin, are commonly known as muscular arteries. These include the femorals, renals, coronaries and cerebrals. Between these two types of arteries (elastic and muscular), there are others which exhibit some characteristics of both the elastic and muscular arteries and they are called transitional arteries i.e. internal carotids. Almost all arteries have three layers (Zhao, et al., 1998): the tunica intima, tunica media and tunica adventitia (see Figure 1.2 for more details). The tunica intima (innermost layer) typically consists of a one cell thick layer of axially oriented endothelial cells with a thin layer of elastin and collagen fibres. The intima is similar in most elastic and muscular arteries. An exception is, however, the aorta, in which the intima also includes a subendothelial layer which contains some smooth muscle cells (often oriented axially) and connective tissue. The internal elastic lamina separates the intima and media, but it is often considered to be part of the latter. The tunica media, which forms the large part of the vessels' wall, consists mainly of smooth muscle cells embedded in a plexus of collagen and elastin and a ground substance gel matrix. The orientation and distribution of each of these fibrous constituents varies with species and location along the vascular tree. The outermost sheet of elastin is called the external elastic lamina; it separates the media and adventitia although it is often considered to belong to the former. The adventitia (outermost layer) consists primarily of collagen fibres with admixed elastin, nerves, fibroblasts and the vasa vasorum. The last mentioned is a vascular network which serves the outer portion of the wall in arteries that are too thick for sufficient diffusion of oxygen from the internal surface. The adventitia constitutes approximately 10 and 50 per cent of the arterial wall in elastic and muscular arteries respectively (Holzapfel, et al., 2000). Composite reinforced 1)y collagen fibers arranged in helical structures Helically arranged fiber- reinforced medial layers Bundles of collagen fibrils External elastic lamina Elastic lamina Elastic fibrils Collagen fibrils Smooth muscle cell lntc tal elastic lamina Endothelial cell 5 Fig. 1.2. Arterial wall layers: Tunica Intima, Tunica Media and Tunica Adventitia (Holzapfel, et al., 2000). 1.2.1.3. Collagen, Elastin and Smooth Muscle Cells Collagen is the most common fibrous protein in our body, constitutes one third of the body's protein, and belongs to a family of structural proteins that build the body (Vincent, 1990). In other words, collagen controls the shape of the cells and helps the regeneration of tissues. It is a constituent of skin, cartilage, bone, arteries and other connective tissues. The word collagen comes from the Greek word "icoXoyOvo", and means "glue producing". Collagen is a triple helix formed by three extended protein chains that bind around one another (Figure 1.3). Many rodlike collagen molecules are cross-linked together to form unextendable collagen fibrils that have high modulus (greater than 1 GPa) and the tensile strength of steel (Vincent, 1990). The striping on the collagen fibril is caused by the regular repeating arrangement of the collagen molecules within the fibril. Individual collagen fibers only deform about 5% in relation to their original length when stretched, so little elasticity is possible (Figure 1.3). RELAX sir■gle elesiTh molecule cross-link1,5 nm (A) (RI Fig. STRETCH 6 elastic fiber 1.3. (A) Collagen and (B) Elastin fibres (Alberts, et al., 2003). On the other hand, elastin is the main elastic protein of vertebrates and is usually found in association with collagen (Faury, 2001). Elastin is very stable when heated and has low modulus (about 1 MPa). Elastin chains are cross-linked together to form rubberlike, elastic fibers (Shadwick, 1999). Each elastin molecule uncoils into a more extended conformation when the fiber is stretched and will recoil spontaneously as soon as the stretching force is relaxed. Thus, elastin is about 1000 times more extensible than collagen. Its elasticity is based on changes in the entropy of the molecular chains, while the material is deformed. An imposed strain increases the order in the molecular network and thus decreases its entropy. According to the thermodynamic laws, the network would try to recover its former shape, increasing their entropy. 1.2.2 MECHANICAL BEHAVIOUR OF ARTERIES 1.2.2.1 Nonlinear Elasticity When a material with linear elasticity is deformed, a plot of the applied stress versus the resulting strain yields a straight line whose slope is the elastic or Young modulus, and it is a measure of the stiffness of the material. This is the simplest rheological behavior of a purely elastic body and can be described by Hooke's law of elasticity. Most metals and rubbers at small strains obey this law. Arteries and most biological tissues exhibit nonlinear elastic properties. When arteries are distended, the elastic modulus is not constant because it increases as the strain changes (Shadwick, 1999). The nonlinear mechanical behavior of arteries is mainly due to /fir - 7 the combination of the properties of its stiff (collagen) and rubbery (elastin) constituents. Roach and Burton (1957) have reported that the initial stiffness at low strains of the arterial wall is due to the elasticity of the elastin, and the higher stiffness at high strains is due to the contribution of fully tensed collagen fibres. An example of nonlinear mechanical properties is obtained in the inflation of a cylindrical element of radius R (Figure 1.4). The expression that relates the tension (T) along the lumen of the arterial wall to the pressure (P) and the radius (R) of the inflated artery can be obtained by using Laplace's law, which is given by: T = PxR . In an inflation experiment, the internal pressure is increased in a step-wise manner, and the arterial radius is measured with the aid of a laser. It is shown that an increase in the circumferential tension means an increase in the internal radius of the tube. Nevertheless, this increment is nonlinear, and more specifically exponential. For each circumferential tension there is only one internal radius possible. If we plot a straight line at each point (T , R) through the origin, then the slope of this line will give us the internal pressure of the cylinder inflated at that point (T ,R). Then, there is only one equilibrium radius for the artery at each internal pressure. Radius, R Fig. 1.4. Mechanical response of an artery to inflation, showing the non-linear wall tension, T, versus internal radius, R (Shadwick, 1999). The earliest study of vascular elasticity appears to be the work of Charles Roy (1881). He constructed an ingenious gravity-driven apparatus that automatically performed Artery Deflation Inflat ion 8 in vitro inflation of blood vessel segments (from humans, rabbits and cats), measured instantaneous pressure and volume change, and plotted the resulting P -V curves. He also performed in vitro uniaxial tests on strips of artery wall using a simple apparatus that plotted the force versus extension curves for the tissue as it was stretched. Using these experimental data, he concluded that the mammalian artery wall exhibits nonlinear elasticity and that the distensibility of the human aorta decreased as a function of age. When an artery is subjected to inflation-deflation cycles, viscoelasticity will cause the pressure-radius curve shown in Figure 1.5 for deflation to fall below that for inflation, forming a hysteresis loop. As was mentioned before, this is due to elastin and collagen chains, which are stretched beyond their linear limit and dissipate some of their elastic energy stored. Thus, they are unable to recover completely. The loop from such cycles represents the energy lost (about 15-20%) in each cycle (Figure 1.5). This means that most of the strain energy is recovered elastically each time the artery wall is distended. The strain energy lost attenuate travelling pressure pulses, which propagates along arteries as waves of circumferential distention of the vessel wall (Fung, 1995). Radius, R Fig. 1.5. Inflation-deflation cycle. 1.2.2.2 Anisotropy of Arteries Arteries are anisotropic materials. Their constituents, such as collagen and elastin, are long chain polymers that are oriented in the arterial wall in order that they function most 9effectively. Due to this orientation, arteries react differently to stresses applied in different directions. Properties in the longitudinal direction are different for those in the circumferential and radial directions. Collagen is the main constituent that gives the anisotropy to the artery, because its chains are more oriented than elastin chains inside the layers of the artery. Cox (1975) showed that the arterial wall is stiffer in the circumferential direction than axially. Arterial tissue has been identified as a curvilinear orthotropic material with two axes of symmetry (also known as transverse isotropic) In such kind of materials the stresses in circumferential, longitudinal and radial directions produce only normal strains (Lally, et al., 2004). 1.2.3. EXPERIMENTAL TESTING 1.2.3.1. Basic Experiments As discussed above, the structural mechanical response of the arterial wall to forces is complex. In fact, the mechanical properties of arteries are determined by the mechanical properties of their individual components (collagen and elastin mostly); the relative proportions of their individual components (volume fractions); their structural geometry and orientation; and the coupling between these individual components (Cox, 1984; Wells, et al., 1999). The mechanical behavior of arteries also depends on physical and chemical environmental factors, such as temperature, osmotic pressure, pH, partial pressure of carbon dioxide and oxygen, ionic and monosaccharide concentrations. Moreover, the arterial wall is always submitted to loading-unloading forces (pressures) "in vivo", mainly as a consequence of the heart beat and blood flow. This behavior can be imitated "in vitro" preconditioning the samples for up to 10 cycles (Fung, 1993). When no further change occurred in the material stress-strain curve (reduction in the hysteresis loop) after each preconditioning cycle, then a steady state stretching-relaxation cycle is obtained. In "in vitro" conditions (experiments done on cells or tissue outside the body) the mechanical properties are altered due to biological degradation. Therefore, arteries should be tested in appropriate oxygenated, temperature controlled physiological solutions (typically 0.9% saline, ringer, krebs-ringer, or tyrode solutions). The different solutions that can be used 10 contain only the non-living constituents of blood (normally NaC1, NaHCO3, NaH2Pa4, Na2SO4, KC1, MgSO4 and CaC12). Different types of experiments can be employed to obtain the mechanical properties of arteries such as: 1) a classical stress-strain test performed on an isolated segment of the vessel wall, 2) a pressure-diameter test which treats the vessel as a cylinder with variable internal pressure. Uniaxial testing is the simplest procedure that can be performed to obtain the stress- strain response of any material, such as the arterial wall. In order to perform uniaxial tests, the arterial wall is typically cut open along its length and cut out in rectangular samples, either in circumferential or axial directions. The mechanical behavior of samples cut in circumferential and axial directions may be different due to anisotropic effects. Uniaxial tests are not useful to predict the non-uniform stress distribution of the arteries. Then, apart of this kind of tests, biaxial and planar tests may be good choices to have a better understanding of the artery behavior. Figure 1.6 depicts the principles of these types of deformations and some of these were utilized in the present proposed work. This test can be done both in planar and cylindrical configuration. For example, a planar biaxial extension can be done to a square flat sample or to a cylindrical tube (arterial wall) by simultaneously stretching in the axial direction and inflating. Biaxial tests are usually done to square or cruciform samples (Figure 1.7). (C) Fig. 1.6. Schematic representation of (A) uniaxial, (B) biaxial, and (C) Planar extension. Solid lines show the direction of the applied displacement and dash lines show the response of the material in the other principal directions. In (C), the material is prevented to deform in the direction perpendicular to the applied displacement. 11 Fig. 1.7. Different geometries and gripping methods used for biaxial tests: (a) suture attachment or (b) clamps on a square sample, and (c) clamps on a cruciform sample (Sun, et. al., 2003a). Uniaxial and biaxial testing has been used to characterize biological tissues (Lu, et al., 2005; Criscione, et al., 2003a; Criscione, et al., 2003b; Sun, et al., 2003b; Okamoto, et al., 2002; L'Italien, et al., 1994). For example, Prendergast, et al. (2003) performed uniaxial and biaxial extensions on 25 mm square samples from porcine aorta and human femoral arterial tissues. They used a strain rate of 60% per minute for each test. The aortic and femoral specimen thickness was 2.5 ± 0.5 mm and 1.5 ± 0.5 mm, respectively. An image of one biaxial specimen they used is shown in Figure 1.8. The black dots seen in the picture allowed them to measure the circumferential and longitudinal stretches and thus enabled them to plot the corresponding stress-stretch curve. For that purpose, a digital camera was mounted above the test specimen to take pictures at set intervals of the test. The engineering stress was determined in each case by dividing the load by the cross sectional area calculated from each picture shot, and the stretch was obtained by dividing the current length by the original length. Figure 1.9 plots the stress-stretch curve for five pig aorta. Prendergast, et. al. (2003) found that the elastic behavior of all tissues seems to be significantly nonlinear to high strains. Considering the results for the porcine aorta plotted in Figure 1.9, larger extension ratios are achieved in the uniaxial than the biaxial tests and the tissue is stiffer and less nonlinear under biaxial loading. The porcine aorta is much less stiff than the human femoral artery (Prendergast, et. al., 2003), which might be related to the higher proportion of elastin in the aorta (Keshaw, 2001); however, porcine aortas also display significant nonlinear elastic behavior to high strains. 500 I," 400 300 0x00 1.3^1,4^1.5^1.8 Stretch Ratio 12 Fig. 1.8. Biaxial square specimen (Prendergast, et al., 2003). Fig. 1.9. Stress-Stretch curves for pig aorta: (a) Uniaxial tests, (b) Biaxial tests (Prendergast, et. al., 2003). Another experimental technique that could be done to obtain the mechanical properties, such as the axial and circumferential stresses, of an artery wall is cylinder inflation (Figure 1.10). In this "in vitro" experiment, the sample should be at the same conditions that it was in the animal body to obtain properties similar to those reported "in vivo". From this experiment, it is also possible to obtain internal pressure versus radius relationships. Schulze-Bauer et al. (2002) performed inflation test to human femoral arteries (located after the iliacs) to obtain the behavior of the adventitia layer and also performed similar tests to iliac arteries (Schulze-Bauer, et al., 2003b). Dixon et al. (2003) performed inflation test from pressures of 1 to 121 mmHg to porcine coronary arteries. Measurements of pressure and internal diameter were obtained by Chamiot-clerc et al. (1998) from cylindrical segments of internal mammary (runs along the inside edge of the sternum) and radial arteries (located on the inside of the wrist near the side of the thumb). 13 1 Fig. 1.10. Schematic representation of the arterial dimensions: undeformed and deformed configuration (van Andel, et al., 2003). Schulze-Bauer et al. (2002) investigated eleven human femoral arteries within 24 hours from death. The vessel was excised and placed in a 37 ± 1 °C NaC1 solution. After the extension-inflation test was done, the adventitia was then pulled off in "turtle-neck" fashion from the underlying media-intima shell. The separation of the adventitia was "atraumatic" because its adherence to the media layer was weak. The adventitia layer was retracted immediately in the axial direction and was placed in a 37 ± 1 °C NaC1 solution. Extension-inflation tests of separated adventitias were perfoiined at two loading domains: the physiological domain with pressures from 0 to 33 kPa (250 mmHg) and the high pressure domain from 0 to 100 kPa (750 mmHg). An initial axial stretch was performed before the adventitias were inflated (five cycles).Figure 1.11 shows the axial stretch versus the circumferential stretch at 33 kPa. 1.45 - "150%" ^ 1.30-^ c.:43=00p "75%" ":t1 1.25 - "50%" 1.20 ^ 0.80 0.85^0.90^0.95^1,00 Circumferential Stretch (I) Fig. 1.11. Cyclic extension-inflation tests of a preconditioned adventitia specimen. Each of the curves was associated with an initial axial stretch (referred to the difference between the load-free gauge length and the in situ gauge length at 13.3 kPa), which is specified by the labels "50%," "75%," "100%," and "150%" (Schulze-Bauer, et al., 2002). 1.40 I4. ▪ 0 • 1.35 43,,,,s2:ezemmip -1 00%" ?;05 3.0 "E 2,5 • 2-0 - t11114 13P 4c■-'^— c1.5- ro_ 1 0 • Coat* 8P_ ..„„ ZO^40^0^20^40^60^SO Strain (%) Strain (%) 0 14 Wells et al. (1998) performed inflation tests to sheep thoracic aortas from fetal, lamb and adult specimens. Figure 1.12 depicts the circumferential stress-strain curves for fetal, lamb and adult thoracic aortas. They observed that the arterial wall response to the applied strains was stiffer in adult thoracic sheep aortas rather than in fetal or lamb thoracic aortas. Fig. 1.12. Circumferential stress-strain curves for (A) fetal, (B) lamb, and (C) adult thoracic sheep aortas (Wells, et al., 1998). Dashed reference lines show mean wall stresses corresponding to blood pressure conditions obtained in vivo (control, 30% reduced, and 70% increased arterial blood pressure). 0 .5 —,:act% BP I 1 0,0 Another possibility is cutting the artery transversely into two halves. One half could be used to perform the corresponding histological analysis to allow material characterization, and the other half to do uniaxial or planar biaxial extension. On the other hand, if one of the halves is dissected anatomically in its major constituents (intima, media and adventitia), only then, it is possible to obtain the mechanical properties of each artery layer. Holzapfel and collaborators (2002) performed cyclic uniaxial extension tests to stripes of arteries with axial and circumferential orientation. They recorded the tensile force and strip width and length. The thickness they found was 0.98 mm for the intima-media layer and 0.4 mm for the adventitia. Humphrey (1995) proposed an approach to determine the mechanical properties of the media and adventitia by performing inflation tests on "inverted arteries". In other words, he proposed to turn the artery inside out and to perform the inflation tests. From the difference in mechanical behavior with the normal configuration it is possible to extract properties for each individual layer. Unfortunately, corresponding experiments were not published (cited in: Schulze-Bauer, et al., 2002). Until now, the effects of the intima layer on the mechanical properties of arteries have been neglected. 15 1.2.4. MODELLING THE ARTERIAL WALL When a material with linear elasticity is deformed, a plot of the applied stress versus the resulting strain yields a straight line whose slope is the elastic or Young modulus, and it is a measure of the stiffness of the material. This is the simplest rheological behavior of a purely elastic body and can be described by Hooke's law of elasticity. In reality all materials at large deformations deviate from Hooke's law exhibiting viscous as well as elastic behavior. Viscoelasticity describes such kind of materials where the stress-strain relationship depends on time. Whereas elasticity is usually the result of bond stretching in an ordered solid, viscoelasticity is the result of the diffusion of atoms or molecules inside of an amorphous material (Meyer and Chawla, 1999). Some biological tissues exhibit time- dependent behaviour, most of which manifest themselves as an increase of stiffness with increasing deformation rate (see chapter five for a detail discussion). The arterial wall is subjected to unsteady loads due to the pulsatile nature of the flow in the cardiovascular system. Arteries do not meet the definition of an elastic body, which requires that there must be a single-valued relationship between stress and strain. Arteries show hysteresis when they are subjected to cyclic loading and unloading; when held at a constant strain, they show stress relaxation (decreasing stress); when held at a constant stress, they show creep (increasing strain); and when subjected to a sinusoidally varying stress, they exhibit a phase lag in the strain response (Kassab and Molloi, 2001). Since the arterial wall is nonlinear, mechanical models (such as Maxwell and Kelvin-Voigt) based on linear viscoelasticity can not be used. Instead, constitutive equations which describe nonlinearity might be of very complex form and contain many coefficients. For the formulation of elastic properties of arterial walls, several assumptions have been incorporated (Dixon, et al., 2003; Gasser, et al., 2002; Holzapfel and Weizsacker, 1998; Gleason, et al., 2004; Holzapfel, et al., 2004; Prendergast, et al., 2003; Schulze-Bauer et al., 2003a): for example, ideal cylindrical geometry, material homogeneity, incompressibility, cylindrical orthotropism and hyperelasticity. 1.2.4.1. Strain Energy Density Function Most experimental data are generally analyzed with the help of a strain energy density function. It is a measure of the energy stored in the material as a result of 16 deformation. If there is a one-to-one relationship between strain and stress, then the theory of elasticity shows that there exists a strain energy density function W , from which stresses can be computed from the strains as follows (Sacks and Sun, 2003): S —u —^j = 1, 2, 3) 1.—(F,.•F!' —I..) J^2^J^1J^11 Where Sy , Ey , Fu and / if are the components of the second Piola-Kirchhoff stress tensor ( s ), the Green-Lagrange strain tensor ( E ), the finite strain deformation tensor ( F ), and the identity unit tensor (I), respectively. Under small deformations, the Green- Lagrange strain tensor is reduced into the infinitesimal strain tensor, eq : 1 r e,i = —2 [au, /ax; + audax,^ (1.3) Where u, and of are the components of the motion vector, which represents the displacement field of the material's configuration (i.e. the difference between the material's configuration and its original state). In finite deformations the second Piola-Kirchhoff stress tensor ( S ) is used to relate forces in the reference configuration to areas in the same reference configuration. In contrast, the Cauchy stress tensor (a) is used to express the stress relative to the current configuration. Under infinitesimal deformations both stresses are identical. Second Piola- Kirchhoff stress tensor is related to Cauchy and total stress tensors as follows: i . = 1 Fi .S,.F, l^(1.4)J^Il J = —ply. + o (Total stresses) 1 ^IJ (1.5) Where a,/ are the components of the Cauchy stress tensor (a); T., are the components of the total stress (T); the Jacobian (J) is defined as the determinant of the finite strain deformation tensor; and p is the hydrostatic pressure. In incompressible materials the Jacobian is equal to one ( J = det F =1). In the last years, several SEDF (Strain Energy Density Function) were developed and used to predict the mechanical properties of soft tissues. Most of these models make the (1.2) 17 assumption that the material is isotropic, instead of considering the anisotropy (due to the organized arrangement of the collagen components) of the arterial wall. Nevertheless, they can be used to simulate the deformation of the arterial wall in special cases, such as that corresponding to axial extension and inflation of an artery regarded as a thin-walled (or thick-walled) circular cylindrical tube. The most common SEDF used to determine the mechanical properties of an artery are functions of the Finger tensor (B) or more explicitly of its invariants ( I I , /2 ,13 ). The Finger tensor, B , gives us the relative local change in area within the sample and is defined as: B= F • F T^(1.6) The finite strain deformation tensor (F) is defined as F =ax/ax , where x' and x are the initial and current configurations, respectively. The invariants of this tensor are given by: 1.1= trB (1.7) 12 -= (112)ktrB)2 —trB 2 I (1.8) /3 = det B (1.9) 1.2.4.2. Neo-Hookean and Mooney Models Subject to the regularity assumption that W is continuously differentiable infinitely many times with respect to / 1 ,12 , /3 , we may write W ( /„ /2 , /3 ) as an infinite series in powers of 1 1 -3 , / 2 —3 ,13 —1, where c, are material parameters (constants). w(i 1 ,12 ,13 ). ECpqr ( ^ 3)p (I2I 2 - 3 )q (I 3 - O r ^ (1.10) p,q,r=0 14(/1 , /2 ,13 )== Ecp, o (ii —3)P (i 2 _3)q +Ecoor (13 - 1) r^(1.11) p,q=0^ r=0 For an incompressible material 13 =1 and if we let p and q take values 0, 1 then the latter equation simplifies to the two parameter Mooney-Rivlin strain-energy function, which can be written as: w(i,,12 )=--^(ciokli -3 )+c01(12 —3 )^ (1.12) 18 If co , = 0 the latter equation reduces to the Neo-Hookean strain energy function: W(./, )= c1o (/ 1 -3)= —; (I, —3)^ (1.13) In this case, the total stresses should be: aw^ci.(1. 3))Tu=—plu aE+^= plu +^ (1.14) 1.2.4.3. Models Developed to Obtain the Behavior of Arteries Several researchers have developed many mathematical expressions that describe the stress-strain relationship for mechanical tests, but the most common are those based on polynomial or exponential functions. Independently of the function chosen, a three dimensional strain energy function, appropriate for the analysis of thick-walled tubes; or a two dimensional strain energy function should be used to modeling the mechanical properties of arteries. Some of these functions are described below. Delfino et al. (1997) proposed an isotropic rubber-like three dimensional strain energy function for carotid arteries which is able to predict the stiffening effects in the high pressure domain. His equation has two parameters, a stress-like material parameter, "a ", and a non-dimensional parameter, "b ". Both parameters must be greater than zero. W = —a {exp[—b —3)1-1} 2^ (1.15) Ogden (1997) proposed an isotropic, hyperelastic constitutive equation that has six material parameters (three dimensional: A , ,u, ,^, and three non-dimensional: a, , a2 , a3 ), and which is a function of the principal stretch ratios, Al A2 and A3 . The Ogden model strain energy density function has the following form: W =^—3) Another three dimensional strain energy function was developed by Humphrey (1995). For doing that, he had based his development in the two dimensional strain energy function proposed by Fung et al. (1979). Humphrey arrived to the following equation, which has 10 parameters, one of them is a material parameter, c, and the others (bi , i = 1, ,9) are non-dimensional material parameters: (1.16) o Carotid o left iliac^e A.^lower aorta^es' • upper aorta^a' Experimental^Theoretical (polynomial) d d 0 ^Le^t^ I^ i 0.5 0.7 0.9 1.1 Green's Strain, E n 50 10 ID^carotid 0^left iliac A^lower aorta 6^ripper aorta^0' Experimental^Theoretical (polynomial) 100 ;s7E za, 75 I cl) 65 50 25 0,2^0.4 Green's Strain, E00 0.6 0rr 19 1 rW — 2 c[exp(bi E 029 +b2 Ez2z + b3 Er2r +2b4E00E„+2b5 E„E„ + 2b6 E„Eee +b7 E0,2 +b8 Er2z +b9 E,20 )-11^ (1.17) If it is considered that a thick-walled cylindrical tube of incompressible material is deformed in such a way that the strains E re , E, and Eez are zero (extensional flow) and Err is neglected by incompressibility, several two dimensional strain energy functions are developed, such as (Vaishnav, 1973): W = c1 EEB + c2 E00E„ + c3 E,2, + c4 E:,0 + c5 E 20;9E„+c6 E649 E„2 +c7 E'„3^(1.18) This equation is a polynomial expression proposed by Vaishnav et al. (1973) to describe the behavior of canine thoracic aorta. The parameters c l —c, are stress-like material parameters. This potential represents the first attempt to describe the mechanical response of the arterial wall. Figure 1.13 shows a comparison of this polynomial model with experimental data from Fung et al. (1979). Fig. 1.13. Comparison of stress-strain curves obtain for the polynomial strain energy function given by equation 1.18 with experimental data from Fung, et al. (1979). 0.2^0.4 Green's Strain, E N -a- ro -^ 7;e el 0 ^ 0.6^0.5 0.7^0.9 Green's Strain, En o carotid o left diet^o' n lower aorta^a' o upper aorta Experimental^Theoretical (exponential) . 0 1.1 62 25 0 50 E 40 X 30 20 10 • carotid^e • left iliac^e A^lower aorta^a' • upper eoae Experimental^Theoretical (exponential) ,25‘ ^la^fth a' ^ I^I 20 Fung et al. (1979) used the latter function to fit the parameters cc-c, to experimental data from rabbit carotid arteries. An exponential function was used for this having the following form: W = 1 c[exp(b,E; + b2E,2, + 2b4E96,Ezz )-11 ^ (1.19) Where c is a stress-like material parameter and bi b2 , b4 are non-dimensional parameters. This is the strain energy function more commonly used for arteries. Figure 1.14 depicts a comparison of this exponential model with experimental data from Fung, et al. (1979). Fig. 1.14. Comparison of stress-strain curves obtain for the exponential strain energy function given by equation 1.19 with experimental data from Fung, et al. (1979). Until now, the architecture of the arterial wall has not been considered in the formulation of the constitutive equations conventionally used to model the mechanical behavior of the arterial wall. That is why several researchers (Guccione, et al., 1991; Holzapfel, et al., 1996; Holzapfel, et al., 2000; Zulliger, et al., 2004) have formulated constitutive models which incorporate some histological information. Hence, the material parameters involved may be associated with the histological structure of arterial walls. 21 1.2.4.4. The Holzapfel Model This model considers a different strain-energy function (each one with three material parameters) for each of the artery layers. Media and adventitia are the primarily layers which contribute to the mechanical properties of the arterial wall. Since the collagen fibers of arterial walls are not active at low pressures (they do not store strain energy) it was associated ( ) with the mechanical response of the non-collagenous matrix (elastin fibers), which it is assumed to be isotropic. The resistance to stretch at high pressures is almost entirely due to collagen fibers and this mechanical response is therefore taken to be governed by the anisotropic function, Wan.. Therefore, the total strain energy density function is given by: A1, A2 ) Wiso (11 )+ Womso (14 /6 ) (1.20) Here A i =1,2 , are tensors defined by the product of two reference direction vectors ( a0; O ao; ). 11 (B) = trB , 1 4 (B,a01 )= B: A„ 16 (B, a02 ) = B: A2 . The anisotropy arises only through the invariants 1 4 and 16 . The isotropic response in each layer was based on the Neo-Hookean model. The strong stiffening effect on each layer observed at high pressures motivates the use of an exponential function for the description of the strain energy stored in the collagen fibers. w,s0 (.11)= (1 1 -3 ) wan. (1 4 ,1 6)= 1(1 EfexP[k2(1 —1 )2 ]-112k2 z=4,6 (1.21) (1.22) Then, the strain energy function for the media ( Wm ) and the adventitia (WA ) can be written as: ^CM ^\ k^x--41147,14 = m̂^3)+ im^lexpik2m (IoM —1)2]-1}2 2k2M i=4,6 (1.23) ^ WA = eA^ 2k —3)+^ fexp[k2A (/, —1)2 ]-1} 2 2A 1=4,6 Using this model, Holzapfel et al. (2000) have done the simulation depicted in Figure 1.15 of the Cauchy stresses versus the difference r—r, where r; is the inner radius, Az is the axial stretch and p, is the internal pressure for an inflation test. (1.24) 300 8;.( 20() ct) 100 7 500 - t 400 - t - 22 600 Adventitia a = 0.0' tio 0.05^0.10^0.15 r ri [mni] Fig. 1.15. Plots of the Cauchy stresses in the circumferential, a89 , axial, G,, and radial, a r, directions through the deformed media and adventitia layers in the physiological state with p, = 13.33 [kPa] and X. =- 1.7. Holzapfel, et al. (2000). 1.2.4.5. The Guccione Model Guccione et al. (1991) proposed an anisotropic, incompressible hyperelastic equation that takes into account the fiber direction. In their work, simultaneous inflation, extension and torsion of the equatorial region of the canine left ventricle was performed. The mechanical properties of the myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varies linearly across the arterial wall. They found that the stiffness of the myocardium was between 2.4 and 6.6 times greater in the fiber direction than in the transverse direction for different fiber angles assumed. This model is a function of the Lagrangian Green's strain tensor E . Its strain energy density function resembles a Fung type equation, which has the following form: W 2 (e (2^(1.25) Q= b fE,,x +^+ E„)+2kEyz +2b fs (Exy + Ems )^(1.26) —1 00 ^ 0 23 Where c is a material constant that simply scales the stresses, and b f , b, are constants related to the stresses in the fiber and transverse directions, respectively. The rigidity of the material under shear is directly related to bft . All material parameters are nondimensional, with exception of c, which has units of pressure (i.e. MPa). 1.2.5. FINITE ELEMENT MODELING The classic problem in solid mechanics is to compute the motion (displacements) that is induced in a material given a particular applied load. This approach is called "forward problem". Alternatively, one can be interested in determining those loads that cause prescribed motions for a particular material. This is the so called semi-inverse approach. Finally, one can seek to determine the material properties given the motion and the applied loads. This approach is called "inverse problem" (Humphrey, 2002; Kauer, et al., 2002; Feezor, et al., 2001; Laroche, et al., 2004). The optimization or fitting and the finite element models are combined in the concept of inverse modeling. The finite element method, used for the first time by Turner et al. (1956), is able to predict the deformation of a material of known properties, under a known load. In the experiments, the deformation and load are known but not the material properties. If the shape of the testing specimen is simple, there often is an analytical solution to the problem of finding the material properties. However, if the shape of the specimen or the boundary conditions are complex, such as cruciform shape for biaxial testing, finite elements can be used to solve the problem. Starting with an initial set of values for the material properties, the finite element model is useful to predict the deformation under the load measured experimentally. If the predicted deformation is different from the measured deformation, an optimization algorithm determines the material properties that best describes the experimentally measured deformation. As an example, the components of the second Piola-Kirchhoff stress tensor (So ) obtained for uniaxial and equibiaxial extensions using the Neo-Hookean model are summarized in Table 1.1. In uniaxial extension, the extension ratio a = dx/dx' is the ratio between the current and initial configuration. The extension ratio is known from the experimental data, so it is possible to calculate the elements of the Finger tensor, B and the 24 strain tensor, E. After that, one can calculate the derivative of the strain energy function with respect to the strain tensor using the chain rule and obtain the constitutive equation for the second Piola-Kirchhoff stress tensor. Table 1.1. Neo-Hookean Model applied to Uniaxial and Equibiaxial Extensions Uniaxial Extension Equibiaxial Extension Extension ratio La =— Lo R^Lal=^ ,a, = ^ ,a = al = a2 Ro Lo -1/2 ^-1/2^'Xi .,--- aX i , X2 = a^x25 x3 = a^x3 -1/2xi = cal , X 2 =- aX 2 , X 3 = a^x3 Finger Tensor B= a2^0^0 0^a'^0 0^0^a-1 B= a2^0^0 0^a2^0 0^0^a' 11 a2 +2ce -1 2a2 +a -1 12 2a + a-2 a4 ±2ce 13 1 1 Strain Energy Function W = — c (ii — 3) 2 W = — c (i i — 3) 2 S 1 , in the principal directions (i = I ) si j aw aa__ 2 aa aEii s,. i c aw aa= 2 a a aE,, S11 )=-2 (1 — (E11 + 1 )-3/2 )s11= 2— c (1- a3 \^c s11 = 4 —c (4 — ce -3 )= L(4-(E11 +1)-312 )4 S22 S22 = +C(a3 - 1)= -4E22 +1)-3 -1) S22 = 4—c. (4 — a-1 = L'(4- (E22 +1 )-3/2 )4 s33 533 = -C(a3 - 1)= -C((E33 +1)-3 -1) S33 = -5-2 (4a 3 —1) = — 2 (4(E33 +1)-3 —1) Assuming an infinitesimal deformation, the Cauchy stress tensor a is identical to the second Piola-Kirchhoff stress tensor, and thus au = . In uniaxial extension, the objective function, S (c) , that needs to be minimized is the one for the stress in the direction of the applied force^In an optimization loop, the parameter c is iterated until the predicted stress, f (En„ c) , matches the experimental stress, all by the aid of numerical methods like the Gauss-Newton. N is the number of experimental data. 25 e N (c)^— f (Eni , c)) 2^ (1.27) i=4 The equations for 6 22 and 633 are not used in uniaxial extension because in this test is not possible to measure experimentally the compression of the material in the second and third directions. In analyzing an in-plane biaxial extension test of a thin specimen, a first approximation can be done if data referring to deformation of the central region of the sample, which can be assumed homogeneous, is considered. In this case, the calculations are straightforward using an analytical method. Using the equations for a-, and 622 (see Table 1.1) simultaneously, the minimization of the objective function is done to obtain the fitted parameters as follows: S(c) N r , = Kai - f^C))2 (ani f (E22i C.) 1=1. (1.28) Nevertheless, if one is interested in the entire domain, including sites near the attachments where the loads are applied, then one must use a finite element model. Such a method was used as part of this PhD work. In the finite element model one divides the domain of interest into a finite number of non-overlapping but adjacent sub-domains, called elements, each of which is of finite size. One then seeks approximate solutions within each finite element, requiring only that the continuity in deformations and stresses be respected between neighboring elements at selected points called nodes and that the boundary conditions are satisfied identically. Finite elements method reduces the partial or ordinary differential equations to a system of algebraic equations that are solved simultaneously for values of the parameters until the objective function is minimized. A typical finite element model will be based on the conservation of momentum which requires that the time rate of change of the total momentum (i. e. mass times velocity for all particles) of a body must balance all the forces (body plus surface) that act on the body: —d f pvdV = f pgdV + fr•Tidv^ (1.29) dt 26 where v is the velocity, g and T are the body and surface (or traction) forces that act on the body in the current configuration CI , and thereby the local field equation is given by: apv rV • T i= I^.1+ pgat (1.30) Neglecting the acceleration term, body and pressure forces, considering an incompressible material and taking into account the definition of the total stress tensor (T = —pI + c ); this equation can be reduced to: [v . 0-] = 0 (1.31) where o- represents the stress tensor. Solution of the later equation by finite elements requires the development of the components of the second Piola- Kirchhoff stress tensor and the modulus matrix (D), which are summarized in Appendixes E and F for some of the constitutive equations used in this work (i.e. Mooney-Rivlin, Ogden, Guccione). Biaxial testing of biological tissues are performed using thin specimens and act on only by in-plane loads; the components of equation 1.31 for biaxial extension in Cartesian coordinates are: a611 + a612 = 0 and a621 + a622 , 0 al^ax2^al^ax2 (1.32) In biaxial experiments the shear strain and stress is nearly zero (Sacks and Sun, 2003). In this case, the above equations are reduced to: a611 =0 and ' 622 = 0 ax,^ax2 (1.33) Depending on the choice of the weighting or basis function, w , one can obtain different finite element methods, for example, the Galerkin method, where the test functions are derivatives of the trial functions (commonly polynomial interpolation functions). Appropriate boundary conditions should be applied based on the particular experiment to be simulated. These boundary conditions can be defined simply as those values or quantities of interest that are imposed on the boundary of the body. The two most common boundary conditions are displacement and traction conditions. 27 1.3. REPERENCES ALBERTS, B., JOHNSON, A., LEWIS, J., RAFF, M., BRAY, D., HOPKIN, K., ROBERTS, K., WALTER, P., 2003. Essential Cell Biology. 2nd Ed. Garland Science/Taylor & Francis Group. ARONOW, W. S., SCHWARTZ, K. S., KOENIGSBERG, M., 1987. Correlation of serum lipids, calcium, and phosphorus, diabetes mellitus and history of systemic hypertension with presence or absence of calcified or thickened aortic cusps or root in elderly patients. Am. J. Cardiol., 59, 998 — 999. CHARMIOT-CLERC, P., COPIE, X., RENAUD, J.F., SAFAR, M., GIRERD, X., 1998. Comparative reactivity and mechanical properties of human isolated internal mammary and radial arteries. Cardiovascular Research, 37, 811 — 819. COULSON, R. J., CIPOLLA, M. J., VITULLO, L., CHESLER, N. C., 2004. Mechanical properties of rat middle cerebral arteries with and without myogenic tone. Journal of Biomechanical Engineering, 126, 76 — 81. COX, R.H., 1975. Pressure dependence of the mechanical properties of arteries in vivo. Am. J. Physiol., 229 (5), 1371 — 1375. COX, R.H., 1984. Viscoelastic properties of canine pulmonary arteries. Am. J. Physiol. Heart Circ. Physiol., 246, H90 — H96. CRISCIONE, J. C., SACKS, M. S., HUNTER, W. C., 2003a. Experimentally tractable, pseudo-elastic constitutive law for biomembranes: I. Theory. Journal of Biomechanical Engineering, 125, 94 — 99. CRISCIONE, J. C., SACKS, M. S., HUNTER, W. C., 2003b. Experimentally tractable, pseudo-elastic constitutive law for biomembranes: II. Application. Journal of Biomechanical Engineering, 125, 100 — 105. DAYAL, R., FARIES, P. L., LIN, S. C., BERNHEIM, J., HOLLENBECK, S., DeRUBERTIS, B., TROCCIOLA, S., RHEE, J., McKINSEY, J., MORRISSEY, N. J., KENT, K. C., 2004. Computer simulation as a component of catheter-based training. J. Vasc. Surg., 40, 1112 —1117. DIXON, S. A., HEIKES, R. G., VITO, R. P., 2003. Constitutive modeling of porcine coronary arteries using designed experiments. Journal of Biomechanical Engineering, 125, 274 — 279. DUMOULIN, C., COCHELIN, B., 2000. Mechanical behaviour modelling of balloon- expandable stents. J. Biomech., 33, 1461 — 1470. 28 ELLIS, S. G., AND MULLER, D. W. M., 1992. Arterial injury and the enigma of coronary restenosis. Journal of the American College of Cardiology, 19, 275 — 277. ELTCHANINOFF, H., CR1BIER, A., KONING, R., CHAN, C., JOLLY, N., TAN, A., LETAC, B., 1996. Effects of prolonged sequential balloon inflations on results of coronary angioplasty. Am. J. Cardiol., 77, 1062 — 1066. ETAVE, F., FINET, G., BOIVIN, M., BOYER, J. C., RIOUFOL, G., THOLLET, G., 2001. Mechanical properties of coronary stents determined by using finite element analysis. J. Biomech., 34, 1065 —1075. FAURY, G. 2001. Function-structure relationship of elastic arteries in evolution: From microfibrils to elastin and elastic fibres. Pathol. Biol., 49, 310 — 325. FEEZOR, C., TAJADDINI, A., KILPATRICK, D., VINCE, D. G., JACOBSON, D., 2001. Integration of animal and human coronary tissue testing with finite element techniques for assessing differences in arterial behavior. BED-Bioengineering Conference ASME 2001, 50, 135 — 136. FUNG, Y. C., 1993. Biomechanics: Mechanical properties of living tissues. 2nd Ed. New York: Springer-Verlag. FUNG, Y. C., FRONEK, K., PATITUCCI, P., 1979. Pseudoelasticity of arteries and the choice of its mathematical expression. Am. J. Physiol., 237, H620 — H631. FUNG, Y. C., LIU, S. Q., 1995. Determination of the mechanical properties of the different layers of blood vessels in vivo. Proc. Natl. Acad. Sci. USA, 92, 2169 — 2173. GASSER, T. C., SCHULZE-BAUER, C. A. J., HOLZAPFEL, G. A., 2002. A three dimensional finite element model for arterial clamping. Journal of Biomechanical Engineering, 124, 355 — 363. GASSER, T. C., HOLZAPFEL, G. A., 2007. Modeling plaque fissuring and dissection during balloon angioplasty intervention. Annals of Biomedical Engineering, 35, 711 — 723. GLEASON, R. L., TABER, L. A., HUMPHREY, J. D., 2004. A 2-D model of flow- induced alterations in the geometry, structure, and properties of carotid arteries. Journal of Biomechanical Engineering, 126, 371 — 381. GOURISANKARAN, V., SHARMA, M. G., 2000. The finite element analysis of stresses in atherosclerotic arteries during balloon angioplasty. Crit. Rev. Biomed. Eng., 28, 47 — 51. GUCCIONE, J. M., McCULLOCH, A. D., WALDMAN, L. K., 1991. Passive material properties of intact ventricular myocardium determined from a cylindrical model. Journal of Biomechanical Engineering, 113, 42 — 55. 29 HOLMES, D., FITZGERALD, P., GOLDBERG, S., LaBLANCHE, J. M., LINCOFF, A. M., SAVAGE, M., SERRUYS, P. W., WILLERSON, J., GRANETT, J. R., CHAN, R., SHUSTERMAN, N. H., POLAND, M., 2000. The PRESTO (Prevention of restenosis with tranilast and its outcomes) protocol: A double-blind, placebo-controlled trial. Am. Heart J., 139, 23 — 31. HOLZAPFEL, G. A., EBERLEIN, R., WRIGGERS, P., WEIZSACKER, H. W., 1996. Large strain analysis of soft biological membranes: Formulation and finite element analysis. Computer Methods in Applied Mechanics and Engineering, 132, 45 — 61. HOLZAPFEL, G. A., GASSER, T. C., OGDEN, R. W., 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity, 61, 1 — 48. HOLZAPFEL, G. A., GASSER, T. C., OGDEN, R. W., 2004. Comparison of a multi-layer structural model for arterial walls with a Fung-type model, and issues of material stability. Journal of Biomechanical Engineering, 126, 264 — 275. HOLZAPFEL, G. A., GASSER, T. C., STADLER, M., 2002. A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis. European Journal of Mechanics A/Solids, 21, 441 — 463. HOLZAPFEL, G. A., STADLER, M., SCHULZE-BAUER, C. A. J., 2002. A layer-specific three-dimensional model for the simulation of balloon angioplasty using magnetic resonance imaging and mechanical testing. Annals of Biomedical Engineering, 30, 753 — 767. HOLZAPFEL, G. A., WEIZSACKER, H. W., 1998. Biomechanical behavior of the arterial wall and its numerical characterization. Computers in Biology and Medicine, 28, 377 — 392. HUMPHREY, J. D., 1995. Mechanics of the arterial wall: Review and directions. Critical Reviews in Biomedical Engineering, 23, 1 — 162. HUMPHREY, J. D., 2002. Cardiovascular solid mechanics: Cells, tissues, and organs. 1 st Ed. New York: Springer-Verlag. KASSAB, G. S., MOLLOI, S., 2001. Cross-sectional area and volume compliance of porcine left coronary arteries. Am. J. Physiol. Heart Circ. Physiol., 281, H623 — H628. KAUER, M., VUSKOVIC, V., DUAL, J., SZEKELY G., BAJKA, M., 2002. Inverse finite element characterization of soft tissues. Medical Image Analysis, 6, 275 — 287. KESHAW, K., 2001. Microstructure of human arteries. J. Anat. Soc. India, 50 (2), 127 — 130. 30 KIOUSIS, D. E., GASSER, T. C., HOLZAPFEL, G. A., 2007. A numerical model to study the interaction of vascular stents with human atherosclerotic lesions. Annals of Biomedical Engineering, 35(11), 1857 — 1869. LALLY, C., DOLAN, F., PRENDERGAST, P. J., 2005. Cardiovascular stent design and vessel stresses: A finite element analysis. Journal of Biomechanics, 38, 1574 — 1581. LALLY, C., REID, A. J., PRENDERGAST, P. J., 2004. Elastic behavior of porcine coronary artery tissue under uniaxial and equibiaxial tension. Ann. Biomed. Eng., 32 (10), 1355 — 1364. LAROCHE, D., DELORME, S., DIRADDO, R., 2004. Computer simulation of balloon angioplasty: Effect of balloon deployment on contact mechanics. ASM, Proc. Mater. Proc. Med. Dev. Conf, 2003. 501 — 505. LIANG, D. K., YANG, D.-Z., QI, M., WANG, W.-Q., 2005. Finite element analysis of the implantation of a balloon-expandable stent in a stenosed artery. International Journal of Cardiology, 104, 314 — 318. LINDROOS, M., KUPARI, M., HEIKKILA, J., TUILVIS, R., 1993. Prevalence of aortic valve abnormalities in the elderly: An echocardiographic study of a random population sample../. Am. Coll. Cardiol., 21, 1220 — 1226. L'ITALIEN, G. J., CHANDRASEKAR, N. R., LAMURAGLIA, G. M., PEVEC, W. C., DHARA, S., WARNOCK, D. F., ABBOTT, W. M., 1994. Biaxial elastic properties of rat arteries in vivo: Influence of vascular wall cells on anisotropy. Am. J. Physiol. Heart Circ. Physiol., 267, 11574 — H579. LU, S. H., SACKS, M. S., CHUNG, S. Y., GLOECKNER, D. C., PRUCHNIC, R., HUARD, J., DE GROAT, W. C., CHANCELLOR, M. B., 2005. Biaxial mechanical properties of muscle-derived cell seeded small intestinal submucosa for bladder wall reconstitution. Biomaterials, 26, 442 — 449. MEYER, M. A., CHAWLA, K. K., 1999, Mechanical Behavior of Materials, Prentice-Hall, Upper Saddle River, NJ, 98 — 103. MIGLIAVACCA, F., PETRINI, L., COLOMBO, M., AURICCHIO, F., PIETRABISSA, R., 2002. Mechanical behavior of coronary stents investigated through the finite element method. J. Biomech., 35, 803 — 811. MIKETIC, S., CARLSSON, J., AND TEBBE, U., 1998. Influence of gradually increased slow balloon inflation on restenosis after coronary angioplasty. American Heart Journal, 135(4), 709 — 713. NEEQUAYE, S. K., AGGARWAL, R., VAN HERZEELE, I., DARZI, A., CHESHIRE, N. J., 2007. Endovascular skills training and assessment. J. Vasc. Surg., 46, 1055 — 1064. 31 OGDEN, R. W., 1997. Non-Linear Elastic Deformations. 1st Ed. Dover Publications. OH, S., KLEINBERGER, M., MCELHANEY, J. H., 1994. Finite-Element analysis of balloon angioplasty. Med. & Biol. Eng. & Comput., 32, S108 - S114. OHMAN, E. M., MARQUIS, J. F.,RICCI, D. R., BROWN, R. I., KNUDTSON, M. L., KEREAIKES, D. J., SAMAHA, J. K., MARGOLIS, J. R., NIEDERMAN, A. L., DEAN, L. S., 1994. A randomized comparison of the effects of gradual prolonged versus standard primary balloon inflation on early and late outcome. Results of a multicenter clinical trial. Perfusion balloon catheter study group. Circulation, 89, 1118 — 1125. OKAMOTO, R. J., WAGENSEIL, J. E., DELONG, W. R., PETERSON, S. J., KOUCHOUKOS, N. T., SUNDT III, T. M., 2002. Mechanical properties of dilated human ascending aorta. Ann. Biomed. Eng., 30 (5), 624 — 635. OTTENSMEYER, M. P., KERDOK, A. E., HOWE, R.D., DAWSON, S. L., 2003. The effects of testing environment on the viscoelastic properties of soft tissues. In press, pp. 1 — 10. OTTO, C. M., LIND, B. K., KITZMAN, D. W., GERSH, B. J., SISCOVICK, D. S., 1999. Association of aortic-valve sclerosis with cardiovascular mortality and morbidity in the elderly. N. Engl. J. Med., 341, 142 — 147. OURIEL, K., 2001. Peripheral Arterial Disease. Lancet, 358, 1257 — 1264. PRENDERGAST, P. J., LALLY, C., DALY, S., REID, A. J., LEE, T. C., QUINN, D., DOLAN, F., 2003. Analysis of prolapse in cardiovascular stents: A constitutive equation for vascular tissue and finite-element modelling. Journal of Biomechanical Engineering, 125, 692 — 699. PUGSLEY, M. K., TABRIZCHI, R. 2000. The vascular system. An overview of structure and function. Journal of Pharmacological and Toxicological Methods, 44, 333 — 340. ROY, C. S., 1881. The form of the pulse-wave: As studied in the carotid of the rabbit. J Physiol., 2, 66 — 90. ROACH, M. R., AND BURTON, A. C., 1957. The reason for the shape of the distensibility curves of arteries. Can J Biochem Physiol., 35, 681 — 90. ROGERS, C., TSENG, D. Y., SQUIRE, J. C., EDELMAN, E. R., 1999. Balloon-artery interaction during stent placement. A finite element analysis approach to pressure, compliance, and stent design as contributors to vascular injury. Circ. Res., 84, 378 — 383. SACKS, M. S., SUN, W., 2003. Multiaxial mechanical behavior of biological materials. Annu. Rev. Biomed. Eng., 5, 251 — 284. 32 SCHULZE-BAUER, C. A. J., HOLZAPFEL, G. A., 2003a. Determination of constitutive equations for human arteries from clinical data. Journal of biomechanics, 36, 165 — 169. SCHULZE-BAUER, C. A. J., MORTH, C., HOLZAPFEL, G. A., 2003b. Passive biaxial mechanical response of aged human iliac arteries. Journal of Biomechanical Engineering, 125, 395 — 406. SCHULZE-BAUER, C. A. J., REGITNIG, P., HOLZAPFEL, G. A., 2002. Mechanics of the human femoral adventitia including the high-pressure response. Am. J. Physiol. Heart Circ. Physiol., 282, H2427 — H2440. SCHWARTZ, R. S., HOLMES, D. R. JR., TOPOL, E. J., 1992. The restenosis paradigm revisited: an alternative proposal for cellular mechanism. Journal of the American College of Cardiology, 20, 1284 — 1293. SHADWICK, R. E., 1999. Mechanical design in arteries. The Journal of Experimental Biology, 202, 3305 — 3313. STEWART, B. F., SISCOVICK, D., LIND, B. K., GARDIN, J. M., GOTTDIENER, J. S., SMITH, V. E., KITZMAN, D. W., OTTO, C. M., 1997. Clinical factors associated with calcific aortic valve disease. Cardiovascular health study. L. Am. Coll. Cardiol., 29, 630 — 634. SUN, W., SACKS, M. S.,SCOTT, M. J., 2003a. Numerical simulations of the planar biaxial mechanical behavior of biological materials. 2003 Summer Bioengineering Conference, Florida, pp. 1 — 2. SUN, W., SACKS, M. S., SELLARO, T. L., SLAUGHTER, W. S., SCOTT, M. J., 2003b. Biaxial mechanical response of bioprosthetic heart valve biomaterials to high in-plane shear. Journal of Biomechanical Engineering, 125, 372 — 380. TENAGLIA, A.N., QUIGLEY, P.J., KEREIAKES, D.J., ABBOTTSMITH, C.W., PHILLIPS, H.R., TCHENG, J.E., RENDALL, D., OHMAN, E.M., 1992. Coronary angioplasty performed with gradual and prolonged inflation using a perfusion balloon catheter: Procedural success and restenosis rate. Am. Heart J. 124, 585 — 589. THERIAULT, P., TERRIAULT, P., BRAILOVSKI, V., GALLO, R., 2006. Finite element modeling of a progressively expanding shape memory stent. Journal of Biomechanics, 39, 2837 — 2844. TIMMIS, S. B. H., HERMILLER, J. B., BURNS, W. H., MEYERS, S. N., DAVIDSON, C. J., 1999. Comparison of immediate and in-hospital results of conventional balloon and perfusion balloon angioplasty using intracoronary ultrasound. American Journal of Cardiology, 83, 311 — 316. TURNER, M. J., CLOUGH, R. W., MARTIN, H. C., IOPP, L. J., 1956. Stiffness and deflation analysis of complex structures. J. Aeronauts Sci., 23(9), 805 — 824. 33 UMEDA, H., IWASE, M., KANDA, H., IZAWA, H., NAGATA, K., ISHIKI, R., SAWADA, K., MUROHARA, T., YOKOTA, M., 2004. Promising efficacy of primary gradual and prolonged balloon angioplasty in small coronary arteries: A randomized comparison with cutting balloon angioplasty and conventional balloon angioplasty. American Heart Journal, 147, 1 — 8. VAN ANDEL, C. J., PISTECKY, P. V., BORST, C., 2003. Mechanical properties of porcine and human arteries: Implications for coronary anastomotic connectors. Ann. Thorac. Surg., 76, 58 — 65. VINCENT, J. F. V., 1990. Structural biomaterials, revised edition, London, UK: Princeton University Press. WEISS, T., LEIBOVITZ, D., KATZ, I., DANENBERG, C., VARSHITSKY, B., LOTAN, C., 2007. The value of computerized angioplasty in patients undergoing coronary stenting: a prospective, randomized trial. Am. J. Cardiol., 100, 118L. WELLS, S. M., LANGILLE, B. L., ADAMSON, S. L., 1998. In vivo and in vitro mechanical properties of the sheep thoracic aorta in the perinatal period and adulthood. Am. J. Physiol. Heart Circ. Physiol., 274, H1749 — H1760. WELLS, S. M., LANGILLE, B. L., LEE, J. M., 1999. Determinants of mechanical properties in the developing ovine thoracic aorta. Am. J. Physiol., 277, H1385 — H1391. WU, W., QI, M., LIU, X.-P., YANG, D.-Z., WANG, W.-Q., 2007. Delivery and release of nitinol stent in carotid artery and their interactions: A finite element analysis. Journal of Biomechanics, 40, 3034 — 3040. YANG, F., HOLZAPFEL, G. A., SCHULZE-BAUER, C. A. J., STOLLBERGER, R., THEDENS, D., BOLINGER, L., STOLPEN, A., SONKA, M., 2003. Vascular MR segmentation: Wall and plaque. Proceedings of SPIE, 5032, 1667 — 1675. ZHAO, S. Z., XU, X. Y., COLLINS, M. W., 1998. The numerical analysis of fluid-solid interactions for blood flow in arterial structures. Part 1: A review of models for arterial wall behaviour. Proceedings of the Institution of Mechanical Engineers — Part H — Journal of Engineering in Medicine, 212, 229 — 240. ZULLIGER, M. A., FRIDEZ, P., HAYASHI, K., STERGIOPULOS, N., 2004. A strain energy function for arteries accounting for wall composition and structure. Journal of Biomechanics, 37, 989 — 1000. 34 CHAPTER 2 2. SCOPE OF WORK 2.1. INTRODUCTION The arterial wall is a complex biological tissue that is always extended and compressed due to the heart beating and blood flow through the lumen. Measuring the mechanical properties while testing the arteries in vivo is very difficult, which is the main reason several researchers (Lally, et al., 2004; Prendergast, et al., 2003; Van Andel, et al., 2003) have opted to carry on in vitro tests instead. In addition, human arterial samples are commonly tested several days after death, assuming that the mechanical properties of the tissues were preserved by refrigerating and by using chemical solutions mimicking the physiological environment (Okamoto, et al., 2002; Vande Geest, et al, 2004; Mohan and Melvin, 1983; Van Andel, et al., 2003; Schulze-Bauer, et al., 2003). The discrepancies in preservation conditions used in different studies and the lack of clear demonstration that the mechanical properties were not altered by such preservation, preludes the study of the effects of several preservation methods on the mechanical properties of arteries. Knowledge of the deformation rate effects of soft tissue is important for accurate predictions of biomechanical behavior. It is well known (Gay, et al., 2008; De Smet, et al., 2007; Doehring, et al., 2004; Hu and Desai, 2004; Provenzano, et al., 2001; Pioletti and Rakotomanana, 2000; Hsieh, et al., 1999; Wang, et al., 1999; Yingling, et al., 1997) that some biological tissues such as bone, vertebrae, ligaments, tendons and heart valves exhibit time-dependent behavior. However, the mechanical properties of arteries has been studied using only a single deformation rate (Lally, et al., 2004; Okamoto, et al., 2002; Prendergast, et al., 2003; Waldman and Lee, 2002), which preludes a detail analysis of deformation rate effects on arterial behavior. The different kind of fixtures employed to attach an specimen to the actuators of an extensional apparatus preludes the measure of the arterial wall anisotropic behavior and the comparison of the differences in the mechanical behavior between three different types of geometries (rectangular, square and cruciform samples) using grips or hooks to attach the boundary of the sample to the actuators. Constitutive models, either isotropic or anisotropic, have been used to get the material properties of the arterial tissue. However, several sample geometries and ways to 35 attach the sample into the arms of the testing machine are employed. Therefore, simulation of different meshes and thus loading conditions would result in small variations in the material parameters for a particular constitutive equation. The extensional tests performed during the making of this thesis were simulated with the aim of an inverse finite element software. The present work is intended to contribute to a better understanding of the arterial wall behavior, in particular deformation rate effects and preservation method effects. The objectives of this work are discussed in detail in this chapter. 2.2. OBJECTIVES The main goal of the current investigation is the characterization and modelling of the mechanical properties of porcine thoracic aorta arterial walls. This goal has been accomplished by meeting the following specific objectives: 1) Measure the mechanical properties of porcine arteries with uniaxial, planar biaxial and tube inflation tests in vitro. The mechanical properties of specimens of thoracic aortas are measured by performing uniaxial and biaxial experiments. Rectangular samples were cut in the axial and in the circumferential directions for uniaxial testing. Square and cruciform-shaped specimens were also prepared for biaxial testing. The idea is to perform first this type of experiments with aorta specimens, because they are large and easy to work with. Then inflation tests are performed to carotid and coronary arteries. 2) Select several constitutive models, such as Mooney-Rivlin, Ogden and Guccione, to mathematically represent the behavior of arteries, with particular emphasis on the constitutive equation that gave better fitting results (Mooney-Rivlin five-parameter model). The inverse modeling results obtained with Ogden and Guccione constitutive equations can be found in Appendix A. Mooney-Rivlin five-parameter inverse modeling results are discussed in detail in chapters 3 and 5. For this purpose, an optimization method is used to fit the adjusted parameters so that the available experimental data are optimally represented. More specifically: • Use an inverse finite element method to fit each constitutive model to uniaxial experimental data. Although the stress distribution is uniform and the geometry 36 of the sample is simple in uniaxial tests, allowing the employment of an analytical method, the inverse modeling approach was preferred to obtain the material properties. • Use an inverse finite element method to fit each constitutive model to the planar biaxial experimental data. In this case, the stress distribution is not uniform and the specimen geometry is much more complicated than in uniaxial tests. In order to obtain the model parameters from the experimental data, an optimization method coupled with the two dimensional finite element model is used. • Validate the uniaxial and biaxial fit with inflation experimental data gathered from cylindrical arteries (coronaries and carotids). In an inflation test, the specimen is loaded in the circumferential direction but prevented to deform in the axial direction. The inflation test was simulated using the Mooney-Rivlin five-parameter constitutive equation used in uniaxial and biaxial testing. A brief description of the equipment, experimental and modeling results can be found in Appendix B. 3) To freeze cruciform samples in different mediums (in solution or not) at different storing temperatures during two months to study the effect of freezing on the mechanical properties of the arterial wall. 4) To perform uniaxial and biaxial test on rectangular and cruciform samples, respectively, at different deformation rates. Differences in the force-stretch curves at each deformation rate are compared. The stress-relaxation of rectangular samples held at a constant stretch ratio were obtain in preliminary tests (Appendix C). 5) Develop a model, based in Mooney-Rivlin five-parameter model, which is able to predict the effect of deformation rate on the aorta samples. 6) To perform preliminary tests in order to study the active response of arteries. The contraction and relaxation of smooth muscle cells was achieved using KCl and EGTA, respectively. A brief description of the methodology and results obtained can be found in Appendix D. 37 2.3. THESIS ORGANIZATION Chapter 1 of the thesis presents an introduction to the subject of study, as well as the literature related to the characterization of arteries. In this chapter, the factors that greatly influence the mechanical properties of biological tissues, and the need of constitutive models that reflects the actual experimental data over a wide range of deformations, are remarked. The physiology of arteries (i.e. constituents) and their mechanical behaviour (i.e. nonlinearity, anisotropy) are discussed in detail. The experimental testing and modelling (using analytical models or the finite element model) performed to obtain the mechanical properties of arteries are also commented in this chapter. Chapter 2 first discusses the scope of the present work. It also includes and describes the objectives and the organization of this thesis. In chapter 3 the comparison in mechanical testing of three sample geometries is studied. Uniaxial tests were performed to rectangular samples attached with grips, and biaxial experiments to cruciform and square samples attached with grips and hooks, respectively. The material parameters were fitted in an optimization loop until the difference between the simulation predictions and the experimental results is minimized. Two hyperelastic isotropic models (Ogden and Mooney-Rivlin five-parameters) and one anisotropic model (Guccione) were implemented in the finite element code to obtain thoracic aorta material parameters under uniaxial and biaxial testing. The constitutive equation that better predict the "J" shape curve of the arterial tissue was the Mooney-Rivlin five-parameter model. This model simulation results are discussed in detail through this chapter, which is based on a journal paper that has been submitted for publication (Virues- Delgadillo, J. 0., Delorme, S., DiRaddo, R., Hatzikiriakos, S. G., Thibault, F., 2007, "Mechanical Characterization of Arterial Wall: Should Cruciform or Square Sample Be Used in Biaxial Testing?," Journal of the Mechanical Behavior of Biomedical Materials, submitted). The results obtained with the Ogden and Guccione models are summarize in Appendix A. A study of the effect of freezing on the mechanical properties of cruciform samples tested biaxially is developed in chapter 4. Differences in the force-stretch curves between fresh and thawed samples are compared. Isotonic saline solution, Krebs supplemented with 1.8 DMSO and dipping in liquid nitrogen are the different storing mediums used in this 38 work. Samples were frozen at -20 or -80 °C during two months before re-testing. This chapter is based on a journal paper which has been submitted for publication (Virues- Delgadillo, J. 0., Delorme, S., El_Ayoubi, R., DiRaddo, R., Hatzikiriakos, S. G., 2007, "Effect of Freezing on the Biaxial Mechanical Properties of Arterial Samples," Biorheology, submitted). Chapter 5 focuses on the effect of deformation rate on the mechanical properties of the arterial wall. Deformation rates between 10 and 200%/s were used. One material parameter in the five-parameter Mooney-Rivlin model was found to be a function of the deformation rate. This chapter is based on a journal paper that has been submitted for publication (Virues-Delgadillo, J. 0., Delorme, S., Diraddo, R., Hatzikiriakos, S. G., Mora, V., 2007, "Effect of Deformation Rate on the Mechanical Properties of Arteries," Journal of the Mechanical Behavior of Biomedical Materials, submitted). The conclusions and contributions to knowledge are discussed in Chapter 6. A general summary of the most significant aspects resulted from this work and some recommendations for future work are also presented here. Finally the Mooney-Rivlin model was also used to obtain the mechanical properties of cylindrical arteries (i.e. porcine left coronary and carotid arteries) submitted to inflation tests. A brief discussion of the results obtained can be found in Appendix B. 39 2.4. REFERENCES DE SMET, E., JAECQUES, S. V. N., JANSEN, J. J., WALBOOMERS, F., VANDER SLOTEN, J., NAERT, I. E., 2007. Effect of constant strain rate, composed of varying amplitude and frequency, of early loading on peri-implant bone (re)modeling. Journal of Clinical Periodontology, 34, 618 — 624. DOEHRING, T. C., CAREW, E. 0., VESELY, I., 2004. The effect of strain rate on the viscoelastic response of aortic valve tissue: A direct-fit Approach. Annals of Biomedical Engineering, 32(2), 223 — 232. GAY, R. E., ILHARREBORDE, B., ZHAO, K., BOUMEDIENE, E., AN, K.-N., 2008. The effect of loading rate and degeneration on neutral region motion in human cadaveric lumbar motion segments. Clinical Biomechanics, 23, 1 — 7. HSIEH, Y.-F., WANG, T., TURNER, C. H., 1999. Viscoelastic response of the rat loading model: Implications for studies of strain-adaptive bone formation. Bone, 25(3), 379 — 382. HU, T., DESAI, J. P., 2004. Soft-tissue material properties under large deformation: Strain rate effect. Proceedings of the 2e Annual International Conference of the IEEE EMBS, San Francisco, CA, USA (September 1 — 5, 2004), 2758 — 2761. LALLY, C., REID, A. J., PRENDERGAST, P. J., 2004. Elastic behavior of porcine coronary artery tissue under uniaxial and equibiaxial tension. Ann. Biomed. Eng., 32 (10), 1355 — 1364. MOHAN, D., MELVIN, J. W., 1983. Failure properties of passive human aortic tissue. II Biaxial tension tests. Journal of Biomechanics, 16, 31 — 44. PIOLETTI, D., RAKOTOMANANA, L. R., 2000. Non-linear viscoelastic laws for soft biological tissues. Eur. J. Mech. A/Solids, 19, 749 — 759. PRENDERGAST, P. J., LALLY, C., DALY, S., REID, A. J., LEE, T. C., QUINN, D., DOLAN, F., 2003. Analysis of prolapse in cardiovascular stents: A constitutive equation for vascular tissue and finite-element modelling. Journal of Biomechanical Engineering, 125, 692 — 699. PROVENZANO, P., LAKES, R., KEENAN, T., VANDERBY Jr., R., 2001. Nonlinear ligament viscoelasticity. Annals of Biomedical Engineering, 29, 908 — 914. OKAMOTO, R. J., WAGENSEIL, J. E., DELONG, W. R., PETERSON, S. J., KOUCHOUKOS, N. T., AND SUNDT III, T. M., 2002. Mechanical properties of dilated human ascending aorta. Ann. Biomed. Eng., 30(5), 624 — 635. SCHULZE-BAUER, C. A. J., MORTH, C., HOLZAPFEL, G. A., 2003. Passive biaxial mechanical response of aged human iliac arteries. Journal of Biomechanical Engineering, 125, 395 — 406. 40 VAN ANDEL, C. J., PISTECKY, P. V., BORST, C., 2003. Mechanical properties of porcine and human arteries: Implications for coronary anastomotic connectors. Ann. Thorac. Surg., 76, 58 — 65. VANDE GEEST, J. P., SACKS, M. S., VORP, D. A., 2004. Age dependency of the biaxial biomechanical behavior of human abdominal aorta. Journal of Biomechanical Engineering, 126, 815 — 822. VIRUES-DELGADILLO, J. 0., DELORME, S., DIRADDO, R., HATZIKIRIAKOS, S. G., THIBAULT, F., 2007, "Mechanical Characterization of Arterial Wall: Should Cruciform or Square Sample Be Used in Biaxial Testing?," Journal of the Mechanical Behavior of Biomedical Materials, submitted. VIRUES-DELGADILLO, J. 0., DELORME, S., EL_AYOUBI, R., DIRADDO, R., HATZIKIRIAKOS, S. G., 2007, "Effect of Freezing on the Biaxial Mechanical Properties of Arterial Samples," Biorheology, submitted. VIRUES-DELGADILLO, J. 0., DELORME, S., DIRADDO, R., HATZIKIRIAKOS, S. G., MORA, V., 2007, "Effect of Deformation Rate on the Mechanical Properties of Arteries," Journal of the Mechanical Behavior of Biomedical Materials, submitted. WALDMAN, S. D., AND LEE, J. M., 2002. Boundary conditions during biaxial testing of planar connective tissues: Part 1: Dynamic behavior. Journal of Materials Science. Materials in Medicine, 13, 933 — 938. WANG, J. L., PARNIANPOUR, M., SHIRAZI-ADL, A., ENGIN, A. E., 1999. Rate effect on sharing of passive lumbar motion segment under load-controlled sagittal flexion: Viscoelastic finite element analysis. Theoretical and Applied Fracture Mechanics, 32, 119 — 128. YINGLING, V. R., CALLAGHAN, J. P., McGILL, S. M., 1997. Dynamic loading affects the mechanical properties and failure site of porcine spines. Clinical Biomechanics, 12(5), 301 — 305. 41 CHAPTER 3 3. MECHANICAL CHARACTERIZATION OF ARTERIAL WALL: SHOULD CRUCIFORM OR SQUARE SAMPLE BE USED IN BIAXIAL TESTING? 3.1. INTRODUCTION Due to the difficulty of testing soft tissues such as arteries in vivo, mechanical testing of arteries in vitro is often preferred. Uniaxial and biaxial testing has been used to characterize anisotropic materials such as arteries, although methodological aspects are still debated (Sacks and Sun, 2003). For instance, microsurgical or fishing hooks, crocodile grips, grip clamps and other kind of fixtures have been employed to attach the sample to the actuators of the uniaxial or biaxial apparatus. According to the Saint-Venant principle, stress distribution in a body is uniform away from the points of application of the load. This implies that local stress concentration phenomena occur close to the point of application of the load (edges of the sample), depending on the way the load is applied. Therefore, strain is typically measured far from the edges of the sample. This approach has been demonstrated both on uniaxial and biaxial experiments. In biaxial mode, hooks or sutures are typically attached along the four borders of a square sample of arterial wall tissue (Choudhury, 2005; Lally, et al., 2004; Okamoto, et al., 2002; Prendergast, et al., 2003), so as to allow stretch of the borders along their length. However, local stress concentration can cause the sample to fail at the hook puncture site. Cruciform samples clamped with grips are often used for testing industrial materials when sample thickness distribution can be controlled (Hjelm, 1994; Zinov'ev, et al., 1972). However, this type of sample geometry creates high stresses in the curved parts of the sample boundaries, as was demonstrated with computer simulations on various biaxial sample shapes and boundary conditions (Sun, et al., 2005). Inverse modeling has been used to obtain material properties from experiments where an analytical solution is difficult to obtain (Seshaiyer and Humphrey, 2003). Because inverse modeling can take into account the well-defined boundary conditions of biaxial cruciform samples, the main objective of this study is to evaluate the feasibility of inverse modeling for biaxial testing of cruciform samples over a wider range of strains than is possible with square samples. A secondary objective is to investigate whether data • A version of this chapter has been submitted for publication to the Journal of the Mechanical Behavior of Biomedical Materials (Virues-Delgadillo, J. 0., Delorme, S., Diraddo, R., Hatzikiriakos, S. G., Thibault, F., 2007, Mechanical Characterization of Arterial Wall: Should Cruciform or Square Sample Be Used in Biaxial Testing?) 42 obtained from uniaxial tests perfollned in 2 orthogonal directions can accurately predict biaxial behavior, and vice versa. 3.2. MA 1ERIALS AND METHODS 3.2.1. Experimental Setup Seven thoracic aortas were harvested within the day of death of pigs from a local slaughterhouse. Upon arrival, any visible connective tissue was dissected away from the external wall of the artery. Then the artery was cut open along its length, and cut out in rectangular, square and cruciform shapes (Fig. 3.1). The rectangular samples were tested uniaxially while the square and cruciform samples were tested biaxially. A total of 16 samples were obtained from the 7 aortas: 4 cruciform samples from 2 aortas along with 8 rectangular and 4 square samples from 5 aortas. Thickness was measured with a vernier caliper. Samples were stored in saline solution at 4 °C for no longer than 4 days prior to testing. (a) ^40^ 74,_ ^20 Î^1\ Grip Zone (b)^Grip Zone (C) 4015 55 Hook Zone Fig. 3.1. Sample dimension in millimetres for (a) uniaxial testing: rectangular sample clamped with grips; and (b, c) biaxial testing: (b) cruciform sample clamped with grips, (c) square sample attached with hooks. 43 The biaxial test bench (ElectroForce® LM1, Bose Corporation, Minnetonka, MN), shown in Fig. 3.2, includes four linear actuators capable of applying a peak force of 200 N and a maximum velocity of 3.2 m/s, over a displacement range of 24 mm (12 mm for each actuator). Samples can be mounted in horizontal configuration inside a saline bath heated at body temperature (37 °C). Grip clamps (for cruciform and rectangular samples) or fishing hooks (for square samples) can be used to attach the sample to arms extending from the actuators over the top of the bath. Displacement transducers are built in each of the four actuators. Two of the actuators (in orthogonal directions) are equipped with a 222 N (50 lbs) load cell. An optical strain extensometer includes a CCD camera located above the sample to record the sample deformation during the test, and software capable of tracking the position of four ink dots on the sample and extracting online the strains in the two principal directions ( e22) and the shear strain (e12). (a) Fig. 3.2. (a) Biaxial test bench: (b) cruciform setup and (c) square setup 44 Eight rectangular samples were tested uniaxially: four in the circumferential direction and four in the axial direction. Preliminary destructive tests revealed that failure rarely occurred below a nominal stretch ratio of 2.00. Therefore a 1.80 nominal stretch ratio was applied (8.00 mm displacement by each actuator). The forces were measured with one load cell. The original length of the sample was taken as the distance between the pair of grips (Fig. 3.1.a). Four cruciform samples were tested biaxially. An equal displacement on each of the four grips was applied. The forces at the grips were measured with the two load cells. In cruciform samples, the maximum displacement allowed by each actuator (12 mm) was applied, creating a non uniform strain distribution in the sample. This corresponds to an average nominal stretch ratio of about 1.60, calculated using the distance between facing grips. Four square samples were also tested biaxially. Threads were used to attach four fishing hooks to each side of the sample. The threads were connected to the arms of the actuators with a pulley system that allowed maintaining an equal tension in each thread. A specially designed jig was used to ensure that the hooks were always placed at the same position on the sample. The forces at the arms of the actuators were measured with the two load cells. Four dots, each one located 8.5 mm apart of the axis center, were printed in a square pattern on the center of the top surface of the sample using a water resistant oil- based quick drying ink The deformation at the center of the sample was measured using the optical strain extensometer. In preliminary tests, rupture of at least one hook puncture hole due to the high stress concentration occurred between 18 and 22 mm displacement. Therefore, the displacement was limited to 16 mm, corresponding to an average nominal stretch ratio of 1.64, calculated from the distance between the hooks on the parallel edges of the sample. It is important to clarify that the nominal stretch ratio calculated using the distance between the ink dots (i.e. 1.45) is different than the nominal stretch ratio between the hook sites. For all tests, a triangular wave form displacement at a 1Hz frequency was applied for 10 pre-conditioning cycles (Fung, 1993) and 10 steady-state cycles. The sampling rate of the force measurements was set so that forces were measured at the same stretch values in each cycle, therefore allowing the calculation of an average force-stretch curve over the 45 last 10 cycles. For all the samples tested, the experimental data obtained (such as sample thickness and the material load force-stretch data), in uniaxial and biaxial testing, were summarized as a mean value ± the standard deviation. 3.2.2. Constitutive Equations In the present work, a hyperelastic constitutive model was modified and fitted to the experimental data: the Mooney-Rivlin (Mooney, 1940) model. 00 W(/p / 2 )=^(/, —3)`(/ 2 —3)'^ (3.1) g, J =0 The Mooney-Rivlin model is a function of the 1 st and 2nd invariants of the Green- Cauchy tensor (right), where II and I2 can be written as a function of the principal stretch ratios. 11=4+222+2;^ (3.2) 4=4/1. -Fig4 (3.3) The strain energy density function used in this study has been used successfully elsewhere (Lally, et al., 2004; Prendergast, et al., 2003) to obtain the arterial behavior, and has only five material parameters (a10 , a01 , a11, a20 , a30 ): W = a10(11 —3) + ao1(12 —3) + (Ii —3)(12 —3) + a20(11 —3)2 + a30(11 — 3) 3^(3.4) The corresponding stress component of the stress tensor can be obtained when the strain density function is differentiated with respect to the corresponding strain component. 3.2.3. Inverse Modeling In order to adjust the parameters of this model, the inverse modeling technique was used. In this approach, finite element simulations of the experiment are performed by applying either displacement or force conditions, on a mesh of the same size and shape as the sample. In the present work, the Industrial Materials Institute (IMI, Boucherville, QC) large deformation finite element software was used, with a convergence criterion of 0.01 mm on nodal displacements, of 1% on nodal forces. This software has been validated for predicting molten polymer forming processes such as blow molding and thermoforming (Debergue and Laroche, 2001; Laroche, et al., 1999). Although the thoracic aorta has been 46 shown to be anisotropic (Vande Geest, et al., 2006; Vande Geest, 2005), for the purpose of demonstrating the inverse modeling approach the aorta behavior was modeled using an isotropic constitutive model, the Mooney-Rivlin (Mooney, 1940) model, as did others (Prendergast, et al., 2003; Tezduyar, et al., 2007; Scotti and Finol, 2007; Wang, et al., 2001; Raghavan and Vorp, 2000). A set of initial values for the material properties is input. The simulation predicts the reaction forces at the boundaries (i.e. uniaxial and biaxial cruciform tests) or the displacement of the nodes corresponding to the ink dots in the center of the sample (i.e. biaxial square test). The material properties are then adjusted using an optimization technique until a set of force-displacement experimental data matches the values calculated by the model. In this work, the sequential quadratic programming (SQP), a second order optimization method of Design Optimization Tools (DOT) software (VanderPlaats, 1999), was used. For the case of biaxial cruciform extension (Fig. 3.3), displacement boundary conditions were applied to the mesh and reaction forces in the circumferential ( x ) and axial (y) directions were simultaneously calculated by the finite element software. In the optimization loop, the material properties were iteratively adjusted until the following objective function was minimized: S(c) =^[(F'x — fx (Lx ,c))2 +(Fy — fy (Ly ,c))2 1^(3.5) i.1 where S(c) is the objective function, c is the vector of unknown material properties; Lx and L y are the applied displacements; F and Fy are the experimentally measured reaction forces at the grips; and fx (Lx ,c) and fy (Ly ,c) are the reaction forces predicted by the finite element model. N is the total number of data points (L, F) gathered in the experiment. Similarly as in biaxial tests with cruciform samples, in uniaxial extension tests, displacement conditions were applied and reaction forces were predicted at the grips. However, two objective functions were defined, one for each direction: N r S(c) = ZKF x f x (Lx ,c))2 1^ (3.6) i=i Material^C, Parameters Co Reaction Force I ^ imm"600- I Fx, , Fy2 ,^Fy„ at Lyi , t,2 , Fy, , Fy2 ,^Fr,^Ly „, Ly , ,^Lyn , Reaction Force IF^at CI, 1,2, Cni Fy , , Fy2,...^ Ly2,^Lyn ! I Equi-Biaxial I Deformation' 1-.1 , 1 LYI' 1-r2^1-t. Simulation c„. Material synimowasommem* Parameters Cam, Fig. 3.3. Schematic representation of the inverse modeling technique used to obtain the material properties for the biaxial experiments with a cruciform sample. 47 S (c) =^— f ,(L),,c))2 ]^ (3.7) Therefore, different material properties were obtained for the circumferential and for the axial directions, from the uniaxial extension experimental data. In biaxial tests with square samples, forces were applied to each one of the nodes corresponding to the location of the hook sites, and strains were predicted at nodes corresponding to the location of the ink dots. Hence, the following objective function was minimized: N S (C) E Kex - (F,,,c)) 2 +(i'y — ey (Fy ,c))2 ] (3.8) where ex , ey, are the measured strains at the ink dot locations, and ex (F,,k), ey (Fy ,k) are the predicted strains at the ink dot locations. Table 3.1 describes the applied boundary conditions and the predicted values used for each type of test. Table 3.1 Shape, attachment method, applied condition and predicted values for each type of experimental test performed. Type of test Sample shape Sample attachment Applied condition Predicted values Uniaxial Biaxial Rectangular Cruciform Grips Grips Displacement Displacement Reaction forces at grips Reaction forces at grips Biaxial Square Hooks and sutures Forces Strain at center of sample 48 The finite element meshes used for simulation of uniaxial, biaxial cruciform and biaxial square tests are shown in Fig. 3.4. Only the part of the sample free to deform between the pair of grip clamps was meshed for uniaxial (Fig 3.4.a) and biaxial cruciform (Fig 3.4.b) simulations. Because of symmetry, only one quarter of the cruciform sample area (Fig 3.4.b) was simulated, which reduced computational time. In simulations of uniaxial extension tests and of biaxial tests with cruciform samples, the experimentally measured displacements were applied to the nodes located on the appropriate boundaries of the mesh. Predicted reaction forces distributed on the boundary nodes were summed for each tab of the sample and multiplied by 2 to account for symmetry, for comparison with experimental data. The use of reaction forces at the boundaries allows considering the deformation of the whole sample rather than just a sub-domain. The mesh used for the simulation of biaxial tests with a square sample is the square bounded by the perimeter traced by the hook puncture sites (Fig 3.4.c). In simulations of the biaxial tests with square samples, experimentally measured forces divided by the number of hooks per sample edge (4) were applied to the nodes located at the hooks puncture sites. The ink dots at the center of the square samples define a sub-domain, as described in (Seshaiyer and Humphrey, 2003), in which stresses and strains are considered uniformly distributed. (a)^ (b) yryr d F^F F Fig. 3.4. Undeformed meshes and boundary conditions for (a) uniaxial extension tests, (b) biaxial tests with cruciform samples, and (c) biaxial tests with square samples. 1.2 1.4 Stretch Ratio 1.6 1.8 0 8 -u_ 2'12-- 16 4 ^ - p- • Sample 1 Sample 2 - -0-- - Sample 3 - -0-- • Sample 4 —a— Sample 5 — 11--Sample 6 - Sample 7 Sample 8 Cruciform Data (b) 20 •-a- - Sample g -- Sample 10 - -0 - Sample 11 -0- Sample 12 --*-- Sample 9 —II-- Sample tO Sample 11 --.--SAmple 12 1.1 ^ 1-2^1.3^1.4 Stretch Ratio 16 8 LL 49 3.3. RESULTS 3.3.1. Experimental Results The mean thickness of all tested specimens tested was 2.4 ± 0.4 mm. Figure 3.5 shows the experimental force data plotted against the "nominal" stretch ratio. In the rectangular samples, the stretch ratio is nearly uniformly distributed because the sample's length is five times greater than its wider and therefore edge effects can be neglected. However, in the cruciform samples the stretch ratio is not uniformly distributed due to sample's shape. This makes it very difficult to compare between experimental results from different tests. Figure 3.5 shows the variability of experimental results between samples as well as the anisotropic behavior of the arterial tissue. For all three types of tests, nonlinear behavior was observed, and differences between samples were small. In uniaxial extension tests (Fig. 3.5.a) and biaxial tests with square samples (Fig. 3.5.c), the samples were consistently stiffer in the circumferential direction than in the axial direction. Differences between samples and between directions were smallest in biaxial tests with cruciform samples (Fig. 3.5.b). Higher nominal stretch ratios were reached in the cruciform samples than in the square samples. (a) 20 -^Uniaxial Data (c) 20 Square Data - Sample 13 16 - -0-- Sample 14 •0- - Sarrple 15 2. 12 - • -0- - Sample 16 Sariple 13 `0 Sample 14 u. T Sample 15 --■-- 4 ^Sample 16 1.2^1.3 Stretch Ratio 1.4^1.5 Fig. 3.5. Force — stretch ratio curves for (a) uniaxial extension, (b) biaxial testing with cruciform samples, and (c) biaxial testing with square samples. Circumferential data are represented by dashed lines and hollow symbols. Axial data are represented by solid lines and solid symbols. 50 3.3.2. Inverse Modeling Results Table 3.2 summarizes the adjusted Mooney-Rivlin parameters for each type of extensional test. For uniaxial tests, one set of parameters was obtained for each direction. For biaxial tests with cruciform and square samples, one set of parameters for both directions were obtained by simultaneously fitting circumferential and axial experimental data. Each parameter value was bound in the range of -10 4 kPa to 104 kPa. Repeated optimization with different initial parameter values consistently converged towards the same solution. Table 3.2 Moone -Rivlin fitted parameters Mooney- Rivlin Parameters (kPa) Uniaxial Circumf. Uniaxial Axial Biaxial Cruciform Biaxial Square a10 18.15 8.13 1.32 127.52 a01 16.80 8.94 0.63 -95.24 all 2.48 0.01 2.90 0.08 a2() 1.96 0.00 3.85 49.73 a30 12.27 9.32 30.94 1.60 The computed circumferential and axial forces obtained using the Mooney-Rivlin fitted parameters from Table 3.2 in uniaxial, biaxial cruciform and biaxial square are shown in Figure 3.6. For each type of test, a close fit was obtained between the model and the experimental data. In biaxial tests, the model was adjusted in between the circumferential and the axial data points. (a)^ Uniaxial 6z – Uniaxial Data — Uniaxial5- w § 4- I' 73 3 -47, 41)'' 2 - 1 - a 0 1^1.2^14^1.6 ^1.8 Stretch Ratio 1.2 1.4 Stretch Ratio Uniaxial — Wiaxial Data ^ U,iaxial Fit Biaxial Cruciform 1 (c ) 1.6 1.8 51 24 - 20 - 16 - 4) 12- ▪ 8 - 4 - 0 + Circumferential Data Axial Data ^ Biaxial Cruciform FR -Y- 1.2^1.4 ^ 1.6^1.8 Stretch Ratio (d) 24 20 Z 16 t) 12 IL • 8 4 0 1 Biaxial Square ■ Circumferential Data ^ Axial Data ^ Biaxial Square Fit 1.6 ^ 1.81.2 ^ 1.4 Stretch Ratio Fig. 3.6. Mooney-Rivlin experimental and computed reaction forces versus stretch ratios: (a) circumferential and (b) axial directions for uniaxial tests, (c) biaxial tests with cruciform samples and (d) biaxial tests with square samples. Mooney-Rivlin Fitted Parameter Mooney - Rivlin Fitted Parameter (a) (b) Mooney - Rivlin Fitted ParameterMinimization of the Objective Function Mooney - Rivlin Fitted Parameter 1.005 6^0^1^2^3 Iteration (d) 42^3 Iteration 1E+00 1.E-01 1.0+0 ' 0 4 5 642^3 Iteration 2^3 iteration (e) t.0-04 6^0 if) Case_A - - Case E1 —Case_C —C ase_ A - - - • Case —Case_C 10-01 7.5 a. 1E- 10-03 in 1 0-04 1 E-05 1.E+05 0 0 c 1 E+03 u -eu -19 1.E+01 11.1-o• 0 1 E-01 (c) Mooney -Rivlin Fitted Parameter O. 1 0-02 1.0-03 10-04 1. 0+00 1.E-01 • • - .Case_B —Case_C —Case_A -^• , CaseB Case_C 1.E+001 1 E -0 gy 1.0-02 11-0 at 1 0-04 1 G5 6^0^1^2^3 Iteration ^Case_A .CaseB —••^Case_C 1.0+00 10-01 1E-02 1.E-03 1 0-04 1.E-0`5 0 75" o_ c1-3 2^3 Iteration 52 Figure 3.7.a shows the value of the objective function plotted against the iteration number in the optimization loop, for three different initial guesses. The objective function was evaluated 50 times in iterations 0 (initial guess) to 6. Figure 3.7.b illustrates optimization of one of the material parameter, a 30 , using three different initial guesses. Similar figures (Figs. 3.7.c-f) were obtained for the rest of the model parameters ( au) 6/01 ail a20) . Fig. 3.7. Optimization of the Mooney-Rivlin parameters for a biaxial test with a cruciform sample. (a) Objective function, and (b) a30 , (c) a lc, , (d) a01 , (e) all , (t) a 20 parameter values as a function of iteration number, for three different initial guesses for material parameters: Case_A (all parameters = 0.1 MPa), Case_B (all parameters = 1.0 MPa), and Case_C (all parameters = 0.0001 MPa). +1.2^1A^1 6^1.8 Stretch Ratio 53 3.3.2.1. Uniaxial Fit Figure 3.8 shows that the stress and stretch ratio distributions are non uniform close to the free edges of the sample. The local stretch ratio ranges from 1.70 to 1.82, for a nominal stretch ratio of 1.80. (a) Stress (MP a) I 0.6485 0.6344 0.6202 0.6061 0.5920 0.5779 - 0.5638 0.5497 0.5355 0.5214 0.5073 (b) Stretch Ratio 1.8211 1.8099 17986 1.7873 1.7760 I 1.76481.75351.74221.73091.71961.7084 Fig. 3.8. Deformed configurations at 1.80 nominal stretch ratio for rectangular mesh in uniaxial tension: distributions of (a) 6 stress and (b) stretch ratio in the principal direction. Mooney-Rivlin model parameters adjusted to the experimental uniaxial data were used to simulate the uniaxial and biaxial tests. The force-stretch curves for the experimental and simulated data are plotted in Figure 3.9. Although an excellent fit is observed between the uniaxial data and uniaxial simulation, the Mooney-Rivlin model adjusted to the uniaxial data does not predict well the biaxial behavior, especially in the nonlinear part of the curve at high stretch ratios. (a) 24 - £ 20 - f 0 16 U- -.2 12 - w ! 2i.7) 4 - 0 1 - Uniaxial -+ Biaxial Cruciform ■ Biaxial Square ^ Uniaxial-fit ^ Biaxial Cruciform-mod - - - - - Biaxial Square-mod Mooney - Rivlin 5 Parameters Model 054 (13) 24 - ^Mooney - Rivlin 5 Parameters Model Uniaxial ▪ Biaxial Cruciform ▪ Biaxial Square ^ Biaxial Cruciform-mod Biaxial Square-mod - • + 4 1^1.2 1.4 Stretch Ratio 1.6 1_8 Fig. 3.9. Simulated uniaxial and biaxial tests using Mooney-Rivlin parameters fitted to uniaxial experimental data: (a) circumferential and (b) axial force-stretch curves. 3.3.2.2. Biaxial Cruciform Fit Figure 3.10 shows that the stress distribution is non-uniform in the cruciform sample, highest stresses being located at the curved boundaries, and lowest stresses close to the center of the full sample (upper left corner of quarter-sample). Stretch ratios range from 1.24 to 1.65 for a nominal stretch ratio of 1.60. (a) Stress Via) 0.4277 0.3923 0.3568 0.3214 0,2859 0.2504 0.2150 0.1795 0.1441 01086 0.0732 (b) Stretch Ratio 1.6464 1.6060 1.5657 1.5253 1.4849 1.4445 1.4041 1.3637 1.3233 1.2829 1.2425 Y Fig. 3.10. Deformed configurations at 1.60 nominal stretch ratio for cruciform mesh in biaxial stretching: distribution of (a) Q ^and (b) stretch ratio in the principal direction. In Figure 3.11, the parameters adjusted to the cruciform biaxial tests were used to simulate all types of tests. Since the Mooney-Rivlin model is isotropic and a biaxial test was used, the parameters were adjusted simultaneously to the axial and circumferential Mooney- Rivlin 5 Parameters Model 24 - — Uniaxial fi 20 _^+ Biaxial Cruciform ■ Biaxial Square - - - - - Uniaxial-rrod Biaxial Cruciform-fit /Biaxial Square-mod^+ ■ + 55 experimental data points, contrarily to Figure 3.9 where different parameters were obtained for both directions. In this case, the agreement between cruciform experimental and simulated data is good, with the best predictions under 1.5 stretch ratios. The modeled curve fitted slightly better axial rather than circumferential forces below 1.4 stretch ratios. The Mooney-Rivlin model adjusted to the cruciform biaxial data was able to fit the uniaxial data up to stretch ratios of 1.5, but underestimated the biaxial behaviour of the square samples. (a) 1^1.2^1. 4^t6^1.8 Stretch Ratio (b)^Mooney - Rivlin 5 Parameters Model 24 - 20 - 2 16 0 12 8 - —^iaxia + Biaxial Cruciform ■ Biaxial Square - - - - - Uniaxia I-mod ^ Biaxial Cruciform-fit ^ Biaxial Square-mod ■, fa. - -^- —^7 _^- 1.2^IA 1.6^1.8 Stretch Ratio Fig. 3.11. Simulated uniaxial and biaxial tests using Mooney-Rivlin parameters fitted to biaxial experimental data with cruciform samples: (a) circumferential and (b) axial force-stretch curves. 4- 0 NON (a) Ink-dot 56 3.3.2.3. Biaxial Square Fit In Figure 3.12, the stress and stretch ratio distributions are presented for the square sample under biaxial stretching. The stress distribution in biaxial extension is non-uniform and maximal close to the hook punctures sites, and mostly unifotui near the center of the sample. The maximum stresses in this simulation are almost three times the values of the maximum stresses for the simulated rectangular and cruciform geometries (square: 1.90 MPa; cruciform: 0.43 MPa; rectangular: 0.65 MPa). Stretch ratios span from 1.18 to 1.95 for a nominal stretch ratio of 1.45 calculated from the ink dots. Stress (MPa) 1.8984 (b) Stretch Ratio 1.9501 1.7142 1.8734 1.5301 1.7968 1.3459 1.7201 1.1617 1.6435 0.9775 1.5668 0.7934 1.4902 0.6092 1.4135 0.4250 1.3369 0.2408 Ink-dot 1.2603 0.0567 1.1836 Fig. 3.12. Deformed configurations at 1.64 nominal stretch ratio (at the puncture sites) for square mesh in biaxial stretching: distribution of (a) c stress and (b) stretch ratio in the principal direction. Figure 3.13 shows the simulation results for all types of tests, using the Mooney- Rivlin parameters fitted to the square biaxial test results. The predictions of crucifonn biaxial behavior and uniaxial behavior were largely overestimated compared to experimental results. 3.4. DISCUSSION Small sample variability between the axial force-stretch curves of the samples tested uniaxially and biaxially was found in this work. Circumferential data had a slightly higher variability, particularly in the biaxial square data. Prendergast et al. (2003) found higher 57 variability between square samples tested biaxially than was observed in this study, although the location of the samples along the aorta was not specified in their study. In the present study, higher forces were consistently observed in the circumferential direction in all 16 thoracic aorta samples tested, regardless of the test method (uniaxial, biaxial cruciform, biaxial square). This would justify further investigation of the inverse modeling technique with anisotropic constitutive models. (a) 24 - 20 - ▪ 16 - TO' 1 12 - 41) • - E • 4 - 0 1 Mooney - Rivlin 5 Parameters Model - Uniaxial + Biaxial Cruciform ■ Biaxial Square - - - - - Uniaxial-mod ^ Biaxial Cruciform-mod Biaxial Square-fit ---..-E±- - .1.-±---4-ĝ - --- Ili- ---+^ •.='.. ' -I 1.2 1.4 Stretch Ratio t 6^1.8 (b) 24 Mooney - Rivlin 5 Parameters Model - Uniaxial + Biaxial Cruciform ^ 20^■ Biaxial Square - - - - - Uniaxial-mod ^ Biaxial Cruciform-modS 6 Biaxial Square-fitC.) LL4F5 12 :Cc< 8 1A Stretch Ratio 1-6 Fig. 3.13. Simulated uniaxial and biaxial tests using Mooney-Rivlin parameters fitted to biaxial experimental data with square samples: (a) circumferential and (b) axial force-stretch curves. The differences observed in the mechanical behavior of thoracic aortas (Fig. 3.5) under the different extensional test performed in this study suggest that the apparent 58 anisotropy in the material is dependent on the specimen shape and gripping method (sutures or grips). Waldman and Lee (2002) observed similar differences in bobine pericardial tissue using different sample gripping methods on the same specimens. It is hypothesized (Sun, et al., 2003; Waldman and Lee, 2002; Waldman and Lee, 2005) that when the sample is attached with clamps (i.e. rectangular and cruciform samples), all the fibers residing near the sample edges are loaded. However, when sutures are used to attach the sample, only the fibers within the vicinity of the suture sites are directly loaded. The larger proportion of fibers loaded when the sample is attached with grips, such as cruciform samples, might be a closer representation of in vivo behavior. The inverse modeling technique was demonstrated to be a useful tool to tune constitutive models to experimental data. However, Mooney-Rivlin parameters obtained from each type of test vary greatly in this study. They are also very different from those reported by Prendergast et al. (2003). This might be due to the high number of parameters in the Mooney-Rivlin model and the fact that multiple combinations of widely different parameter values can produce similar stress-strain curves for a single deformation mode. Also, the Mooney-Rivlin model fitted to the cruciform biaxial data predicted well the uniaxial behavior up to stretch ratios of 1.5. This suggests that additional information on the material behaviour at higher stretch ratios than can be reached in biaxial tests might be obtained from uniaxial tests. Therefore, the inverse modeling method could be further improved by simultaneously fitting the results of complementary uniaxial and biaxial tests. The agreement between biaxial and uniaxial simulations using the same material parameters was better when the parameters were obtained from the biaxial experiments with cruciform samples, than from the biaxial experiments with square samples. This might be due to the fact that the biaxial tests with the square samples do not take into account the high stretch ratios outside of the square area defined by the four ink dots. Because of stress concentration, failure occurred at the hook puncture sites before a stretch ratio of 1.5 could be reached at the center of the specimen. In contrast, the inverse modeling technique used with cruciform biaxial tests allows taking into account the whole behavior of the sample, including high stresses on curved borders, therefore allowing to reach maximum stretch ratios up to 1.64 in the sample. 59 Waldman and Lee (2002) have observed a higher apparent stiffness when testing square samples of soft tissue with grip clamps than with sutures. In both types of tests, the stretch ratios were measured by tracking the position of points near the center of the specimen. They concluded that existing biaxial tests cannot reflect the properties of tissues in vivo and that more work needs to be done to standardize the boundary conditions in order to produce a uniform stress distribution in the center of the sample, where the deformations are measured. Our study shows that the inverse modeling technique allows taking into account the boundary conditions of the test and the sample geometry, circumventing the need to make measurements of stretch ratio away from the boundaries of the sample where the stress distribution is assumed to be uniform. Advantages of cruciform samples over square samples include: 1) characterization over a higher deformation range, hence better prediction of behavior at large deformations; 2) simpler data acquisition through the use of actuator displacement rather than extensovideometry; 3) easier attachment of sample with grip clamps rather than sutures. However, the nonuniform stress distribution in cruciform samples requires the use of inverse modeling adjustment of constitutive model parameters. 3.5. SUMMARY The mechanical behavior of porcine thoracic aortas in vitro under uniaxial and biaxial testing was studied. We found that inverse modeling of biaxial experiments over a wider range of strains was more feasible using cruciform samples rather than square samples. Uniaxial and biaxial simulations were better predicted using the material parameters from biaxial cruciform experiments. The material parameters obtained from biaxial square experiments might not predict well uniaxial behavior because the high stretch ratios outside of the square area defined by the four ink dots were not taken into account. Inverse modeling can be used to fit the material parameters of constitutive models, such as Mooney-Rivlin, to biaxial data measured on cruciform samples. A five-parameter Mooney-Rivlin model was found suitable in representing experimental data obtain using various tests, however, more work is required in order to define the optimal set of experiments in order to use the inverse modeling method to fit anisotropic, viscoelastic or active (muscle tone) models of soft tissues. 60 3.6. REFERENCES CHOUDHURY, N. Z., 2005. Characterization of healthy and diseased human ascending aorta tissue. Ms. Eng. degree thesis, McGill University, Montreal, Quebec. DEBERGUE, P., AND LAROCHE, D., 2001. 3D finite elements for the prediction of thermoforming processes 4th Int. ESAFORM Conf. on Mater. Forming, 1, 365 — 368. FUNG, Y. C., 1993, Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., Springer-Verlag, New York, NY, 261 — 262. HJELM, H. E., 1994. Yield surface for gray cast iron on biaxial stress. J. Eng. Mater. Tech., 116, 148 — 154. LALLY, C., REID, A. J., AND PRENDERGAST, P. J., 2004. Elastic behavior of porcine coronary artery tissue under uniaxial and equibiaxial tension. Ann. Biomed. Eng., 32(10), 1355 — 1364. LAROCHE, D., KABANEMI, K.K., PECORA, L., AND DIRADDO, R.W., 1999. Integrated numerical modeling of the blow molding process. Polymer Eng Sci, 39, 1223 — 1233. MOONEY, M. A., 1940. Theory of large elastic deformation. J. Appl. Phys., 11, 582 — 592. OGDEN, R. W., 1982, Elastic Deformations in Rubberlike Solids, In: Mechanics of solids: The Rodney Hill 60th anniversary volume, Pergamon Press, Oxford, 499 — 537. OKAMOTO, R. J., WAGENSEIL, J. E., DELONG, W. R., PETERSON, S. J., KOUCHOUKOS, N. T., AND SUNDT III, T. M., 2002. Mechanical properties of dilated human ascending aorta. Ann. Biomed. Eng., 30(5), 624 — 635. PRENDERGAST, P. J., LALLY, C., DALY, S., REID, A. J., LEE, T. C., QUINN, D., AND DOLAN, F., 2003. Analysis of prolapse in cardiovascular stents: A constitutive equation for vascular tissue and finite-element modelling Journal of Biomechanical Engineering, 125, 692 — 699. RAGHAVAN, M. L., AND VORP, D. A., 2000. Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: Identification of a finite strain constitutive model and evaluation of its applicability. Journal of Biomechanics, 33, 475 — 482. SACKS, M. S., SUN, W., 2003. Multiaxial mechanical behavior of biological materials. Annu. Rev. Biomed. Eng., 5, 251 — 284. SCOTT!, C. M., AND FINOL E. A., 2007. Compliant biomechanics of abdominal aortic aneurysms: A fluid—structure interaction study. Computers & Structures, 85, 1097 — 1113. 61 SESHAIYER, P., AND HUMPHREY, J. D., 2003. A sub-domain inverse finite element characterization of hyperelastic membranes including soft tissues. Journal of Biomechanical Engineering, 125, 363 — 371. SUN, W., SACKS, M. S., AND SCOTT, M. J., 2003. Numerical simulations of the planar biaxial mechanical behavior of biological materials. 2003 Summer Bioengineering Conference, Florida, pp. 1 — 2. SUN, W., SACKS, M. S., AND SCUFF, M. J., 2005. Effects of boundary conditions on the estimation of the planar biaxial mechanical properties of soft tissues. Journal of Biomechanical Engineering, 127, 709 — 715. TEZDUYAR, T. E., SATHE, S., CRAGIN, T., NANNA, B., CONKLIN, B. S., PAUSEWANG, J., AND SCHWAAB, M., 2007. Modelling of fluid—structure interactions with the space—time finite elements: Arterial fluid mechanics. Int. J. Num. Meth. Fluids, 54, 901 — 922. VANDE GEEST, J. P., 2005. Towards an improved rupture potential index for abdominal aortic aneurysms: Anisotropic constitutive modeling and noninvasive wall strength estimation. University of Pittsburgh, PhD Thesis. VANDE GEEST, J. P., SACKS, M. S., VORP, D. A., 2006. The effect of aneurysm on the biaxial mechanical behavior of human abdominal aorta. Journal of Biomechanics, 39, 1324 — 1334. VANDERPLAATS, G.N., 1999. Numerical optimization techniques for engineering design. VanderPlaats Research & Development Inc. WALDMAN, S. D., AND LEE, J. M., 2002. Boundary conditions during biaxial testing of planar connective tissues: Part 1: Dynamic behavior. Journal of Materials Science. Materials in Medicine, 13, 933 — 938. WALDMAN, S. D., AND LEE, J. M., 2005. Effect of sample geometry on the apparent biaxial mechanical behaviour of planar connective tissues. Biomaterials, 26, 7504 — 7513. WANG, D. H. J., MAKAROUN, M., WEBSTER, M. W., VORP, D. A., 2001. Mechanical properties and microstructure of intraluminal thrombus from abdominal aortic aneurysm. Journal of Biomechanical Engineering, 123(6), 536 — 539. ZINOV'EV, M. V., IL'ICHEV, V. YA., RYKOV V. A., AND SAVVA, S. P., 1972. Method of testing samples in a biaxial stressed state at low temperatures. Strength Mater, 4, 637 — 639. 62 CHAPTER 4 4. EFFECT OF FREEZING ON THE BIAXIAL MECHANICAL PROPERTIES OF ARTERIAL SAMPLES 4.1. INTRODUCTION The most common method for investigating the mechanical behavior of soft tissues consists of conducting mechanical tests on animal tissue explants, i.e. harvested within a day after the death of the animal (Waldman and Lee, 2002; Prendergast, et al., 2003; Lally, et al., 2004). There is however few data on in vitro mechanical behavior of human arterial tissues, in part because of the logistical difficulty of performing mechanical tests on fresh human tissues. Some studies (Okamoto, et al., 2002; Vande Geest, et al., 2004; Mohan and Melvin, 1983; Van Andel, et al., 2003; Schulze-Bauer, et al., 2003, Raghavan, et al., 1996, Carmines, et al., 1991, Langewouters, et al., 1984, Hudetz, et al., 1981) have reported testing human tissues several days after death, assuming that mechanical properties of the tissues were preserved by refrigerating and by using chemical solutions mimicking the physiological environment, such as saline, Ringer's, Tyrode's, Hank's, Krebs and variations based on these solutions. Cryopreservation methods (Han and Bischof, 2004) developed to preserve artery samples for reimplantation could also provide a means to preserve arteries for days and weeks before mechanical testing. Artery mechanical properties have been shown to depend on the relative proportion and arrangement of the arterial wall constituents such as collagen, elastin and smooth muscle cells (Gamero, et al., 2001; Bujan, et al., 2000). Smooth muscle cells contribute to structural, mechanical and functional changes in the arterial wall through several processes, including cell growth, elongation and reorganization of cells, and alteration of extracellular matrix composition (Intengan and Schiffrin, 2000). The viability of smooth muscle cells, as well as the integrity of elastin and collagen fibers contribute to the arterial wall mechanical behaviour (Almassi, et al., 1996, Pascual, et al., 2002). Damage caused to smooth muscle cells by ice formation and fragmentation of extracellular matrix fibers might affect the mechanical properties of the artery. Cryoprotective agents (CPA), such as dimethyl sulfoxide (DMSO), have been used in several studies to protect the cells from cryoinjury (Schenke-Layland, et al., 2006, Brockbank, 1989, Miles, 1999). These CPAs are typically added to the storing solution in order to reduce ice formation in both intra- and extracellular • A version of this chapter has been submitted for publication to Biorheology (Virues-Delgadillo, J. 0., Delorme, S., E1_Ayoubi, R., Diraddo, R., Hatzikiriakos, S. G., 2007, Effect of Freezing on the Biaxial Mechanical Properties of Arterial Samples) 63 space by preventing water movement out of the tissue (Miles, 1999). The effect of cryopreservation has been investigated with respect to cell viability (Devireddy, et al., 2003; Neidert, et al., 2004) and histological changes (Song, et al., 1995; Cui, et al., 2002; Pacholewicz, et al., 1996; Hunt, et al., 1994, Wusteman and Pegg, 2001), but little is known about the effect of cryopreservation on mechanical properties of the arterial wall. It has been found (Cooper, et al., 1971; Pascual, et al., 2001) that major arteries can withstand freezing and thawing without subsequent rupture. Fresh and cryopreserved behaviour of arteries has been investigated using inflation (Blondel, et al., 2000) and uniaxial tests (Adham, et al., 1996; Venkatasubramanian, et al., 2006). Blondel (2000) and Venkatasubramanian (2006) observed significant stiffening of femoral arteries cryopreserved at -80 °C and -150 °C respectively compared to fresh arteries. Adham (1996) observed no difference in high strain modulus of aortas preserved at +4 °C for 1 month compared to cryopreservation with DMSO at -135 °C for 4 months. The goal of the present study was to evaluate the effect of several conservation methods on the biaxial mechanical properties of arteries. The investigated conservation methods include freezing for two months at either -20 °C or -80 °C, in the presence of isotonic saline solution or Krebs-Henseleit solution with DMSO, or without solution but dipped in liquid nitrogen. This study describes the storing conditions that best preserves the mechanical properties of arteries in axial and circumferential directions. 4.2. MATERIALS AND METHODS 4.2.1. Experimental Setup Eighteen thoracic aortas were harvested within the day of death of pigs from a local slaughterhouse and cleaned of remaining connective tissue. One cruciform-shaped sample was cut out from each aorta (Fig. 4.1.a) for equibiaxial testing. The average thickness of all tested specimens, measured with a vernier caliper, was 2.3 ± 0.2 mm. Samples were stored in isotonic saline solution at 4 °C for up to 8 hours prior to testing. Cruciform samples were mechanically tested on a planar biaxial test bench (ElectroForce® LM1, Bose Corporation, Minnetonka, MN), shown in Fig. 4.1.b and capable of applying a peak force of 200 N over a displacement range of 12 mm per actuator. Samples were mounted in horizontal configuration inside a saline bath heated at 64 body temperature (37 °C). Grip clamps were used to attach the four tabs of the cruciform sample to the arms extending from the actuators over the top of the bath, accordingly to a previously described method (Chapter 3 of this thesis; Virues-Delgadillo, et al., 2007). (a)^Grip Zone 40 15 ^ 55 7.5 Fig. 4.1. Sample dimension in millimetres for biaxial testing and (b) biaxial test bench used for cruciform sample clamped with grips. 65 Equibiaxial testing was done by applying a 12 mm displacement (which corresponds to an average nominal stretch ratio of 1.55) on each one of the four grips, creating a non uniform strain distribution in the sample. The nominal stretch ratio was calculated using the distance between facing grips. The 1.55 stretch ratio was selected based on earlier experiments (Chapter 3 of this thesis; Virues-Delgadillo, et al., 2007) because it allows capturing the nonlinear part of the stress-stretch curve, while avoiding rupture of the samples. Triangular displacement wave forms were applied at a deformation rate of 110 %/s, which corresponds to a frequency of 1.0 Hz. Displacements were applied for 20 cycles. The first 10 cycles were used for pre-conditioning. The force-stretch data was averaged over the last 10 cycles. After the 18 fresh samples had been tested biaxially, they were randomized into three groups for storing: Group I, samples were put in a polypropylene tube filled with isotonic saline solution; Group II, samples were put in a polypropylene tube filled with Krebs-Henseleit solution, supplemented by 1.8 M DMSO; and group III, samples were dipped in liquid nitrogen and then put in a polypropylene tube without any solution. For each group, half of the samples were stored at -20 °C and the other half at -80 °C. Samples in group II were stored at 4 °C for 20 minutes prior to freezing. Three samples were stored under each of the six different storage conditions, as shown in Table 4.1. Table 4.1. Number of samples for each storage condition Fresh Tissue Thawed Tissue Group Storage temperature -20 °C -80 °C 18 I 3 3 II 3 3 HI 3 3 After two months, the samples were thawed at 4 °C for 24 hours and rinsed in isotonic saline solution at body temperature for one minute. The same mechanical testing procedure and conditions were then repeated. 66 4.2.2. Statistical Analysis Medians and ranges (percentile between 75% — 25%) were calculated for circumferential and axial force-stretch curves. Thawed/Fresh force ratio ( FinTmhxawed P„F'X'h ) at 1.55 stretch ratio were compared between all groups using the ANOVA Krustal-Wallis statistic test for independent variables with a 0.05 level of significance ( p = 0.05 ). The p value quantifies the probability of concluding that the storing condition used has an effect on the mechanical properties when in reality it did not. Krustal-Wallis test (non-parametric) was selected because the small sample size does not allow verifying the hypothesis of normal distribution, which is required to perform a Student's t-test. A statistical power analysis was also used to know the probability of falsely reject the null hypothesis using a threshold of 80%. 4.3. RESULTS 4.3.1. Experimental Results The medians of the force-stretch curves in the circumferential and axial directions for fresh specimens (n =18) are plotted in Figure 4.2. The difference in mechanical behavior below a stretch ratio of 1.5 is small. Force-Stretch Curve Median & Percentile (75% -25%) - Circumferential - - - - Axial 12 as o 8 IL 1.2^1.3^1.4^1.5 ^ 1.6 Stretch Ratio Fig. 4.2. Medians of the force-stretch behavior of fresh cruciform samples tested at 110 %/s ( n = 18). Open triangles and closed circles represent axial and circumferential mean forces, respectively. The percentile of the data points is also shown. Fresh - {3--- Frozen (-20T)/ Thawed Frozen (-80°C)/ Thawed 1.1^1.2^1.3^1.4^1.5^1.6 Stretch R atio 67 The medians of the force-stretch curves for the samples in group I, II and III are presented in Figures 4.3, 4.4 and 4.5 respectively. In groups I, II and III, preserved samples appeared to be stiffer than fresh samples in the circumferential direction, but these differences were not statistically significant. Preserved sample forces in the axial direction appeared to be similar to fresh samples. (a) Group I Fresh - Frozen (-20'C) / Thawed ----A--- Frozen (-80"C)/ Thawed 20 w 16 w • 12 8 • 4 0 (b) 1.0^1.1^1.2^1.3^1.4^1.5 ^ 1.6 Stretch Ratio Group I Fig. 4.3. Medians of cruciform samples tested at a deformation rate of 110 %/s, and stored in saline solution (n = 3 ): Arterial wall behavior in (a) circumferential and (b) axial directions. 1.2^1.3^1.4 Stretch Ratio 0 ^ 1.0 1.1 1.5 1.6 4 - (a) Group II 68 20 - 16 12 - co- 8 - E - 4 - U c-) • Fresh Frozen (-20°C) / Thawed Frozen (-80°C) / Thawed 0 I^I^I^I 1.0^1.1^1.2^1.3^1.4^1.5^1.6 Stretch Ratio (b) Group II 20 - - Fresh 16 - - {3- Frozen (-20°C) / Thawed Frozen (-80°C) / Thawed Fig. 4.4. Medians of cruciform samples tested at a deformation rate of 110 %/s, and stored in Krebs solution with dimethyl sulfoxide, DMSO ( n = 3 ): Arterial wall behavior in (a) circumferential and (b) axial directions. 20 --•-- Fresh Frozen (-20'C) / Thawed Frozen (-80°C) / Thawed 1.2^1.3^1.4^1.5^1.6 Stretch Ratio 1.1 (b) ---4-- Fresh Frozen (-20°C) / Thawed ----A--- Frozen (-80°C) / Thawed (a) Group III Group III 1.0 ^ 1.1 ^ 1.2^1.3^1.4^1.5^1.6 Stretch Ratio Fig. 4.5. Medians of cruciform samples tested at a deformation rate of 110 %/s, dipped in liquid nitrogen and stored in air ( n = 3 ): Arterial wall behavior in (a) circumferential and (b) axial directions. 69 -ow 1.4 o 1 2 p = 0.64 ^ Median ^ 25%-75% I Min-Max 1 0 0.8 0.6 0.4 0.2 II^III Group 3.0 2.8 2.6 2.4 2.2 0 Gs 2 0 70 ha /FMS"Figures 4.6.a and 4.6.b shows the Thawed/Fresh ratios (FTax wed^) of themaxm circumferential forces measured at the maximum stretch ratio (2 =1.55) for all the samples stored at -20 and -80 °C, respectively. The loading forces in the circumferential direction of all thawed samples (Groups, I, II and III) were almost two times higher than the loading forces of fresh samples (median of the ratio FmTahxawedFmFla:sh a 2.0 ). On the other hand, difference in axial forces between fresh and thawed tissue was lower than 40% for all groups (Figures 4.6.c and 4.6.d shows axial results at -20 and -80 °C, respectively). The medians, percentiles (75% — 25%) and the ANOVA Krustal-Wallis test p values were also included in Figure 4.6. (a)^ Biaxial Cruciform: Circumferetial Data (-20 °C) (b) Biaxial Cruciform: Circumferetial Data (-80 °C) 71 III11 Group Biaxial Cruciform: Axial Data (-20 °C) p = 0.64 o Median ^ 25%-75% I Min-Max 0.4 (c) 2.6 2.4 2.2 2.0 0 74 1 8 1 0 0.8 0.6 2.0 1.8 1.6 0.8 0.6 0.4 III P =— 0 64 ^ Median ^ 25%-75% I Min-Max II Group 72 (d)^ Biaxial Cruciform: Axial Data (-80 °C) 1 8 1 6 0 1 4 0 0 0.8 0.6 0.4 II Group Fig. 4.6. Comparison of Thawed/Fresh force ratios per storage group at the maximum stretch ratio applied ( =1.55 ): (a, b) circumferential and (c, d) axial direction FmThawed I p max Fresh ax^ratios of samples/ A stored at -20 and -80 °C. 4.4. DISCUSSION The influence of freezing in different solutions on the arterial wall mechanical properties was investigated in this study. Results were reported in terms of force vs. stretch ratio rather than stress vs. stretch ratio. This is due to the fact that stress and strain distributions of a cruciform sample subjected to biaxial extension experiment are not homogeneous, even when an equibiaxial stretch is applied. It was shown in a previous study (Chapter 3 of this thesis; Virues-Delgadillo, et al., 2007) that the highest and lowest stresses can be found near the curved boundaries and near the center of the sample, respectively. The differences observed in the mechanical behavior (i.e. medians of force data points) of thawed samples were not significant (p =0.64), independently of the storing medium used (saline, Krebs with DMSO and dipping in liquid nitrogen). In order to III p = 0.64 0 Median ^ 25%-75% Min-Max 73 confirm that indeed there is not differences in fresh and thawed behavior; a statistical power analysis (http://www.dssresearch.com/toolkit) was performed using the force data points obtained at 1.55 stretch ratio for each group between the fresh and the thawed specimens. The probability of rejecting a false null hypothesis (i.e. the storing condition used has no effect on the mechanical properties when in reality it might have an effect) and thus minimizing the occurrence of a /3 error (which occurs if it is conclude that there is not difference in fresh and thawed specimen behavior when in reality there might be a difference) increases as the statistical power increases. Statistical powers between 13% and 93% were obtained in circumferential direction. In axial direction, the power was lower than 47% and higher than 5%. The statistical power obtained for mostly all groups was not high enough to verify how significant the difference was between thawed and fresh tissue. In addition, the software used to obtain the power assumes that the data is parametric. Non- parametric analysis will likely estimate a bit less power. Future studies with a higher sample size may further support the results obtained in this work. The observed lack of significant differences in this study was likely due to the small number of samples tested per storing group (n = 3 ). However, these results are consistent with findings reported by Adham et al. (1996) and by Venkatasubramanian et al. (2006), who reported no significant difference between mechanical behavior of fresh artery samples and of samples cryopreserved with a cryoprotective agent such as DMSO, with up to 13 samples in each group. The thawed samples appeared to be stiffer than the fresh samples at high stretch ratios. The differences in the load force-stretch curves might be the result of a modification in structure due to crosslinking or a change in fiber alignment. Elder et al. (2005) stated that the increased tissue stiffness after cryopreservation may be related to the thermal change (i.e. drop of temperature) that occurs during preservation, catalyzing the thermal crosslinking of collagen fibers within the extracellular matrix. Collagen fibers are mainly oriented in the circumferential direction (Humphrey, 1995), thus the collagen matrix would develop stiffer interconnections in the circumferential than in axial direction during freezing. After thawing, the collagen matrix might have been reinforced in the circumferential direction, which is reflected in the stiffer response observed. 74 In this study the effect of freezing on cell injury and functionality was not investigated. However previous studies (Song, et al., 1994; Ku, et al., 1990; Bateson and Pegg, 1994) have shown that contractile function of both endothelial and smooth muscle cells were preserved in arteries of a variety of species following freezing and thawing in DMSO solutions. DMSO reduces the mass transport of water and solutes throughout cell membrane while freezing. When water moves out of the cytoplasm, DMSO dissolves the suspended electrolytes, reducing the harmful effects of high solute concentration. DMSO interacts and partially replaces water molecules in such a way that the freezing point in the solution is lowered during cooling and the intracellular ice is reduced. Ice crystal formation and growth is prevented when the cryoprotectant-water mixture solidifies in a glass-like structure; and thus, preserving cell viability and mechanical properties. Song et al. (1995) found that the maximum concentration needed to prevent damage in the tissue is 15% (wt/wt). In the present study, the DMSO concentration was 1.8 M, i.e. < 12% wt/wt. This concentration is expected to maintain cell viability within the tissue. Evidence has shown that after thawing, both biochemical and functional activities of arterial tissue cryopreserved at low temperatures in Krebs solution containing 1.8 M of DMSO were comparable to fresh tissues (Ellis and Muller-Schweinitzer, 1991). Freezing and storing arteries in saline solution do not conserve the mechanical properties. As described by Lovelock (1953), during the freezing process, the volume of liquid water in the cytoplasm decreases due to extracellular ice formation, leading to cellular dehydration. If the dehydration is too severe, the high electrolyte concentration inside the cytoplasm could result in cell death (Mazur, et al, 1981). DMSO reduces the mass transport of water and solutes through the cell membrane while freezing. Ice crystal formation and growth is prevented when the cryoprotectant-water mixture solidifies in a glass-like structure, thus preserving cell viability and mechanical properties. The concentration of DMSO used in this study is expected to maintain cell viability within the tissue, albeit this study was not focused on freezing effects on cell viability. The better results obtained with DMSO suggest that cell death might play a role in changes of tissue stiffness, although this study could not unveil the exact mechanism involved. Also, during thawing, the deformation of the cytoplasm due to volume increase could have permanently modified the orientation of the intracellular fibers, and as a result permanently changed the 75 mechanical behaviour of the cells. Therefore our results suggest that, as concluded by Venkatasubramanian et al. (2006), the changes in mechanical properties could be due to cell loss, to damage to the extracellular matrix or to a combination of both. Stachecki et al. (1998) suggested that the reduction or removal of sodium from the storing solution is of primary importance to freeze cells efficiently, and proved that is possible to replace sodium with a bigger molecular size ion, which encounters more obstacles to penetrate the cell membrane. Dipping the arterial sample in liquid nitrogen could not preserve the mechanical properties of the specimens evaluated in this study. When exposing the specimens to a very rapid decreasing temperature, ice crystal formation might have occurred faster than cell dehydratation, resulting in intracellular ice formation (Toner, et al., 1990) and cell death. The differences observed between fresh and thawed samples suggest that the formation of intracellular ice crystals might have and effect on tissue stiffness. Vitrification is an alternative to obtain an amorphous glassy state matrix and thus minimizing ice nucleation and growth. However, it requires high concentrations of cryoprotective agents (Fahy, et al., 1984; Thakrar, et al., 2006). For example, Song et al. (2000) were able to avoid crystallization by using a combination of CPAs (DMSO, formamide and 1,2-propanediol) and rapidly cooling the vessel to -196 °C. Jimenez Rios and Rabin (2006) have improved cell viability by using a specific freezing rate, pressurized liquid nitrogen, and specific type and concentration of CPA. Non-toxic cryoprotectants, like glyerol (Gao, et al., 1995) or fetal bovine serum (Yiu, et al., 2007) could also be used. 4.5. SUMMARY In the present study, we have found that the mechanical properties of arteries were not significantly changed after preservation of arteries for two months independently of the storing medium (saline, Krebs with DMSO and dipping in liquid nitrogen) or freezing temperature (-20 °C or -80 °C) used. The lack of significant differences observed in this study was likely due to the small number of samples tested per storing group (n = 3 ). Further studies are required to clarify the impact of cryopreservation on extracellular matrix architecture to help tailor an optimized approach to preserve the mechanical properties of arteries. 76 4.6. REFERENCES ADHAM, M., GOURNIER, J. P., FAVRE, J. P., DE LA ROCHE, E., DUCERF, C., BAULIEUX, J., BARRAL, X., POUYET, M., 1996. Mechanical characteristics of fresh and frozen human descending thoracic aorta. Journal of Surgical Research, 64, 32 — 34. ALMASSI, G. H., FARAHBAKHSH, B., WOOLDRIBGE, T., RUSCH, N. J., AND OLINGER, G. N., 1996. Endothelium and vascular smooth muscle function in internal mammary artery after cryopreservation. Journal of Surgical Research, 60, 355 — 360. BATESON E. A. J., PEGG, D. E., 1994. Cryopreservation of arteries: Selection of a model for human small elastic arteries and preliminary results of preservation of ring-segments with dimethyl sulphoxide. CryoLetters, 15, 15 — 26. BLONDEL W. C. P. M., LEHALLE, B., MAURICE, G., WANG, X., STOLTZ, J. F., 2000. Rheological properties of fresh and cryopreserved human arteries tested in vitro. Rheol Acta, 39, 461 — 468. BROCKBANK K. G. M., 1989. Basic principles of viable tissue preservation. In: Clarke D.R. (ed) Transplantation techniques and use of cryopreserved allograft cardiac valves and vascular tissue. Adams Publishing Group, Boston, 9 — 23. BUJAN, J., PASCUAL, G., GARCIA-HONDUVILLA, N., GIMENO, M. J., JURADO, F., CARRERA-SAN, M. A., AND BELLON, J. M., 2000. Rapid thawing increases the fragility of the cryopreserved arterial wall. Eur. J. Vasc. Endovasc. Surg, 20, 13 — 20. CARMINES, D. V., McELHANEY, J. H., AND SNACK, R., 1991. A piece-wise non- linear elastic stress expression of human and pig coronary arteries tested in vitro. J. Biomech., 24, 899 — 906. COOPER, I. S., SAMRA, K., WISNIEWSKA, K., 1971. Effects of freezing on mayor arteries. Stroke, 2, 471 — 482. CUI, X., LABARRERE, C., HE, L., CHENG, S.,SIDERYS, H., KOVACS, R., GAO, D., 2002. Cryopreservation and microsurgical implantation of rabbit carotid arteries. Cell Preservation Technology, 1(2), 121 — 128. DEVIREDDY, R. V., NEIDERT, M. R., BISCHOF, J. C., TRANQUILLO, R. T., 2003. Cryopreservation of collagen-based tissue equivalents. I. Effect of freezing in the absence of cryoprotective agents. Tissue Engineering, 9(6), 1089 — 1100. ELDER, E., CHEN, Z., ENSLEY, A., NEREM, R., BROCKBANK, K., SONG, Y., 2005. Enhanced tissue strength in cryopreserved, collagen-based blood vessel constructs, Transplantation Proceedings, 37, 4525 — 4629. 77 ELLIS, P., AND MULLER-SCHWEINITZER, E., 1991. Maintenance of functional activity of human pulmonary arteries after cryopreservation. Br. J. Pharmacol., 103(2), 1377 — 1380. FAHY, G. M., MACFARLANE, D. R., ANGELL, C. A., MERYMAN, H. T., 1984. Vitrification as an approach to cryopreservation. Cryobiology, 21, 407 — 426. GAMERO, L. G., ARMENTANO, R. L., BARRA, J. G., SIMON, A., LEVENSON, J., 2001. Identification of arterial wall dynamics in conscious dogs. Exp. Physiol, 86, 519 — 528. GAO, D. Y., LIN, S., WATSON, P. F., AND CRITSER, J. K., 1995. Fracture phenomena in an isotonic salt solution during freezing and their elimination using glycerol. Cryobiology, 32, 270 — 284. HAN, B., AND BISCHOF, J. C., 2004. Engineering challenges in tissue preservation. Cell Preservation Technology, 2(2), 91 — 112. HUDETZ, G., MARK, G., KOVACH, A. G. B., KERENYI, T., FODY, L., AND MONOS, E., 1981. Biomechanical properties of normal and fibrosclerotic human cerebral arteries. Atherosclerosis, 39, 353 — 365. HUMPHREY, J. D., 1995. Mechanics of the arterial wall: Review and directions. Critical Reviews in Biomedical Engineering, 23, 1 — 162. HUNT, C. J., SONG, Y. C., BATESON, E. A. J., PEGG, D. E., 1994. Fractures in cryopreserved arteries. Cryobiology, 31, 506 — 515. INTENGAN, H. D., SCHIFFRIN, E. L., 2000. Structure and mechanical properties of resistance arteries in hypertension. Role of adhesion molecules and extracellular matrix determinants. Hypertension, 36, 312 — 318. JIMENEZ RIOS, J. L., RABIN, Y., 2006. Termal expansion of blood vessels in low cryogenic temperatures Part I: A new experimental device. Cryobiology, 52, 269 — 283. KU, D. D., WILLIS, W. L., CAULFIELD, J. B., 1990. Retention of endothelium- dependent vasodilatory responses in canine coronary arteries following cryopreservation. Cryobiology, 27, 511 — 520. LALLY, C., REID, A. J., AND PRENDERGAST, P. J., 2004. Elastic behavior of porcine coronary artery tissue under uniaxial and equibiaxial tension. Ann. Biomed. Eng., 32(10), 1355 —1364. LANGEWOUTERS, G. J., WESSELING, K. H., AND GOEDHARD, W. J. A., 1984. The static elastic properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model. J. Biomech., 6, 425 — 435. 78 LOVELOCK, J. E., 1953. The haemolysis of human red blood-cells by freezing and thawing. Biochim. Biophys. Acta, 10(3), 414 — 426. MAZUR, P., RALL, W. F., RIGOPOULOS, N., 1981. Relative contributions of the fraction of unfrozen water and of salt concentration to the survival of slowly frozen human erythrocytes. Biophysical Journal, 36, 653 — 675. MILES, C. A., 1999. Polymer-in-a-box mechanism for thermal stabilization of collagen molecules in fibers. Biophys. J., 76, 3243 — 3252. MOHAN, D., MELVIN, J. W., 1983. Failure properties of passive human aortic tissue. II Biaxial tension tests. Journal of Biomechanics, 16, 31 — 44. NEIDERT, M. R., DEVIREDDY, R. V., TRANQUILLO, R. T., BISCHOF, J. C., 2004. Cryopreservation of collagen-based tissue equivalents. II. Improved freezing in the presence of cryoprotective agents. Tissue Engineering, 10(1/2), 23 — 31. OKAMOTO, R. J., WAGENSEIL, J. E., DELONG, W. R., PETERSON, S. J., KOUCHOUKOS, N. T., AND SUNDT III, T. M., 2002. Mechanical properties of dilated human ascending aorta. Ann. Biomed. Eng., 30(5), 624 — 635. PACHOLEWICZ, J. K., ADKINS, M. S., BORIS, W. J., GU, J., XENACHIS, C., KLABUNDE, R. E., JASIONOWSKI, T., MCGRATH, L. B., 1996. Evaluation of cryopreserved internal thoracic artery as an alternative coronary graft: Evidence for preserved fuctional, metabolic and structural integrity. Eur. J. Cardio-Thorac. Surg., 10, 20 — 25. PASCUAL, G., GARCiA-HONDUVILLA, N., RODRiGUEZ, M., TUREGANO, F., BUJAN, J., BELLON, M., 2001. Effect of the thawing process on cryopreserved arteries. Annals of Vascular Surgery, 15(6), 619 — 627. PASCUAL, G., JURADO, F., RODRiGUEZ, M., CORRALES, C., LOPEZ-HERVAS, P., BELLON, J. M., AND BUJAN, J., 2002. The use of ischaemic vessels as prostheses or tissue engineering scaffolds after cryopreservation, Eur J. Vasc. Endovasc. Surg., 24, 23 — 30. PRENDERGAST, P. J., LALLY, C., DALY, S., REID, A. J., LEE, T. C., QUINN, D., AND DOLAN, F., 2003. Analysis of prolapse in cardiovascular stents: A constitutive equation for vascular tissue and finite-element modelling. Journal of Biomechanical Engineering, 125, 692 — 699. RAGHAVAN, M. L., WEBSTER, M. W., AND VORP, D. A., 1996. Ex vivo biomechanical behavior of abdominal aortic aneurysm: assessment using a new mathematical model. Ann. Biomed. Eng., 24, 573 — 582. SCHENKE-LAYLAND, K., MADERSHAHIAN, N., RIEMANN, I., STARCHER, B., HALBHUBER, K. J., KONIG, K., AND STOCK, U. A., 2006. Impact of cryopreservation 79 on extracellular matrix structures of heart valve leaflets. The Annals of Thoracic Surgery, 81,918-926. SCHULZE-BAUER, CH. A. J., MORTH, CH., HOLZAPFEL, G. A., 2003. Passive biaxial mechanics response of aged human iliac arteries. Journal of Biomechanical Engineering, 125, 395 — 406. SONG, Y. C., KHIRABADI, B. S., LIGHTFOOT, F., BROCKBANK, K. G. M., TAYLOR, M. J., 2000. Vitreous cryopreservation maintains the function of vascular grafts. Nature Biotechnology, 18(3), 296 — 299. SONG, Y. C., HUNT, C. J., PEGG, D. E., 1994. Cryopreservation of the common carotid artery of the rabbit. Cryobiology, 31, 317 — 329. SONG, Y. C., PEGG, D. E., HUNT, C. J., 1995. Cryopreservation of the common carotid artery of the rabbit: Optimization of dimethyl sulfoxide concentration and cooling rate. Cryobiology, 32, 405 — 421. STACHECKI, J. J., COHEN, J., WILLADSEN, S., 1998. Detrimental effects of sodium during mouse oocyte cryopreservation. Biology of Reproduction, 59, pp. 395 — 400. THAKRAR, R. R., PATEL, V. P., HAMILTON, G., FULLER, B. J., SEIFALIAN, A. M., 2006. Vitreous cryopreservation maintains the viscoelastic property of human vascular grafts. The FASEB Journal, 20(7), 874 — 881. TONER, M., CRAVALHO, E. G., AND KAREL, M., 1990. Thermodynamics and kinetics of intracellular ice formation during freezing of biological cells. J. Appl. Physics, 67, 1582 — 1593. VAN ANDEL, C. J., PISTECKY, P. V., BORST, C., 2003. Mechanical properties of porcine and human arteries: Implications for coronary anastomotic connectors. Ann Thorac Surg, 76, 58 — 65. VANDE GEEST, J. P., SACKS, M. S., VORP, D. A., 2004. Age dependency of the biaxial biomechanical behavior of human abdominal aorta. Journal of Biomechanical Engineering, 126, 815 — 822. VENKATASUBRAMANIAN, R. T., GRASSL, E. D., BAROCAS, V. H., LAFONTAINE, D., BISCHOF, J. C., 2006. Effects of freezing and cryopreservation on the mechanical properties of arteries. Annals of Biomedical Engineering, 34(5), 823 — 832. VIRUES-DELGADILLO, J. 0., DELORME, S., DIRADDO, R., HATZIKIRIAKOS, S. G., THIBAULT, F., 2007. Mechanical characterization of arterial wall: Should cruciform or square sample be used in biaxial testing? Journal of the Mechanical Behavior of Biomedical Materials, submitted. 80 WALDMAN, S. D., AND LEE, J. M., 2002. Boundary conditions during biaxial testing of planar connective tissues: Part 1: Dynamic behavior. Journal of Materials Science. Materials in Medicine, 13, 933 — 938. WUSTEMAN, M. C., PEGG, D. E., 2001. Differences in the requirements for cryopreservation of porcine aortic smooth muscle and endothelial cells. Tissue Eng., 7(5), 507 — 518. YIU, W. K., CHENG, S. W. K., AND SUMPIO, B. E., 2007. Direct comparison of endotelial cell and smooth muscle call response to supercooling and rewarming. Journal of Vascular Surgery, 46, 557 — 564. 81 CHAPTER 5 5. EFFECT OF DEFORMATION RATE ON THE MECHANICAL PROPERTIES OF ARTERIAL SAMPLES 5.1. INTRODUCTION The knowledge of the viscoelastic properties is important to predict the biomechanical behaviour of soft tissues. To model their viscoelastic behaviour, first one performs appropriate mechanical tests to characterize deformation-rate effects, and then one selects a constitutive equation capable of representing those effects. Material parameter estimation is fundamental for posterior simulation of soft tissue at boundary conditions not selected in the experimental protocol. The effect of deformation rate on the mechanical properties of soft biological tissues has been investigated, in particular for ligaments (Thornton, et al., 2007a; Thornton, et al., 2007b; Provenzano, et al. 2001; Pioletti and Rakotomanana, 2000; Pioletti, et al., 1999; Kwan, et al., 1993), tendons (Pioletti and Rakotomanana, 2000; Haut., 1983; Haut and Little, 1972), spines (Gay, et al., 2008; Wang, et al., 1999; Yingling, et al., 1997; Tran, et al., 1995), bones (De Smet, et al., 2007; Ebacher, et al., 2007; Hsieh, et al., 1999; Schaffler, et al., 1989), liver (Hu and Desai, 2004), heart valves (Doehring, et al., 2004; Wells and Sacks, 2002) and myocardium (Demer and Yin, 1983; Giles, et al., 2007). Most biological tissues stiffen with increasing deformation rate (Haut and Little, 1972; Pioletti and Rakotomanana, 2000; Pioletti et al., 1999; Wang, et al., 1999; Yingling, et al., 1997; Tran, et al., 1995; Schaffler, et al., 1989). This time-dependent behavior has been described by viscoelastic constitutive models (Giles, et al., 2007; Nekouzadeh, et al., 2007; Pioletti, et al. 1998; Sarver, et al., 2003; Vena, et al., 2006; Zhang, et al., 2007). However, it was recently demonstrated that some biological tissues, such as liver, myocardium and skin, soften with increasing deformation rate (Giles, et al., 2007; Hu and Desai, 2004). Deformation rate effects of arteries, in particular thoracic aorta, were not included in previous studies. Overstretch injury to the arterial wall during an angioplasty or stenting procedure has been shown to be correlated to the incidence of restenosis, i.e. in-growing tissue re- blocking the artery lumen (Ellis and Muller, 1992; Schwartz, et al., 1992). Based on the hypothesis that lower deformation rate results in lower intramural stresses, slow balloon • A version of this chapter has been submitted for publication to the Journal of the Mechanical Behavior of Biomedical Materials (Virues-Delgadillo, J. 0., Delorme, S., Diraddo, R., Hatzikiriakos, S. G., Mora, V., 2007, Effect of Deformation Rate on the Mechanical Properties of Arteries) 82 inflation has been proposed as a means to minimize vascular injury and reduce restenosis incidence (Tenaglia, et al., 1992). Early studies did not conclude there was any difference in restenosis rates between conventional and slow balloon inflation (Miketic, et al., 1998; Tenaglia, et al., 1992; Timmis, et al., 1999) while some observed better immediate results (Eltchaninoff, et al., 1996; Ohman, et. al., 1994). In more recent studies, significantly lower restenosis rates were clinically observed with slow balloon inflation (Umeda, et al., 2004; Weiss, et al., 2007). Slow stent deployment has also been proposed as a means to minimize arterial injury (Theriault, et al., 2006). Finite element simulation of angioplasty and stenting can be used to optimize angioplasty procedure parameters, such as inflation pressure (Gasser and Holzapfel, 2007; Holzapfel, et al., 2002; Lally, et al., 2005; Liang, et al., 2005; Oh, et al., 1994; Wu, et al., 2007). Optimization of inflation pressure rate requires accurate constitutive modeling of artery behavior including the effect of deformation rate. Numerous experimental studies have been performed to characterize the mechanical behaviour of arteries in vitro (Lally, et al., 2004; Okamoto, et al., 2002; Prendergast, et al., 2003; Waldman and Lee, 2002). However, only a single deformation rate was used. The objective of this study is thus to measure and model the effect of deformation rate on the tensile behavior of the arteries, namely thoracic aortas from pigs. 5.2. MATERIALS AND METHODS 5.2.1. Experimental Setup Five thoracic aortas were harvested within the day of death of pigs from a local slaughterhouse and cleaned of remaining connective tissue. Then each artery was cut open along its length, and cut out in rectangular and cruciform-shaped specimens. The thickness of all specimens was measured with a vernier caliper (mean 2.4 mm, standard deviation 0.2 mm). Twelve samples were obtained from the five aortas: 8 rectangular samples (4 were cut in circumferential direction and 4 in axial direction) and 4 cruciform samples. Samples were stored in isotonic saline solution at 4 °C for no longer than 24 hours prior to testing. The methodology has been detailed in a previous paper (Chapter 3 of this thesis; Virues-Delgadillo, et al., 2007) and will be summarized here. Briefly, rectangular and cruciform samples were used for uniaxial and biaxial testing, respectively. Rectangular 83 samples were 40 mm long and 4 mm wide, but only 20 mm of the sample were free to deform. The distance between grips in cruciform samples was 40 mm. A nominal stretch ratio of 1.5 was applied to avoid permanent deformation on the tissue. Uniaxial and biaxial testing was carried on a planar biaxial test bench (ElectroForce® LM1, Bose Corporation, Minnetonka, MN) capable of applying a peak force of 200 N over a displacement range of 12 mm per actuator. A saline bath maintained at body temperature (37 °C) was used. Samples were mounted in horizontal configuration with the help of grip clamps to the arms extending from the actuators over the top of the bath. Each sample was subjected to triangular wave form displacements of 1.5 stretch ratio of amplitude, and deformation rates of 10, 50, 100, 120, 140, 160, 180 and 200 %/s, which correspond to frequencies of 0.1, 0.5, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0 Hz. Half of the samples were subjected to deformation rates in the following order: 160, 120, 50, 200, 140, 100, 10 and 180 %/s (randomly tested), and the other half were tested from 10 to 200 %/s in ascending order in order to be confident that the deformation rate effect observed do not depend on the testing procedure order. In vivo, the artery is constantly submitted to stresses while the tissue is inflated-deflated due to blood pressure. In order to mimic in vivo loading conditions during in vitro tests, pre-conditioning cycles are required. Each test lasted for ten cycles. The first five cycles were used to pre-condition the tissue in order to reach a steady- state behavior. The force-stretch data was averaged over the last five cycles. 5.2.2. Statistical Analysis The experimental data obtained in uniaxial and biaxial testing are represented in mostly all figures in this chapter as the mean value ± the standard errors (i.e. Fig. 5.3). The medians rather than the means are plotted only in Fig. 5.6. As shown in Table 5.1, twenty loading-unloading cycles were used to obtain the circumferential and axial force-stretch mean curves per deformation rate and testing condition. Circumferential and axial forces at 1.5 stretch ratio from both uniaxial and biaxial tests were used to calculate the medians, ranges (percentile between 75% — 25%) and p values with the help of the statistical tests selected. A multiple comparison between forces belonging to all groups (i.e. experimental forces obtained at each deformation rate belong to one group) was performed to study the 84 significance of the effect of deformation rate using the ANOVA Friedman statistic test for dependent variables. The Wilcoxon test was also used to compare all deformation rate forces by pairs (i.e. forces at the highest deformation rate, 200%/s, with forces at the lowest deformation rate, 10%/s). In order to identify if the effect of deformation rate is more significant than the effect of anisotropy, circumferential and axial forces at each deformation rate were also compared using the Wilcoxon test. A 0.05 level of significance ( p = 0.05) were selected for both, the Friedman and Wilcoxon tests. Table 5.1. Number of force-stretch experimental data averaged per deformation rate Testing Condition Specimen Direction Number of Specimens Number of Cycles per Specimen Total Number of Cycles Uniaxial Circumferential 4 5 20Axial 4 5 20 Biaxial 4 5 20 5.2.3. Inverse Modeling An inverse modeling technique was used to adjust the parameters of the proposed constitutive model (described below in the results section). A detail description of the technique can be found elsewhere (Chapter 3 of this thesis; Virues-Delgadillo, et al., 2007). Briefly, the reaction forces at the boundaries (grips) were predicted by finite element simulation of the experiment consisting of applying displacement conditions on a mesh of the same size and shape as the sample. In an optimization loop, the material properties were iteratively adjusted until the following objective function was minimized, which occurs when a set of force-displacement experimental data matches the values calculated by the model: S (C) = S (C) Biaxial ± 5'^Uniaxial ^ (5.1) The objective function to be minimized is the sum of uniaxial and biaxial cruciform objective functions. N r S(C)Biaxial = E [( x (dx )— fx (dx ,c))2 +(Fy (dy )— fy (dy ,c)yi^(5.2) where c is the vector of unknown material properties; dx and d, are the applied displacements; Fx (dx ) and Fy (dy ) are the experimentally measured reaction forces at the grips; and fx (dx ,c) and fy (dy ,c) are the reaction forces predicted by the finite element 85 model in biaxial cruciform extension; N is the total number of data points gathered in the experiments. N r S(c)untax,./ Et(F(d)-f (d 0)2 1 i =1 Uniaxial extension predicted forces (f(d,c)) were fitted to the average of circumferential and axial reaction forces ( F (d)). d is the applied displacement in uniaxial extension. In contrast with the method described in Chapter 3 and in Virues-Delgadillo et al., (2007), here a combined uniaxial-biaxial mesh was used to simultaneously simulate uniaxial and biaxial testing conditions (Fig. 5.1). Only one quarter of the rectangular and the cruciform sample area free to deform between the grips was meshed due to symmetry. The uniaxial and biaxial cruciform mesh-sections are linked together at the origin. Boundary conditions are shown in Fig. 5.1. In some cases, a small stretch shift was applied to the experimental results to obtain better agreement between uniaxial and biaxial fits. #.°'12. vAvAmmmmu Av vArAimunVfil P2,2911rArigiU° OAmen ... -irA aVAKir.MgearAvAIP OpRtmea,AirAlvAnrAVAra lpre44404 tiltOr AntrATtk7A7A7A74. APO 41000 1-24:44AVOYAnnt4110- $121,-,02101PAO* °."0/4 Sy* 1011" itir1/40 04120 41.1101ral■ ArtAll• 441 .4190 1,0111■411,- Eiroar 4,147 0wit1P-VraUff-darerd tirtA 40120b100AVAVq■ "NOritalpik"411-WPAVA4k. 0r4101411k raittleatIONOOVAVA4-pi iri -IF ,41 rA■PgurAIP YS-1214 0 4704,000,40OP A N. -*AUT.p200%**441,04.00-41% f Vit 4411014 moi,IA WM-Iworai&vivAU■ 0WAVAZAVATATtiwe LVAYAk d Fig. 5.1. Mesh and boundary conditions used in simultaneous simulation of uniaxial and biaxial testing. (5.3) 86 As a first approach, all the material parameters of the constitutive model selected were allowed to vary; then, the parameters that do not appear to diverge (i.e. lower slope of the linear regression calculated with each fitted parameter at all deformation rates) were kept constant in the subsequent simulations. Other fits were also performed assuming that some material parameters can take values equal to zero in order to see if the model can be reduced to a simpler form (section 5.3.3 summarizes the modeling results). 5.3. RESULTS 5.3.1. Experimental Results Figure 5.2 shows cycle-to-cycle variability between cycles 6 and 10, for a typical sample tested biaxially at a deformation rate of 100%/s. Cycle-to-cycle variability was observed to be small in all cases. Biaxial Cruciform Raw Data, 100%/s Stretch Ratio Fig. 5.2. Loading-unloading circumferential force-stretch cycles of a typical biaxial cruciform sample: Five steady-state cycles at a deformation rate of 100%/s. Figure 5.3 shows the standard error bars (sample-to-sample variability) of circumferential and axial forces at the lowest (10 %/s) and highest (200 %/s) deformation rates. Similar standard errors were obtained with the other deformation rates. 6 - 87 (a)^ Experimental Data 7 z 6 - a) 5-cou. $4-- • 3 _ 2- m • 1 - U o 10 %/s ^ 200 %/s Biaxial^L,j 8a a 8 — UniaxialitO Ea^Ett0 1.4^1.5 Stretch Ratio (b)^ Experimental Data 7- 1.3 o 10 %/s ^ 200 %Is Biaxial ---.... ,7s, Ill g a^,t^a^8 O o^0^12,^ti^8^'s- Uniaxial I 1 1^ I 1.2^1.3^1.4^1.5 I0 1 Stretch Ratio Fig. 5.3. Sample to sample variability of thoracic aorta. Open diamonds and squares represent the mean force-stretch behavior at 10 and 200 %/s, respectively ( n = 4 ): (a) circumferential, and (b) axial directions. Standard error bars are also shown. o 10 %is ©50' %is A 100 %/s x 120 %is x 140 %/s — 160 %/s o 180 %/s + 200 %/s Increasing Deformation Rate z 4 0 C C •▪ 3 -a) E a Decreasing Force (20% Reduction) Biaxial 88 Figure 5.4 shows the effect of deformation rate on the circumferential behavior of thoracic aorta from uniaxial and biaxial testing. Lower forces were observed at higher than at lower deformation rates. In particular, a 20% lower force was observed at 1.5 stretch ratio with the lowest deformation rate (10%/s) compared to the highest deformation rate (200%/s). The same phenomenon was observed in the axial direction (Fig. 5.5). Experimental Data Iii 1^1.1^1.2^1.3 Stretch Ratio 1.4^1.5 Fig. 5.4. Effect of deformation rate on the force vs. stretch ratio curve of arteries: circumferential force vs. stretch ratio curves in uniaxial and biaxial testing. Experimental Data 89 0 %/s ^ 50 %/s A 100 %it X 120 %/s x 140 %is — 160 %/s o 180 %/s + 200 %/s Increasing Deformation Rate Decreasing Force (20% Reduction) Biaxial 0 u. 3 Uniaxial I 0^e 1.2^1.3 Stretch Ratio oi 1.4 1.5 Fig. 5.5. Effect of deformation rate on the force vs. stretch ratio curve of arteries: axial force vs. stretch ratio curves in uniaxial and biaxial testing. Figure 5.6 shows the median, percentiles (75% — 25%) and the ANOVA Friedman test p value obtained using the forces at 1.5 stretch ratio for each deformation rate. Significant differences (p < 0.05) were observed, with the highest significant differences under biaxial testing ( p 5 0.002 ). (a) 1 5 Effect of Deformation Rate: Uniaxial Data 1.4 1.3 z a) P. 1 2 0 U- 1 1 E) E 1.0 U 0.9 0.8 0.7 p = 0.02 ^ Median 0 25%-75% I Min-Max10^50^100^120^ 140^160^180 200 Deformation Rate (%/s) 90 10^50^100^120^140^160^180 200 Deformation Rate (%/s) p = 0.011 ^ Median ^ 25%-75% I Min-Max Effect of Deformation Rate: Uniaxial Data 1.2 1.1 0.8 0.7 0.6 0.5 (c) 12 Effect of Deformation Rate: Biaxial Data 1 1 10 z N 2 9aLL 8 E 7 U 6 5 4 p = 0.002 ^ Median ^ 25%-75% I Mm-Max10^50^100^120^140^160^180^200 (d) 8.0 Effect of Deformation Rate: Biaxial Data Deformation Rate (%/s) 7.5 7,0 z 6.5 U 0LL 6.0 5.5 5.0 4.5 10^50^100^120 140^160^180 200 Deformation Rate (%/s) Fig. 5.6. Deformation rate effect on (a, b) uniaxial and (c, d) biaxial forces at maximum stretch ratio (^1.5 ): (a, c) circumferential and (b, d) axial direction. p =0.001 ^ Median [71 25%-75% I Min-Max 91 6.5 - 4.0 - 3.5 6.0 - 5.5 a 5.0 - cu. 4.5 - Circumferential Force 92 Table 5.2 shows Wilcoxon test p values of uniaxial and biaxial data of some of the deformation rate pairs analyzed (i.e. comparison of the force at the maximum deformation rate with the forces at other deformation rates). Marginally significant differences were observed in the forces of mostly all deformation rates. Similar p values were obtained in the other comparisons (i.e. 50%/s vs. all deformation rates). Table 5.2. Effect of deformation rate: Wilcoxon test p values Force %/s 10 50 100 120 140 160 180 Uniaxial Circumferential 200 0.07 0.07 0.14 0.07 0.07 0.07 0.27 Axial 0.07 0.07 0.07 0.07 0.07 0.07 0.07 Biaxial Circumferential 0.07 0.07 0.07 0.07 0.07 0.07 0.72Axial 0.07 0.07 0.07 0.07 0.07 0.07 0.47 In Figure 5.7, circumferential and axial forces at 1.5 stretch ratio of a typical sample are plotted against deformation rate. In this figure, one can observe that the effect of deformation rate (20 % difference in force between smallest and highest deformation rate) is approximately twice as large as the effect of anisotropy (difference in force between circumferential and axial directions). This supports the hypothesis of modelling deformation-rate effects with an isotropic model. Effect of Deformation Rate vs Effect of Anisotropy: Biaxial Extension 0^50^100^150^200 Deformation Rate Fig. 5.7. Comparison of the effect of deformation rate versus the effect of anisotropy. Typical sample forces at maximum stretch ratio (1.5). Table 5.3 shows p values calculated using the Wilcoxon test comparing the circumferential force to the axial force at 1.5 stretch ratio per each deformation rate. 93 Table 5.3. Effect of Anisotropy (Circumferential vs. axial forces, n= 4 ): Wilcoxon test p values 10 %/s 50 %/s 100 %/s 120 %/s 140 %/s 160 %/s 180 %/s 200 %/s Uniaxial 0.47 0.14 0.27 0.14 0.27 0.14 0.14 0.14 Biaxial 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 5.3.2. Constitutive Modeling Based on the experimental results, we propose a new rate-dependant isotropic hyperelastic model, based on the Mooney-Rivlin model (Mooney, 1940) given by the following strain energy density function: 00 0/ 1 , / 2 )= Ea u (I i —3)`(/2^(5.4) The Mooney-Rivlin model is a function of the 1 st and 2nd invariants ( / 1 , /2 ) of the right Green-Cauchy tensor: ± 22 ± 223^ (5.5) /2 = 22 +AZ/g + 23A 12^(5.6) where A, , A.2 and A.3 are the principal stretch ratios. In its most common expression, only five parameters are selected alo ao1 1 a20 a30) and the strain energy function reduces to: W aw (11 3) + aw (12 3 ) + all (11 —3)(I2 3 ) + a20 (11 —3)2 + a30 (11 3) 3^(5.7) The underlying hypothesis for using the Mooney-Rivlin model is that one or some of the five parameters vary with deformation rate. This can be modelated by making these parameters functions of the deformation rate. Those relationships are explored in this work. 6.3.3. Inverse Modeling Results 6.3.3.1. Uniaxial & Biaxial Cruciform Fit (Simultaneous) Initially, all five Mooney-Rivlin parameters were allowed to vary while being bounded in the range of 10 -4 kPa to 104 kPa. Table 5.4 summarizes the fitted parameters obtained for the uniaxial and biaxial cruciform force vs. stretch ratio curve averaged for each deformation rate. In a second simulation only parameter a 30 was allowed to vary because this was the parameter with the highest slope (absolute value of B) in the linear regressions reported in 94 Table 5.4. The remaining parameters were maintained constant by using the average of the fitted values obtained for all deformation rates. The fitted parameters obtained are summarized in Table 5.5. Table 5.4. First simulation results (material parameters) of uniaxial and biaxial cruciform test M-R Fitted Parameter Values (kPa) SUM / nDeformation Rate (%/s) alo am an am a30 (n = 24) 10 9.25 7.57 5.77 6.94 4.43 2.69E-03 50 9.21 7.45 5.45 6.58 3.64 2.09E-03 100 8.90 7.23 5.51 6.66 3.86 1.76E-03 120 9.46 7.66 5.38 6.52 3.03 1.66E-03 140 9.96 8.07 5.30 6.45 2.36 3.45E-03 160 9.22 7.47 5.29 6.40 3.24 2.72E-03 180 9.07 7.32 5.34 6.49 2.95 2.80E-03 200 8.86 7.17 5.24 6.36 2.80 1.90E-03 Linear Regression Coefficients & R 2 aii =A + BA) A 9.32 7.60 5.69 6.85 4.27 B -7.00E-04 -9.00E-04 -2.40E-03 -2.50E-03 -8.20E-03 R2 0.01 0.04 0.80 0.78 0.65 Table 5.5. Second simulation results (material parameters) of uniaxial and biaxial cruciform tests M-R Fitted Parameter Values (kPa) SUM / nDeformation Rate (%/s) a10 a01 an azo a30 (n = 24) 10 4.99 2.64E-03 50 3.66 2.18E-03 100 3.74 1.88E-03 120 140 9.24 7.79 5.41 6.55 3.16 2.75 1.68E-03 3.56E-03 160 3.06 2.83E-03 180 2.72 2.98E-03 200 2.26 1.98E-03 Linear Regression Coefficients & R2 (ad = A + BA) A - - - - 4.76 B - - - - -1.22E-02 R2 - - - - 0.88 The sum of least squares (last column in Tables 5.4 and 5.5) were almost unchanged when the first four parameters were kept constant, supporting the hypothesis that only a30 varies with deformation rate. Other fits were performed assuming that an = a20 = 0 to see if the model could be reduced to a simpler form (Table 5.6). First, parameters a m, , a 01 and a30 were allowed to vary. The highest regression coefficient (absolute value of B) and the highest correlation 95 coefficient ( R 2 ) were obtained for a30 . Thus parameters a 10 and a 01 were assumed to be constant (i.e. a10 = 12.59 kPa , a01 = 9.54 kPa) which were calculated as the average value of the fitted results obtained for all deformation rates. Parameter a 30 was fitted again (Table 5.7). The sum of least squares remained small although higher than in the Table 5.4. Repeated optimization with different initial parameter values consistently converged towards the same solution. Table 5.6. Third simulation results (material parameters) of uniaxial and biaxial cruciform tests M-R Fitted Parameter Values (kPa) (n = 24) Deformation Rate (%/s) a10 SUM /n an a20 a30 10 12.90 9.72 10.76 2.61E-03 50 12.35 9.31 9.82 2.68E-03 100 12.39 9.35 9.86 2.61E-03 120 140 12.73 13.38 9.65 10.14 0.00 0.00 8.91 7.92 3.04E-03 5.17E-03 160 12.37 9.36 9.12 3.40E-03 180 12.27 9.35 8.74 4.96E-03 200 12.35 9.41 8.17 4.09E-03 Linear Regression Coefficients & R2^.. = A + BA) A 12.76 9.61 - - 10.66 B -1.50E-03 -6.00E-04 - -1.25E-02 R2 0.07 0.02 - - 0.73 Table 5.7. Final simulation results (material parameters) of uniaxial and biaxial cruciform tests M-R Fitted Parameter Values (kPa) (n = 24)DeformationRate (%/s) a10 SUM / n an a20 a30 10 10.91 2.62E-03 50 9.45 2.78E-03 100 9.53 2.70E-03 120 140 12.59 9.54 0.00 0.00 8.87 8.41 3.07E-03 5.33E-03 160 8.76 3.50E-03 180 8.38 5.13E-03 200 7.85 4.21E-03 Linear Regression Coefficients & R2 (ad = A+ BA) A - - - - 10.66 B - - - - -1.37E-02 R2 - - - - 0.88 Uniaxial and biaxial simulations fitted values of parameter a30 (Table 5.7) are plotted against deformation rate in Fig. 5.8. This parameter decreased from 11 to 8 kPa when the deformation rate increased from 10 to 200 %/s, respectively. A quadratic 7- 6 8- 96 polynomial was also used to model the variation of a 30 with deformation rate but its correlation coefficient was similar to that obtained with a linear relationship, i.e. 0.89 and 0.88 respectively. Therefore, the linear fit was preferred over the polynomial fit due to its simplicity. The following linear equation is proposed for parameter a 30 : = f (A) = A + n(^ (5.8) where A is the deformation rate in s -1 , AN is the nominal stretch ratio (AN = 1.5), A and B are the linear regression coefficients. The strain rate was normalized by the nominal stretch ratio, in order to account for the biological variability in peak stretch of aorta samples. Table 5.8 summarizes the adjusted parameters of equation 5.8. Table 5.8 Linear regression adjusted parameters Parameter Units Uniaxial & Biaxial Cruciform A B kPa kPa*s 10.66 -2.06 Parameter Dependency on Deformation Rate 0^02^0.4^0.6^0.8^1^1.2^1.4 (1/S) Fig. 5.8. Parameter a30 dependency on deformation rate for both, uniaxial and biaxial extensions. 1.2^1.31 1.1 1.4 1.5 10 %/s ^ 10 %/s, Model A 200 %/s ^ 200 %/s, Model BiaxialIncreasing Deformation Rate 6 5 4 L.) ^1 0u. 2 0 Uniaxial 97 Figure 5.9 shows the computed forces at the lowest (10 %/s) and highest (200 %/s) deformation rates. Circumferential and axial forces were averaged in order to clearly exemplify the model prediction of the arterial behavior when the deformation rate is increased. Average Experimental Data & Model Results Stretch Ratio Fig. 5.9. Mooney-Rivlin computed reaction forces versus stretch ratios in uniaxial and biaxial testing at 10 and 200 %/s. An increase in the deformation rate decreases the reaction forces at a particular stretch ratio. Although an excellent fit is observed between the biaxial data and biaxial simulation, the Modified Mooney-Rivlin model does not predict well the uniaxial behavior, especially in the nonlinear part of the curve at high stretch ratios (> 1.45). Stress Distribution (MPa) 10 %Is 100 vs 200 %Is 10.096 0.093 10.090 0 . 088 10.084 0,082 0.091 0.085 0.080 0.088 0.083 0.078 0.086 0.081 0.076 0.083 0.078 0.074 0.081 0.076 0.072 0.078 0:074 0,070 0.076 0.071 0.068 0.073 0.069 0.066 Stress Distribution (MPa) 10 %/s 100 %/s 200 %/s1 - 0:752 0.676 40.703 0.632 10.833 0.570 , 0.601 0.562 0.507 0.526 0.491 0.444 0.450 0.421 0.381 0.375 0 351 0.318 0.299 0.280 0.255 0.224 0.210 0.192 0.148 0.140 0.129 0.073 0.069 0.066 98 Minimization of the objective function and parameter a 30 estimation, using the data gathered at 100%/s deformation rate, is shown in Fig. 5.10. Figure 5.10.a shows the value of the objective function plotted against the iteration number in the optimization loop, for three different initial guesses. The objective function was evaluated 50 times in iterations 0 (initial guess) to 9. Figure 5.10.b illustrates optimization of material parameter a30 . (a)^Minimization of the Objective Function ^(b)^Mooney- Rivlin Fitted Parameter 1.E+07 C .2 1.E+05 C u- 1.E+03 1.E+01 0 1 E-01 ^Case_A Case 8 Case_C 1.E+01 1 E+00 1.E-01 IUO. 1.E-02 a r4 1.E-03 1.E-04 1.E-05 0^1^2^3^4^5^6^7^0 ^ 10^0^1^2^3^4^5^6^7^0^9^10 ^Iteration Iteration Fig. 5.10. Optimization of the Mooney-Rivlin parameters for both uniaxial and biaxial tests at a deformation rate of 100%. (a) Objective function, and (b) parameter a 30 values as a function of iteration number, for three different initial guesses for material parameter a30 : Case_A ( a30 , = 0.1 MPa), Case_B ( a30 , = 1.0 MPa), and Case_C ( a30 , 0.0001 MPa). Figure 5.11 shows the stress and stretch ratio distributions (at 1.5 stretch ratio) during uniaxial and biaxial testing. Scalar bars of the results at 10, 100 and 200 %/s deformation rates are shown for their comparison: One can observe the overall reduction of the stresses when the deformation rate was increased. Similar to chapter 3 results, the highest non uniform stress distribution was obtained in biaxial testing. (a) Stretch Ratio Distribution 10 %/s 100 %/s 200 %/s 1.897 829111 11.899 1.831 1.909 1.839 1.762 1.763 "1. 770 1 694 1.695 1.700 1.626 1.627 1. 630 1.559 1.559 1. 560 1.491 1.491 1.490 1.423 1.423 1.420 1.356 1.355 1.350 1.288 1.286 1,280 Stretch Ratio Distribution 10 %/s 100 %/s 200 %/s 1.517 1.49811/ 11.520 1.500 11.518 1.499 1.479 1.479 1.480 1.461 1.459 1.461 1.442 1.439 1.442 1.423 1.418 1.423 1,404 1.398 1.404 1.385 1.377 1.385 1.367 1.357 1.366 1.348 1,336 1.347 (b) 99 Fig. 5.11. Deformed configurations at 1.50 nominal stretch ratio for uniaxial and cruciform mesh: Distribution of (a) Crxx stress and (b) stretch ratio in the principal direction. The stress and stretch distributions at 10, 100, and 200 %/s are shown inside the tables. 5.3.3.2. Uniaxial Fit In order to improve uniaxial simulations, two objective functions were defined, one for each direction: S (c) =^kFx (d x ) - f x (d x ,^ (5.9) S (c) =^RF y (d y ) - f y (d y , c))2 1^ (5.10) These objective functions were used to better predict the uniaxial extension experimental data. Different material properties were obtained for the circumferential and for the axial directions. Uniaxial simulations fitted values of parameter a 30 per deformation rates are shown in Fig. 5.12. This parameter was reduced roughly from 33 to 23 kPa when the deformation rate was increased, following the same trend in circumferential and axial directions. It was also observed that material parameter a 30 changed linearly as a function of the deformation rate. 35 - 31 - ea aL 27 23 - 19 - 15 • \A A'N a = 33 6 — 3o = 30.6 - 4 (a) 100 Parameter Dependency on Deformation Rate 0^0.2^0.4^0.6^0.8^1^1.2^1.4 ink,N Clis) (b) Parameter Dependency on Deformation Rate 35 - 31 - as 13--tc 27 - 23- 19 - 15 0 ^ 0.2^0.4^0.6^0.8^1^1.2^1.4 irAN (u s) Fig. 5.12. Parameter a30 dependency on deformation rate for uniaxial extension: (a) Circumferential and (b) axial directions. Table 5.9 summarizes Mooney-Rivlin fitted parameters obtained just for uniaxial tests. The values of a 30 obtained per deformation rate were almost 4 times greater than a 30 values calculated with the combined uniaxial-biaxial simulation. Parameters a10 and a0 , remained almost unchanged in both the uniaxial and combined uniaxial-biaxial simulations, with exception of parameter a01 in circumferential direction (i.e. 18 kPa). Table 5.9 Moone -Rivlin fitted parameters in uniaxial extension Mooney-Rivlin Parameters (kPa) Circumferential Axial alo 9.56 9.56 a01 18.01 7.87 all 0.00 0.00 a20 0.00 0.00 A 33.64 30.62 E -5.32 -4.40 Uniaxia1,100%/s 1 0.8 --- 0.60 8 0.4 0.2 0 •^Circumferential o^Axial ^ Circumferential Fit ^ Axial Fit 101 The computed circumferential and axial forces obtained using the Mooney-Rivlin fitted parameters from Table 5.9 in uniaxial tests, at a deformation rate of 100%/s, are shown in Fig. 5.13. Uniaxial behaviour in both directions was well predicted. 1 ^ 1.1^1.2^1.3 ^ 1.4^1.5 Stretch Ratio Fig. 5.13. Mooney-Rivlin computed reaction forces versus stretch ratios in uniaxial testing at a deformation rate of 100 %/s. Figure 5.14 shows the stress and stretch ratio distribution of rectangular samples. The reduction of the stresses as the deformation rate increases is observed when the stress distribution at each deformation rate is compared (scalar bars at 10, 100 and 200 %/s). (a) (b) Stress Distribution (MPa) 10 %/s 100 %/s 200 %/s 0.171I0.2010. 10.192 0.187 0.1670.191 0.182 0.163 0.186 0.178 0.158 0 181 0.173 0.154 0.176 0.168 0.150 0.171 0.163 0.146 0.165 0.158 0.142 0.160 0.153 0.137 0.155 0.149 0.133 Stretch Ratio Distribution 10 %fs 100 %Is 200 %fs 11.514 1.503 11.516 1.507 11.516 1.507 1.493 1,498 1.498 1 482 1.489 1.489 1 471 1.480 1.480 1.461 1.471 1.471 1.450 1.462 1.462 1.439 1.454 1.453 1.429 1.445 1.444 1.418 1.436 1.435 Fig. 5.14. Deformed configurations at 1.50 nominal stretch ratio for uniaxial mesh using the material parameters of Table 6.3 (Circumferential direction): distribution of (a) Cr xx stress and (b) stretch ratio in the principal direction. The stress and stretch distributions at 10, 100, and 200 %/s are shown inside the tables. 102 5.4. DISCUSSION Contrary to the conventional understanding of biological tissue behaviour, it was discovered in this study that the stiffness of thoracic aorta decreases with deformation rate, which was confirmed by the use of two statistical methods. Overall significant differences (p ^ 0.02) in the mechanical forces of uniaxial and biaxial experiments were found using the ANOVA Friedman statistic test. In addition, the Wilcoxon test help us to observe which particular pair of deformation rates likely had a tendency to be significantly different (i.e. almost all deformation rates were different than 200%/s, p = 0.07 ). An increase in the sample size for further studies will certainly reduce the Wilcoxon test p values obtained. Giles et al. (2007) observed a similar behaviour in load controlled planar biaxial laboratory tests of myocardium and skin samples. Hu and Desai (2004) studied the variation of the elastic modulus of pig liver tissue at different deformation rates under compression tests, and found that at higher strain rates the liver has an apparently lower resistance (i.e. softer) to deformation than at lower deformation rates. However, under uniaxial displacement controlled tests, Pioletti et al. (1999) found that an increase in the deformation rate results in a stiffer material response. In their work, they submitted ligament samples to deformation rates within 0 1 — 40%/s in ascending order. In the present study, thoracic aortas were tested over a 10 — 200%/s range of deformation rates. In both, uniaxial and biaxial tests, the peak force on the arterial samples was around 20% smaller at a deformation rate of 200%/s than at 10%/s (Figs. 5.3 and 5.4). These results were observed on the average of the five steady-state loading cycles, after five-preconditioning cycles for each deformation rate. However, similar results were also observed on the first loading cycle at each deformation rate, eliminating the possibility of this behavior being due to pre- conditioning. The experimental procedure was validated by repeating the same experiments on latex and nitrile samples. As expected, the effect of deformation rate was opposite in these rubbers to what was observed in the arterial samples. An experiment without any sample was performed to measure the combined effects of inertia, friction, and water drag force, which were found to be negligible, i.e. less than 10 % of the effect of deformation rate. Sample variability between circumferential and axial force vs. stretch ratio curves was found to be lower than the effect of deformation rate, supporting the use of an isotropic 103 constitutive equation to predict the behavior of the arterial wall. Isotropic models have also been used to model the artery in other studies (Prendergast, et al., 2003; Raghavan and Vorp, 2000; Scotti and Finol, 2007; Tezduyar, et al., 2007; Virues-Delgadillo, et al., 2007; Wang, et al., 2001). The main constituents of arterial tissues are collagen and elastin. The elastic response of arteries is largely due to elastin because elastin is about 1000 times more extensible than collagen. Elastin chains are cross-linked together to form rubberlike, elastic fibers (Shadwick, 1999). Each elastin molecule uncoils into a more extended conformation when the fiber is stretched and recoils spontaneously as soon as the stretching force is relaxed. Its elasticity is based on changes in the entropy of the molecular chains, while the material is deformed. An imposed strain increases the order in the molecular network and thus decreases its entropy. According to the thermodynamic laws, the network would try to recover its former shape, increasing their entropy. One possible explanation for the observed phenomenon is that when elastin is stretched at high rates it would attain a highly oriented conformation and as a result the cross-links would not be able to bear the load due to slippage. However, as the stretch occurs at lower deformation rates, the elastin molecules would have more time available to adjust in order to prevent slippage of the cross-links and as a result would be able to bear a higher load. This effect might be similar to the stretching behavior of linear and branch polymers (McLeish and Larson, 1998; Sentmanat, et al., 2005). Here elastin at high extensional rates would behave more like a linear polymer where the linear polymers can slip one past the other to exhibit a reduced load mainly due to high oriented conformation imposed by the high stretching rate. On the other hand, elastin at low extensional rates would behave like a branched polymer where the presence of branches prevent cross-linking slippage and as a result bear higher loads. Moreover, Trepat et al. (2007) subjected human airway smooth muscle cells to a transient stretch-unstretch maneuver with zero residual macroscale strain, observing that the cell promptly fluidizes and then slowly re-solidifies. Therefore, is it possible that changes in the alignment and configuration of adjacent fibers within the extracellular matrix occurs as soon as one fiber begins to shear over the other (i.e. fluid-like behavior), making softer the overall response in the tissue when stretching the arterial sample at the 104 highest deformation rate (minimum relaxation in the tissue due to short testing time). At lower deformation rates, the fibers have more time to move back to their original configuration, thus restraining their ability to flow (stiffer response). The constitutive equation selected here is capable of representing the nonlinear elastic behavior of the artery, including the effect of deformation rate, with only four parameters. It produces a unique curve fitting solution to experimental results for each deformation rate. More importantly, it was observed that Mooney Rivlin parameters a lc, and ao , were not a function of the deformation rate. This could imply that they are related to the elastic behavior of the artery. A linear relationship was obtained between parameter a30 and the deformation rate, where the partially elastic contribution is given by coefficient A and the deformation rate response is given by coefficient h . As shown in Figure 5.9, the model better fits the biaxial data than the uniaxial data. Using higher weights for the uniaxial least square differences, it was possible to obtain a better fit for uniaxial data, at the expense of a deterioration of the fit for biaxial data. The use of material parameters an and a20 did not improve the fits significantly enough to justify increasing the complexity of the model with these two extra parameters. The use of anisotropic constitutive equations in future studies might reduce this discrepancy between predicted and experimental forces in uniaxial direction. Clinical results have shown that slow balloon inflation might reduce restenosis. It was hypothesized that the slower deformation rate gives as a result lower intramural stresses and lower arterial injury. This hypothesis is contradictory to our results, which would indicate that a lower inflation rate would result in higher intramural stresses. Other mechanisms might play a role in leading to lower restenosis rates, such as endothelium denudation or artery injury being dependent on stretch ratio rather than on intramural stress. 5.5. SUMMARY In the present study, the effect of deformation rate on the mechanical behavior of arteries in vitro under uniaxial and biaxial extensions was investigated. It was found that the loading force at a stretch ratio of 1.5 is reduced by 20 % when the deformation rate is increased from 10 to 200 %/s, implying that the stiffness of arteries decreases with 105 deformation rate. This behavior might be a consequence of the faster fluidization and small re-solidification that occurs in the cell at higher deformation rates. This effect of deformation rate was observed to be higher than the effect of anisotropy. The development of an isotropic hyperelastic rate-dependent constitutive model, derived from the Mooney- Rivlin hyperelastic model, is capable of representing this behavior. In its proposed form, the model has only 4 parameters, only one of which varies with deformation rate. The small number of material parameters and simple formulation increase the applicability of this model to numerical simulations. The inverse relationship between stiffness and deformation rate raises doubts on the hypothesized relationship between intramural stress, arterial injury, and restenosis. 106 5.6. REFERENCES DEMER, L. L., YIN, F. C. P., 1983. Passive biaxial mechanical properties of isolated canine myocardium. Journal of Physiology, 339, 615 — 630. DE SMET, E., JAECQUES, S. V. N., JANSEN, J. J., WALBOOMERS, F., VANDER SLOTEN, J., NAERT, I. E., 2007. Effect of constant strain rate, composed of varying amplitude and frequency, of early loading on peri-implant bone (re)modeling. Journal of Clinical Periodontology, 34, 618 — 624. DOEHRING, T. C., CAREW, E. 0., VESELY, I., 2004. The effect of strain rate on the viscoelastic response of aortic valve tissue: A direct-fit Approach. Annals of Biomedical Engineering, 32(2), 223 — 232. EBACHER, V., TANG, C., McKAY, H., OXLAND, T. R., GUY, P., WANG, R., 2007. Strain redistribution and cracking behavior of human bone during bending. Bone, 40, 1265 — 1275. ELLIS, S. G., AND MULLER, D. W. M., 1992. Arterial injury and the enigma of coronary restenosis. Journal of the American College of Cardiology, 19, 275 — 277. ELTCHANINOFF, H., CRIBIER, A., KONING, R., CHAN, C., JOLLY, N., TAN, A., LETAC, B., 1996. Effects of prolonged sequential balloon inflations on results of coronary angioplasty. Am. J. Cardiol., 77, 1062 — 1066. GASSER, T. C., HOLZAPFEL, G. A., 2007. Modeling plaque fissuring and dissection during balloon angioplasty intervention. Annals of Biomedical Engineering, 35, 711 — 723. GAY, R. E., ILHARREBORDE, B., ZHAO, K., BOUMEDIENE, E., AN, K.-N., 2008. The effect of loading rate and degeneration on neutral region motion in human cadaveric lumbar motion segments. Clinical Biomechanics, 23, 1 — 7. GILES, J. M., BLACK, A. E., BISCHOFF, J. E., 2007. Anomalous rate dependence of the preconditioned response of soft tissue during load controlled deformation. Journal of Biomechanics, 40, 777 — 785. HAUT, R. C., 1983. Age-dependent influence of strain rate on the tensile failure of rat-tail tandon. Journal of Biomechanical engineering, 105, 296 — 299. HAUT, R. C., LITTLE, R. W., 1972. A constitutive equation for collagen fibers. Journal of Biomechanics, 5, 423 — 430. HOLZAPFEL, G. A., STADLER, M., SCHULZE-BAUER, C. A., 2002. A layer-specific three-dimensional model for the simulation of balloon angioplasty using magnetic resonance imaging and mechanical testing. Ann. Biomed. Eng., 30, 753 — 767. 107 HSIEH, Y.-F., WANG, T., TURNER, C. H., 1999. Viscoelastic response of the rat loading model: Implications for studies of strain-adaptive bone formation. Bone, 25(3), 379 — 382. HU, T., DESAI, J. P., 2004. Soft-tissue material properties under large deformation: Strain rate effect. Proceedings of the 26 th Annual International Conference of the IEEE EMBS, San Francisco, CA, USA (September 1 — 5, 2004), 2758 — 2761. KWAN, M. K., LIN, T. H-C., WOO, S. L-Y., 1993. On the viscoelastic properties of the anteromedial bundle of the anterior cruciate ligament. I Biomechanics, 26, 447 — 452. LALLY, C., DOLAN, F., PRENDERGAST, P. J., 2005. Cardiovascular stent design and vessel stresses: a finite element analysis. Journal of Biomechanics, 38, 1574 — 1581. LALLY, C., REID, A. J., AND PRENDERGAST, P. J., 2004. Elastic behavior of porcine coronary artery tissue under uniaxial and equibiaxial tension. Ann. Biomed. Eng., 32(10), 1355 — 1364. LIANG, D. K., YANG, D.-Z., QI, M., WANG, W.-Q., 2005. Finite element analysis of the implantation of a balloon-expandable stent in a stenosed artery. International Journal of Cardiology, 104, 314 — 318. MIKETIC, S., CARLSSON, J., AND TEBBE, U., 1998. Influence of gradually increased slow balloon inflation on restenosis after coronary angioplasty. American Heart Journal, 135(4), 709 — 713. MOONEY, M. A., 1940. Theory of large elastic deformation. J. Appl. Phys., 11, 582 — 592. McLEISH, T. C. B., LARSON, R. G., 1998. Molecular constitutive equations for a class of branched polymers: the pom-pom model. J. Rheol., 42, 81 — 110. NEKOUZADEH, A., PRYSE, K. M., ELSON, E. L., GENIN, G. M., 2007. A simplified approach to quasi-linear viscoelastic modeling. Journal of Biomechanics, 40, 3070 — 3078. OH, S., KLEINBERGER, M., MCELHANEY, J. H., 1994. Finite-Element analysis of balloon angioplasty. Med. & Biol. Eng. & Comput., 32, S 108 - S 114. OHMAN, E. M., MARQUIS, J. F.,RICCI, D. R., BROWN, R. I., KNUDTSON, M. L., KEREAIKES, D. J., SAMAHA, J. K., MARGOLIS, J. R., NIEDERMAN, A. L., DEAN, L. S., 1994. A randomized comparison of the effects of gradual prolonged versus standard primary balloon inflation on early and late outcome. Results of a multicenter clinical trial. Perfusion balloon catheter study group. Circulation, 89, 1118 — 1125. OKAMOTO, R. J., WAGENSEIL, J. E., DELONG, W. R., PETERSON, S. J., KOUCHOUKOS, N. T., AND SUNDT III, T. M., 2002. Mechanical properties of dilated human ascending aorta. Ann. Biomed. Eng., 30(5), 624 — 635. 108 PIOLETTI, D., RAKOTOMANANA, L. R., 2000. Non-linear viscoelastic laws for soft biological tissues. Eur. J. Mech. A/Solids, 19, 749 — 759. PIOLETTI, D., RAKOTOMANANA, L. R., BENVENUTI, J. —F., LEYVRAZ, P.-F., 1998. Viscoelastic constitutive law in large deformations: Application to human knee ligaments and tendons. Journal of Biomechanics, 31, 753 — 757. PIOLETTI, D., RAKOTOMANANA, L. R., LEYVRAZ, P.-F., 1999. Strain rate effect on the mechanical behavior of the anterior cruciate ligament-bone complex. Medical Engineering & Physics, 21, 95 — 100. PRENDERGAST, P. J., LALLY, C., DALY, S., REID, A. J., LEE, T. C., QUINN, D., AND DOLAN, F., 2003. Analysis of prolapse in cardiovascular stents: A constitutive equation for vascular tissue and finite-element modelling. Journal of Biomechanical Engineering, 125, 692 — 699. PROVENZANO, P., LAKES, R., KEENAN, T., VANDERBY Jr., R., 2001. Nonlinear ligament viscoelasticity. Annals of Biomedical Engineering, 29, 908 — 914. RAGHAVAN, M. L., AND VORP, D. A., 2000. Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: Identification of a finite strain constitutive model and evaluation of its applicability. Journal of Biomechanics, 33, 475 — 482. SARVER, J. J., ROBINSON, P. S., ELLIOTT, D. M., 2003. Methods for quasi-linear viscoelastic modeling of soft tissue: Application to incremental stress-relaxation experiments. Journal of Biomechanical Engineering, 125, 754 — 785. SCHAFFLER M. B., RADIN, E. L., BURR, D. B., 1989. Mechanical and morphological effects of strian rate on fatigue of compact bone. Bone, 10, 207 — 214. SCHWARTZ, R. S., HOLMES, D. R. JR., TOPOL, E. J., 1992. The restenosis paradigm revisited: an alternative proposal for cellular mechanism. Journal of the American College of Cardiology, 20, 1284 — 1293. SCOTTI, C. M., AND FINOL E. A., 2007. Compliant biomechanics of abdominal aortic aneurysms: A fluid—structure interaction study. Computers & Structures, 85, 1097 — 1113. SENTMANAT, M., MULIAWAN, E. B., HATZIKIRIAKOS, S. G., 2005. Fingerprinting the processing of polyethylenes from transient extensional flow and peel experiments in the melt state. Rheologica Acta, 44, 1 — 15. SHADWICK, R. E., 1999. Mechanical design in arteries. J. Exp. Biol., 202, 3305 — 3313. TENAGLIA, A.N., QUIGLEY, P.J., KEREIAKES, D.J., ABBOTTSMITH, C.W., PHILLIPS, H.R., TCHENG, J.E., RENDALL, D., OHMAN, E.M., 1992. Coronary angioplasty performed with gradual and prolonged inflation using a perfusion balloon catheter: Procedural success and restenosis rate. Am. Heart J. 124, 585 — 589. 109 TEZDUYAR, T. E., SATHE, S., CRAGIN, T., NANNA, B., CONKLIN, B. S., PAUSEWANG, J., AND SCHWAAB, M., 2007. Modelling of fluid—structure interactions with the space—time finite elements: Arterial fluid mechanics. Mt. J. Num. Meth. Fluids, 54, 901 — 922. THERIAULT, P., TERRIAULT, P., BRAILOVSKI, V., GALLO, R., 2006. Finite element modeling of a progressively expanding shape memory stent. Journal of Biomechanics, 39, 2837 — 2844. THORNTON, G. M., SCHWAB, T. D., OXLAND, T. R., 2007a. Fatigue is more damaging than creep in ligament revealed by modulus reduction and residual strength. Annals of Biomedical engineering, 35(10), 1713 — 1721. THORNTON, G. M., SCHWAB, T. D., OXLAND, T. R., 2007b. Cyclic loading causes master ruptura and strain rate than static loading in medial collateral ligament at high stress. Clinical Biomechanics, 22, 932 — 940. TIMMIS, S. B. H., HERMILLER, J. B., BURNS, W. H., MEYERS, S. N., DAVIDSON, C. J., 1999. Comparison of immediate and in-hospital results of conventional balloon and perfusion balloon angioplasty using intracoronary ultrasound. American Journal of Cardiology, 83, 311 — 316. TRAN, N. T., WATSON, N. A., TENCER, A. F., CHING, R. P., ANDERSON, P. A., 1995. Mechanism of the burst fracture in the thoracolumbar spine. The effect of loading rate. Spine, 20(18), 1984. — 1988. TREPAT, X., DENG, L., AN, S. S., NAVAJAS, D., TSCHUMPERLIN, D. J., GERTHOFFER, W. T., BUTLER, J. P., FREDBERG, J. J., 2007. Universal physical responses to stretch in the living cell. Nature, 447, 592 — 596. UMEDA, H., IWASE, M., KANDA, H., IZAWA, H., NAGATA, K., ISHIKI, R., SAWADA, K., MUROHARA, T., YOKOTA, M., 2004. Promising efficacy of primary gradual and prolonged balloon angioplasty in small coronary arteries: A randomized comparison with cutting balloon angioplasty and conventional balloon angioplasty. American Heart Journal, 147, 1 — 8. VENA, P., GASTALDI, D., CONTRO, R., 2006. A constituent-based model for the nonlinear viscoelastic behavior of ligaments. Journal of Biomechanical Engineering, 128, 449 — 457. VIRUES-DELGADILLO, J. 0., DELORME, S., DIRADDO, R., HATZIKIRIAKOS, S. G., THIBAULT, F., 2007, "Mechanical characterization of arterial wall: Should cruciform or square sample be used in biaxial testing?," Journal of the Mechanical Behavior of Biomedical Materials, submitted. 110 WALDMAN, S. D., AND LEE, J. M., 2002. Boundary conditions during biaxial testing of planar connective tissues: Part 1: Dynamic behavior. Journal of Materials Science. Materials in Medicine, 13, 933 — 938. WANG, D. H. J., MAKAROUN, M., WEBSTER, M. W., VORP, D. A., 2001. Mechanical properties and microstructure of intraluminal thrombus from abdominal aortic aneurysm. Journal of Biomechanical Engineering, 123(6), 536 — 539. WANG, J. L., PARNIANPOUR, M., SHIRAZI-ADL, A., ENGIN, A. E., 1999. Rate effect on sharing of passive lumbar motion segment under load-controlled sagittal flexion: Viscoelastic finite element analysis. Theoretical and Applied Fracture Mechanics, 32, 119 — 128. WEISS, T., LEIBOVITZ, D., KATZ, I., DANENBERG, C., VARSHITSKY, B., LOTAN, C., 2007. The value of computerized angioplasty in patients undergoing coronary stenting: a prospective, randomized trial. Am. J. Cardiol., 100, 118L. WELLS, S., SACKS, M. S., 2002. Effects of fixation pressure on the biaxial mechanical behavior of porcine bioprosthetic heart valves with long-term cyclic loading. Biomaterials, 23, 2389 — 2399. WU, W., QI, M., LIU, X.-P., YANG, D.-Z., WANG, W.-Q., 2007. Delivery and release of nitinol stent in carotid artery and their interactions: A finite element analysis. Journal of Biomechanics, 40, 3034 — 3040. YINGLING, V. R., CALLAGHAN, J. P., McGILL, S. M., 1997. Dynamic loading affects the mechanical properties and failure site of porcine spines. Clinical Biomechanics, 12(5), 301 — 305. ZHANG, W., CHEN, H. Y., KASSAB, G. S., 2007. A rate-insensitive linear viscoelastic model for soft tissues. Biomaterials, 28, 3579 — 3586. 111 CHAPTER 6 6. CONCLUSIONS, RECOMMENDATIONS AND CONTRIBUTION TO THE KNOWLEDGE 6.1. INTRODUCTION In this work, uniaxial and biaxial extensional tests were performed to porcine thoracic aorta samples in order to obtain the effect that freezing or deformation rate might have on the arterial mechanical behavior. Different sample geometries were tested and compared. The stress distribution of rectangular, cruciform and square samples was obtained through the simulation of uniaxial and biaxial testing. An inverse modeling algorithm was used to get the fitted material parameters of the hyperelastic model selected. A constitutive equation, capable of describing the viscoelastic behavior of the artery when submitted to different deformation rates was also developed. 6.2. CONCLUSIONS Uniaxial and biaxial experimental in vitro data of porcine thoracic aortas were obtained from rectangular, cruciform and square samples. In chapter 3, it was concluded that highly nonlinear behaviour of porcine thoracic aorta under large deformations can easily be measured biaxially using cruciform samples attached with clamps rather than square samples attached with hooks. It was also found that data obtained from biaxial cruciform tests can accurately predict uniaxial behavior performed in two orthogonal directions, proving that no information is gained by performing uniaxial extensions. The aorta response was found to be nearly isotropic, which could be explained by the relatively small number of collagen and smooth muscle cells in relation to the higher amount of elastin fibers within its walls (i.e. the aorta is considered the main elastic artery - Keshaw, 2001). Due to the small anisotropy observed in the specimens tested, a hyperelastic isotropic constitutive equation was selected in this work: The five parameter Mooney Rivlin model (Mooney, 1940). Inverse modeling proved to be a useful technique to obtain the material properties from experiments where an analytical solution is difficult to obtain (Seshaiyer and Humphrey, 2003). The feasibility of inverse modeling for biaxial testing of cruciform samples over a wider range of strains than is possible with square samples was evaluated in this thesis. Inverse modeling takes into account the well-define boundary 112 conditions of biaxial cruciform samples. Using an inverse modeling algorithm, the fitted material parameters of Mooney Rivlin model were obtained. The stress-stretch distribution along the arterial wall was calculated as well. Mooney Rivlin model proved to be accurate and non-sensitive to initial parameter values. The effect of freezing and thawing on biaxial cruciform samples was analyzed in chapter 4. It was observed that the mechanical properties of thoracic aortas were not significantly changed after preservation of arteries for two months independently of the storing medium (saline, Krebs with DMSO and dipping in liquid nitrogen) or freezing temperature (-20 °C or -80 °C) used. However, the lack of significant differences observed in this study was likely due to the small number of samples tested per storing group, and thus only using a higher sample size in future studies might help in the finding of those storing mediums that better preserve the mechanical properties of the arterial wall. For instance, it is believed that a high electrolyte concentration and severe cell dehydratation using a faster freezing temperature rate affect more likely cell viability. In addition, to properly vitrify biological tissues using liquid nitrogen, a higher concentration of cryoprotectant is needed. Therefore cryopreservation might be the method that minimizes histological and mechanical changes in the tissue submitted to biaxial testing. Long-term efficacy of cryopreserved blood vessels is determined by cellular tissue viability, biochemical functions, and mechanical properties. Further studies are required to clarify the impact of cryopreservation on extracellular matrix architecture to help tailor an optimized approach to preserve the mechanical properties of arteries. In chapter 5, the effect of deformation rate on the material properties of porcine aortas in vitro under uniaxial and biaxial extensions was investigated. It was found that the loading force at a stretch ratio of 1.5 is reduced by 20 % when the deformation rate is increased from 10 to 200 %/s, implying that the stiffness of arteries decreases with deformation rate. This behavior might be a consequence of the faster fluidization and small re-solidification that occurs in the cell at higher deformation rates. This effect of deformation rate was observed to be higher than the effect of anisotropy. The development of an isotropic hyperelastic rate-dependent constitutive model, derived from the Mooney- Rivlin hyperelastic model, is capable of representing this behavior. In its proposed form, the model has only 4 parameters, only one of which varies with deformation rate. The 113 relative small number of material parameters and simple formulation increase the applicability of this model in numerical simulations. Inverse finite element simulations using the proposed constitutive relation accurately predict the mechanical properties of the arterial wall. This outcome might be useful in medical techniques such as balloon angioplasty and stent deployment. The inverse relationship between stiffness and deformation rate raises doubts on the hypothesized relationship between intramural stress, arterial injury, and restenosis. 6.3. CONTRIBUTION TO KNOWLEDGE The following contributions to knowledge are the outcomes of this research work: 1. It was demonstrated that an inverse modeling algorithm can be used to fit the material properties by taking into account the non-uniform stress distribution. Inverse modeling also can take into account the well-defined boundary conditions of biaxial cruciform samples. The feasibility of inverse modeling for biaxial testing of cruciform samples over a wider range of strains than is possible with square samples was evaluated. 2. Higher deformation range, simpler data acquisition and easier attachment of the specimen are the main advantages of cruciform samples over square samples. 3. The change in biaxial mechanical behaviour of pig thoracic aorta before and after storage in various solutions and freezing at different temperatures for two months was studied. Samples were stiffer after freezing and thawing than when fresh at stretch ratios higher than 1.4, especially in the circumferential direction. 4. Krebs solution with DMSO, isotonic saline solution and dipping in liquid nitrogen did not significantly change the mechanical properties of the arterial wall after preservation of arteries for two months. However, the lack of significant differences observed in this study was likely due to the small number of samples tested per storing group. 5. Stiffness of porcine thoracic aortas decreases with increase in the deformation rate. 6. The development of a constitutive model derived from Mooney-Rivlin hyperelastic model was capable of representing the viscoelastic behavior of thoracic aortas. 114 Overall this work has contributed to a better understanding of the mechanical properties of the arterial wall. Certainly, more work needs to be done in the future in order to unravel the complications arisen due to biological tissue complex behavior. However, this research has provided the foundations towards a better selection of the geometry and boundary conditions that can be used during in vitro mechanical test analysis, as well as a better understanding of the arterial behavior as a function of the deformation rate. 6.4. RECOMMENDATIONS FOR FUTURE WORK Several important aspects of the arterial wall behavior are yet to be studied. Possible objectives for future work are recommended below: 1. In this work, it was assumed that the artery is a homogeneous material. The development of heterogeneous meshes, with finite elements of different nodal size (i.e. triangles, bricks or hexagonals) might improve the information obtained here. Each element type might represent a different cell type within the artery (i.e. elastin, collagen or smooth muscle cells). 2. Due to the small anisotropy observed in the thoracic aorta samples analyzed, isotropic constitutive equations were used. The aorta is the main elastic artery. Other arterial vessels (i.e. muscular arteries) have a small amount of elastin, thus increasing the anisotropic behaviour. Therefore, to better characterize the material properties of biological tissues, the use or development of anisotropic models using the inverse modeling algorithm derived in this work is essential to obtain a closer approximation of the response the arteries have inside the body. 3. The use of a high molecular weight ion to reduce or fully replace sodium ions from the storing solution (i.e. Krebs with DMSO) might better preserve cell viability, which is of primary importance to freeze cells efficiently (Stachecki, et al., 1998). The cell membrane permeability to the high molecular ion needs to be small; therefore during the freezing process the high molecular ion will not penetrate the membrane into the cytoplasm. 115 4. In order to properly vitrify biological tissues, the use of a specific freezing rate, pressurize the liquid nitrogen phase, and the correct selection of the type and concentration of CPA (Fahy, et al., 1984) should help improve cell viability and thus better preserve the mechanical properties using this technique. 5. The study of the effect of deformation rate on cylindrical arteries, such as coronaries and carotids, can strengthen the results obtained in this thesis. Furthermore, the development of a viscoelastic model based on a hyperelastic anisotropic model (i.e. Fung or Holzapfel) might better represent the in vivo response of the arterial tissue. 6. The inflation of a balloon inside the artery at different deformation rates needs to be performed to confirm that the arterial stress distribution is reduced when the balloon is inflated at a high deformation rate, minimizing injuries and the contact time between the balloon and the inner wall of the artery. 116 6.5. REFERENCES FAHY, G. M., MACFARLANE, D. R., ANGELL, C. A., MERYMAN, H. T., 1984. Vitrification as an approach to cryopreservation. Cryobiology, 21, 407 — 426. KESHAW, K., 2001. Microstructure of human arteries. J. Anat. Soc. India, 50 (2), 127 — 130. MOONEY, M. A., 1940. Theory of Large Elastic Deformation. I Appl. Phys., 11, 582 — 592. SESHAIYER, P., AND HUMPHREY, J. D., 2003. A sub-domain inverse finite element characterization of hyperelastic membranes including soft tissues. Journal of Biomechanical Engineering, 125, 363 — 371. STACHECKI, J. J., COHEN, J., WILLADSEN, S., 1998. Detrimental effects of sodium during mouse oocyte cryopreservation. Biology of Reproduction, 59, pp. 395 — 400. 65z 0 4 <1 0 (a) z 6 5 U- • 4 • 3 4) 2 C 1 Z; 0 UniaxialUniaxial 1.6 1.81.2 ^ 1.4 Stretch Ratio 1.2 1 . 4 Stretch Ratio 1.6 1.8 1.6^1.81.4 Stretch Ratio 1.2 1.2 1.6^1.81.4 Stretch Ratio (c)^ (d)Biaxial Cruciform Biaxial Square + Circurrferential Data x Axial Data - Biaxial Cruciform Fit 4- 0 ^ 24 20 - g 16 12- 0 u. 8- ■ Circurrferential Data o Axial Data - Biaxial Square Fit a 117 APPENDIX A A. Uniaxial-Biaxial Simulations using Ogden & Guccione Constitutive Equations A.1. Ogden Model Isotropic Simulations Table A.1 summarizes the fitted material parameters obtained using the Ogden model. Figure A.1 shows the predicted and experimental forces vs. stretch ratio for each extensional test. The expression of the Ogden model can be found in chapter 1 (Equation 1.16). Table A.1 0 den model fitted parameters Ogden Parameters Uniaxial Circumf. Uniaxial Axial Biaxial Cruciform Biaxial Square /II (kPa) 4.86 0.03 0.52 2.24 1 2 (kPa) 19.17 0.16 0.03 16.11 .t3 (kPa) 0.46 1.02 0.66 0.70 a, 1.00 1.00 1.00 1.00 0(2 5.00 5.00 5.00 5.00 a3 10.00 10.00 16.00 16.00 Fig. A.1. Experimental and Ogden model computed reaction forces versus stretch ratios: (a) circumferential and (b) axial directions for uniaxial tests. Experimental data is represented by the dash figures and modeling data by the solid line. (c) biaxial cruciform and (d) biaxial square. Cruciform experimental data: circumferential (plus) and axial (cross) direction. Square experimental data: circumferential (closed squares) and axial (open squares) direction. Modeling data are represented by solid (circumferential fit) or dash (axial fit) lines. (b) 6 • 4 - o 3- U- 2 - < • 1- 0 - Uniaxial - Uniaxial Data -Uniaxial Fit 24 20 - 16 12 8 4 0 1.2 1.4 Stretch Ratio 1.6^1.8 1.6 1.81.2 ^ 1.4 Stretch Ratio + Circumferential Data x Axial Data -Biaxial Cruciform Fit + ^ ■ Circumferential Data • Axial Data - Biaxial Square Fit 24 20 E 16 12 0 8 4 0 118 A.2. Guccione Model Anisotropic Simulations Using equations 1.25 and 1.26, the Guccione's model material parameters were obtained (Table A.2). In our simulations it was assumed that the fiber was mainly oriented in circumferential direction along the vessel wall. Figure A.2 shows the predicted and experimental forces vs. stretch ratio in uniaxial and biaxial testing. Table A.2 Guccione model fitted parameters Guccione Parameters Uniaxial Circumf. Uniaxial Axial Biaxial Cruciform Biaxial Square C (kPa) bf 13, b f, 518.52 0.21 0.10 0.00 417.98 0.14 0.17 0.01 0.69 6.60 6.60 2.14 517.98 0.24 0.27 0.01 1.2 ^ 1.4 ^ 1.6^1.8^1.2^1.4 ^ 1.6^1.8 Stretch Ratio Stretch Ratio (c)^ (d) Biaxial Cruciform Biaxial Square Fig. A.2. Experimental and Guccione model computed reaction forces versus stretch ratios: (a) circumferential and (b) axial directions for uniaxial tests. Experimental data is represented by the dash figures and modeling data by the solid line. (c) biaxial cruciform and (d) biaxial square. Cruciform experimental data: circumferential (plus) and axial (cross) direction. Square experimental data: circumferential (closed squares) and axial (open squares) direction. Modeling data are represented by solid (circumferential fit) or dash (axial fit) lines. 119 APPENDIX B B. Inflation of Cylindrical Arteries The testing machine LSM — 100 (Laser Measurement System) shown in Figure B.1, is capable of performing inflation tests to arteries. In other words, it can apply an internal pressure to the artery sample and measure its outer diameter. For body temperature tests, the equipment has a heated bath. When using the bath, the artery is inflated with a liquid. Fig. B.1. Inflation test machine. The LMS — 100 includes a variety of clamping devices and attachments that allow blood tissue pressurization either by gas or water. Test data are collected using a computer, and the collected data can be presented in both graphical and tabulated form. VULCANO Intra Vascular Ultra Sound system (IVUS) was used to measure the inner wall diameter. An Eagle Eye Catheter, which diameter is 1 mm, was chosen to get the cross sectional image of the artery (Figure B.2) and to calculate its inner and outer diameters. In this figure, the well define circular ring (in white) represents the position of the Eagle Eye catheter inside the lumen of the artery. 120 Fig. B.2. IVUS Image: Cross sectional area of a left coronary artery. Hearts, coronary and carotid arteries from pigs were harvested from the local slaughterhouse. Upon arrival, a square section (40x40 mm) of cardiac muscle including the left coronary artery was dissected from the pig heart (Fig. B.3.a). Sutures were used to close the branches of the left coronary artery. Any visible connective tissue was dissected away from the external wall of carotid arteries. Plastic ties were used to attach the arterial sample to the deployment tester (Fig. B.3.b). Cylindrical vessels of 40 mm long were selected for their attachment to the deployment tester (Fig. B.3.c). Mean diameter and thickness of carotid arteries were 3.4 and 0.7 mm, respectively. In coronary arteries, the mean diameter and thickness were 2.5 and 0.5 mm, respectively. Using the same inverse modeling algorithm than in uniaxial and biaxial extensions, Mooney-Rivlin hyperelastic constitutive model was fitted to the experimental data. The only difference is that this time the finite element simulation of the inflation experiment was performed by applying pressure conditions on a mesh of the same size and shape as the sample. Only the part of the sample free to deform (20 mm long) between the pair of attachment ties (grips) was meshed (Fig. B.3.d). The simulation can predict the displacement of the nodes corresponding to the circumference of the vessel at the center of the sample. (c) (d) Î 40^ Fig. B.3. Inflation test: (a) Left coronary artery surrounded by cardiac muscle, and (b) carotid artery attached to deployment tester; (c) sample dimension in millimetres, (d) undeformed mesh. 121 Table B.1 summarizes the fitted parameters obtained for Mooney-Rivlin constitutive model and for each type of artery tested. Consistence convergences were obtained with this model. Table B.1 Mooney Rivlin model fitted in Cylindrical Arteries Mooney- Rivlin Parameters (kPa) Left Coronary Artery Carotid Artery a10 8.60 20.67 aoi 11.19 18.07 an 18.22 0.00 a20 22.34 0.00 am, 243.68 16.60 Experimental pressure-stretch curves of coronary and carotid arteries submitted to inflation testing are shown in Fig. B.4. It is well known that carotids and coronaries are classified as elastic and muscular arteries, respectively. Therefore, carotid arteries should possess a higher compliance than coronary arteries. In this figure, it is observed that pressures higher than 0.5 MPa are needed to inflate the coronary artery at 1.4 stretch ratios; meanwhile in the case of carotid arteries, lower pressures are needed (0.2 MPa). Computed Pressure-Stretch curves obtained using the Mooney-Rivlin fitted parameters from Table B.1 are shown in the same figure (Fig. B.4). It is seen that a good correlation between experimental and simulated data was achieved for both types of arteries tested. • EXP - Coronary ■ EXP - Carotid --- MOD - Coronary MOD - Carotid 1.1^1.2^1.3^1.4^1.5^1.6^1.7 Stretch Ratio (a) 122 Inflation Test Fig. B.4. Pressure-Stretch experimental curves of left coronary (diamonds) and carotid (open squares) arteries. Solid (coronary) and dash line (carotid) are the computed results using Mooney fitted parameters. Figure B.5 shows that the stress and stretch ratio distribution is nearly uniform in most of the left coronary artery wall, with lowest stresses being located near the attachment grips (i.e. edges of cylinder), and highest stresses close to the center of the sample. Stretch ratios span from 1.20 to 1.42 and stresses from 0.21 to 0.79 MPa. Stress (IVra)^ Stretch Ratio ^0.793 1.422 ^ 0.728 1.397 0.663^ 1.373 0.598 1.348 0.533 1.323 0.468^i^1.299 0.403 1.274 0.338 1.249 0.273^ 1.225 0.208 1.200 Deformed configurations at the maximum applied pressure for left coronary artery: distribution of (a) stress and (b) stretch ratio in the principal direction. The same stress-stretch distribution tendency was observed in the simulation results obtained with the carotid artery data (Fig. B.6). In this case, the stresses span from 0.16 to 0.77 MPa and the stretch ratios from 1.31 to 1.75. (b) Fig. B.S. Stress (IVPa) Stretch Ratio 0.771 1.749 0.704 1.700 0.636 1.652 0.568 1.604 0.500 1.555 0.433 1.507 0.365 1.458 0,297 1.410 0.230 1.362 0.162 1.313 (a) 123 Fig. B.6. Deformed configurations at the maximum applied pressure for carotid artery: distribution of (a) stress and (b) stretch ratio in the principal direction. The stress distribution in coronary and carotid arteries is nearly the same. This behaviour can be explained using the following relationship: pressure x radiusStress =^ thickness (B.1) Higher stretch ratios, which are related to the vessel radius, were obtained in carotids than in coronaries. In order to maintain the same range of stresses, lower pressures are needed in carotid arteries than in coronary arteries. The thickness was similar in both types of arteries tested. 124 APPENDIX C C. Relaxation Test of Thoracic Aorta Rectangular samples (20 mm long and 4 mm wide) from porcine thoracic aortas were cut along the circumferential axis. Uniaxial extension of the samples at a deformation rate of 10%/s was performed until a user defined stretch ratio was achieved (X = 1.5, 1.7 or 1.9). At this point, the relaxation experiment started: The material samples were held at a constant stretch while the decrease in the circumferential forces as a function of time was recorded (Fig. C.1). Uniaxial Data: Relaxation Experiment ♦1=15 o1,= 1.7 AX= 19 z 12 10U. r. 8^• A A AA, 4E- ED A ALLAA AAAAAA AALAAAAAAAAAAAl 2 4 000000000000D.aaaoco.,:300000000 ti EI 4 • • • • • • • • • • • • • • • ♦ • • • • • • • • • 4 , •^• 0 ^ 450 ^ 900 ^ 1350 ^ 1800 Time (s) Fig. C.1. Relaxation Experiment of Rectangular Porcine Thoracic Aorta Samples (Uniaxial Testing). 16 14 125 APPENDIX D D. Active Response of Thoracic Aorta. In this experimental plan the effect of passive and active mechanical tests on the behavior of the aorta was investigated. Thoracic aortas from rabbits were harvested from the local slaughterhouse. Upon arrival, rings 6 mm long were cut and prepared for uniaxial testing by attaching each ring between two clips clamped to the arm actuators as seen in Fig. D.1. Mean diameter and thickness of aortas were 4.0 and 0.7 mm, respectively. Fig. D.1. Active Pest Setup. At the beginning of the uniaxial extension, the aorta samples were only immersed in isotonic saline solution. At this stage only the passive behaviour of the tissue was recorded. The smooth muscle cells start to contract when the potassium (60 mM K+, Ca+ free) isotonic solution was added to the bath 1200 seconds after the test began, thus showing the active behavior. The relaxation of the smooth muscle cells began when 0.5 mM of EGTA was added to the bath (around 2400 seconds after the test began). The ring was stretched up to 50% deformation for 3600 triangular wave cycles at a frequency of 1.0 Hz. The bath 0.4 0.35 0 .6^0.3 (1 0.25 - Ts 0.2 - E (0 0.15 - E^0.1 -= E5 - 0.05 - • • 11116 1 • • ■ • • • ■ ■ •■ • • • ■ • • • • IIP I• • ■ ■ • • Contraction Phase Relaxation Phase 1800 Time (s) 0 0 600^1200 2400^3000^3600 Passive Behaviour 126 solution was maintained at body temperature (37 °C) and a pH of 7.4 while saturated by vigorous gassing with 95% 02 and 5% CO2 . Peak forces at the beginning (passive behaviour), middle (contraction phase) and final (relaxation phase) sections of the experimental procedure are shown in Fig. D.2. Each data point is the average of the peak forces corresponding to the 20 cycles recorded by the actuator every 200 seconds during uniaxial testing. Rabbit Aorta Passive & Active Behaviour Fig. D.2. Rabbit Aorta Passive and Active Behaviour. 2B/ 3 213 [(Cyy C„ ) /2 Ci (E.5) 2B4213 [( Czz ) — /2 Ci,yy (E.6) Sur = 2A/31/3 [1 — / 1 C 3 S„, = 2A4v3 [1 — —31 // Cid] APPENDIX E E. Second Piola-Kirchhoff Stress Tensor Derived Equations. The second Piola-Kirchhoff stresses are defined by the expression: waw aw ac.S ,; =^ =^ u^ = 2aE, ac„ aE,^ac, The simplest way to calculate the elements of the second Piola-Kirchhoff stress tensor in a finite element program is using its vector form. Only six elements have to be derived due to tensor symmetry. S„„ S„ S„^ (E.2) Su E. 1. Mooney-Rivlin 5 Parameters Model Mooney-Rivlin 5 parameters strain energy density function has the following form: W = (1 3) + am (1; 3) + (/: — 3)(/; — 3) + a 20 (/: — 3) 2 ± a30 (/: —3) 3^(E.3) To obtain the expression of the second Piola-Kirchhoff stress tensor, the following procedure has to be done: Su = 2^= 2 a10^+ a01^+acu^acu^acu aw^[ au: -3)^au-2* — 3)^a(r -3)(12* -3) + a au; -3)2^au; -3)31 a Cu^20 ac,; 30 aci,+a Su = 2[a10 +a11 (1 — 3) + 2a20 (/: — 3) + 3a30 (Ii --3) 2 ]^ + 2[a01^_ 3) ] 2ar acu^ cu S.. = 2A all+ 2B al;^(E.4)aci , The elements of the second Piola-Kirchhoff stress tensor, which were implemented in the finite element code, are the following: 127 (E. 1 ) S = S„ = 2A13 113 [1 — 3 Ci,zz^2B/;213 [(Cxx Cyv ) — 3 /2 Ci =— 3 — Ai i /3 ,^— 4B42/3 [Cxy ± —32 /2 Ci,xy S =--4 AI13 11 -1I3 C. ,yz —4BI3 -213 [C +—3 / Cyz^2 i yz Sa =-3 A/ 1 /;1/3C, — 4B/; 2/3 [C,, + — 2 /2 C, '^3 128 (E.7) (E.8) (E.9) (E.10) E.2. Guccione Model The strain energy density function developed by Guccione has the following form: 1^r W = 2 ClexpkQ — 1j (E.11) Q=bfExx +b,(Eyy +E„)+2b,Eyz 2b.fs (E^Eu )^(E.12) As expressed above, W is a function of the strain tensor E u . To obtain each component of the second Piola=Kirchhoff stress tensor is easier to express W as a function of Cu . Considering the definition of Eu, = (1/ 2)(C y — /u ) and taking into account that Eu is a symmetric tensor and substituting it in the expression for Q, the following equation is obtained: 1 fQ=- 2 112 f xx C * +b,(Cy*y +Czz* )+2b,Cy*,+2bfx (Cx; +CD+(bf +2b,)}^(E.13) The equation of Q in function of ci; is obtained using the expression Cif* = 4 1/3 Cu : I/ 3Q=— { I; [bf Cxx +b,(Cyy +C„)+2b,Cyz +212fx (Cxy +Cu d+(bf +24)1 2 (E.14) The expression of the second Piola-Kirchhoff stress tensor can be obtained as follows: Sy =2 aW =2 a 1 1^1]) = Cexp(Q) aQ acu^ac, C[exp(Q)2 acu Su = --2-1 C exp(Q) 4 2/3{^aacA 23 A45/3 a'3 (E.15) 129 where: aQ = { 13_2,3 aA. ± A a/;213 } ac, 2^ac„^ac,; (E.16) A = bf C„x +br (Cyy +C„)+2b,C yz +2b1s (C ,y + C,)^ (E.17) The elements of the second Piola-Kirchhoff stress tensor, which were implemented in the finite element code, are the following: SL, = —1 C exp(Q)4 2/ 1 [ b fC 3 AC d^ (E.18) 12 S yy = 2 C exp(Q)42/3 [b( C yy^AC ,, yy l Szz 2 = —Cexp(Q)4 2/3 [bt Ca 3-- AC . 1 Sxy = C exp(Q)/;2/3^Cxy 32 ACi , x), S „ = C exp(Q)/3 2/3 [br C„ — 3 AC ,,y=] = Cexp(Q)42/3 [bis C, --32 A (E.19) (E.20) (E.21) (E.22) (E.23) E.3. Fung Model The strain energy density function developed by Fung has the following form: W = —1 bo kxp(Q)-1]^ (E.24) 2 Q = bi E„ + b2 E )2,y + b3 EL + 2b 4 E „E + 2b5 E yy E „ + 2b6E „ + + b7 (E y + E )20+ b8 (E;z +^b9(E, +^(E.25) The value of Q as a function of has the following form: Q = 1—142' 3 — 24113 A2 + A3 } 4 where A1, A2 and A3 are defined as: = b1C X + b2 C y2 y + b3C, + 2b4 C „„C yy + 2b5 C yy C + 2b6 C zzC + + 2b7 C xy2 + 2b8 C + 2b9 C (E.26) (E.27) 130 A, = (b1 + b4 + b6 )Cxx + (b4 +b2 + b5 )Cyy + (b6 + b5 + b3 )Czz^ (E.28) A3 = b, + b2 + b3 + 2b4 + 2b5 + 2b6^(E.29) To obtain the expression of the second Piola-Kirchhoff stress tensor, the following procedure has to be done: aw^a 1r^ aQSu — 2^ = 2^ bo Lexp(Q) — = bo exp(Q)^ac^ac 2 ac, 1^aA, 2 413-5,3 oL - ' 2/- J^+ 2 A^a/3= —bo exp(Q) 4 2/34 ac, 3^acij^acij 3 '2 3 ocu where: (E.30) a Q a42/3 { 2 4 1/3 aA2 + A2 aC113 aCu = 1 1 _213 aA, ± A ' 4 3 aCii^acij^ocu^ac„ (E.31) The elements of the second Piola-Kirchhoff stress tensor, which were implemented in the finite element code, are the following: S 2 bo exp(Q).1;113 f(b,C + b4 C yy + b6 C „)— 3 — A,C, 1CV 3 — (b, +b4 + b6 ) + 31 A, ^(E.32) 1S = — 2 bo exp(Q)/3 1/3 [(b4 Cxx + b2 Cu + b5 C1 ) -3-1 illq, yy j/;1/3 (b4 + b2 + b5 )+ -3-1 A2 Ci, vv SZZ = bo exp(Q)/;1/3 1(b6C, + b5 C yy + b3 C zz ) — 1 Al Cia, IC 1/3 — (b6 b5 + b3 ) + 1 A2q,„} S xy = bo exp(Q)I;113 fb,C xy — 3 Al C ixy 141/3 + 3 A2C S yz = bo exp(Q)4 1/3 {[kCyz — 3 A,Ciyz ]4 1/3 + 3 A2Ci, S = bo exp(Q)11b9 — —1 3 Al Ci' j/3 -' 3 3 A2C (E.33) (E.34) (E.35) (E.36) (E.37) 131 E.4. Holzapfel Model The strain energy density function developed by Holzapfel has the following form: 147 (C*, A , A2 ) = Wiso (/]* ) 4- Wanisofr; 9161^ (E.38) where: Wso (I1*)= L)-2 (/, —3) Waniso (/ *4 , / ^ E le xp[k 2 (/, 2k2 i-4,6 (E.39) (E.40) Then, the strain energy function (Wj) for each artery layer can be written as ( J I,M,A ): W, =(/* —3 )+^ iexPik2,^1-1}± [exPk2, (4/ —1 )2 ]-1 P12^2k2, (E.41) where: = Cx*x + Cy*, + Cu = /; 1/3 /1 (E.42) I *^= C # : A1 j :=1-113I4 j (E.43) I ^C : A2 j =134316 j (E.44) = aol, OO aOlj (E.45) A2 j = a021 0 a021 (E.46) am ) = (0, cos )3.1' sin fi J (E.47) a02 j = (0, cos /3 '— sin ) (E.48) , / *4 and I: are invariants of the Green-Cauchy right tensor; Al, A2 are tensors defined by the product of two (reference) direction vectors (aoi p ao,), and 13j are the angles between the collagen fibers: 0 = cos sin 13i [0 cosf31 sin ]= 0^0^0 0^cost fij^sin A cos fir/ 0 sin /33 cosi ^sin 2 fii (E.49) A2 j = 0 cos^[0 cos^—sinfl.]= — sin 0^0^0 0^COS 2 j6i^— sin fii cos A o — sin fij cos A^sin 2 fli (E.50) /, C, exp [A] + 3^ Y ■^/, . + (76; —1) cos t^+^C .^exp[B]3^,YY = bjI;113 [1 — Co] + 21( IiI;113 4 j — 1) cost J.3; 14 . sin 2 /3.^C.3 exp[A]+ I sin e f3;^Ci zz exp[B]} 3^' 132 14, = 14; =COs 16j = 16] =COS c,^C„,^C„ C ry^C„^Cyz c,^C„^Czz _ 2 Acyy + sin C„ C,Cxx^ _ C„,^C„,^Cy, CZx^C,^Czz 2 igiCyy — sin 0^0 :^0^cos t fij^sin^cos 0^sin^cost;^sin 2 fij cos ,6; (cyz +^) + sin 2 f; Cu 0^0^0 : 0^cos 2 /1 j^— sin^cos 0^— sin A cos flf^sin2fli cos^(Cy, + Czy ) + sin 2 fij Czz (E.51) (E.52) The expression of the second Piola-Kirchhoff stress tensor (m, n = x, y, z) can be obtained as follows, where the term inside each exponential can be defined as A=1c, j (1 *4j —1)2 and B = lc21 (1:j — 1)2 ^aw^411*-3) k11 { a [exp[A]-1]+ a [exp[B]-1]mn ^ S = 2 acm =b; acmn + k2 aCmn^ac„,„ S,„ n =b. a ^+ 2k . ac,„„ ,^,^,^,*(1:; 1)exppl ^ +^—1)exp[Bj^},, ac a1 6jac. m,, (E.53) Thus, the elements of the second Piola-Kirchhoff stress tensor are: S A.„=b j I;113 [1--1—C,„x 1- 3-k1j 1;113 C,,,,{1, j (1 *4j -1)exp[A]+ I6 j (16* — 1)exp[B]}^(E.54) (E.55) S z, = b j I;v3 [1 — 31 C,„] + 2k, jI;v3^* --1 + (I6j — 1 S xy = — ;bjI I I;1/3 Ci,xy --43 k1 j4113Ci,xy {14./ (1 *4./ - 1)exp[A]+ /6j^—1)exp[B]} S yz = -- 2 3 b11 I3 113Ci,yz +4k 1 j 1343 (14* j —lsin^cos^— 3 /4j Ci,yz exp[A] — J l — 1)(sin Jai cos fi. + ^/ .C,./^3^of^ yz exp[B]l ^SzX = — —2^_ibi/i/;' C, — —k, j 13 /3^{/4 " (/* — 1)exp[A]+ I61 (16 ; —1)exp[Bn3 4 3 (E.56) (E.57) (E.58) (E.59) 133 APPENDIX F F. Modulus D Matrix Implemented in the Finite Element Code. The modulus matrix relates the variation of the Piola-Kirchhoff stress tensor with the variations of the Lagrange deformation tensor. D matrix is also a symmetric tensor. D12 D13 D14 D15 D16 D22 D23 D24 D25 D26 D33 D34 D35 D36 D44 D45 D46 D55 D56 D66_ where: S I = Sxx^E1=Exx=(112)(Cxx-1)^C1=Cxx S2 = S yy^E2 E yy = (1/ 2)(C yy - 1)^C2 = C yy 53 = S zZ^E3 = Ezz = (1/2)(Cu -1)^C3 = Czz 54 Sxy = Syx^E4 Exy ±Eyx =Cxy C yx^C4 = Cxy 55 Syz = S zy^E5 = Eyz Ezy =Cyz =Czy^C5 C yz S6 = S„ Sxz^E6 Ezx +Exz =Cz,=Cxz^C6 = Czx In order to obtain the values of the elements of the modulus matrix, the following derivative has to be solved: as.^as ; .^= 2^ , where i, j = 1, 2, ..., 6.^ (F.2)OVE. i As an example, the particular equation of D11 for some of the constitutive equations implemented in IMI's finite element code is shown above. The rest of the elements of the modulus matrix have similar expressions. F.1. Mooney-Rivlin 5 Parameters Model D11 2 a S" = 8 ! A/ 1/3 r / I C, - dC,, + 3 - BI, 213 C l [-5- I^-2(C yy +C)]+•^a x,^3^3^- ^3 + 4213(1-3-1 / 1 C,,,, f(c4 + 3c5 (/: - 3))(1- -3-1 / I C ) + c3 /;v3 ((Cyy + C ) - -3-2 I2C, ,xx D = as, as, as, as4 as5 as, DII aE, aE, aE, aE4 aE5 aE, (F.1) F.2. Guccione Model 134 D„ = C exp(Q)I3213 {b f — 3 C fC xx — AC S ^22Sx,{^xx C exp(Q) 3 C F.3. Fung Model 22S ^bo exp(Q) =^ +^3b1/32/3 —4(bI Cxx^+b6C„)/3-2/3C,,„x + 2(b1 +b4 +b6 )/3u3 C,,xx + bo exp(Q) 3 4 _+ 3 — ^3 — — /3 1/3 Ci ,2xxA2 } F.4. Holzapfel Model = —4 bi /3 1/3 C, xx 3[-2 / C,xx —1] + —4 k i f /3^tV3 C 2^41/3(1 + 2A) exp[A] + 4 1/3 (1+ 2B) exp[B]+3 + [/4J (/41 —1)expki+ I6 j (I6i — 1)eXp[B]j}