COMPUTATIONAL MODELING OF CARDIAC MECHANICS : MICROSTRUCTUAL MODELING & PULMONARY ARTERIAL HYPERTENSION By Ce Xi A DISSERTATION Submitted to Michigan State University in p artial fulfillment of the require ments for the degree of Mechanical Engineering Doctor of Philosophy 2019 ABSTRACT COMPUTATIONAL MODELING OF CARDIAC MECHANICS: MICROSTRUCTUAL MODELING & PULMONARY ARTERIAL HYPERTENSION By Ce Xi Heart diseases, which approximately account for 31 % of all human mortality every year , are the leading cause of death worldwide. Computational cardiac models have gained increasing popularity and become an indispensable and powerful tool in elucidati ng the pathological process of different heart diseases . They can be used to estimate important physiological and clinically relevant quantities that are difficult to directly measure in experiments. The broad goals of this thesis were to develop 1) a micr ostructure - based constitutive model of the heart and 2) patient - specific computational models that would ultimately help medical scientists to diagnose and treat heart diseases. Heart diseases such as heart failure with preserved ejection fraction (HFpEF ) are characterized by abnormalities of ventricular function that can be attributed to, changes in geometry, impaired myocyte (LV) relaxation, cardiac fibrosis and myocyte passive stiffening. Understanding how LV filling is affected by each of the many mic rostructural pathological features in heart diseases is very important and may help in the development of appropriate treatments. To address this need, we have developed and validated a microstructure - based computational model of the myocardium to investig ate the role of tissue constituents and their ultrastructure in affecting the heart function. The model predicted that the LV filling function is sensitive to the collagen ultrastructure and the load taken up by the tissue constituents varies depending on the LV transmural location. This finding may have implication s in the development of new pharmaceutical treatments targeting individual cardiac tissue constituents to normalize LV fil ling function in HFpEF. Pulmonary arterial hypertension (PAH) is a life - threatening disease characterized by elevated pulmonary artery pressure (PAP) and pulmonary artery vascular resistance, with limited survival rate and can affect patients of all ages. The increased pressure o r afterload in the right ventricle (RV) can result in pathological changes in RV mechanics, which are currently not well - understood. this void , we have develop ed patient - specific computational model s to investigate effects of PAH on ventricular mech anics. have quanti fied contractility, and passive tissue stiffness in PAH patient s , and compare them to those found in age - and gender - matched normal controls. Our results showed that RV longitudinal, circumferential and radial strain were depressed in PAH patients compared with controls; RV passive stiffness increased progressively with the degree of remodeling as indexed by the RV and LV end - diastolic volume ratio (RVEDV/LVEDV); Peak co ntractility of the RV was found to be strongly correlated, and had an inverse relationship with RVEDV/LVEDV. These results provide the mechanical basis of using RVEDV/LVEDV as a clinical index for delineating disease severity and estimating RVF W contractil ity in PAH patients. iv ACKNOWLEDGEMENTS First of all, I would like to thank my advisor, Dr. Lik Chuan Lee, who gave me tremendous support and guidance during my entire Ph.D. research study. During those hard times when I me up and instead encouraged me and helped me to discover my potentials to overcome various difficult challenges . Also, I would like to thank my committee members, Dr. Seungik Baek, Dr. Sa ra Roccabianca, Dr. Tamara Reid Bush and Dr. Mohsen Zayernouri for providing valuable advice and insights for this research. Second, I would like to thank my collaborators in National Heart Center Singapore ( Hua Zou , Xiaodan Zhao and Liang Zhong), École Polytechnique (Martin Genet) and Simula Research Laboratory (Henrik Finsberg and Samuel Wall) for their extraordinary contributions and continuous support in this research. Third, I would like to thank all my families who always believe in me and care abo ut me. Especially, I would like to thank my grandpa who cultivate and educate me since my childhood. Although he passed away during my first year in Michigan State University, I will never forget his goodness to me and will continue to move forward to make him feel proud of me. Last but not least, I would like to thank my lifetime friend Ziguang Lee and my girlfriend Mengqi Tu for their companion and warmhearted help during th ose lonely and dark times. v TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ......... viii LIST OF FIGURES ................................ ................................ ................................ .......... ix CHAPTER 1 OVERVIEW ................................ ................................ ................................ 1 1.1 Background ................................ ................................ ................................ .......... 2 1. 2 Objectives of this dissertation ................................ ................................ ............ 7 CHAPTER 2 LITERATURE REVIEW ................................ ................................ ............. 8 2.1 Heart anatomy and microstructure ................................ ................................ ..... 9 2.1.1 Organ structure ................................ ................................ .......................... 9 2.1 .2 Histology ................................ ................................ ................................ ... 10 2.2 Cardiovascular diseases ................................ ................................ ................... 13 2. 2 .1 Pulmonary arterial hypertension ................................ ............................ 13 2.2.2 Heart failure with preserved ejection fraction ................................ ........ 15 2.3 Cardiac imaging techniques ................................ ................................ ............. 17 2. 3 .1 Echocardiography ................................ ................................ .................... 1 7 2.3.2 Cardiac magnetic resonance imaging ................................ .................... 18 2.4 Cardiac computational modeling ................................ ................................ ...... 19 2. 4 .1 Phenomenological models ................................ ................................ ...... 20 2.4.2 Microstructural models ................................ ................................ ............ 2 1 CHAPTER 3 MICROSTRUCTURE - BASED FINITE ELEMENT MODEL OF LEFT VENTRICLE PASSIVE INFLATION ................................ ................................ ............. 23 3.1 Introduction ................................ ................................ ................................ ........ 25 3.2 Mater ials and m ethods ................................ ................................ ...................... 27 3.2.1 Microstructural constitutive model formulation ................................ ... 27 3.2.2 Parameter estimation ................................ ................................ .............. 3 3 3.2.3 Finite element simulation of left ventricular passive filling ................ 3 5 3.2.4 Statistical a nalysis ................................ ................................ .................. 38 3.3 Results ................................ ................................ ................................ ................ 39 3.3.1 Biaxial tests ................................ ................................ .............................. 39 3.3.2 Finite element simulation of passive filling ................................ .......... 43 3.4 Discussion ................................ ................................ ................................ .......... 53 3.5 Limitations ................................ ................................ ................................ .......... 57 CHAPTER 4 QUANTIF ICATION OF THREE - DIMENSIONAL BI VENTR ICULAR STRAINS IN PULMONARY ARTERIAL HYPERTENSION USING HYPERELASTIC WARPING ................................ ................................ ................................ ..................... 59 4.1 Introduction ................................ ................................ ................................ ........ 61 4.2 Materials and m ethods ................................ ................................ ...................... 63 4.2.1 Study population ................................ ................................ ..................... 63 4.2.2 CMR image acquisition ................................ ................................ ........... 65 vi 4.2.3 Right heart catheterization ................................ ................................ ..... 67 4 .2.4 Overall framework of hyperelastic warping registration ...................... 67 4.3 Results ................................ ................................ ................................ ................ 73 4.3.1 Patient demographics ................................ ................................ ............. 73 4.3.2 Peak systolic circumferential, longitudinal and radial strains ............ 73 4.4 Discussion ................................ ................................ ................................ .......... 74 4.4.1 RV strains ................................ ................................ ................................ 75 4.4.2 Septum and LV strains ................................ ................................ ............ 75 4.5 Limitations ................................ ................................ ................................ ......... 76 CHAPTER 5 PATIENT - SPECIFIC COMPUTATIONAL ANALYSIS OF VENTRICULAR MECHANICS IN PULMONARY ARTERIAL HYPERTENSION ................................ .... 77 5.1 Introduction ................................ ................................ ................................ ........ 80 5.2 Methods ................................ ................................ ................................ .............. 81 5.2.1 Acquisition ................................ ................................ ............................... 81 5.2.2 Biventricular geometry ................................ ................................ ........... 82 5.2.3 Pressure - volume loops ................................ ................................ ........... 82 5.2.4 Strain analysis of magnetic resonance images ................................ .... 83 5.2 .5 Biventricular finite element model ................................ ......................... 83 5.3 Results ................................ ................................ ................................ ................ 89 5.3.1 Pressure - volume loops ................................ ................................ ........... 89 5.3.2 Regional circumferential strain ................................ .............................. 90 5.3.3 Myofiber stress ................................ ................................ ........................ 91 5.3.4 Model parameters ................................ ................................ .................... 92 5.3.5 Se ptum c urvature ................................ ................................ .................... 93 5.4 Discussion ................................ ................................ ................................ .......... 96 5.4.1 Circumferential strain ................................ ................................ ............. 97 5.4.2 Myofiber stress ................................ ................................ ........................ 97 5.4.3 Fitted model parameters ................................ ................................ ......... 98 5.4.4 Septum curvature ................................ ................................ .................... 99 5.5 Model limitations ................................ ................................ .............................. 1 01 CHAPTER 6 EFFICIENT ESTIMATION OF PERSONALIZED BIVENTRICULAR MECHANICAL FUNCTION EMPLOYING GRADIENT - BASED OPTIMIZATION ...... 1 03 6.1 Introduction ................................ ................................ ................................ ...... 1 05 6.2 Methods ................................ ................................ ................................ ............ 1 08 6.2.1 Data acquisition and pre - processing ................................ ................... 108 6.2.2 Mechanical m odeling ................................ ................................ ............ 1 11 6.2.3 PDE - constrained optimization ................................ ............................. 1 14 6.2.4 Parameter estimation ................................ ................................ ............ 1 17 6.2.5 Implementation details ................................ ................................ .......... 1 20 6.3 Results ................................ ................................ ................................ .............. 1 21 6.3.1 Data assimilation, validation and solver performance ................. 1 21 6.3.2 Mechanical analysis ................................ ................................ .............. 1 24 6.4 Discussion ................................ ................................ ................................ ........ 1 27 6.4.1 Data compatibility and multi - objective optimization .......................... 1 28 vii 6.4.2 Contractility ................................ ................................ ........................... 1 29 6.4.3 Fiber stress ................................ ................................ ............................ 1 29 6.4.4 Fiber angle sensitivity ................................ ................................ ........... 1 30 6.5 Limit ations ................................ ................................ ................................ ........ 1 32 CHAPTER 7 PATIENT DATA ASSIMILATION ANALYSIS SHOWS AN INVERSE LINEAR RELATIONSHIP BETWEEN RV CONTRACTION AND REMODELING IN PULMONARY ARTERIAL HYPERTENSION ................................ ............................. 1 35 7.1 Introduction ................................ ................................ ................................ ...... 1 37 7.2 Methods ................................ ................................ ................................ ............ 1 41 7.2.1 Patient cohort and data processing ................................ ..................... 1 41 7.2.2 Con struction of personalized models ................................ ................. 1 43 7.3 Results ................................ ................................ ................................ .............. 1 46 7.3.1 Patient data and regional strains ................................ ....................... 1 46 7.3.2 Model results: contractility, stiffness and stress ............................... 1 48 7.4 Discussion ................................ ................................ ................................ ........ 1 55 7.5 Limitations ................................ ................................ ................................ ........ 1 58 7.6 Conclusion ................................ ................................ ................................ ........ 1 59 CHAPTER 8 CONCLUSIONS ................................ ................................ .................... 1 60 APPENDICES ................................ ................................ ................................ ............. 1 63 APPENDIX A: Plane stress formulation ................................ ............................... 1 6 4 APPENDIX B: Results of biaxial tests for other specimens ............................... 1 66 APPENDIX C: Muscle fiber stress and stiffness modulus ................................ .. 171 APPENDIX D: Strain as a function of volume change ................................ .. 172 APPENDIX E: Mesh sensitivity and computational efficiency ........................... 174 APPENDIX F: Parameter sensitivity analysis ................................ ...................... 17 7 BIBLIOGRAPHY ................................ ................................ ................................ ......... 179 viii LIST OF TABLES Table 3.1 . List of model parameters with the corresponding lower/upper bounds according to literature references ................................ ................................ .................. 3 5 ................................ ................................ .. 4 1 Table 3.3 . Comparison of fitted parameters with measurements of isolated myocytes and collagen fibers ................................ ................................ ................................ ............... 4 2 Table 3.4 . Average correlatio n matrix of fitted model parameters ................................ . 4 3 Table 3.5 . Percentage of absolute change in at a LV pressure of 20 and 30 mmHg with 100% change in each model parameter about its mean fitted value ...................... 5 2 Table 4.1 . Baseline demographic and CMR characteristics of P A H patients and Controls. The control group did not undergo right heart catheterization ................................ ...... 64 Table 4.2 . Average circumferential, longitudinal and radial strains in RV, LV and septum ................................ ................................ ................................ ................................ ...... 74 Table 5.1 . Model parameters. References from which the parameter values were obtained are denoted by square bracket ................................ ................................ ....... 87 Table 5.2 . Regional model parameters units: and in kPa; in kPa ms/mL ................................ ................................ ................................ .................... 93 Table 6.1 . Average LV and RV fiber stress (kPa) at end - diastole and end - systole ..... 1 26 Table 7.1 . Demographics of the PAH patient and control groups .............................. 1 40 Table E1 . Running time of passive inflation simulation for a FE mesh of 3565 elements with different number of processors (Intel ® Xeon (R) CPU E5 - 1660 v3@ 3.00GHz x 16) ................................ ................................ ................................ ................................ .... 176 Table A1 . Sensitivity analysis of model parameters in the normal case ..................... 1 77 ix LIST OF FIGURES Figure 2.1. Cross - section of the heart, showing atria, ventricles and valves . Adapted from ( https://www.texasheart.org ) ................................ ................................ ........................ 1 0 Figure 2.2 . Illustration showing the comprising layers of the heart wall [65] ................. 1 1 Figure 2.3. Schematic representation of the fibrillar collagen matrix of the myocardium (Adapted from Weber et al. [1] ) ................................ ................................ ..................... 1 2 Figure 3.1 . (a) Unloaded canine LV FE geometry (all dimensions are in cm) . (b) Muscle fiber orientation with a linear transmural variation from at the endocardium to at the epicardium. (c) Schematic representation of the local arrangement of muscle fiber and collagen fiber network. The muscle fiber defines an axis of s ymmetry for the spatial arrangement of collagen fiber that attached to it. Collagen fiber dispersion is prescribed . An azimuthal angle corresponds to the axial fiber family whereas a circumferential angle corresponds to the radial fiber family ................................ ................................ ................................ ..................... 3 3 Figure 3.2 . (a) Transmural division of mesh into 7 layers with the same wall thickness. (b) Circumferential direction. (c) Radial directio n. (d) Longitudinal direction ................. 38 Figure 3.3 . (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0118. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxial test of specimen 0118 ................................ ................................ ...... 4 0 Figure 3.4 . Comparison of the inflation pressure - volume curves d erived from the 6 sets of fitted model parameter values (Table 3.2) and their mean values with measurements (plotted as mean ± SD) ................................ ................................ ................................ 4 4 Figure 3.5 . Transmural distributions of the strain components at a LV pressure of 8 mmHg. Colored solid lines: model predictions using best - fit and mean parameters in Table 2. Measurements of strains at LV pressure of 8 ± 4 mmHg [2] are plotted as means ± SD. Note: shear strain results from experimental data are not sho wn as the measured values are small and difficult to digitiz e ................................ ................................ ......... 4 6 Figure 3.6 . Transmural distributions of the stresses at a ventricular pressure of 8 mmHg. Colored solid lines: model predictions using best - fit and mean parameters in Table 3.2 ................................ ................................ ................................ ................................ ...... 47 Figure 3.7 . Transmural variation of the contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total normal stress in x the local muscle fiber direction at an LV pressure of 8 mmHg using the mean fitte d parameter values from Table 3.2 ................................ ................................ .................. 49 Figure 3.8 . Collagen fiber tortuosity as a function of LV pressure. Colored solid lines: model predictions using the best - fit and mean parameters in Table 2. Black scattered dots: measurements of the perimysial collagen fiber tortuosity in rat hearts [149] . Black square marker with error bars (mean SD): measurements of collagen tortuosity in normal rat LV [150] ................................ ................................ ................................ ........ 5 0 Figure 3.9 . Sensitivity of (a) collagen vol ume fraction , (b) collagen waviness , (c) collagen waviness deviation , (e) collagen azimuthal angle , (f) collagen azimuthal angle deviation , (g) muscle stiffness and (h) muscle fiber helix angle to the pressure - volume curve ................................ .................. 5 1 Figure 4.1 . CMR images including short - axis image from apex to basal, and 2 - chamber, 3 chamber and 4 - chamber long - axis images ................................ ................................ 65 Figure 4.2 . Overall framework of quantifying the biventricu lar circumferential, longitudinal and radial strains ................................ ................................ ................................ ........... 66 Figure 4.3 . Reconstr uction of biventricular mesh: (A ) segmentation of contours of LV, RV and septum; ( B ) Surface reconstruction; ( C ) Bi - ventricular geometry ( D ) Biventricular mesh; ( E ) Partition of region (Red: RV free wall; Green: Septum; Blue: LV free wall) .. 69 Figure 4.4 . Measurement of the septal displacement in 4 - chamber view as the boundary condition: ( A ) End of systole; ( B ) End of diastole; ( C ) Apply the displacement of septal on the model ................................ ................................ ................................ ...................... 69 Figure 4.5 . Registration the meshed model with images for strain - time curves ( A ) meshed model with CMR image; ( B ) circumferential strain orientation; ( C ) longitudinal strain orientation; ( D ) radial strain orientation; ( E ) circumferential strain - time curve ; ( F ) longitudinal strain - time curve; ( G ) radial strain - time curve ................................ ............ 72 Figure 5.1 . (a) Biventricular geometry reconstructed from MR images with LV (right, light region) and RVFW (left, dark region) material regions. (b) Myofiber orientation prescribed using LDRB method [3] with a transmural variation of 60 degree (endo) to - 60 degree (epi) for the entire biventricular model. (c) Coupling the biventricular model to three - element Windkessel mo dels. (d) Polar coordinate used to calculate local cu endocardium ................................ ................................ ................................ ................. 88 Figure 5.2 . Measurements and model predictions of the PV loops for the (a) PAH patient - systole) ................................ ............. 89 Figure 5.3 . Circumf erential strain for (a) PAH patient and (b) normal subject. (c) Comparison of regional peak negative between the PAH patient and normal subject ................................ ................................ ................................ ................................ ...... 91 xi Figure 5.4 . Peak myofiber stress in the PAH patient (left) and normal subject (right) 92 Figure 5.5 . normal subject. (c) Comparison of normalized septum curvature between PAH patient and normal subject ................................ ................................ ................................ ........ 94 Figure 5.6 . Effects of afterload on septum curvature wit hin a cardiac cycle. (a) Time - systolic curvature versus end - systolic transseptal pressure gradient. (c) Corresponding PV loops ............................. 95 Figure 5.7 . Effects of preload on septum curvature within a card iac cycle. (a) Time course - diastolic curvature versus early - diastolic transseptal pressure gradient. (c) Corresponding PV loops ................................ .......... 95 Figure 6.1 . Left: finite element mesh of a biventricular geometry reconstructed from MR images separated into 3 material regions, namely, LVFW (blue), septum (green) and RVFW (right). Right: myocardial fiber orientation are assigned using the LDRB algorithm [4] with an angle + prescribed on the endocardium and epicardium, respectively. Here showing the fiber architectu re for ranging from 30 to 80 with increments of 10 , where the absolute value of the fiber angle is used as color - map ................................ ................................ ................................ ................................ .... 1 10 Figure 6.2 . To the left we see the model - personalization pipeline . To the right we show a comparison of the unloaded geometry for CASE3. The upper figure shows the resulting unloaded, zero pressure geometry in red and the original image - based geometry in transparent, while the bottom figure shows the unloaded geometry, i nflated to the target pressure in the image - based geometry, and the original image - based geometry in transparent for comparison ................................ ....................... 1 18 Figure 6.3 . Results of the gradient - based minimization of model - data mismatch for different choice of fiber angles. Left: s imulated (color lines) and measured (black circles) PV loops in the LV (top row) and RV (bottom row). Center: simulated (color lines) and measured (black circles) circumferential strain in the LV (top row), RV (middle row) and septum (bottom row). Right: a verage values of cost functional for the volume (top row) and strain (bottom row), for each choice of fiber angle ........ 1 22 Figure 6.4 . Validation of the model - personalization process using simulated and measured longitudinal strain which was not used in the optimization. Upper, middle and lower panel show the longitudinal strain curves for different choice of fiber angles in the LV, RV and septum respectively ................................ ................................ ..... 1 23 Figure 6.5 . To the left, average traces of the active strain parameter in Eq. (6.3) in the LV (top) and RV (bottom) for different choice of fiber angle. To the right average traces of Cauchy fiber stress in the LV (top) and RV (bottom) for different choice of fiber angle. The fiber angles were defined symmetrically across the wall with a negative a ngle on the epicardium and a positive angle on the endocardium ranging from 30 with xii increments of 10 . On the x axis we plot the normalized time with respect to end - diastole (ED) and end - systole (ES). Horizontal dotted lines indicate timings of open ing of the aortic and mitral valve ................................ ................................ ................................ ........... 125 Figure 6.6 . Comparison of fiber stress and cardiac contraction using the active strain and active stress approach using 60 fiber angle. To the left, average traces of the active stress (top) parameter in Eq. (6 .8), and the active strain (bottom) parameter in 3. The active stress parameter, with unit kPa, is plotted on a logarithmic scale for easier comparison with the active strain parameter. To the right, estimated Cauchy fiber stress using the active stress (t op) and active strain formulation (bottom). On the x axis we plot the normalized time with respect to end - diastole (ED) and end - systole (ES). Horizontal dotted lines indicate timings of opening and closing of the aortic valve ...................... 1 26 Figure 6.7. Snap shots of the end - diastolic and end - systolic configuration and the estimated fiber stresses shown as color - map ................................ ............................ 1 27 Figure 7.1 . Comparison of peak regional (a) circumferential and (b) longitudinal strains between control (unstriped) and PAH (striped) groups with (* ) denoting that the latter value is statistically lower (P<0.05) than the former. Time traces of RVFW longitudinal strain of the (c) PAH patients and (d) control subjects. Thick line denotes average and error bar denotes standard deviation ................................ ................................ ........... 1 47 Figure 7. 2. Examples of simulations results from a control (left column) and an extensively remodeled (RVDEV/LVEDV = 1.75) PAH patient (right column). A) PV loop data (open markers) and simulations (solid lines) for the LV (red) and RV (blue). B) Calculated fiber str esses at end diastole, and C) Calculated fiber stresses at end systole ................................ ................................ ................................ ................................ .... 1 49 Figure 7. 3. Overall data assimilation errors in the control (circle) and PAH patient (diamond) populations. Comparison between measured and simulated (a) volumes for both the RV (blue) and LV (red); (b) for the LVFW (red), SEPT (green), and RVFW (blue) at all cardiac time points. A y = x line is also plotted to show the zero error reference ................................ ................................ ................................ ................................ .... 1 50 Figure 7.4 . Analysis of peak RVFW contractility for control and PAH patient groups. (a) Graphical depiction of for both controls (diamonds) and patients (black dots) as a function of the ratio of RVEDV/LVEDV. Dashed line shows a linear fit to the PAH patient group ( = - 0.13(RVEDV/LVEDV) +0.44, R 2 = 0.77). (b) Average for controls patients with RVEDV /LVEDV < 1.5 (similar to controls) and patients with RVEDVi ( = - 0.001(RVEDVi) +0.32, R 2 = 0.40) (d) Linear fit of with RVEF ( = - 0.39(RVEF) +0.09, R 2 = 0.50) ................................ ................................ ..................... 1 51 Figure 7.5 . Analysis of peak LVFW contractility for control and PAH patient groups. (a) Graphical depiction of peak LVFW contractility for both controls (diamonds) and patients (black dots) as a function of the ration of RVEDV/LVEDV. (b) Average xiii for controls, patients with RVEDV/LVEDV< 1.5 ................................ ................................ ................................ ............................... 1 52 Figure 7. 6. Comparison of passive tissue stiffness parameter between control and the PAH groups in the (a) LVFW + SEPT and (b) RVFW regions ............................... 1 53 Figure 7. 7. Comparison of peak myofiber wall stress between control and the PAH groups in the (a) LVFW and (b) RVFW regions ................................ .................. 1 54 Figure B1 . (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0119. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxial test of specimen 0119 ................................ ................................ .... 166 Figure B2 . (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0124. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial f luid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxial test of specimen 0124 ................................ ................................ .... 167 Figure B3 . (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and cons tant = 1.1, 1.15 and 1.2 tests of specimen 0217. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directio ns in the equi - biaxial test of specimen 0217 ................................ ................................ ... 168 Figure B4 . (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0303. (b) Contribution of the collagen fiber network , muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxial test of specimen 0303 ................................ ................................ .... 169 Figure B5 . (a) Comparison of the fitted and exp erimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0330. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in th e fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxial test of specimen 0330 ................................ ................................ ... 170 Figure D1 . Normal strain components vs. LV cavity volume change at the sub - endocardial, midwall and sub - epicardial regions. Colored solid lines: model predictions using best - fit parameters and their mean values (Table 3.2 of the main text). Experimental data [2] are plotted as means ± SD. Note: sub - epicardial, midwall and sub - endocardial regions are reported as 2.0 ± 0. 3%, 40.6 ± 5.5%, 79.1 ± 10.6% of the total wall thickness with respect to the epicardium in the experiments, which in our case, roughly corresponds xiv to layer 1 (0 - 14%), layer 3 (29 - 43%) and layer 6 (71 - 86%) of the LV FE model shown in Figure 3.2 in the main text ................................ ................................ ....................... 172 Figure D2 . Shear strain components vs. LV cavity volume change at the sub - endocardial, midwall and sub - epicardial regions. Colored solid lines: model predictions using the best - fit parameters and their mean values in Table 3.2 of the ma in text. Experimental data [2] are plotted as means ± SD. Refer to Figure D1 for details of the transmural locations ................................ ................................ ................................ .... 173 Figure E1 . Effects of number of elements on the passive pressure volume relation using the mean fitted model parameters from biaxial tests ................................ ................... 174 Figure E2 . Effects of mesh density (number of cell elements) on the estimated transmural fiber stress at LV pressure of 8mmHg or 1.07kpa using the mean fitted model parameters (Table 3.2) from the biaxial tests ................................ ................................ ................. 175 Figure A1 . PV loops for perturbation of different model parameters in the normal case ................................ ................................ ................................ ................................ .... 1 78 Figure A2. PV loops of the normal case using fitted PAH model parameters ............ 178 1 CHAPTER 1 OVERVIEW 2 1.1 Background Heart diseases, which approximately account for 31 % of all human mortality every year , are the leading cause of death worldwide. With the recent advancement in both experimental and imaging techniques, patient - specific computational cardiac modeling using the finite element (FE) method has gained increasin g popularity and has become an indispensable and powerful tool in elucidating the effects of various heart diseases and clinical interventions such as passive left ventricular (LV) constraint [5], [6] and cardiac support device (CSD) [7] . Cardiac computational model can be used to estimate important physiological and clinically relevant quantities t to directly measure in experiments. For example, myocardial stress distribution s , which are important determinant s of myocardial oxygen consumption [8], [9] and ventricular hypertrophy [10] , are not clinically or experimentally measurable. Com putational heart models can also provide deformation map, stress and strain fields that supplement clinical data. They also enable one to quantify the relative contribution of underlying mechanisms in the human heart towards some phenomenon without incurring the typical high costs associated with conducting new experiments [11] . In spite of significant advancements associated with computational cardiac models, there remains unresolved issues and aspects that need to be addressed and improved. Here, we list 3 issues and limitations as described below: A . Lack of microstructure - based computational model to quantify the contribution of individual tissue constituent to LV function Construction of a constitutive law for the myocardium is of fundamental importance for accurate computational simulation and analysis of the mechanical behavior of 3 ventricular myocardium. Two traditionally employed approaches for modeling the mechanical behavior of soft t issues at the continuum level are, namely, th e phenomenological or structure - based approaches. The phenomenological approach in soft tissue mechanics was pioneered by Fung [12] , who found that the elasticity of living soft tissue is highly nonlinear, and that the elastic stress of elongation test is approximately an exponential function of the stretch ratio. This result lays the foundation for many future phenomenological constitutive models in soft tissues, including those of the myocardium [13] [16] , a rteries [17] , skin [18] . Although phenomenological models can represent the bulk cardiac tissue mechanical behavior, their model parameters lack phy sical interpretation and cannot elucidate the underlying mechanisms of tissue behavior. Microstructural constitutive models attempt to integrate information on tissue composition and microstructure to avoid ambiguities in material characterization. This class of models can potentially offer insights into the mechanical effects due to changes in function, structure, and mechanics of individual tissue components [19] . Structural constitutive models have been developed for a variety of intact tissues and tissue components including lung [20] , collagen [2 1], [22] , cartilage [23] , heart valves [24] , and maturing skin [25] . Lanir developed the first comprehensive, multidimensional structural constitutive model formulation [26] . With some modifications made to the original formulation , Horowitz et al. proposed a three - dimensional constitutive law for the passive myocardium [27] and implemented it in a two - dimensional finite element framework for the simulation of the passive - state mechanics of a thin myocardial strip [28] , together with three - dimensional formul ation for a thick ventricular wall segment [29] . Microstructural 4 approach, however, requires a detailed description of the complex microstructure, thereby rendering it mathematically involved and signific antly more difficult to employ than the phenomenological approach. It remains to be implemented for the realistic 3D LV geometry with detailed measured structure information to enhance understanding of how changes in the tissue ultrastructure and micromech anical behaviors (e.g., collagen distribution and waviness, myofiber o rientation) affect LV functions. B. Difficulty in quantifying RV strain Quantification of RV function is important for the clinical management of PAH patients. Recent studies have dem onstrated that the RV strain is strongly associated with RV function and is a potentially useful prognostic marker of PAH patients [30], [31] . Compared to quantifying strain in the LV, in vivo quantitative assessment of the strain in the right ve ntricle is challenging because of complexity of ventricular shape, the thinness of the RV free wall. Tissue Doppler Echocardiography (TDE) is a relatively new real - time ultrasound technique that offer an objective means to quantify global and regional left and right ventricular function. Although it can improve the accuracy and reproducibility of conventional echocardiography, significant limitations of TDE include angle dependency [32] . Speckle - Tracking Echocardiography (STE) is t he most widely available imaging modality for quantification of myocardial deformation and is angle - independent. Its image quality of the RV is also often poor [33] . Meanwhile, cardiac magnetic resonance (CMR) imaging has emerged as the gold standard for quantification of LV and RV strains [34] [36] , it can provide tomographic images of the right ventricle with superior spatial resolution. In addition, CMR was demonstrated to have superior reproducibility over two - dimensio nal (2D) echocardiography [37] . 5 Hyperelastic warping is a deformable image registration technique that has been used to obtain strain measurements from medical images such as magnetic resonance imaging (MRI), positron emission tomography (PET) and computed tomography (CT) [38] [40] . It integrates nonlinear continuum mechanics with image - based data to find a compatible deformation map between a target image and a template image. Application of hyperelastic war ping approach in cardiac motion has been primarily confined to quantifying the LV strains [40] , which has been verified by tagged MRI [41] and 3D CSPAMM MR images [42] . Littl e has been done, however, to further this efficient and robust image registration technique to estimate regional myocardial strains in both the LV and RV, which may help serve as a potential prognostic marker for the severity of heart diseases such as PAH as well as a reference for validating computational heart models. C. Lack of patient - specific computational model that investigates right ventricular mechanics Computational models have primarily been used to analyze LV mechanics in heart diseases (e .g. myocardial infarction [43] , left branch bundle block [44] , and mitral valve regurgitation [45], [46] ) as well as heart failure tr eatments [47] . In comparison, the role of the right ventricular (RV) function has been overlooked for many years. Restricted in its role to pumping blood through a single organ (the lung) that is in close proximity, and less frequently involved compared to the left ventricle in epidemic heart diseases, the RV is less muscular and has generally been considered as a mere passive chamber [48] . Emerging data have, however, shown that RV function is the major determinant of functional state and prognosis in ma ny heart diseases such as pulmonary arterial hypertension (PAH) [49] [52] . 6 Affecting the RV function, PAH is a disease associated with vasoconstriction of the pulmonary arterial vessels that results in an elevated pulmonary arterial pressure. The clinical diagnosis for PAH is an elevated mean pul monary artery (PA) pressure greater than 25 mmHg with normal wedge pressure [53] . Without treatment, PAH can quickly lead to decompensated right heart failure and death. The current prognosis of PAH remains poo r, with about 15% mortality within 1 year on modern therapy [54] and a low 3 - year survival rate of 67% [55] . High afterload in the right ventricle caused by PAH can lead to changes in ventricular mechanics [56] . For example, an elevated RV pressure can lead to abnormal leftward motion of the septum into the left ventricle (LV) cavity. Moreover, PAH is also associated with long - term functional, structural, and geometrical changes in the RV [57] . These changes and their impacts on ventricular mechanics are currently not well - understood [58] . Although animal models have been used to understand ventricular mechanics associated with RV remodeling in PAH [59], [60] , similar studies in humans are lacking and most clinical investigations have been confined to quantifying global ventricular mechanics through pressure volume (PV) loop measurements [61], [62] . On the other hand, the i ncreased size and pressure overload in RV caused by PAH can also impair LV filling and systolic performance through ventricular interdependence [63], [64] , as well as through direct mechanical interactions via the interventricular septum and myofibers that encircle both chambers. Thus, it is of increasing importance to develop a biventricular computational model to understand and quantify biventricular mechanics in PAH. 7 1.2 Objectives of this dissertation In this dissertation, we seek to overcome the issues and limitations discussed earlier, namely: 1. Implement a microstructure - based computational modeling framework to quantify the roles and contributions of each myocardial tissue constituent to LV passive mechanics. 2. Implement and extend an effective and robust image registration technique based on hyperelastic warping to qua ntify both LV and RV strains in PAH. 3. Develop patient - specific computational model s to investigate biventricular mechanics in PAH. Specifically, we seek to quantify regional ventricular myofiber stress, myofiber strain, contractil ity, and passive tissue stiffness as well as the effects of trans - septal pressure gradients on the septum curvature based on patient data . 8 CHAPTER 2 LITERATURE REVIEW 9 2.1 Heart a natomy and m icrostructure 2. 1.1 Organ structure In humans and other mammals, the heart is a hollow muscular organ that pumps blood through vascular vessels to provide the whole body with oxygen and nutrients and to remove metabolic wastes. The heart ca n be divided into four chambers . The upper receivi ng chambers are called the left and right atria, which collect blood that returns to the heart and facilitate filling of the ventricles . T he lower discharging chambers are called the left and right ventricles ( Figure 2.1 ) , which serve as a pump for the systemic circulation and pulmonary circulation, respectively. The left ventricle is much thicker and more muscular than the right ventricle as a consequence of the need to support a higher pressure in the systemic circulation. T he cavity of the left vent ricle is conical in shape during diastole and characterized by a more spherical shape as interventricular pressure rises at the end of isovolumic contraction [65] , whereas the right ventricle possesses a more irregular crescent shape when viewed from its cross - section. The left and right sides of the heart are separated by a wall of muscle called the septum . T he heart has four valves. The mitral valve connects the left ventricle and the left atrium, which prevents backflow of blood into the left atrium when the left ventricle contract s and ejects blood to the aorta through the aortic valve. Correspondingly, the tricuspid valve connects the right atrium and the right ventricle, which prevents backflow of blood into the right atrium when the right ventricle contracts and ejects blood to the pulmona ry arte ry through the pulmonary valve. 10 Figure 2.1 . Cross - section of the hear t, showing atria, ventricles and valves. Adapted from ( https://www.texasheart.org ) 2. 1.2 Histology The heart, together with a small amount of fluid, is surrounded by a noncompliant fibrous sac called the pe ricardium, whose outer surface is called the parietal pericardium. The heart wall comprises of three distinct layers: an inner layer (the endocardium), a middle layer (the myocardium) and an outer layer (the epicardium) ( Figure 2.2 ). 11 Figure 2.2 . Illustration showing the comprising layers of the heart wall [66] The epicardium (also called visceral pericardium) is a layer of squamous cells that overlies a network of fibrillar connective tissue. The endocardium lines with the inside of the four chambers and consists mainly of epimysial collagen, elastin and a layer of endothelial cells. The myocardium is the th ickest layer of the heart wall, which is a composite material consisting of myocytes that are tethered and supported by an extensive fibrillar connective tissue network c omposed largely of collagen. T he myocytes are the most prominent type of fibers, which occupy about 70 percent of the myocardium in volume . T he remaining 30 percent consists of various interstitial components [67] , and 2 - 5 percent of the interstitial volume is occupied by the collagen fibers [68] . The collagen 12 fiber network i s arranged in several levels of organization: epimysium, the sheath of connective tissue that surrounds muscles; perimysium, tendon - like extensions of the epimysium that aggregate myocytes into bundles ; endomysium, collagen fibrils that connect adjacent myocytes to one another ( Figure 2.3 ) [69] . Figure 2.3 . Schematic representation of the fibrillar collagen matrix of the myocardium (Adapted from Weber et al. [1] ) Because of its spatial configuration and strong tensile strength relative to the myocytes , the collagen matrix is a major determinant of diastolic and systolic myocardial stiffness , structure integrity and mechanical properties of the myocardium . Also, the remodeling of fibrillar collagen matrix of the myocardium plays an active role in the hypertrophy that 13 accompanies various forms of chron ic pressure overload, in which an accumulation and enhanced dimension of the collagen fibers and their realignment relative to muscle fibers, have been observed [1] . 2.2 Cardiovascular diseases Cardiovascular disease (CVD) is a class of diseases that affect the structure and functi on of the heart and/ or blood vessels, and is the leading cause of death worldwide [70] . About 17.1 million people die each year from CVDs, which account for 31% of all death globally and impose a hea vy financial burden on the human society [71] . Various CVDs include myocardial infarction ( i.e., heart attack), stroke, cardiomyopathy, valvular heart disease and etc. Here we focus on two types of heart diseases: (1) pulmonary arterial hypertension (PAH) that causes right heart failure, and (2) heart failure wi th preserved ejection fraction that is also prev iously referred to as diastolic heart failure . 2.2.1 Pulmonary arterial hype rtension Pulmonary arterial hypertension (PAH) is a disease resulting from the constriction of pulmonary arteries that produces an elevated pulmonary vascular resistance , which ultimately leads to right heart failure [53] . The clinical diagnosis for PAH is sustained elevation of pulmonary arterial pressure at more than 25 mmHg at rest or 30 mmHg with exercise and with normal mean pulmonary - capillary wedge pressure and left ventricular end - dia stolic pressure of less than 15 mmHg [72] . While previously considered a rare disease, the most recent evidence suggests the prevalence of PAH is about 15 per million in France [73] . This disease also affects more females than males [74] . 14 P ulmonary arterial hypertension can be idiopathic in which its exact cause is unknown, or familial, or associated with several conditions or diseases, such as congenital heart disease, HIV infectio n, exposure to toxins and drugs [73] . Common symptoms of PAH include shortness of breath during exertion, dizziness, swelling of ankles and legs, chest p ain as well as a fast heartbeat . As the condition worsens, its symptoms may limit all physical activity. T here is no known cure for this disease and supportive measures suc h as oxygen therapy, diuretics and blood - thinnin g medicines are used to manage this disease [75] . The current prognosis of PAH remains poor, with about 15% mortality within 1 - year on modern therapy [54] and a low 3 - year survival rate of 67% [76] . Image - based computational models are increasingly used to analyze ventricular mechanics in heart disea s es (e.g., myocardial infarction [43] , left branch bundle block [44] , and mitral valve reg urgitation [45] , [77] ) as w ell as heart failure treatments [78] . The focus of these computational analyses is, however, on the LV that has historically received more atten tion than the RV. The RV, whose function is a major determinant of prognosis in PAH [79] , has largely been overlooked. evelop an image - based computational model to investigate effects of PAH on ventricular mechanics. contractility, and passive tissue stiffness in a PAH patient, and compa re them to those found in an age - and gender - matched normal subject. We also want to investigate and quantify the effects of transseptal pressure gra dients on the septum curvature. 15 2. 2.2 Heart failure with preserved ejection fraction It is well established that approximately half of patients with heart failure (HF) have preserved systolic function (LV ejection fraction 45 - 50%) (i.e., heart failure with preserved ejection fraction , HFpEF), while the rest have a reduction in ejection f raction, (i.e., heart failure with re duced ejection fraction , HFrEF) [80] . The prevalence of HFpEF is increasing at an alarming rate at about 1% per year, with mortalit y rates comparable to HFrEF [81] . Taken together with the aging world population, HFpEF is expected to become the predominant form of H F, where 65% of patients hospitalized with HF by the year 2020 are projected to have EF > 40 % [82] . Besides, in contrast to improved prognosis of HFrEF through proof of effective medications, the prognosis of HFpEF have remained poor and u nchanged over the past 20 years [80] . These trends underscore the importance of this growing public health problem. C ompared to HFrEF, epidemiological studies have shown that HFpEF patients are typically older , consist of more women and has hypertension as the most common etiology [83] . Other risk factors for HFpEF include hyperlipidemia, dia betes, renal dysfunction, obesity and the pre sence of ischemic heart disease [84] . Clinical manifestations of HFpEF include shortness of breath, exercise intolerance, fatigue [85] . Historically, HFpEF was abnormalities of ventricular diastolic function such as impaired LV relaxation and i ncreased LV diastolic stiffness . This increased LV stiffness has been associated with many micro - level remodeling features found in HFpE F patients such as interstitial fibrosis (e.g. progressive collagen accumulation, enhanced collagen cross - linkin g and collagen phenotype shift) [86] , myocyte stiffening [87] and increased myocyte diameter [88] that 16 leads to increased LV mass and thickened LV wall , which is referred as concentric hypertrophy. On the oth er hand, HFpEF patients typically also have subtle abnormalities in systolic performance, which become more exacerbated during e xercise. Very recent studies [33], [89] [91] have discovered that global longitudinal and circumferential strains are reduced in HFpEF patients. These results suggest that systolic function may also be impaired. Moreover, ongoing investigations suggest that, although diastolic abnormalities may be p resent in HFpEF patients, other aspects of pathophysiology are likely to be responsible for the symptoms, such as chronotropic incompetence, atrial dysfunction, pulmonary hypertension and RV dysfunction, impaired vasodi lation, endothelial dysfunction [92] . Clinical trials have yet to identi f y an effective treatment for HFpEF [93] . This issue is largely because of the pathophysiological heterogeneity that exists within the broad spectrum of HFpEF. Large randomized trials in broad populations of patients with HFpEF have generally shown neutral results [94] . Recently, substantial attention has been paid to the cardiac interstitium as a promising therapeu tic target in HFpEF , since studies have shown a correlation between cardiac collagen and diastolic dysfunction in this syndrome [95] . Our current understanding of the impact of LV microstructural remodeling on its function in HFpEF patients is largely based on inference drawn by correlating simplified indices such as LV relaxa tion time and stiffness constant with microstructural features such as extra cellular matrix volume fraction [96], [97] and force measur ements in single cardiomyocytes [87] . While useful, the lack of a physics - based connection between these 17 data makes it very difficult to quantify the role and contribution of microstructural chang es in tissue constituents and their mechanical behavior to changes in LV diastolic function in HFpEF. Thus, our second goal is to develop a microstructural computational model by taking into account mechanics and microstructural features of individual tis sue constituents (e.g. collagen waviness, spatial orientation) in order to i) isolate roles and contributions of each tissue constituent to ventricular dysfunction ii) enhance our understanding in pathophysiology of HFpEF so that targeted therapies can be developed based on a sound scientific foundation. 2.3 Cardiac imaging techniques Cardiac imaging techniques play a significant role i n diagnosis and management of heart failure , quantitative analysis of heart function and myocardial deformation, and development of better treatment strategies [98] . Various imaging modalities are increasingly used in the diagnostic evaluation of cardiovascular diseases, among which echocardio graphy and cardiac magnetic resonance (CMR) imaging represent two of the most commonly performed imaging examinations in routine clinical practice. 2. 3.1 Echocardiography With the merits of availability, safety and low cost, echocardiography is the first and most widely used exam for the diagnosis, prognosis and reassessment of heart failure patients [98] . Over the past 50 years, this imaging technique has evolved from a simple M - code tracing to a family of modalities including two - dimensional (2D) imaging, spectral Doppler, color flow Dop pler and tissue Doppler imaging [99] . They can measure 18 multipl e clinically important parameters of cardiac function, including LV ejection fraction, cavity volumes and mass. In most studies, ventricular strain is measured using tissue Doppler imaging, which can provide an objective means to quantify global and regional ventricular function and improve the accuracy and reproducibility of co nventional echocardiography [32] . However, tissue Doppler - based assessment of ventricular function is angle - dependent and typically as sess only mitral annular motion [33] . More recently, B - mode speckle tracking has allowed for quantitative a ssessment of ventricular deformation and is angle - independent. However, the image quality of RV is often poor, thus limiting its accuracy and reproducibility in evaluation of RV strain, which is a useful prognostic marker for PAH patients [31] . 2. 3.2 Cardiac magnetic resonance imaging Cardiac magnetic resonance (CMR) imaging, also known as cardiac MRI, is a key imaging technique for noninvasive assessment of myocardial structure and function [100] . It has emerged as the gold standard for quantific ation of myocardial deformation [35], [101] , which can provide tomographic images of both ventricles with superior spatial resolution. In addition, CMR was demonstrated to have superior reprod ucibility over echocardiography [37] . Hyperelastic warping is a deformable image registration technique that has been used to obtain strain measurements from medical images such as magnetic resonance imagi ng (MRI), positron emission tomography (PET) and computed tomography (CT) [38] [40] . It integrates nonlinear continuum mechanics with image - based data to find a compatible deformation map between a target image and a template image. Application 19 of the hyperelastic warping approach in cardiac motion has been primarily confined to quantifying the LV strains [40] , which has been verified by tagged MRI [41] and 3D CSPAMM MR images [42] . Little has been done, however, to further this efficient and robust image registration technique to estimate regional myocardial strains in both LV and RV , which may help serve as a potential prognostic marker for the severity of heart diseases such as PAH and provide clinical reference for validating computational heart models. Therefore, o ur third goal is to implement this image registration technique to quantify biventricular strains associated with PAH patients for quantitative evaluation of cardiac function . 2. 4 Cardiac computational modeling The combination of cardiac computational models and biophysical simulations can help interpret experimental data and promote understanding , diagnosis and treatment of heart failure [102] . For this reason, cardiac computational modeling has become a rising field of research. With the recent advancement in both experimental and imaging techniques, cardiac computational modeling has become an indispensable and powerful tool in elucidating ef fects of various heart diseases and clinical interventions such as passive left ventricular (LV) constraint [5], [6] and cardiac sup port device (CSD) [7] . Cardiac computational models can be used to estimate important physiological experiments. Of crucia l importance is the ventricular wall stress [9], [103] , which is a n important determinant o f myocardial oxygen consumption and key driver for pathological 20 ventric ular remodeling and hypertrophy [104] . Thin - walled [105] or thick - walled [106] simplified ellipsoidal formulas generalized from to estimate stress in idealized geometries, however, these methods severely underestimate myofiber stress in realistic patient - specific geometries and cannot accu rately estimate regional stress [107] . In order to use cardiac computational models for accurately evaluation of ventricular stress, it is necessary to develop accurate constitutive model s describing the mechanical behavior of the myocardium. Two categories of constitutive models have been used traditionally to model the mechanical beh avior of the cardiac tissues . The first category is phenomenological models , which are based on experimental measurements of the tissue mechanical behaviors . The second category is microstructural models , which directly model the inherent microstructur e an d composition of ventricular constituents. 2.4.1 Phenomenological models The phenomenological approach in describing soft tissue mechanics was pioneered by Fung [12] , who found that the elasticity of living soft tissue is highly nonlinear, and that the elastic stress for the rabbi is approximately an exponential function of the stretch ratio. This result lays the foundation of many future phenomenological constitutive models for ventricular myocardium. Most of the phenomenological models of myocardium were proposed based on the assumption material properties of myocardium are symmetric about the myofiber orientation. These models are typically known as transversely isotropic models, of which typical examples inc lude models of Humphrey and Yin [13] , Humphrey et al. [108] , 21 Guccione et al. [14] , Costa et al. [109] and Kerckhoffs at el. [110] . While transversely isotropic models are proposed based on biaxial tests of cardiac tissues, the results from shear experiments showed that the resistance of myocardial tissue to simple shear loading in different planes is noticeably different [111] , which suggests that myocardial tissue is actually an orthotropic material with distinct material properties in orthonormal planes of symmetry. M ore recently, orthotropic models of passive myocar dium have been proposed including the p ole - zero model of Hunter et al. [112] , Holzapfel & Ogden [16] and Schmid et al. [113] . Although phenomenological models h ave been widely used to describe cardiac mechanics and can represent the bulk cardiac tissue mechanical behaviors, their model parameters typically lack physical interpretation and cannot elucidate the underlying mechanisms of tissue behavior. 2.4 .2 Microstructural models Microstructural constitutive models attempt to integrate information on tissue composition and microstructure to avoid ambiguities in material characterization. This class of models can potentially offer insights into the mecha nical effects due to changes in function, structure, and mechanics of individual tissue components [114] . Structural constitutive models have been developed for a vari ety of intact tissues and tissue components including lung [20] , collagen [21], [22] , cartilage [23] , heart valves [24] , and maturing skin [25] . Lanir developed the first comprehensive, multidimensional structural constitutive model formulation [26] . With some modifications made to the original formulation , Horowitz et al. proposed a three - dimensional constitutive law for the passive myocardium [27] and implemented it in a two - dimensional finite element framework for 22 the simulation of the passive - state mecha nics of a thin myocardial strip [28] , together with three - dimensional formulation for a thick ventricul ar wall segment . Recently, M. S. Sacks im plemented a structural constitutive model into a finite element framework for membrane tissues, However, finite element implementation of structural models that takes into account the realistic LV geometry and muscle fiber architecture is still lacking, wh ich is an essential step to promote our understanding of how changes in the tissue ultrastructure and micromechanical behaviors (e.g., collagen distribution and waviness, myofiber orientation) affect LV functions. 23 CHAPTER 3 MICROSTRUCTURE - BASED FINITE ELEMENT MODEL OF LEFT VENTRICLE PASSIVE INFLATION 24 Abstract Isolating the role(s) of microstructural pathological features in affecting diastolic filling is important in developing targeted treatments for heart diseases. We developed a microstructure - based constitutive model of the myocardium and implemented it in an efficient open - source finite element modeling framework to simulate passive inflation of the left ventricle (LV) in a representative 3D geometry based on experim entally measured muscle fiber architecture. The constitutive model was calibrated using previous tissue - level biaxial mechanical test data derived from the canine heart and validated with independent sets of measurements made at both the isolated constitue nt and organ level. Using the validated model, we investigated the load taken up by each tissue constituent and their effects on LV passive inflation. The model predicts that the LV compliance is sensitive to the collagen ultrastructure, specifically, the collagen fiber azimuthal angle with respect to the local muscle fiber direction and its waviness. The model also predicts that most of the load in the sub - epicardial and sub - endocardial regions is taken up, respectively, by the muscle fibers and collagen f iber network. This result suggests that normalizing LV passive stiffness by altering the collagen fiber network and myocyte stiffness is most effective when applied to the sub - endocardial and sub - epicardial regions, respectively. This finding may have imp lication for the development of new pharmaceutical treatments targeting individual cardiac tissue constituents to normalize LV filling function in heart diseases. 25 3. 1 Introduction pumping performance [115], [116] . Heart failure with preserved ejection fraction (HFpEF) (referred previously as diastolic heart failure) is associated with abnormalities in LV filling that can be detected fro m clinical imaging [117], [118] . Because HFpEF accounts for about one - half of all chronic HF pa tients [119] , understanding how LV filling is affected by the many microstructural pathological features associated with this syndrome (e.g., cardiac fibrosis [120] , myocyte passive stiffening [87] , and collagen u ltrastructure alterations [121] is very important. As such, the development of a 3D LV mechanics model that includes myocardial tissue constituents and their microstructure will enable one to isolate and quantify the impact of each constituent on LV filling. Current constitutive models used to describe the mechanical behavior of the myocardium can be broadly categorized into two types: i.e., phenomenological and behavior is usually described by a mathematical expression of its strain energy functi on (SEF) with the form of a polynomial [108] , an exponential [14], [16] or a zero - pole function [122] of either the invariants or components of the strain ten sor. While able to represent the bulk cardiac tissue mechanical behavior, these phenomenological models cannot and ultrastructure (e.g., collagen waviness) to the LV me chanics and function. Moreover, model parameters in the phenomenological models can only be obtained by fitting with experimental measurements of stress - strain relationships and have no direct physical meaning. 26 First developed by Lanir, microstructural c onstitutive models take into account the well as their intrinsic mechanical behavior [123] . One advantage of microstructural constitutive models is that their morphometric parameters, in addition of being physiologically m eaningful, can be directly measured. With the significant advancement in microscopy imaging (e.g., multiphoton microscopy) techniques that enable the tissue measured, these models can be exploited to directly isolate and quantify the effects of microstructural changes on the overall tissue mechanical behavior and organ function. Microstructural constitutive models have been applied to describe the mechanics of various t issues, such as skin [124] , heart valves [125] , arteries [126] as well as the passive myoc ardium [27] . The solution of the mechanical equilibrium boundary value problem using these model s, however, were obtained semi - analytically and their applications were therefore confined to simple idealized geometries such as thick - walled cylinder [127] . Quantification of three - dimensional tissue mechanics in a realis tic organ geometry requires an efficient finite element (FE) implementation of the microstructural constitutive model, which is, to the best of our knowledge, currently lacking and limited to tissue mechanics of a thin myocardial strip [28] , a thick single - element myocardial wall segment [29] , as well as a thin membrane [128] . To address this gap, we developed a microstructural FE model of the LV with a representative 3D geometry and muscle fiber ar chitecture to describe the passive mechanics of myocardial tissue during inflation. The model was implemented using an open - source FE library FEniCS [129] . Additionally, we estimated the microstructural 27 constitutive model parameters using previous biaxial mechanical test measurements of the canine myocardial tissue. These model parameters were also validated using independent sets o f measurements made on isolated cardiac tissue constituents and the intact LV during passive filling. Based on the calibrated parameters, we investigated the load taken up by each constituent during passive filling. The FE model shows that the global LV fi lling function is sensitive to the tissue microstructure. 3. 2 Materials and m ethods 3. 2 .1 Microstructural constitutive model formulation The myocardium has two major load - bearing elements: collagen fibers and muscle fibers . Arranged in a fibrous networ interconnecting adjacent myocytes and fibrous weaves surrounding groups of myocytes [130], [131] . Both types of fibers are embedded in an interstitial fluid matrix. The fibers carry tensile forces, while the fluid matrix sustains o nly hydrostatic pressure. On the other hand, the myocardium also contains a non - fibrous ground matrix consisting of non - load bearing substances such as fibroblasts, plasma cells, as well as a gel - like ground substance composing of glycosaminoglycans and gl ycoproteins. Following Horowitz et al . [27] , we developed a microstructure - based constitutive model to describe the passive mechanical behavior of the myocardium based on the following assumptions: (a) The myocardium is treated as an incompressible pseudo - hyperelastic composite material [132] , namely, there exists a strain energy function from which the stresses are derived. The viscoelastic aspect of the tissue is not considered in our 28 study due to the relatively short duration of the cardiac cycle compared with the characteristic relaxation time of the myocardium. (b) The fibers are thin and perfectly flexible. They can only carry tensile loads and have no compressive strength and if contracted will buckle under zero load. (c) Each fiber is subjected to a uniaxial strain that is the tensorial transformation of the by the numerous interconnections between muscle and collagen fibers [27], [114], [133] . Correspondingly, the total strain - energy function of the myocardium can be represented by the volume - weighted summation of the strain energy function of its constituents, i.e.: (3.1) , interstitial fluid matrix . , where is the displacement field and is the identity tensor . 29 ( 3. 2) where is the first invariant of the right Cauchy - Green deformation tensor , and is a material constant. The resulting second Piola - Kirchhoff (PK2) stress tensor associated with the non - fibrous ground matrix is then given by: ( 3. 3) where is the right Cauchy - Green strain tensor. 3.2.1.2 Muscle fibe rs. Because the isolated myocyte exhibits a non - linear stress - strain relationship of an exponential form [135] , we use d an SEF [13] that is given as: ( 3. 4) to describe the muscle mechanical behavior. In Eq. (4), is the uniaxial stretch of the muscle fiber, is a unit vector describing the local muscle fiber direction in the undeformed configuration, whereas and are the material constants. The corresponding PK2 stress tensor associated with the muscle fibers is given by : ( 3. 5) 30 3.2.1.3 Collagen fiber network For simplicity, different hierarchies of collagen fibers ( perimysial , epimysial and endomysial or struts) are l umped into one inclusive system in the model. Because straightened individual collagen fiber exhibit s a linear stress - strain relation [136] , uniaxial stress of a single straightened collagen fiber is prescribed to be a linear function of its strain taken with respect from its initial straig htened configuration with a slope or elastic mod ulus . Collagen fibers, however, are observed to be undulated and slack in the unst ressed state in the myocardium [130] . Because of their crimped structural feature, taken with respect from its initial straightened configuration is given by : (3.6) where is the total strain of the collagen fiber taken with respect from its initial wavy configu ration and is the strain at which the collagen fiber first straightens; i.e., straightening strain. Assuming that the collagen fiber do show any resistance to stretch before becoming straightened ( or reaching their straightening strain ), the resulting SEF for a single wavy collagen fiber is given by : ( 3.7 ) T he extent of collagen fiber undulation can vary , however, across collagen fibers in the m yocardium [114] . To stochastically account for the gradual recruitment of wavy collagen fibers, we assume a truncated normal distribution with the density function : (3.8) 31 where is a truncated parameter with denoting the cumulative normal distribution function, wh ereas, and denote the mean and variance of the collagen fiber straightening strain, respectively. Thus, the resulting SEF for a uniaxial ensemble of collagen fibers over all possible waviness (weighted by the waviness distribution is given by: (3.9) The mechanical behavior of the overall collagen fiber ensemble is the sum of the contributions from all uniaxial en sembles over all collagen fiber orientation s . Following Horowitz et al . [27] , we assume that the collagen fibers are symmetrically distributed around the muscle fiber to which they are attached, with the muscle fiber serving as an axis of symmetry ( Fig ure 3. 1 ). Following this assumption, the collagen fibers are ascribed a uniform distribution in the circumferential direction ( , and a bimodal normal distribution in the azimuthal direction ) . Thus, the overall spatial distribution function is given by: ( 3. 10) where ( ) are the means and ( , are the standard deviations of the bimodal distribution of the azimuthal angle . Since the arrangement of the collagen fibers is referred to the muscle fibers to which they ar e attached, the uniaxial strain of a local collagen fiber oriented with respect to the local muscle fiber that it is attached to can be related to the local muscle fiber strains by a tensorial transformation as: (3.11) 32 where and is the Green - Lagrange strain tensor in the local material coordinate system. The components of are related to the Green - Lagrange strain tensor in the global coordinate system by: (3.12) with the subscript denoting the muscle fiber , cross muscle fiber , tissue sheet normal directions. If the waviness density distribution function is assumed to be homogeneous over all spatial orientations, the total collagen fiber SEF is giv en by: (3.13) C orresponding ly, the PK2 stress tensor associated with the collagen network is given by: (3.14) We note that the approach here to describe the collagen network stress tensor in Eq. ( 3. 14) differs from the generalized structure tensor (GST) approach [137] , which assumes that the dispersion of fiber orientation (as described by a structure tensor) is separable from the dispersion of tortuosity. No such assumptio n is made in this approach, which is [137] . For a detail comparison of the GST and AI ap proaches, refer to Holzapfel and Odgen [138] . From Eq. ( 3. 1), the total PK2 stress tensor of the myocardium is given as: (3.15) with the components defined in Eqs. (3.3), (3.5) and (3.14). 33 Figure 3.1 . (a) Unloaded canine LV FE geometry (all dimensions are in cm). (b) Muscle fiber orientation with a linear transmural variation from at the endocardium to at the epicardium. (c) Schematic representation of the local arrangement of muscle fiber and collagen fiber network. The muscle fiber defines an axis of symmetry for the spatial arrangement of collagen fiber that attached to it. Collagen fiber dispersion is prescribed . An azimuthal angle corresponds to the axial fiber family whereas a circumferential angle corresponds to the radial fiber family. 3. 2 .2 Parameter estimation We used the biaxial mecha nical test data obtained from 6 canine mid - wall myocardium in a previous experimental study [139] to estimate the model parameters . In that study, equi - biaxial stretching and constant tests , where cross - fiber stretch was varied at a constant muscle fiber stretch of , were performed on thin slab of tissues taken from the mid ventricular wall . We fixed the volume fraction of the tissue constituent s to be based on previous experimental measurements [67], [140] . The collagen spatial distribution variance was also f ixed at 0.1 rad (or 34 equivalently 6 ) based on a quantitative analysis of the collagen orientation in the canine LV [121] . This leave s the remaining 7 unknown m odel parameters to be estimated, namely, material constants of the constituents , , , , mean value of the collagen spatial distribution function , and the mean and variance of the collagen waviness distribution . These model parameters were fitted to the experimental data with an objective function defined as the sum of squared residuals (SSEs) between the model predictions and experimental measurements, i .e., 3. 16 In Eq. (16), is the total number of data points and are the measured values of the Cauchy stresses . W e also imposed constraints to our model parameters, especially those related to the microstructural features, to va ry within physiological ranges based on previous studies ( Table 3. 1 ) . MATLAB (The Mat hworks, Inc., Natic k , MA, USA) was used to find parameters that lie between the imposed physiological lower and upper bounds , and minimize the objective function. An optimal solution was obtained when the change in parameters or the objective function was less than . Dif ferent sets of initial guesses were randomly generated between the lower bounds and upper bounds in order to eliminate parameter dependence on the initial guess values. Details of the plane stress formulation for fitting the biaxial test data is given in t he APPENDIX A . 35 Table 3.1 . List of model parameters with the corresponding lower/upper bounds according to literature references Type Parameter Lower Bound Upper Bound References Ground matrix k P a 0.01 10 [141], [142] Muscle fibers k P a 0.1 1.0 [143] 1 50 Collagen fibers M P a 0.1 100 [124] >0 0.3 [27], [127] >0 0.1 rad 0 we estimated the asymptotic correlation matrix R associated with these parameters using the scaled inverse of the hessian matrix H evaluated at the optimal solution [142] . Specifically, components of the estimated correlation matrix R was defined as : ( not summed). ( 3. 17) The closer the absolute value of the off - diagonal terms to 1, the stronger the correlation between the th and th parameters. 3. 2 .3 Finite element simulation of left ventricular passive filling 3.2.3.1 Unloaded geometry and microstructure The unloaded LV geometry was modeled as a half prolate ellipsoid with geometric dimensions that are based on measurements of isolated arrested canine hearts ( Fig ure 3. 1 ) [2] . Specifically, the unloaded LV cavity volume is about 20 ml. The geometry is meshed with 17600 quadratic tetrahedral elements and 83799 nodes. The local muscle fiber orientation in the LV is prescribed based on the mean values reported in an 36 ex periment [2] , which shows that the helical angle vari ed linearly in the transmural direction from about at the endocardium to at the epicardium. 3.2.3.2 Finite element formulation Passive filling of the LV was simulated by incrementally increasing the pressure at endocardium as a Neumann boundary condition. The weak form of the mechanical equilibrium equation is given by: ( 3. 18) where is the first Piola - Kirchhoff stress tensor, is the prescribed LV cavity pressure at the endocardial surface with unit outward normal of , whereas and are the variation of the displacement field and the Lagrange multiplie r , respectively, Thus, the Euler - Lagrange problem becomes one of finding that satisfies . The basal deformation is constrained to be in - plane (i.e., ) and the endocardial basal nodes are fixed in all directions to account for the relative stiff valve annuli [144] . The nonlinear FE problem was solved using Newton - Raphson method and implemented based on the parallel open - source FE library FEniCS. The code is freely available at https://cexi@bitbucket.org/cexi/microstructure_lv.git . Convergence of the numerical solution for the nonlinear FE problem was accepted when the normalized two - norm of the residual vector is smaller than . Analyses of the mesh sensitivity and computational efficiency were also performed. 37 3.2.3.3 Post - processing of myocardial strains and stresses To compare with strain measurements and estimate myocardial stress, we computed strains and stress components in the circumferential , longitudinal and radial directions, as well as fiber and cross - fiber directions ( Fig ure 3. 2 ). We also divided the LV FE mesh into 7 layers with equal thickness. The strain and stress components were averaged over the volume of each layer to determine their transmural variations ( Fig ure 3. 2 ). 3.2.3.4 Post - processing of collagen fiber tortuosity Volume - averaged 2D tortuosity (fiber arc length/midline length) of the collagen fiber network was computed at a given ventricular pressure as: ( 3. 19) where is the initial tortuosity for a collagen fiber with an initial straightening strain , is the tortuosity of deformed local collagen fiber with spatial orientation ( ) at vent ricular pressure and is the whole material volume at the reference configuration. 38 Figure 3.2 . (a) Transmural division of mesh into 7 layers with the same wall thickness. (b) Circumferential direction. (c) Radial direction. (d) Longitudinal direction. 3. 2 .4 Statistical a nalysis Categorical variables are expressed as number and percentage, and continuous variables as mean ± standard deviation (SD). Nonlinear regression analysis was performed to correlate model - predicted biaxial stresses with corresponding stress - strain measurements in canine LV midwall myocardium ( n = 6). 39 3.3 Results 3. 3 . 1 Biaxial tests Table 3. 2 lists the best fitted model parameters for 6 specimens with corresponding mean square errors. About eighty iteration s were required to obtain the optimized model parameters for each specimen. In general, our microstructural model shows very good fit to the measurements. Except for , the spreads of all other parameter values are relatively small across the specimens. The large SD of may be related to the fact that the stiffness modulus and volume fraction for the ground matrix are much smaller than other the myocardial constituents [141] . The average correlation coefficient of the fit betwee n the measurements and model - predicted stresses in the muscle fiber and cross - fiber directions are 0.980 and 0.999, respectively. The maximum root mean square error is 0.152 kPa, which is less than 2% of the range of stress reached in the biaxial tests. We also note that increasing the lower and upper bounds of the mean collagen waviness to , which corresponds to the collagen tortuosity range of 1.2 1.6 as found in the skin [124] , led to a larger mean square error and increased both the mean value of and by about 2 times. A comparison between the meas urements and model predictions of the equi - biaxial and constant test using the best - fit material parameters for a representative specimen is shown in Fig ure 3a . Corresponding stresses in the muscle fiber and cross - fiber directions associated with each co nstituent in the equi biaxial test of specimen 0118 is shown in Figure 3b . 40 Figure 3.3 . (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0118. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxial test of specimen 0118. Taking all the specimen s into account, we found that the collagen fiber network accounts, on average, for 57% and 99.4% of the total stress in the muscle fiber and cross - fiber 41 directions, respectively, at a stretch of 1.32. The muscle fiber, on the other hand, accounts on averag e for about 43% of the total stress in the fiber direction at that stretch value. At stretch lower than 1.1, however, the collagen fiber network contributes very little to the total stress. For the results of all other specimens, refer to the APPENDIX B . Table 3. 2 . Best - fit model parameters and root mean square errors ( RMSE ) for biaxial tests of six canine mid - wall LV myocardium (kPa) (kPa) (MPa) (rad) Table 3. 3 compares the microstructural model parameters that are directly related experiments conducted on the isolated constituents. Specifically the average fitted values of the collag en fiber elastic modulus , mean collagen fiber straightening strain , are within the range of reported values [135], [145] [147] . We computed the uniaxial stress and the resulting stiffness modulus (see APPENDIX C for details) at an engineering strain , which is associated with the stretch imposed on the muscle fibers in the experiments. Measurements of the mean inclination angle of the collagen 42 fibers with respect to the muscle fibers are not available, however, for comparison with the fitted value of 53.1 . Table 3. 3 . Comparison of fitted parameters with measurements of isolated myocytes and collagen fibers. Description Fitted model parameters Reported Values Cardiac muscle stiffness modulus 13.49 ± 1.92kPa* Guinea pig:16.4 ± 11.0 kpa [145] ; Hamster: 7.48 ± 1.73 kpa [135] ; Cardiac muscle uniaxial stress 1.36 ± 0.28 kPa* Guinea pig: 2.1 ± 1.4 kpa [145] ; Hamster: 0.88 ± 0.48 kpa [135] ; Collagen fiber stiffness 5.1 ± 1.7MPa Skin: 0 ~ 50 MPa [146] Collagen fiber mean tortuosity 1.116 ** Canine: 1.01~1.2 [147] *Cardiac muscle stiffness and uniaxial stress were computed at 2.2 sarcomere length. Based on a resting sarcomere length of 1.9 , this translates at an engineering strain ** The collagen fiber mean tortuosity was computed from the collagen fiber straightening strain by where Table 3. 4 shows the estimated average correlation matrix R of the model paramet ers from all the specimens. Parameters and describing the muscle fiber mechanical behavior has the greatest interaction and are inversely related with correlation coefficient - 0.93 . Large positive correlation between the collagen elastic modulus and mean straightening strain is also found ( correlation coefficient 0.75 ). Interactions between the parameter pairs , and are moderate; their largest c orrelation coefficient is 0.55. 43 Table 3.4 . Average correlatio n matrix of fitted model parameters 1 - 0.52 0.33 0.04 0.49 - 0.09 - 0.35 1 - 0.93 0.27 - 0.04 0.09 0.07 1 - 0.37 - 0.03 - 0.07 0.23 1 0.75 0.54 - 0.22 1 0.55 - 0.04 1 0.07 1 3. 3 . 2 Finite element simulation of passive filling Differences of the transmural fiber stress distribution and pressure - volume relationship are found to be < 2% between FE meshes containing 17,600 and 38,985 elements in the mesh sensitivity analysis. ( APPENDIX E ). With 16 processors (Intel ® Xeon (R) CPU E5 - 1660 v3@ 3.00GHz x 16), the computing time for simulating the inflation of an LV model with 17,600 elements is about 13 hours. The FE implementation (that utilizes the domain decomposition method in FEniCS) also exhibits an approximately linear scalability ( APPENDIX E ), which indicates that the computing time can be reduced substantially with more processors. An estimation of the simulation time for solving active mechanics over a cardiac cy cle is also provided in APPENDIX E . 44 3.3.2.1 Inflation pressure - volume relations . Figure 3.4 shows the relationships between the change in LV cavity volume with respect to the LV pressure using the fitted parameters in Table 3. 2 for all the 6 specimens. Compared to the measurements from the experiments on isolated canine hearts [2], [148] , the pressure - volume curves derived from all 6 sets of parameter values fell largely within 1 SD of the mean measurements. Figure 3.4 . Comparison of the inflation pressure - volume curves derived from the 6 sets of fitted model parameter values (Table 3.2) and their mean values with measurements (plotted as mean ± SD). 45 3.3.2.2 Local ventricular wall strains Figure 3. 5 shows the comparison of transmural distribution of normal strains at a LV pressure of 8 mmHg with measurements [2] . Consistent with the experiments, the model predicted both circumfer ential strain and longitudinal strain to increase from the epicardium to endocardium, and the radial strain to be negative and decrease from epicardium to endocardium ( Figure 3. 5a ). Model prediction of the transmural distributi on is closest to the experiments, whereas the predicted transmural distribution of has an offset of ~ 0.08 with respect to the measured mean values. Both muscle fiber strain and cross fiber strain are predicted to increase from the e picardium to the endocardium, and both are largely within 1 SD of the measurements ( Figure 3. 5b ). In - plane shear is negative and its magnitude increases from the epicardium to the endocardium. Compared to the normal strains, the magnitude and trans mural gradient of shear strains are small ( Figure 3. 5c ). We also compared model predictions of the normal and shear strains as a function of the change in LV cavity volume with measurements at the sub - epicardial, midwall, and sub - endocardial [2] ( APPENDIX D ). Overall, model predictions of the relationship between strains and are comparable to the measurements. Of all the normal strains, the relationship between and is closest to the measurements at all 3 transmural locations, whereas a t = 20 ml in the sub - endocardial region has the largest discrepancy with the measurements (measured: 0.126; model: 0.257). The predicted relationship between the shear strains and are largely within the 1 SD of the measurements. 4 6 Figure 3.5 . Tra nsmural distributions of the strain components at a LV pressure of 8 mmHg. Colored solid lines: model predictions using best - fit and mean parameters in Table 2. Measurements of strains at LV pressure of 8 ± 4 mmHg [2] are plotted as means ± SD. Note: shear strain results from experimental data are not shown as the measured values are small and difficult to digitiz e. 47 3.3.2.3 Local ventricular wall stresses Figure 3. 6 shows the transmural distribution of stresses corresponding to a LV pressure of 8 mmHg. The results show that both circumferential stress and longitudinal stress increase from the epicardium to the endocardium ( Figure 3. 6a ). F igure 3.6 . Transmural distributions of the stresses at a ventricular pressure of 8 mmHg. Colored solid lines: model predictions using best - fit and mean parameters in Table 3.2. 48 The magnitude of is larger than at all transmural depth. Radial stress is negative and increases in magnitude towards the endocardium. Both fiber stress and cross - fiber stress increase from the e pi cardium to the endo cardium ( Figure 3. 6b ). Magnitude of the shear stress components are small compared to the normal stresses. In - plane shear stress has a larger transmural gradient than the other transverse shear stress ( Figure 3. 6c ). 3.3.2.4 Contribution of the tissue constituents to the wall stress Figure 3. 7 shows the transmural variation of the contribution of each tissue constituent to the total normal stress in the mean muscle fiber direction at an LV pressure of 8 mmHg as predicted by the model using the mean value of the fitted parameters in Table 3. 2 . The contribution of the interstitial fluid matrix to is simply The results show that the contribution of muscle fiber is substantial at all transmural locations, accounting between 40 70% of the total stress. The contribution of collagen fibers to the total stress increases from 22% at the epicardium to 96% at the e ndocardium. Interestingly, the results also show that at 60% transmural depth, where or fiber stretch , the contribution of collagen fibers starts to exceed that of the muscle fibers; i.e., the collagen fibers become the dominant load - bear ing constituent at a transmural depth > 60%. We also note that the contribution to by each constituent remains largely unchanged using model parameters fitted from the biaxial test data with larger lower and upper bounds of the mean collagen wavines s at . 49 Figure 3.7 . Transmural variation of the contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total normal stress in the local muscle fiber direction at an LV pressure of 8 mmHg using the mean fitted parameter values from Table 3.2. 3.3.2.5 Collagen fiber tortuosity Figure 3. 8 shows the comparison of the model predictions of the relationship between collagen fiber tortuosity and LV pressure with the measurements. Our model predicts that the collagen fiber tortuosity decreases from 1.123 ± 0.009 to 1.066 ± 0.01 when LV pressure is increased from 0 to 25 mmHg. The model predictions are mostly within the measured ranges when the LV pressure is less than 10 mmHg. Discrepancy between model prediction and measurement is generally larger at higher pressure. 50 Figure 3.8 . Collagen fiber tortuosity as a function of LV pressure. Colored solid lines: model predictions using the best - fit and me an parameters in Table 2. Black scattered dots: measurements of the perimysial collagen fiber tortuosity in rat hearts [149] . Black square marker with error bars (mean SD): measurements of collagen tortuosity in normal rat LV [150] . 3.3.2.6 Parameter sensitivity analysis is very sensitive to the collagen volume fraction , collagen waviness , collagen stiffness modulus , collagen azimuthal angle and muscle fiber stiffness (Fig ure 3.9 ). is On the other hand, the relationship between the LV chamber stiffness and mean collagen azimuthal angle is non - monotonic for 51 the range 0 90 . The effects on the pres sure - volume curve are less for the other parameters. Figure 3.9 . Sensitivity of (a) collagen volume fraction , (b) collagen waviness , (c) collagen waviness deviation , (e) collagen azimuthal angle , (f) collagen azimuthal angle deviation , (g) muscle stiffness and (h) muscle fiber helix angle to the pressure - volume curve. 52 Table 3. 5 shows the percentage of absolute change in at LV pressures of 20 and 30 mmHg when each model parameter is chan ged 100% with respect to the corresponding mean value (baseline). Our results show that the collagen ultrastructure (i.e., waviness and azimuthal angle) has a substantial impact on the . Collagen volume fraction and the stiffness of individual collagen fiber roughly have the same effects on . On the other hand, the muscle fiber stiffness and orientation have lesser influence on compared to the collagen network. The sensitivities of to the model parameters are relatively the same at both press ures. Table 3.5 . Percentage of absolute change in at a LV pressure of 20 and 30 mmHg with 100% change in each model parameter about its mean fitted value. Model parameters Percentage of absolute change in at 20 mmHg Percentage of absolute change i n at 30 mmHg Collagen Azimuthal Angle 48.3 44.7 Collagen Waviness 46.7 39.9 Collagen Volume Fraction 29.5 31.5 Collagen Stiffness 29.6 31.6 Collagen Waviness Deviation 11.1 11.6 Muscle Stiffness 9.83 6.5 Muscle Helix Angle 4.4 7.5 Collagen Angle Deviation 0.84 1.1 *Sensitivity analysis of the muscle fiber helix angle was evaluated for a change between a symmetric transmural variation of helix angle of (30 / - 30 ) (baseline) to (60 / - 60 ). 53 3. 4 Discussion We have developed and validated a microstructure - based constitutive model of the passive myocardium in a three - dimensional FE modeling framework, and have shown that the LV filling function is sensitive to the collagen ultrastructure and the load taken up by the tissue constituents varies depending on the LV transmural location. To the best of our knowledge, this is the first FE implementation of a microstructural constitutive model to simulate passive filling of a LV i n a representative 3D geometry with ex perimentally measured transmural muscle fiber orientation. Calibrated against the tissue - level biaxial test data derived from the mid - wall of the canine myocardium [139] , we show that the fitted parameter values are consistent with microscale measurements made on individual cardiac tissue constituents and the FE model predictions using these values are largely in agreement with independent measurements from experiments on the intact canine LV. This study, therefore, p rovides a rigorous comparison of the microstructural constitutive model predictions with measurements made across multiple scales; i.e., at the constituent, tissue and organ levels. tension at a sarcomere length of 2.2 are comparable to the reported values of other species such guinea pig [145] and hamster [135] since we are not able to find any passive mechanical tests co nducted on canine myocytes. For the collagen fiber network, the calibrated mean elastic modulus of a straightened collagen fiber is ~ 5 MPa, which is within the reported values of 0 50 MPa for collagen fibers in the skin tissue [146] and comparable to the mean fitted v alue of 3.76 MPa in the rat right ventricle [142] . Direct measurements of the collagen fiber elastic modulus are, to the best of our knowledge, not 54 available. In terms of the collagen fiber network mic calibrated mean collagen fiber tortuosity of 1.116 is well within the histological measurements of 1.01 1.2 [147] . While the calibrated mean inclination angle of 0.927 rad or 53.1 cannot be quantitatively compared with measurements, histological studies have rep orted qualitatively that the collagen fibers are inclined relative to the muscle fibers [131] . At the tissue level, the constitutive model is able to fit the measurements derived from all the biaxial testing protocols very well for each canine specime n (fiber direction: R=0.98; cross - fiber direction: R=0.999). An assessment of the relative contribution by each constituent to the total stress reveals that the collagen fiber network bears most of the load in the cardiac tissue, accounting for about 57% o f the total load when cardiac tissue is stretched beyond 1.3. The cardiac muscle fibers, on the other hand, still accounts for a significant portion (~ 43%) of the total load. This finding is consistent with a recent study which shows that the mechanical c ontribution of muscle fibers is significant at all physiological ranges of stretch [142] . At the organ level, the FE implementation of the microstructural constitutive model in a representative 3D LV geometry enables comparison with measurements of the pressure - volume curves, strains and collagen tortuosity during inflation. The pressure vs. vo of the measurements in canine hearts [2], [148] . Predictio n of the relationship between collagen fiber tortuosity and LV pressure is also comparable with the measurements [149], [150] , showing that the collagen fibers are not all fully stretched over the range of diastolic filling pressures (0 25 mmHg). Moreover, model predictions of the strains as a function 55 of LV cavity volume change and their trans mural distribution across the LV wall are also largely in agreement with the measurements [2] . The generally good agreement between model predictions of tissue - level strain, collagen fiber tortuosity and pressure - volume relationship of the LV with measurements using the fitted parameters from biaxial mecha nical data provide confidence of the More significantly, the consistency of the fitted parameters with microscale mechanical and structural measurements of isolated tissue constitue nts substantiate, to some level, the validity of the model prediction of the contribution of each constituent to the total stress during LV inflation. Specifically, the model predicts that the contribution of muscle fibers to the total stress is significan t at low strain, but its contribution diminishes at higher strain. Conversely, the contribution of collagen fiber network to the total stress is small at low strain but its contribution increases due to the gradual recruitment of wavy collagen fibers at hi gher strain. As a result, the collagen fiber becomes the dominant load - bearing constituent at hig h strain, accounting for more than half of the normal stress in the local muscle fiber direction at a stretch of 1.32. This result supports the suggestion made in a previous study [151] that myocytes, where titin is the primary contributor to its passive stress response, may be the major component of the myocardial stiffness at low LV pressure. Interestingly, this result is also reflected in the transmural variation of the constituen Fig. 3.7 ). At a LV pressure of 8 mmHg, the simulations show that the cardiac muscle fiber accounts for most of the load in the local muscle fiber direction between the transmural depth of 0% (epicardium) 60% of 56 the ventricular wall. The collagen fiber, on the other hand, is the dominant load - bearing constituent between the transmural depth of 60% - 100% (endocardium) (i.e., the sub - endocardial region). This finding has implications on pharmaceutical therapies unde r investigation for treating HFpEF that targets specific tissue individual constituent; e.g., attenuating myocardial fibrosis [152] and reducing myocyte titin stiffness [153] . In particular, our finding suggests that reducing LV passive stiffness by altering the collagen fiber network may be most effective when applied to the sub - endocardial region. Conversely, altering the myocyte stiffness to reduce LV passive stiffness may be most effective when applied to the sub - epicardial region. Although we have demonstrated that the microstructural model can be applied to predict load sharing of the constituents that cannot be experimentally measured, caution must be exercised in direct ly applying these results for optimizing treatments as they were obtained based on 1) normal LV geometry, 2) assumptions that the LV has homogeneous material properties and collagen fiber waviness and 3) omission of residual stresses. The model needs to be calibrated using more data, especially those from humans that consider gender - differences in the LV passive behavior [154] , before predictions can be reliably applied in the clinics. Through a parameter sensitivity study, we show that the LV passive stiffness is not only sensitive to the stiffness of the individual tissue constituents (pa rticularly, muscle and collagen fiber), but it is also sensitive to the collagen fiber network ultrastructure; i.e., the mean waviness , the mean azimuthal angle with respect to the muscle fibers. These findings underscore the importance of quanti fying changes in the collagen fiber network ultrastructure during remodeling, particularly in diseases where fibrosis is a key feature such as HFpEF [155] . 57 3.5 Limitations The microstructural FE LV model is not without limitations. First, we adopt a linear transmural variation of the muscle fiber helix angle without any planar splay. This framework can be extended to include physiological distributions of muscle fiber splay in future. Second, the assumptions of homogene ous 1) axisymmetrical distribution of collagen fibers around local muscle fiber direction, 2) bimodal normal distribution to describe the collagen fiber spatial orientation and 3) truncated normal distribution to describe the collagen fiber waviness need t o be validated against microscopy measurements of the myocardium, which to the best of our knowledge, is not available. Third, due to the lack of histological or microscopy measurements of the myocardial tissue ultrastructure, we had to fit most of the mod including morphometric parameters that should, in principle, be prescribed using values directly measured in experiments. This is not ideal as some of the parameters are highly correlated to each other ( Table 3. 4 ). Nevertheless, we have attempted to minimize this and validating our fitted model parameters against independent measurements of the constituents and organ - scale LV inflation experiments. Fourth, we have, for simplicity, ignored the presence of residual strains and stresses in simulating passive filling of the intact LV that may produce a more uniform transmural distribution of stress [14] . The presence of residual stresses and residual strains in the LV may also be responsible for the transmural differences of collagen waviness as observed in some histological studies [149] . Finally, the microstructural model formulation ignores any direct mechanical or 58 physical interactions between the myocytes and collagen fibers, which a recent study suggests to be present in the LV [156] . In conclusion, we have developed a microstructural constitutive model in an efficient FE framework, and used it to investigate the contribution of individ ual tissue constituent to LV mechanics during passive inflation. This framework can be readily biventricular geometry and an active contraction constitutive model of the my ocyte to simulate active mechanics in the LV as well as by calibrating the model using human data. 59 CHAPTER 4 QUANTIFICATION OF THREE - DIMENSIONAL BI VENTRICULAR STRAINS IN PULMONARY ARTERIAL HYPERTENSION USING HYPERELASTIC WARPING 60 Abstract Pulmonary arterial hypertension (P A H) is a life - threatening disease characterized by elevated pulmonary artery pressure (PAP) and pulmonary artery vascular resistance, with limited survival rate and can affect patients of all ages. Advanced imaging of multiple strain components may aid in t he diagnosis and understanding of this disease. In this study, we use a model - to - image registration technique for rapid comprehensive assessment of biventricular circumferential, longitudinal and radial strains. 20 P A H patients and 20 age - and gender - compa rable Controls underwent cardiac magnetic resonance (CMR) imaging. A three - dimensional (3D) heart model including the left ventricle (LV), right ventricle (RV) and septum was reconstructed from these CMR images. The hyperelastic warping method was used to reference the segmented model with the target images and biventricular circumferential, longitudinal and radial strains were obtained. Results show that RV longitudinal, circumferential and radial strain were significantly depressed compared with Controls ( - 11.00±3.65% vs. - 14.79±2.18%, p <0.001; - 7.56±2.83% vs. - 9.47±1.88%, p <0.05; 18.48±9.56% vs. 24.81±7.02%, p <0.05); Septum circumferential and radial strain were significantly depressed compared with Controls ( - 4.35±1.77 vs. - 5.89±1.74, p <0.05; - 10.10 ± 5.00 vs. 13.87±4.73, p <0.05); LV circumferential strain was depressed significantly compared with Control ( - 11.60±2.52 vs. - 14.44±1.97, p <0.001). Our results show that not only RV, but also septum and LV impairment may exist in P A H patients although other CMR parameters in the LV show normal values. Our methodology holds the potential for characterizing 3D strains in P A H patients that may facilitate developing in - depth understanding of the interventricular influence in this disease. 61 4.1 I ntroduction Pulmo nary arterial hypertension (P A H) is a progressive disorder characterized by elevated pulmonary artery pressure (PAP) and pulmonary artery vascular resistance, which increase right ventricular (RV) afterload and lead to RV dysfunction and ultimately right - s ided heart failure [157] [159] . PAP measurement by right heart catheterization (RHC) is the current golden standard for P A H, but it is invasive [160] . Echocardiography is the modality of choice in evalua ting P A H because of its availability and low cost, and three - dimensional echocardiography (3DE) is one technique that can be used to quantify RV volumes, image reconstructions and strains. 3DE overcomes the limitations of two - dimensional techniques, potent ially allowing volume assessment without the use of any geometrical assumptions. However, the endocardial definition assessed by 3DE is sometime difficult to discern in the RV, owing to the abundance of aberrant papillary muscles. This p oor lateral resolut ion in RV can result in underestimated RV cavity volumes using 3DE [161] . CMR has recently been considered as a potential one - stop - shop imaging modality in PH [162] [164] . It is the gold standard for measuring ejection fraction, cavit y volume and mass to non - invasively quantify global heart function [165] [167] . It also provides superior resolution and reproducible results for quantifying ventricular curvedness, deformation, systolic and diastolic function in diverse heart diseases [168] [170] as well as for quantifying in vivo myocardial properties through inverse finite element (FE) modeling [171], [172] . The most commonly used features in cine MRI are the endocardial and epicardial contours, or specific feature points on these contours. Due to its high resolution, motion tracking can be performed over the cardiac cycle. Most motion - tracking 62 methods aim to extract strains in the heart, as myocardial strain encapsulates the basic mechanical function of the myocardium and has the potential to become an important clinical index of regional ventricular function. Regional myocardial strains have direct and indirect relationships w ith cardiac diseases [173] . Normal strains are widely used i n radial, circumferential and longitudinal directions, with circumferential strain the type reported most frequently [174] . Longitudinal strain is the main strain component in assessment of RV function. It has been found to be significantly associated with hemodynamic parameters of RV function [160], [175] [177] . In addition, recent studies have shown that RV longitudinal strain may be a useful prognostic maker in patients with pulmonary hypertension [178], [179] . The hyperelastic warping approach is a deformable image registration technique used to obtain strain measurements from medical images such as magnetic resonance imaging (MRI) and computed tomography (CT). Veress et al. [39] proposed hyperelastic warping to determine left ventricular strain from mid - diastole to end diastole using multiple short - axis (SA) cone slices from MR images. In the hyperelastic warping method, a FE model of the region of interest is deformed by a body force that depends on the difference of image intensities between the template image and the target image. Hyperelastic strain energy based on continuum mechanics is applied to constrain and regularize the deformation [180], [181] . Other regularizers have also been proposed, such as incompressibility [182] , or equilibrium gap [183], [184] . Application of the hyperelastic warping approach in cardiac motion and function has focused primaril y on quantifying LV strains [38], [185] that has been verified by tagged MRI [186] and 3D CSPAMM MR 63 images [181] . This method has also been applied to quantify circumferential strain in individual patient with HFpEF [187] and pulmonary hypertension [188] . In this research, we aimed to quantify the 3D ci rcumferential, longitudinal and radial strains in PH patients and normal controls using the hyperelastic warping method. The goals of the presented study were t o introduce a novel framework to characterize 3D circumferential, longitudinal and radial strain s simultaneously and t o investigate the abnormalities in RV, LV and septum strains and interventricular interactions of P A H patients . 4.2 Materials and methods 4.2.1 Study population Twenty paired sub - groups of subjects were enrolled and underwent CMR scans. For each sub - group, one Control, one PH patient were recruited. They were age - comparable and gender - matched. Normal controls had no known cardiovascular disease or other co - morbidi ties. Patients with a clinical history of PH were recruited as PH patients. All enrolled participants gave written informed consent. The demographics of the study groups are summarized in Table 4. 1 . The protocol was approved by the Local Institutional Review Board, and informed consents were obtained from all subjects. 64 Table 4. 1 . Baseline demographic and CMR characteristics of P A H patients and Controls. The control group did not unde rgo right heart catheterization Variables Control (n = 20) P A H (n = 20) P value § Demographics Age, years old 48 ± 15 47 ± 15 0.824 Gender, Male/Female 5/15 5/15 N/A Weight, kg 60 ± 14 58 ± 12 0.590 Height, cm 159 ± 10 160 ± 11 0.895 Cardiac magnetic resonance LV ejection fraction, % 65 ± 6 61 ± 9 0.177 LVEDV index, ml/m 2 71 ± 8 71 ± 24 0.991 LVESV index, ml/m 2 25 ± 6 28 ± 14 0.361 LVSV index, ml/m 2 45 ± 6 42 ± 12 0.334 LV mass index, g/m 2 39 ± 9 45 ± 14 0.180 RV ejection fraction, % 59 ± 6 43 ± 12 <0.001 RVEDV index, ml/m 2 75 ± 12 114 ± 44 <0.001 RVESV index, ml/m 2 31 ± 8 66 ± 31 <0.001 RVSV index, ml/m 2 44 ± 7 48 ± 21 0.498 Hemodynamics Heart rate, bpm 71 ± 13 83 ± 15 0.012 Diastolic blood pressure, mmHg 78 ± 13 77 ± 15 0.819 Systolic blood pressure, mmHg 131 ± 25 128 ± 23 0.615 Cardiac output, L/min N/A 4.4 ± 2.2 N/A Cardiac index, L/min/m2 N/A 2.7 ± 1.1 N/A Right atrial pressure, mmHg N/A 8.3 ± 7.5 N/A Mean pulmonary artery pressure, mmHg N/A 45.8 ± 15.2 N/A Pulmonary capillary wedge pressure, mmHg N/A 12.8 ± 6.5 N/A Systemic vascular resistance, dynes/cm5 N/A 1774.9 ± 619.2 N/A Pulmonary vascular resistance, dynes/cm5 N/A 745.9 ± 520.3 N/A Pulmonary systemic flow ratio N/A 1.24 ± 0.4 N/A Pulmonary and systemic resistance ratio N/A 0.4 ± 0.2 N/A Data are mean ± SD . EDV: end - diastolic volume; ESV: end - systolic volume; SV: stroke volume; § Independent Sample t Test. 65 4.2.2 CMR image acquisition Cardiac MR scans were performed using steady - state free precession (SSFP) cine gradient echo sequences. All subjects were imaged in a 3.0T Philips scanner (Philips - Adrglam; Philips Healthcare, Netherlands) with a 12 - element body matrix coil. SSFP end - expiratory breath - hold cine images were acquired in multi - planar short - and long - axis views. Typical imaging parameters were as follows: TR/TE 34/1 ms, flip angle 45 o , slice thickness 8 mm for both short and long - axis, echo time 1.404 ms, repetition time 2.808 ms, pixel bandwidth 1796 Hz, percent phase field of view 100 mm, temporal resolution ~35 ms, in - plane spatial resolution 1.6 × 1.6 - 1.8 × 1.8 mm, 40 frame /cardiac cycle for normal control and 30 for other enrolled subjects. Both short - axis images and long - axis images were acquired, showed in Figure 4.1 . Figure 4.1 . CMR images including short - axis image from apex to basal, and 2 - chamber, 3 chamber and 4 - chamber long - axis images. 66 Figure 4. 2 . Overall framework of quantifying the biventricular circumferential, longitudinal and radial strains. 67 4.2.3 Right heart catheterization Right heart catheterization (RHC) was performed at rest using standard techniques. Invasive hemodynamic measurements were not available in the control group. 4.2.4 Overall frame work of hyperelastic warping registration The core of the method was based on the hyperelastic warping technique to register a meshed biventricular model reconstructed from the CMR images at end of systole (ES), along with the CMR images at other time poi nt during a cardiac cycle, so as to track the motion of heart and obtain the strain - time curves. The short axis images were used as the target images. As there is an excursion in long - axis direction of the ventricle, we measured the displacement of septal manually, and assigned this displacement as a boundary condition in long - axis direction. The whole framework was illustrated in Figure 4. 2 , and it wa s implemented using a combination of open - source software: MeVisLab (MeVis Medical Solution AG, Bremen, Germany), Gmsh [189] , FEniCS [129] and in - house code [181] . 4.2.4.1 Model reconstruction Figure 4. 3 shows the reconstruction of the model. Contours of the LV, RV and Epicardium were manually segmented in both short - axis and long - axis images, as shown in Figure 4. 3 (A) . The full image stack was evaluated and the frame with aortic valve closure had been identified as end - systole for all the short axis lo cations [190] . Papillary muscles were included as part of LV volume. Depending on the size of heart and quality of image, 4~9 short - axis images and 4 - chamber long - axis images were used for each of the 40 subjects. These contours were then used t o reconstruct the LV endo surface, RV 68 endo surface and epicardium surface, as shown in Figure 4. 3 (B) . Using a plane to cut off the cover of the model , we obtained the biventricular mode Figure 4. 3 (C) . Gmsh was used for generation of the meshes for each model, with a mesh size of 0.3, as shown in Figure 4. 3 (D) . The points of these 40 models ranged from 2313~3298, and the finite elements ranges from 7527~11485. Each model was partitioned into th ree regions: LV, RV and Septum, as shown in Figure 4. 3 (E) . 4.2.4.2 Image preparation and boundary condition assignment Short axis images were interpolated to double the number of short - axis slices for better registration, as the full short - axis image stack was sparse compared with the model. This step was implemented using a Lanczos filter in MeVisLab. Meanwhile, excursion of the ventricles in the long - axis direction was measured, as shown in Figure 4.4 . A sinusoidal function of this displacement was imposed as a constraint in the biventricular model in the hyperelastic warping method. 69 Figure 4. 3 . Reconstruction of biventricular mesh: (A) segmentation of contours of LV, RV and septum; (B) Surface reconstruction; (C) Bi - ventricular geometry (D) Biventricular mesh; (E) Partition of region (Red: RV free wall; Green: Septum; Blue: LV free wall). Figure 4. 4 . Measureme nt of the septal displacement in 4 - chamber view as the boundary condition: (A) End of systole; (B) End of diastole; (C) Apply the displacement of septal on the model. 70 4.2.4.3 Hyperelastic warping registration Hyperelastic warping is a deformable image registration technique that can be used to measure cardiac strain derived from analysis of medical images such as MRI, ultrasound and microPET imaging [38] . In hyperelastic warping, the meshed heart model is deformed to a target image (T) set during the registration process. Deformation of model is defined as , where u is the displacement fi eld, is the position, and the deformation gradient is defined as ( 4.1 ) The forces responsible for registrat ion deformation are derived from the difference in image intensity of two volumetric image data sets by minimizing the energy expression (4 .2) Here, W is the hyperelastic strain energy related to the material model of myocardium and is the Cauchy - Green deformation tensor. A Neo - Hookean strain energy function is used to define W. The other energy term produces an image force field responsible for the local registration of the discretized reference image R to the target image T , and is given as (4.3) where is the penalty parameter enforcing the alignment of the reference image to the target image. The hyperelastic warping registration was implemented by aligning the meshed model with all frames of short - axis images. Lagrange - Green strain component was obtained at different time frames during a cardiac cycle. 71 4.2.4.4 Post - processing of strains Since the deformation gradient was defined with ES as the reference configuration, the local Green - Lagrange strain tensor with end - diastole (ED) as the reference a more commonly used metric was defined as : (4.4) where is the identity tensor and is the deformation gradient tensor at ED. Normal strains in the circumferential , longitudinal , and radial directio ns were computed by projecting onto these directions using with T he circumferential , longitudinal and radial were prescribed using a Laplace - Dirichlet Rule - Based (LDRB) algorithm [3] with myofiber angle prescribed to be zero. Figure 4. 5 (A) shows the meshed model of three regions (LV, RV and septum) with the CMR images. Figure 4. 5 (B) and (C) represents the circumferential strain direction and circumferential strain - time curve for LV, RV and septum respectively. Figure 4. 5 (D) and (E) shows the longitudinal direction and the longitudinal strain - time curve, and Figure 4. 5 (F) and (G) the radial direction and radial strain - time curve. 72 Figure 4. 5 . Registration the meshed model with images for strain - time curves (A) meshed model with CMR image; (B) circumferential strain orientation; (C) longitudinal strain orientation; (D) radial strain orientation; (E) circumferential strain - time curve; (F) longitudinal strain - time curve; (G) radial strain - time curve. 73 4.3 Results 4.3.1 Patient demographics Study subjects consisted of 5 males and 15 females in each group, with a mean ± SD age of 48±15 and 47±15 years for Controls and P A H patients, respectively. Demographic and clinical characteristic of study subjects were given in Table 4. 1 . Compared with normal Controls, P A H patient had a comparable LV ejection fraction (EF), but RV EF was lower than the controls ( 43±12 vs. 59±6 , p <0.001). RV end - diastolic volume (EDV) index and RV end - systolic volume (ESV) index were higher in PH patients than controls (114±44 vs. 75±12 and 66±31 vs. 31±8 respectively, p <0.001), but RV stroke volume (SV) was comparable. PH patients were comparable to Contro ls with respect to height, weight, body surface area (BSA), body mass index (BMI), systolic blood pressure (SBP), diastolic blood pressure (DBP), LV EDV, LV ESV, LV SV, LV mass index. Heart rate of PH patients was higher than Controls (83±15 vs. 71±13, p=0 .012). Invasive hemodynamic parameters were reported in Table 4. 1 too. P A H patients had a mean pulmonary artery pressure (mPAP) of 45.8±15.2 mmHg, righ t atrial pressure (RAP) of 8.3±7.5 mmHg, pulmonary capillary wedge pressure of 12.8±6.5 mmHg, Cardiac output of 4.4±2.2 L/min, Cardiac index of 2.7±1.1 L/min/m2, pulmonary and systemic resistance ratio of 0.4±0.2, and pulmonary systemic flow ratio of 1.24 ±0.4. Systemic vascular resistance of PH patients was 1774.9±619.2 dyne s/cm5, and pulmonary vascular resistance of 745.9±520.3 dyne s/cm5. 4.3.2 Peak systolic circumferential , longitudinal and radial strains Table 4 . 2 showed the peak circumferential, longitudinal and radial strains in different regions (RV, LV and septum) for Controls and P A H patients respectively. PH patients had significant depress ed RV circumferential ( - 7.56 ± 2.83% vs. - 9.47 ± 1.88%, p <0.05), longitudinal ( - 11.00 ± 3.65% vs. - 14.79 ± 2.18%, p <0.001) and radial (18.48 ± 9.56% vs. 24.81 ± 7.02%, p <0.05) strains. LV strains were also lower in P A H patients, but only LV circumferential strain ( - 11.60 ± 2.52% vs. - 14.44 ± 1.97%) had significant difference between patients and Controls ( p <0.001). Septum strains in P A H patients were 74 lower in Controls, but only circumferential and radial strains had statistics significance ( p <0.05). Table 4 . 2 . Average circumferential, longitudinal and radial strains in RV, LV and se ptum Variables Control (n = 20) PH (n = 20) P value § (%) - 9.47 ± 1.88 - 7.56 ± 2.83 0.016 (%) - 14.79 ± 2.18 - 11.00 ± 3.65 <0.001 (%) 24.81 ± 7.02 18.48 ± 9.56 0.022 (%) - 14.44 ± 1.97 - 11.60 ± 2.52 <0.001 (%) - 15.10 ± 2.14 - 13.31 ± 3.68 0.068 (%) 34.36 ± 9.25 27.84 ± 11.7 0.058 (%) - 5.89 ± 1.74 - 4.35 ± 1.77 0.008 (%) - 16.42 ± 2.54 - 14.05 ± 4.70 0.057 (%) 13.87 ± 4.73 10.10 ± 5.00 0.019 Values are mean ± SD. : Right ventricular peak circumferential strain; : left ventricular peak circumferential strain; : Septum peak circumferential strain. : Right ventricular peak longitudinal strain; : Left ventricular peak longitudinal strain; : Septum peak longitudinal strain; : Right ventricular peak radial strain; : Left ventricular peak radial strain; : Septum peak radial strain. P AH: Pulmonary hypertension; § Independent Sample t Test. 4.4 Discussion P A H is a serious disease often accompanied by RV dysfunction and failure. Survival in patients with P A H is strongly related to RV function [191] . In P A H, RV free wall strain emerged as an important predictor of outcome [192] . However, LV performance may be affected because both ventricles share the interventricular sep tum within the same pericardial sac, although P A H is defined by normal LV filling pressure. Keeping this concept of interventricular independence in mind, we performed this study to investigate 75 the 3D RV, LV and septum strains by treating the heart as a whole and to study the bi - ventricular motion simultaneously. 4.4.1 RV strains Circumferential, longitudinal and radial strains in RV for P A H patients were significantly reduced compared with th e controls, demonstrated the comprehensive impairment in RV motion for this type of heart disease. This is consistent with the reduced RVEF in P A H patients. R educed RV longitudinal strain has been widely reported in P A H patients by MRI [193], [194] , speckle - tracking on 2D echocardiography [175], [192] as well as 3D echocardiography [195] . Reduction in RV circumferential and radial st rain may be due to circumferential stretch and subsequent reduction in contractility. Because the myocardium of RV free wall is oblique in the superficial submyocardium, this finding may signify the failure of outer layer to contract circumferentially [196] . MRI studies had highlighted the value of considering RV circumferential strain for some pathology such as arrhythmogenic RV cardiomyopathy or RV afterload changes [197] . 4.4.2 Septum and LV strains The finding that septal circumferential and radial strains are depressed in P A H patients is interesting and important. Under normal circumstances, the septum behaves much like a thick - walled cylinder subject to internal and external pressu re, with the resulting stresses being circumferential tension and radial compression, just like the remainder of the left ventricle [198] . In P A H, however, this pressure gradient reduced or reversed. The septum withstanding this pressure load becomes more flatted, and is threatened by substantial compressive wall stress, which impedes septal blood flow an d 76 affects septal performance [198] . In a study by Rico et al . [199] , P A H was associated with impaired RV free wall longitudinal strain and septal circumferential strain. Our study also revealed that the LV circumferential strain was reduced, but longitudinal and radial strain was preserved in P A H patients. Although no signi ficant difference was observed in LV CMR parameters ( Table 4. 1 ), LV circumferential strain may reveal an early impairment of LV. This is consistent with the findings of Puwanant el al [200] , who found both septum and LV circu mferential strains were decreased in P A H patients, but the longitudinal strain in the LV was preserved. A potential explanation could be that a reverse septal curvature and more D - shaped LV in patients affect more than longitudinal myocardial fiber shorten ing. The finding that both circumferential and radial strains were reduced in septum and only the circumferential strain in LV was reduced supports the hypothesis that P A H influences the septum more than the LV free wall . Two studies on the finite element modelling can help us understand this physiology [201], [202] . 4.5 Limitations There are limitations in this study, including a small number of enrolled subjects. In order to increase the statistics power, a larger sample size should be included. Besides, longitudinal strain was calculated by imposing a basal longitudinal displacement that var ies sinusoidally with time. Despite a reasonable peak longitudinal strain obtained, the longitudinal strain - time curve did not show exact dynamic function of the heart. A higher out - of - plane resolution (smaller slice thickness) may obviate the need to impo se such an assumption. 77 CHAPTER 5 PATIENT - SPECIFIC COMPUTATIONAL ANALYSIS OF VENTRICULAR MECHANICS IN PULMONARY ARTERIAL HYPERTENSION 78 Abstract Patient - specific biventricular computational models associated with a normal subject and pulmonary arterial hypertension (PAH) patient were developed to investigate the disease effects on ventricular mechanics. These models were developed using geometry reconstructed from magnetic resonance (MR) images, and constitutive descriptors of pa s sive and activ e mechanics in cardiac tissues. Model parameter values associated with ventricular mechanical properties and myofiber architecture were obtained by fitting the models with measured pressure volume loops and circumferential strain calculated from MR images using a hyperelastic warping method. Results show that the peak right ventri cle (RV) pressure was substantially higher in the PAH patient (65 mmHg versus 20 mmHg), who also has a significantly reduced ejection fraction (EF) in both ventricles (left ventric le (LV): 39% v s. 66% and RV: 18% v s. 64% ). Peak systolic circumfer ential strain was comparatively lower in both the left ventricle (LV) and RV free wall (RVFW) of the PAH patient (LV: 6.8% vs. 13.2% and RVFW: 2.1% v s. 9.4%). Passive stiffness, contractility , and myofiber stress in the PAH patient were all found to be substantially increased in both ventricles, whereas septum wall in the PAH patient possessed a smaller curvature than that in the LV free wall. Simulations using the PAH model revealed an approx imately linear relationship between the septum curvature and the transseptal pressure gradient at both early - diast ole and end - systole. These find ings suggest that PAH can induce LV remodeling, and septum curvature measurements may be useful in quantifying transseptal pressure gradient in PAH patients. 79 Nomenclature Regional myofiber passive stiffness (kPa) Pulmonary arterial compliance (mL/kPa) Systemic arterial compliance (mL/kPa) Regional circumferential strain (%) EDP End - diastolic pressure EDV End - diastolic volume ESV End - systolic volume LVSB Left ventricular septal bow Aortic valve opening pressure (mmHg) Pulmonary valve opening pressure (mmHg) PA Pulmonary artery PAH Pulmonary arterial hypertension Pulmonary aortic resistance (kPa ms/mL) Systemic aortic resistance (kPa ms/mL) Pulmonary peripheral resistance (kPa ms/mL) Systemic peripheral resistance (kPa ms/mL) Regional myocardial contractility (kPa) RVFW myofiber helix angle (deg) Regional curvature at the midventricular level (cm - 1 ) Normalized septum curvature (ratio of septum curvature to LVFW curvature) End - systolic septum curvature (cm - 1 ) Early - diastolic septum curvature (cm - 1 ) 80 5.1 Introduction Pulmonary arterial hypertension (PAH) is a disease associated with vasoconstriction of the pulmonary arterial vessels, resulting in an elevated pulmonary arterial pressure. The clinical diagnosis for PAH is an elevated mean pulmonary artery (PA) pressure greater than 25 mmHg with normal wedge pressure [1]. Without treatment, PAH can quickly lead to decompensated right heart failure and death. The current prognosis of PAH remains poor, with about 15% mo rtality within 1 y ea r on modern therapy [2] and a low 3 - y ea r survival rate of 67% [3]. High afterload in the right ventricle (RV) caused by PAH can lead to changes in ventricular mechanics [4]. For example, an elevated RV pressure (RVP) can lead to abnorm which is a left - ward motion of the septum into the left ventricle (LV) cavity. Moreover, PAH is also associated with long - term functional, structural, and geometrical changes in the RV [5]. These changes and their impacts on ventricular mechanics are currently not well - understood [6]. Although animal models have been used to understand ventricular mechanics associated with RV remodeling in PAH [7, 8], similar studies in humans are lacking and most clinical investigations have been confined to quantifying global ventricular mechanics through pressure volume (PV) loop measurements [9,10]. Image - based computational models are increasingly used to analyze ventricular mechanics in heart diseases (e.g., myocardial infarction [11], left branch bundle block [12], and mitral valve regurgitation [13, 14 ] ) as well as heart failure treatments [15]. The focus of these computational analyses is, however, on the LV that has historically recei ved more attention than the RV. On the other hand, computational modeling analyses of PAH have 81 mostly been performed on pulmonary arteries [16, 17]. The RV, whose function is a major determinant of prognosis in PAH [18], has largely been overlooked. To fi ll the void in the current research of PAH, here we describe an image - based computational analysis of the ventricular mechanics in a PAH patient. Specifically, we seek to quantify regional ventricular myofiber stress, myofiber strain, contractility, and pa ssive tissue stiffness in a PAH patient, and compare them to those found in an age - and gender - matched normal subject. We also use the developed biventricular computational model of the PAH patient to investigate and quantify the effects of transseptal pre ssure gradients on the septum curvature. 5.2 Method 5.2.1 Acquisition A 3 - T Philip s scanner with electrocardiogra phy gating was used to acquire cine magnetic resonance (MR) images from one PAH patient as well as one gender - and age - matched normal human subject. The patient was a 42 - y ea r - old male who had been diagnosed with severe PAH for 2.5 y ea r at the time of MR imaging and right heart catheterization for the study. Left and right ventricular, atria, and arterial pressures were also acquired from the P AH patients by right heart catheterization as part of their clinical care. All data were acquired at the National Heart Center of Singa pore, and all enrolled participants gave written informed consent . 82 5.2.2 Biventricular g eometry Three - dimensional bive ntricu lar geometries of the PAH patien t and normal subject were recon structed by manually delineating the contours of the LV endocardium, RV endocardium, and epicardium in different short - and long - axis views of the cine MR images. These contours were then fitted to obtain 3D surfaces of the LV endocardial, RV endo cardial, and epicardial surfaces using a medical image analysis software MEVISLAB Finite element (FE) meshes for the two cases were generated in the volume enclosed by these surfaces using GMSH [1 9]. The meshes consist of approximately 4000 quadratic tetrahedral elements and 7000 nodes. 5.2.3 Pressure volume l oops Cavity volumes of the LV and RV were measured in the PAH pa tient and normal subject at dif ferent time frames within a cardiac cycle usi ng the biventricular geometries described in Sec.2.2. These measured volumes were paired with the acquired pressure data to reconstruct the LV and RV PV loops of the PAH patient. To synchronize the MR images with the acquired pressure data, we first adjust ed for any differen ces in heart rate by scaling the heartbeat duration in the MR images to that of the pressure data. Thereafter, the pressure data were synchronized with the MR images (which were acquired starting at end - diastole) using the end - diastolic pressure, which corresponds to the pressure at the beginning of a steep pressure rise. For the normal subject, n ormal RVP waveform from a previ ous study [20] and a scaled normal left ventricular pressure (LVP) waveform (with end - systolic pressure equal to 0.9 of the measured cuff pressure [21]) were paired with the cavity volume measure ments to reconstruct the PV loops. 83 5.2.4 Strain a nalys is of magnetic resonance i mages Regional circumferential strain was estimated from the PAH patient and the normal subject using a hyperelastic warping method that is implemented by a plugin (FEWarp) in FEBio [22]. Briefly, in the hyperelastic warping method, an FE model of the biventricular geometry from the template image was deformed into alignment with the c orresponding object in the target image via a computed image - based local body force term that depends on (1) the difference in image intensity between the template and target images, (2) the target image intensity gradient, and (3) a prescribed penalty fac tor [23]. Because previous studies [23, 24] have shown that strain predictions from hyperelastic warping method are insensitive to changes in material coefficients, a com pressible neo - Hookean strain en ergy function with unit material constant was used in b oth cases, and the penalty factor was increased incrementally until the strain field converged. Spatially averaged was computed in the LV and RV free wall (RVFW) at different cardiac time points using end - diastole as the reference configuration. Cir cumferential direction was prescribed using the Laplace Dirichlet rule - based (LDRB) algorithm [25 ] with myo fiber angle set to zero. 5.2.5 Biv entricular finite element m odel Biventricular geometries reconstructed from MR images corresponding to the time po int at which LVP and RVP were at their lowest (in early dias tole) were used as the reference state for the FE models of the PAH patient and normal subject ( Figure 5.1(a) ). Myocardial fiber orientation was prescribed in the models using an LDRB algorithm [2 5]. Based on the previous experimental measurements [26], myofiber helix angle was prescribed in the LV with a linear transmural variation from 60 deg at the endocardium to 84 - 60 deg at the epicardium (i.e., 60 deg/ - 60 deg) ( Figure 5.1(b) ). Given the lack o f human RV myofiber data, myofiber orientation in the RVFW was described similarly using a linear transmural variation deg/ deg, with as a model parameter that is fitted to the measurements. 5.2.5.1 Cardiac m echanics. A cellular - based cardiac electro - mechanics model [27, 28] was used to model the mechanical behavior of the biventricular unit over a cardiac cycle. Because the focus here is on biventricular mechanics, we have simplified the model by prescribing a stimulus current everywhere in the model to describe homogeneous activation of the biventricular unit. The resultant model can therefore be described by the following system of ordinary differential equations and partial differential equations: (5.1a) (5.1b) (5.1c) Briefly, Eqs. (5.1a) and (5.1b) consist of a system of ordinary differential equations that describe the local coupling between cellular electrophysiology [29] and cross - bridge cycling process [30]. Here, , , and denote a vector of state va riables consisting of various membrane channels and intracellular ionic concentration, the total ionic current that is scaled with the membrane capacitance, the transmembrane potential, and the myofiber stretch, respectively. Eqs. (5.1c) describes the quas i - static mechanical equilibrium of the cardiac tissue with 85 denoting the Cauchy stress tensor. As shown previously [31], the inertia effect on stresses is not significant in a beating heart. An active stress formulation was used to model contraction of t he tissue. In this formulation, was decomposed into a passive component and an active component that accounts for the stiffening of the tissue in the myofiber direction during cross - bridge cycling, i.e . , ) (5.2) The active stress tensor depends on the time evolution of the state variables , myofiber stretch and rate of elastic myofiber stretch . The contractile stress was scaled by a reference tension , a model parameter that is associated with the tissue contractility. Although previous studies have suggested the presence of a secondary active stress acting transverse ly to the muscle fiber [32], we did not include this feature in our model for simplicity. The passive stress was described by a Fung - type transversely isotropic hyperelastic constitutive model with the following strain energy function: (5.3a) where (5.3b) In the above equation, is the Green - Lagrange strain tensor, J is the Jacobian of the deformation gradient tensor , and with are the stra in components with and , respectively, denoting the fiber, sheet, and sheet normal 86 directions. The passive material parameters are , C, and . The passive stress is related to the strain energy function as follows: (5.4) 5. 2.5.2 Simulation of a cardiac c ycle. A full cardiac cycl e was simulated based on an open - loop circulatory model. Specifically, passive filling was simulated by incrementally applying pressure to the LV and RV endocardial surfaces until the prescribed end - diastolic pressures (EDP) were reached. The biventricular model was then stimulated, and the LV and RV cavity volumes were con strained to remain constant during the isovolumic contraction phase. The ejection phase was simulated by coupl ing the LV and RV cav ity volumes each to a three - element Windkessel model whe n the LVP and RVP exceeded the aortic and pulmonary valve opening pressures, respectively ( Figure 1(c) ). The isovolumic relaxation phase was simulated by constraining the cavity volumes and began when outflows from the LV and RV became negative. The base of the biventricular model was constrained from moving out of the plane, and the epicardial surface was imp osed with a spring boundary con dition (with a distributed spring constant of 0.5 kPa/cm) to simulate the presence of pericardial fluid surrounding th e biventricular unit. 5.2.5.3 Model p arameterization. Biventricular FE models of the PAH patient and normal subject were each divided into two material regions, namely, the LV and the RVFW with different passive stiffness and contractility ( Fi gure 1(a) ). Together with the peripheral resi stance of the Windkessel models (i.e., ) and the RVFW myofiber helix angle (i.e., ), t hese material parameters were manually adjusted to fit the (1) experi mentally measured LV 87 and RV PV loops, and (2) the regional cir cumferential strain measurements from hyperelastic warping in each case. Specifically, the passive stiffness parameter in both LV and RV was first adjusted to fit the corresponding measured end - diastolic volumes and pressur es. Thereafter, regional myocar dial contractility, peripheral resistance (i.e., ), a nd RVFW myofiber helix angle (i.e., ) were adjusted to fit the sys tolic portion of PV loop and the circumferential strain in both LV and RV . All other material parameters were prescribed with the same values in both cases. The model parameters are summarized in Table 5.1 . Table 5.1 . Model parameters. References from which the parameter values were obtaine d are denoted by square bracket 5.2.5.4 Septum c urvature. The fitted PAH model was used to quantify the effects of varying RV afterload and preload on the regional curvature of the LV endo cardial surface. Regional curva ture was calculated at the midventricular level (half - way between the apex and base) from the biventricular FE model at different time points within a cardiac cycle. Specifically, the con - Fitt ed Prescribed Passive stress model , [30] Active stress model , Other parameters [29] Windkessel model Myofiber orientation LV helix angle 60 deg/ - 60 deg [25] 88 tour of an LV endocardial slice at the midventricular level was parameterized as in a polar coordinate system ( Figure 1(d) ) and was computed at each point in the contour using (5.5) where . Septum and LV free wall (LVFW) curvature were computed by averaging for all the points loca ted in their respective region. Figure 5.1 . (a) Biventricular geometry reconstructed from MR images with LV (right, light region) and RVFW (left, dark region) material regions. (b) Myofiber orientation prescribed using LDRB method [3] with a transmural variation of 60 degree (endo) to - 60 degree (epi) for the entire biventricular model. (c) Coupling the biventricular model to three - the LV endocardium. 89 5.3 Results 5.3.1 Pressure volume l oops The PV loops of the RV and LV predicted by the model wer e in agreement with the measure ments in both PAH and normal cases ( Figure 5.2 ). The average error, which was calculated by averaging the percentage difference of (1) the predicted and measured pressure during the filling and ejection phases, and (2) the pr edict ed and measured volume dur ing the two isovolumic phases, was within 4% for the two cases. The maximum error was 10% and occurred during the LV ejec tion phase in the PAH case. Figure 5.2. Measurements and model predictions of the PV loops for the (a) PAH p atient - systole) Peak RVP was about 3.5 times higher in the PAH patient than that found in the normal subject (65 mmHg v s. 20 mmHg), whereas peak LVP was only slightly higher in the PAH patient (110 mmHg v s. 100 mmHg). End - diastolic volume (EDV) and end - systolic volume (ESV) of the LV and RV were also larger in the PAH patient than in the 90 normal subject with the largest differ ence found the RV (RV EDV: 275 ml v s. 100 ml; RV ESV: 225 ml v s. 36 ml). Correspondin gly, the ejection fraction (EF) of the PAH patient was substantially lower in both ventricles when compared to that of the normal subject (LV EF: 39% v s. 66% and RV EF: 18% v s. 64%). Work done by the LV and RV was also computed based on the areas enclosed in the PV loops, which show a slightly lower LV work in the PAH patient (5692 mmHg mL v s. 5870 mmHg mL) and a significantly increased RV work in the PAH patient (2438 mmHg mL v s. 1304 mmHg mL). 5.3.2 Regional circumferential s train Circumferential strain measured using hyperelastic warping was substantially depressed in both LV and RVFW of the PAH patient when compared to the normal subject ( Figure 5.3(a) and 5.3(b) ). Specifically, spatially aver aged peak circumferential strain in a cardiac cycle was 2.1% and 9.4% in the RVFW of the PAH patient and the normal subject, respectively. Left ventricular peak was also depressed in the PAH patient when compared to the normal subject (6.8% v s. 13.2%). In both cases, peak in the LV was found to be larger than that in the RVFW. The model predictions agreed reasonably well with these strain measurements, with a maximum difference between the model - predicted peak strain and the measurements at about 5% in the LV of the normal subject. We also found that the is very sensitive to the myofiber orientation, and in order to capture differences in between the LV and RVFW, transmural variation of myofiber angle in the RVFW has to be larger than that in the LV. 91 Figure 5.3 . Circumferential strain for (a) PAH patient and (b) normal subject. (c) Comparison of regional peak negative between the PAH patient and normal subject. 5.3.3 Myofiber s tress Myofiber stress in the PAH patient was substantially higher in the entire biventricular region compared to the normal subject (Figure 5. 4). Spe cifically, the peak average myo fiber stress in the RVFW and LV of the PAH patient was almost twice the values found in the normal subject (LV: 99.7 kPa v s. 51.2 kPa; RVFW: 73.2 kPa v s. 43 kPa). For both PAH patient and normal subject, the myofiber stress in the LV was higher than that in the RVFW. The myofiber stress was also comparatively lower at the thicker RV insertion points than that found in the thinner LVFW and septum in both ca ses. 92 Figure 5.4 . Peak myofiber stress in the PAH patient (left) and normal subject (right) 5.3.4 Model p arameters Table 5.2 shows the fitted model parameter values for the PAH patient and normal subject. Model parameters and reflect the tis con and contractility were both higher in the LV and RVFW of the PAH patient than those in the no rmal subject. Peripheral resist ance in the RV was also substantially higher in the PAH patient than in the normal subject, but was similar in the LV of both. A sensitivity analysis (see APPENDIX F ) on the parameters , and shows that the difference in parameter values between the normal and PAH cases cannot be attributed to the uncertainty associated with the measurements of RVP (4 mmHg) [20]. We also found that the transmural variation of myofiber angle was steeper in the RVFW with the fibers oriented more longitudinal ly than in the LV in both cases. This regional variat ion was 93 found necessary to pro duce a smaller circumferential strain in the RVFW (compared to the LV) as found in the hyperelastic warping measurements ( Figure 5.3 ). Table 5.2 . Regional model parameters u nits: and in kPa; in kPa ms/mL Case LV RVFW (deg) PAH 1.68 262.5 40 1.96 160 80 26 Normal 0.035 123 40 0.12 73.8 75 5 5.3.5 Septum c urvature Two - dimensional curvature lated from Eq. (5) on the LV en docardial surface at the midven tricular short - axis slice revealed a substantially lower in the septum of the PAH patient ( Figure 5(a) and 5 ( b) ). Compared to the normal subject, who has a more homogeneous cu rvature in the LV, the septum curvature in the PAH patient was substantially lower than that found in the LVFW. To eliminate the effects of LV diameter on , the septum curvature was normalized against the average LV curvature to give a normalized curvatur e . We found that the normalized curvature was lower in the septum of the PAH patient than that in the normal subject ( Figure 5(c) ). 94 Figure 5.5 . normal subject. (c) Comparison of normalized septum curvature between PAH patient and normal subject. 5. 3.5.1 Effects of a fterload The septum curvature decreased wi th increasing RV afterload ( Figure 5.6(a) ), which was adjusted in the PAH model by simultaneously increasing the pulmonary peripheral resistance and decreasing the RVFW contractility so that the stroke volume was relatively unchanged ( Fig ure 5.6(c) ). The largest variation of septum RV afterload occurred during systole, with maximum curvature varying almost linearly ( ) with the end - systolic transseptal pressure gradient between the LV and RV (i.e., LVP RVP) ( Figure 5.6(b) ). On the other hand, septum curvature was less than the normal case ( Figure 5 .5 (b) ) even after reducing the afterload to a value close to the normal physiological RVP of 30 mmHg. 95 Figure 5.6 . Effects of afterload on septum curvature within a cardiac cycle. (a) Time - systolic curvature versus end - systolic transseptal pressure gradient. (c) Corresponding PV loops. Figure 5.7. Effects o f preload on septum curvature within a cardiac cycle. (a) Time course - diastolic curvature versus early - diastolic transseptal pressure gradient. (c) Corresponding PV loops. 96 5.3.5.2 Effects of p reload The septum curvature also decreased with increasing RV preload ( Figure 5. 7(a) ), which was simulated by increasing the prescribed EDP. Because of the length - dependent effects of the contraction model, RVFW contractility was also simultaneously adju sted to maintain a constant stroke volume in each of the preload cases with respect to the baseline PAH case ( Figure 5. 7(c) ). In contrast to the effects of afterload, the largest varia tion of septum curvature occurred during early diastole, whereas the curvature was little chan ged in systole. The maximum dia stolic curvature also varied linearly ( ) with the transseptal pressure gradient between LV and RV at early dias tole. Specifically, we found that an increase in transseptal gradient by 1 mmHg produces an average increase in by 0.0156 cm - 1 . 5.4 Discussion We have described the development o f patient - specific biven tricular computational models using pressure measurements and MR data from a PAH patient and a normal human subject. In each case, we have reconstructed th e LV and RV PV loops using pres sure data, and cavity volumes quantified from the MR images. A hyperelastic warping method was also used to compute the LV and RVFW circumferential strains from cine MR images. These strain measurements and PV loops were used to individually parameterize the biventricular computational models. Using these patient - specific models, we have compared the in vivo regional ventricular mechanics between the PAH patient and normal subject, which inc ludes regional myofiber stress distri bution, mechanical properties, a nd regional curvature. 97 Addition ally, we have also quantified the effects of RV preload and afterload on the septal curvature using the PAH model. As most previous computational studies of PAH have focused on the arteries [16, 17], this study, to the best of our knowledge, is the first patient - specific computational analysis of ventricular mechanics associated with PAH, even though the results and con clusions presented here are based on one PAH patient and normal subject that is clearly insufficient to represent a statistically suffi cient sample size. 5.4.1 Circumferential s train Compared to the LV, quantifica tion of RV strain is significantly more challenging because of its complex geometry and thin wall. Nevertheless, RV strain analysis has been performed on PAH patients using echocardiographic images with speckle tracking [33], as well as strain - encoded MR imaging [34]. In these analyses, RVFW of PAH patients was found to be significantly depressed compared to normal subjects. Our results, which were obtained using a hyperelastic warping method [23] on cine MR images, agree with these studies. Besides a reduction in RVFW , we also found that the LV was reduced in the PAH pat ient compared to the normal sub ject (6.8% in PAH versus 13.2% in normal). This feature was also found in another echocardiography study [ 3 5] in which LV lateral wall and septum circumferential strains were all found to be significantly reduced in pulmonary hypertensive patients. 5.4.2 Myofiber s tress RV wall stress is complicated by its irregular geometry [36]. We have overcome this complication in our computational analysis and have found that myofiber stre ss in the RVFW of the PAH patient was about twice as high as that in the normal subject. The larger RVFW myofiber stress can be attributed to the 98 combination of a higher afterload (3 X of normal) and a substantially larger RV (2 X of normal) in the PAH pat ient. Additionally, we found that LV myofiber stress was also about twice as high as that in the nor mal subject despite having a normal afterload. The higher LV myofiber stress could be attributed to the larger LV cavity (1.5 X of normal). Given that eleva ted myofiber stress is a feature that is associated with ventricular remodeling [37], this result suggests that remodeling associated with PAH may not only be confined to the RV but could also be found in the LV. 5.4.3 Fitted model p arameters 5.4.3.1 Ventricular p roperties. Fitted model parameters associ ated with regional biventricular mechanical properties also sug gest significant remodeling in the PAH patient. Specifically, we found that the diastolic passive stiffness and contractility of the PAH patient were larger than that of the normal subject in both LV and RVFW. These findin gs were consistent with the pre vious experimental studies. By comparing the passive tension found in RV tissue of PAH patie nts undergoing heart/lung trans planta tion with that from donors having non - failing heart, Rain et al. [38] found that RV diastolic stiffness is increased in PAH patients and is closely associated with the disease sev erity meas ured using stroke volume and a 6 - min walk test. They attributed thi s increase to RV fibrosis and the intrinsic stiffening of RV myo cytes. A similar finding was also re ported in rats subjected to pul monary artery banding [7]. Interestingly, the study by Rain et al. [38] also found an increase in the RV end - systolic elastan ce and the force - generating capacity in isolated RV cardiomyocytes from PAH patients. An increase in RV end - systolic elastance was found in rats induced with PH via monocrotaline injection compared to the controls [39]. 99 As suggested in these studies , the i ncrease in RV contractil ity associated with PAH is most likely a compensatory mechanism attempting to cope with the increased afterload in the RV. As with our findings for a nd myofiber stress, the substan tially higher LV diastolic passive stiffness and contractility found here suggests the presence of remodeling in the LV of the PAH patient. Although we did not find any studies that measure LV mechanical properties that are directly associated with PAH, it was observed that patients with right heart failure often have relaxation abnormalities that may affect the mechanical properties of the LV [40]. Moreover, recent studies have also found evidence of electrophysiological remodeling in the LV of PAH patients [41]. 5.4.3.2 Myofiber o rientation. Histol ogical studies have found that compared to the LV that has muscle fiber running obliquely toward the apex, the deep muscle fibers of the RV are longitudi nally aligned from base to apex [42]. Our result also suggests that the myofiber orientation is more lo ngitudinal in the RV than in the LV in both cases, an outcome that is in part due to the smaller cir cumferential strain found in the RVFW in both cases. On the other hand, studies on rats [7] have fou nd that both myofiber and colla gen fiber orientations in the RVFW became more longitudinal as a result of RV remodeling in PAH. We, however, did not find this feature in our study. 5.4.4 Septum c urvature The fact that abnormal septal geom etry and motion is a well - recognized feature in PAH patients [43] has motivated us to quantify this fe ature and investigate its sensi tivity to RV preload and afterload. Our results show that septum curvature from the short - axis view at midventricular level was substantially lower in the PAH pati ent compared to that 100 in the no r mal subject. Physically, this implies that the septum is more PAH patient. With varying preload (and constant afterload in the RV), we found that septum curvature at early - diastole increased linearly with increasing diastolic transsepta l pressure gradient (LVP RVP) ( ). Septum curvature in systole is, however, not highly affected by changes in the preload. Similar findings were also reported in other studies [44, 45], which found that the septum shape and motion at early - diastole are sensitive to the transseptal pressure gradient between the LV and RV. Specifically, Tanaka (negative curvature) in PAH patients resulted from a negative transseptal pressure gradient, wh ereas Beyar et al. [ 45] reported that the bowing occurs only at a transseptal pressure gradient lower than 5 mmHg. Although we did not observe any bowing for the range of transseptal pressure gradient between 1.5 and 2.5 mmHg, our finding of a linear varia tion between the pressure gradient and septum curvature is consistent with these studies. Similarly, we found that the end - systolic septum curvature increased almost linearly with incre asing systolic transseptal pres sure gradient ( ), and the curvat ure at diastole was insen sitive to this variation. Our result is quantitatively in good agreement with an echocardiogra phic study on children with pul monary hypertension [46], which f ound a highly linear correlation ( ) between septum cur vature and R V end - systolic pres sure. This direct relationship between septum curvature and end - systolic pressure would suggest that the curvature can potentially be used as a noninvasive marker for quantifying RV pressure in PAH patient. Also, we note that the septal shape was quantified here using 2D curvature, which does not consider the exten t of surface bend ing. The 3D surface curvedness h as been used in ischemic cardio myopathy [47], 101 heart failure [48], and repaired tetralogy of fallot [49]. Septal 3D curvedness ma y al so be potentially useful in pre dicting pressure gradient in PAH patients. 5. 5 Model l imitations. Overal l, the predictions of the biven tricular computational models are generally consistent with the results from clinical and experimental studies of PA H. Our results have also suggest that PAH may produce significant LV remodel ing though caution must be exercised in extrapolating these results, which are based from only one constructed patient model and control. Additionally, there a re also some limitati ons associ ated with our models. First, we have used the biventricular geometry that corresponds to the lowest pressure at the filling phase as the zero - stress state, and the values of the fitted mechanical properties may be affected by this assumption. Of course, this is an approximation as the zero - stress state does not exist in vivo. Other investigators have used the end - systole [50] or mid - systole [51] as the zero - stress state, or have estimated the zero - stress state from the end - diastole geometry using prescribed mechanical properties [12]. To the best of our knowledge, there appears to be no consensus on the best approach to address this issue. Second, we have not sampled th e entire parameter space in fit ting our model. Specifically, we h ave kept the pa ssive tissue ani sotropy fixed based on values obtained in normal humans [52] and the LV myofiber orientation fixed at 60 deg/60 deg. Because of the sensitivity of mechanical models, the use of a rule - ba sed algo rithm instead of actual fiber data is also a simplification that could introduce error. However, we have chosen to focus on the key fea tures found in the RV that are 102 as sociated with PAH. Although fur ther study is needed, we do not expect these limitations and simplifications to highly impact the resu lts. 103 CHAPTER 6 EFFICIENT ESTIMATION OF PERSONALIZED BIVENTRICULAR MECHANICAL FUNCTION EMPLOYING GRADIENT - BASED OPTIMIZATION 104 Abstract Individually personalized computational models of heart mechanics can be used to estimate important physiological and clinically - relevant quantities that are difficult, if not impossible, to directly measure in the beating heart. Here we present a novel and efficient framework for cre ating patient - specific biventricular models using a gradient - based data assimilation method for evaluating regional myocardial contractility and estimating myofiber stress. These simulations can be performed on a regular laptop in less than two hours and produce excellent fit between measured and simulated volume and strain data through the entire cardiac cycle. By applying the framework using data obtai ned from three healthy human bi ventricles, we extracted clinically important quantities as well as explored the role of fiber angles on heart function. Our r esults show that steep fiber angles at the endocardium and epicardium are required to produce simu lated motion compatible with measured strain and volume data. We also find that the contraction and subsequent systolic stresses in the right ventricle are sign i ficantly lower than in the left ventri cle. Variability of the estimated quantities with respect to both patient data and modeling choices are also found to be low. Because of its high efficiency, this framework may be applicable to modeling of patient - specific cardiac mechanics for diagnostic purposes. 105 6.1 Introduction Cardiac computational modeling has emerged as both a powerful method to provide basic insight into cardiac function / dysfunction, and as a sup port tool to improve current clinical practice. Its development is in part driven by significant advancements in medical imaging techniques [203] [205] , which now provide a wealth of information about cardiac structure and kinematics . Merging this infor mation with bio physical descriptions of cardiac behavior allows for the creation of powerful patient specific models of the heart [11], [206], [207] . Such models can be used to predict the outcome of different treatment strategies [208] or to extra ct useful indicators of mechanical function, such as myocardial contractility [209], [210] and myofi ber stress [188], [211] , potential biomarkers which are currently difficult, if not impossible, to measure directly using imaging techniques [212] . Of particular importance is ventricular myofiber stress [107] , which is hypothesized to be a key driver of pathological remodeling processes in cardiac diseases [213] . Correspondingly, quantifying stress and determining how cardiac interventions may reduce abnormal stress is considered a useful avenue in developin g treatments for heart failure [214] . However, while measurements of heart motion are possible using an array of imaging techniques, no direct measurements of the load experienced by myocytes are currently possible in vivo and estimates are used instead. One widely used method is the law of Laplace, a simplified model that takes into account pressure, wall thickness and curvature, and can be used to evaluate str ess in idealized geometries. However, despite its wide use, it has been shown that this law severely under estimates myofiber stress in largely irregular 106 patient - s pecific ventricular geometries [215] . Furthermore, regional stresses also cannot be accurately estimated using this idealized law. In order to overcome these li mitations, patient specific simulation using finite element modeling is widely accepted as a viable way to accurately estimate myofiber stresses in the complex geometry of the heart, and has been used in designing heart failure treatments to reduce myocard ial stress [214], [216], [217] . However, one of the many challenges faced by researchers developing patient specific models is to efficiently and accurately incorporate individual data into the them, which oft en requires determining model parameters that best reproduce the observations i.e., data assimilation [218], [219] . Typically, one defines a cost function representing the mismatch between simulated and observed data, and searches for model parameters that minimize this cost functi on. Several techniques have been employed to solve this minimization problem. Global methods, using parameter sweeps [42], [220], [221] or genetic algorithms [222], [223] , are attractive because they can cover the entire parameter space, and are theref ore more li kely to retrieve the global mini mum of the cost function. However, such methods require an extensive number of functional evaluations, which in the case of heart mechanics can be computationally expensive. Local optimization methods, on the othe r hand, typically start at some given initial guess, and iteratively search the local neighborhood for better candidates in the minimization of the cost function. These methods are typically f aster than global methods, but have the drawback that the soluti on may depend on the initial guess. One example of a local method is the reduced order unscented Kalman filtering (ruKF) approach [224] , which has been applied to personalize cardiac 107 electromechanical models from cine MRI [225] . Another class of local optimization me thods are the gradient - based methods, which successively reduce the cost functional by searching along the gradient descent direction. While these methods may substantially improve the convergence towards the minimum, estimating the gradient in these metho ds, however, introduces significant additional computational costs. Specifically, estimating the gradient by standard finite differ ences typically requires as many functional evaluations as the number of control parameters (N + 1 evaluations for N paramete rs). Gradient - based methods are therefore impractical if the number of control parameters is large. Nevertheless, gradient based approaches have been applied to per sonalize cardiac mechanics models in several studies [218], [226] [228] . For e xample in [227] a sequential quadratic programming (SQP) optimization technique was utilized to estimate passive material param eters , while in [229] a Levenberg Marquardt method was used to estimate material parameters from shear data. In [228] the minimization was performed using a quasi - Newton BFGS - B method, where the gradient was computed using the adjoint method. In this study, however, the adjoint equation was derived analytically, which may be challenging in more complex problems of cardiac mechanics. More recently, a new approach based on automated derivation of functional gradients via solving the corresponding adjoint system have emerged [230] . Overcoming the issue that gradient - based methods face in dealing with a large n umber of parameters, this approach enables one to compute the functional gradient at a computational expense that does not depend on the number of control parameters [226] . 108 Here, we apply such a gradient - based data assimilation framework in order to fuse clinical imaging data from a cohort of healthy subjects to a bi ventricular mechanics model accurately and efficiently. By relating physical processes to the kinematics observed in medical images, we extracted c linically important quantities from these subject - fitted models and evaluated the sensitivity of these quantities to modeling choices such as fiber architecture and model for active contraction. The paper is organized as follows. In s ection 6.2 we present the pipeline for data assimilation that includes an outline of the underlying ventricular mechanics model and solution methods. Section 6. 3 presents the results of applying the framework to imaging data acquired from three healthy subjects, including a co mparison of model prediction with the observed data, analysis of mechanical parameters extracted from the model, and a sensitivity analysis of model parameters to the input data. Finally, in sections 6. 4 and 6. 5 we discuss the performance of the framework and draw conclusions about its applicability in clinical settings. 6.2 Methods 6.2.1 Data acquisition and pre - processing Cine magnetic resonance (MR) images of 3 healthy subjects, referred here as CASE1, CASE2 and CASE3, were acquired at the National Heart Center of Singapore and written informed consent was obtained from all participants. Three - dimensional biventricular geometries of each subject were manually segmented from the MR images at multiple cardiac time points using the medical image analysi s software MeVisLab ( http://www.mevislab.de ). 109 Cavity volumes of the left ventricle (LV) and right ventricle (RV) were computed from the segmented geometries at different time points in a cardiac cycle in each subjec t . Using a method described in [188] , these volumes were paired with normal left and right ventricular pressure wa ve forms from previous studies [ 231] to construct pressure - volume loops of the LV and RV. Based on a previous empirical study [232] , LV pressure for each subject was also scaled so that the end - systolic pressure is 90% of the measured cuff pressure. The observed regional circumferential and longitudinal Green - Lagrange strains in the LV free wall (LVFW), septum and RV free wall (RVFW) were estimated fr om the MR images in each subject using an hyperelastic warping technique [40], [181] . The output of this pre - processing step is used to calculate the estimated regional strain - time data. Briefly, a bi ventricular finite element model reconstructed from the end - systolic (template) image was registered to all other cine (target) images acquired in the cardiac cycle by minimizing the squared difference between the target and template image intensities. This ill - posed correlation problem is regularized b y also minimizing a prescribed strain energy function over the mesh. We note that other regularization approaches hav e also been proposed, such as regularization based on incompressibility [182] or on equilibrium [184], [233] . Hyperelastic warping offers a good balance between regula rization and strain estimation [181] . The implementation of the image correlation procedure is based on FEniCS [129] , and is freely available. Three - dimensional biventricular meshes of the three normal subjects were created using Gmsh with the number of elements ranging from 4000 - 8000 110 tetrahedral elements. The chosen reference geometries were reconstructed from MR images in late diastole, and all meshes were uni formly refined in order to perform a convergence analysis . Rule based fibers were assigned using the Laplace Dirichlet Rule - Based (LDRB) algorithm [3] . Although previous histological studies [234] suggest that myofiber fiber helix angle varies transmurally from +60 at the epicardium, variability in fiber angle, nevertheless, exists between individuals. Therefore, we seek to also test how different fiber angle gradient alters the param eter estimation and the extracted outputs. More specifically, an angle + / is prescribed on the endo - /epicardium for ranging from 30 to 80 at increments of 10 . If not otherwise specified, an angle of +60 on the endo - and epicardium respectively is prescribed. In Figure 6.1 , we show this range of fiber fields for one of the subjects. Figure 6.1 . Left: finite element mesh of a biventricular geometry reconstructed from MR images separated into 3 material regions, namely, LVFW (blue), septum (green) and RVFW (right). Right: myocardial fiber orientation are assigned using the LDRB algorithm [4] with an angle + prescribed on the endocardium and epicardium, respectively. Here showing the fiber architectu re for ranging from 30 to 80 with increments of 10 , where the absolute value of the fiber angle is used as color - map. 111 6.2.2 Mechanical m odeling We consider a configuration of a biventricular continuum body and denote the reference and current configu rations by and , respectively. Letting and be the coordinates of a given material point in the reference and current configuration, respectively, we have the corresponding displacement field , and the deformation gradient is given by: (6.1) Mechanics of the heart wall was described using an active strain formulation ( [235] ) that assumes a multiplicative decomposition of the deformation gradient, (6.2) Here, is associated with an inelastic deformation resulting from the actively contracting muscle fibers, whereas is associated with the elastic deformation that preserves compatibility in the tissue , and passively carrying the mechanical load. We choose to have the specific form as follows: (6.3) Where the parameter is associated with the relative active shortening along the muscle fibers. The same form of the active deformation gradient has previously been applied in e.g. [226], [236] . We consider the transversely Holzapfel a nd Ogden hyperelastic material [16] model that has the strai n energy density function (6.4) Where the invariants are given by 112 (6.5) Here is the right Cauchy Green tensor, and denotes the unit fiber vector field in the reference configuration. Within the active strain formulation, the strain energy depends only on the elastic deformations, and so the modified strain energy function was used instead For comparison, we also test the more frequently used active stress formulation [237] . In this formulation, the total Cauchy stress tensor is additively decomposed into a passive and an active component i.e., (6.6) Where the passive stress tensor is given by (6.7) And the active stress tensor is given by (6.8) Here is the magnitude of the active stress and controls the amount of transverse activ e stresses. Although active stresses, in principle, develop along the fiber directio n, studies have shown [238] that active stresses in the transverse direction are non - negligible due to imperfect alignment of the muscle fibers. We therefore set = 0 . 2 [239] , and note that transverse active stresses are naturally embedded in the active strain formulation by re quiring . Myocardium was assumed to be incompressible. The incompressibility was enforced in the model using a two - p ( J 1) was added to the total strain energy with p denoting a Lagrange multiplier that represents the hydrostatic pressure. The deviatoric and volumetric mechanical responses were 113 also uncoupled by multiplicatively decomp osing the deformation gradient [240] : (6.9) and letting the strain - energy be a function of only isochoric deformations, namely, . Ventricul ar base was fixed in the longitudinal direction and the biventricular geometr y was anchored by constraining the epicardial surface using a Robin - type boundary condition with a linear spring of stiffness [188] . Measured cavity pressure in the LV ( ) and RV ( ) were applied as a Neumann condition at the endocardial surfaces. The Euler - Lagrange equations in the Lagrangian form reads: Find such that for all ( ) and (6.10) w ith (6.11) Here , completed with homogeneous Dirichlet boundary data, , is th e outward pointing unit normal and is the first Piola - Kirchhoff stress tensor. For an incompressible, hyperelastic, continuum body, the total Cauchy stress tensor is given by (6.12) With the decoupling of the isochoric and volumetric deformation according to (6.9), the first term in Eq. (6.12) represents the deviatoric stresses and is the hydrostatic 114 pressure. Myofiber stress w as computed by first a push forward of the fiber field to the current configura tion, , and then an inner product with the stress tensor . The average fiber stress in a given region was computed by integrating the fiber stress over that region an d dividing by the volume, i.e., . 6.2.3 PDE - constrained optimization The ventricular mechanics model outlined in Section 6. 2.2 contains model parameters that may vary from individual to individual. Calibration of these model (or control) parameters was achieved by solving a PDE - constrained optimization problem, where we minimized a cost functional representing the mismatch between the simulated and observed data, subject to the constraint of satisfying Eqs. ( 6. 10) - ( 6. 11). The minimization problem can be formally stated as minimize (( ), ) subject to (6.13) Here is the objective functional that we want to minimize, which depends on the st ate variable ( ) and the control parameter(s) . The state variables may also depend on the control parameters . To ease notation, this dependency is not explicitly stated here. Minimization of the cost functional should bring the simulated results closer to the clinical observations. Therefore, should reflect a distance between the simulated results and the observed data. Given a measurement point , let be the simulated state variables at that point, and let represents any generic model parameter, that we want to estimate. The cost functional is then given by 115 (6.14) The first two terms represent the mismatch between simulated and observed volumes and strains , respectively, whereas is a regularization term that penalizes non - smooth values of the control parameter for numerical stability. The weights ed in the optimization. The simulated cavity volume was given by (6.15) This equation holds as long as the base remains flat and is located at the plane. We let be the sum of the squared relative volume error in each chamber: (6.16) Here ( ) and ( ) are the simulated and measured cavity volumes, respectively. Volumetric averaged strains were computed in each material region (i.e., LVFW , RVFW and septum) using end - diastolic (ED) as a reference. Letting be the deformation gradient tensor associated with ED, the Green - Lagrange strain tensor w ith ED as reference was given by . Averaged normal strains along the circumferential direction in material region was defined by (6.17) Correspondingly, the str ain mismatch functional was given by the total squared error between the simulated circumferential strain and the measured circumferential 116 strain over all material regions: (6.18) Finally, the regularization term was defined as the total squared distance from the mean value, that is if , then with (6.19) As noted above, the purpose of this term is to avoid numerical instabilities by penalizing large variations in the control parameters. The functional gradient (6.20) Points in the direction of steepest descent and is required in gradient - based optimization methods. While the terms and are typically straightforward to compute, the term cannot be computed easily, since the state variable can only be determined by solving the force - balance equation Eq . (6.10). Specifically, estimating this term with N control parameters using a finite difference approach will require one to solve the (typically computational expensive) force - balance equation N+1 times, becoming impractical when N is large. Instead, it is possible to transform the system of equations into its adjoint system, whe re the gradient is given by (6.21) With referring to the adjoint (for Hermitian transpose) and is the soluti on of (6.22) We can therefore solve the adjoint equation first Eq. (6.22), and then compute the 117 functional gradient by plugging the solution into Eq. (6.21). Hence, computing the functional gradient using the adjoint approach requires only one additional solve of a linearized system that is independent of the number of control parameters. 6.2.4 Para meter estimation The pipeline for fitting the model to patient data was divided into two sequential phases; a passive phase where we estimated the material parame ters that define the passive behavior of the myocardium, and an active phase where we estimat ed the amount of active contraction. In both cases, the control parameters were spatially resolved. During the passive phase the control parameter was allowed to vary spatially on the LV (LVFW + septum) and RV segments, while in the active phase the LV was separated into LV free wall and septal segments, which provided additional degrees of freedom to allow for non - homogeneous LV contraction. Geometries used in the simulation were reconstructed from medical images. These geometries are, in principle, not load - free. Hence, we need to estimate the unloaded (zero pressure) geometries, which will revert back to the original reconstructed geometries when loaded with the measured pressure. Several methods exists for estimating the unloaded geometry [241], [242] . Among the mo st simplest ones is the backward displacement method [243], [244] that can also be used to incorporate residual stresses into the finite element mod els by simulating tissue growth [245] . Nevertheless, this inverse problem (of finding the unloaded geometry) has been shown to produce non - unique solutions, espec ially when buckling is present [241] , although relaxation techniques can be used to imp rove convergence and stability [246] . For the case of a bi ventricular geometry, buckling 118 may occur due to the thin RVFW and a high RV pressure. For this reason, we choose a simpler approach to estimate the unloaded configu ration. A s shown in the left of Figure 6. 2 , we start by applying one iteration of the backward displacement method with initial values prescribed for the material parameters followed by the material parameter estimation as outlined below. This will result in a deflated geometry as shown in the right of Figure 6. 2 . Four material parameters, i.e., , , and from Eq. (6.4) have to be estimated in the passive phase. Due to the sparsity of passive data used for the optimization, if we let all these parameters vary freely, we may end up in a situation where multiple parameter sets will equally minimize the cost functional, and the optimal control will depend heavily on the initial guess of the optimization. Figure 6.2 . To the left we see the model - personalization pipeline . To the right we show a comparison of the unloaded geometry for CASE3. The upper figure sh ows the resulting unloaded, zero pressure geometry in red and the original image - based geometry in transparent, while the bottom figure shows the unloaded geometry, inflated to the target pressure in the image - based geometry, and the original image - based g eometry in transparent for comparison. 119 We therefore restricted our control parameter to be only the linear isotr opic parameter with an initial g uess , and have the remaining parameters fixed according to [220] . The weights were set to and in Eq. (6.14) so that only ED volumes were used for fitting. Since fitting the left and right ventricular end - diastolic volumes might require different material properties of the left and right ventricular wall, the parameter was spatially resolved wit h one parameter associated with the LV (LVFW + septum) and one parameter associated with the RVFW. In the active phase. The optimized passive material parameters were fixed and the relative active fiber shortening in Eq. (6.3) was chosen as control p arameter. For this phase, the weights in Eq. (6.14) were set to and , so that both strain and volume are considered in the optimization. This choice of weighting was made ad hoc, reflecting the relative size of the different terms in t he cost functional. It should also be noted that the volume functional in Eq. ( 6. 16) is a relative error while the strain functional in Eq. ( 6. 18) represents a total error. The cost function parameter values were taken from [226] , where they were chosen based on an L - curve type analysis. For each time point, we estimated locally in the LVFW, RVFW and septum. The initial guesses for the optimization were set to zero in the first iteration. In subsequent iter ations, the initial guesses were set to the optimized values found in the previous iteration. Note that in the case when active stress formulation was used instead, the parameter in Eq. ( 6. 8) was used as the control parameter and estimated in a similar fashion. A schematic illustration of the full optimization pipeline 120 is provided to the left in Figure 6.2 . The control parameter in the active phase represents an index of contractility ( [247] ), meaning that the higher the value of the control parameter, the more forcefull y the myocardium is trying to contract against the external loads. To separate between the LV and RV contrac tility, we extracted the average value of this control parameter in these two segments. 6.2.5 Implementation details The force - balance equations o f this incompressible nonlinear elasticity problem were solved using the finite element method with Lagrange ele ments. More specifically, the displacement and hydrostatic pressure fields were interpolated using piecewise quadratic and linear Lagrange basis functions, respectively. These mixed elements, known as t he Taylor - Hood finite elements [248] , are known to satisfy the discrete inf - sup condition [249] and leads to a stable discretization. The solver was implemen ted in FEniCS , which is an open - source platform for solving PDEs using the finite element method. Nonlinear syst ems of equations were solved using and a distributed memory parallel LU solver [250] was used to solve the linear systems. To solve the optimization problem Eq. ( 6. 13) we applied a sequential quadrat ic programming algorithm (SQP) [251] . This gradient - based optimization algorithm requires the functional gradient Eq. ( 6. 20). This gradient was computed by solving an automatically derived adjoint equation using dolfin - adjoint [230] . The full source code is publicly available. 121 6.3 Results In this section we present the results from the model personalization process. Results of the data matching are presented in s ection 6. 3.1, together with a validation of the model and analysis of the solver performance. To validate the model we compare the simulated and measured longitudinal strain which was not used in the optimization. In s ection 6. 3.2 we present the results of the extracted mechanical features such as indices of contrac tility and fiber stress. We also investigated the efficiency of the algorithm and the effect of mesh refinement, and found that the chosen refinement level was sufficient to yield convergent solutions. 6.3.1 D ata assimilation, validation and solver performance The simulated and measured pressure - volume (PV) loops of the RV and LV, as well as circumferential strain in the LV, septum and RV are shown in Figure 6.3 . We found that the fit of the data was highly de pendent on the choice of fiber angles, which affects both volume change and circumferential shortening. Plotting the average value of the volume cost functional (as defined in Eq. ( 6. 16)) for each choice of fiber angles (see the upper right panel in Figur e 6. 3 ) revealed that the optimal value of lies in the range 70 for all 3 cases. 122 Figure 6.3 . Results of the gradient - based minimization of model - data mismatch for different choice of fiber angles. Left: simulated (color lines) and measured (black circles) PV loops in the LV (top row) and RV (bottom row). Center: simulated (color lines) and measured (black circles) circumferential strain in the LV (top row), RV (middle row) and septum (bottom row). Right: average values of cost functional for the volume (top row) and strain (bottom row), for each choice of fiber angle. Although available, we chose not to use longitudinal strain data in the optimization. Therefore, the comparison of model - predicted longitudinal strain with the measurements serves as a validation of the model - personalization process. The simulated and meas ured LV longitudinal strain curves are shown in Figure 6. 4 . We note that the fit in all regions was again highly sensitive to the choice of fiber angle. Choosing = 70 produced the best fit for the LV longitudinal strain for CASE1 and CASE3, while an ang le 60 gave the best fit for CASE2. The Septal and RV longitudinal strain was best fitted with = 80 . 123 Figure 6.4 . Validation of the model - personalization process using simulated and measured longitudinal strain which was not used in the optimization. Upper, middle and lower panel show the longitudinal strain curves for different choice of fiber angles in the LV, RV and septum respectively. 124 6.3.2 Mechanical analysis 6.3.2.1 Cardiac contraction The estimated active strain parameter in Eq. ( 6. 3), which served as the control parameter during the optimization in the active phase, is plotted for various fiber angles to the left in Figure 6.5 . This parameter varies regionally in the LVFW, septum and RVFW, but is shown here as an average in the LV cont aining LVFW + septum (top) and RV containing only RVFW (bottom). As shown in the figure, time - variation and magnitude of were similar in the 3 cases and insensitive to the prescribed fiber angles. Time traces of the active strain parameter found in th e LV and RV are plotted together for the 60 fiber angle case in the left of Figure 6.6 in order better compare their differences. A similar plot of the active stress parameter in logarithmic scale is also shown in the same figure. As shown in the fi gure, the time - variations of and , which are indices of cardiac contractility, were largely similar between the LV and RV with peak values located approximately at end - systole. Peak values found in the RV were, however, lower than those found in the LV. These findings were consistent across all the 3 cases. 6.3.2.2 Fiber stress Time traces of the average Cauchy fiber stress are shown on the right of Figure 6.5 for different fiber angle variations. Only very small variations in the average fiber stress were found with respect to the choice of fiber angle in the optimization process. Snap shots of the fiber stress distribution at ED and ES are plotted in Figure 6 . 7 for the 60 fiber angle case. Regional variation of fiber stress was largely consistent between the 3 125 cases with pockets of high and low stresses found, respectively, at the apex - epicardial and endocardial regions at ES. In Figure 6.6 , we compare the a verage fiber stress time variation found using the two different active contraction formulations, either active strain or active stress. As shown in the figure, fiber stress in the RV computed using active stress and active strain formulations behaved simi larly with time. Similar regional variation was also found where both formulations predicted higher fiber stress in the LV than the RV. Peak fiber stress predicted in the LV, however, was different in the two formulations with higher stress occurring at is ovolumic relaxation in the active stress formulation. The specific average value of the end - diastolic and end systolic fiber stress for the case of a 60 fiber angle are displayed in Table 6.1 , showing small variability between patients, despite difference s in individual PV loops . Figure 6. 5 . To the left, average traces of the active strain parameter in Eq. (6.3) in the LV (top) and RV (bottom) for different choice of fiber angle. To the right average traces of Cauchy fiber stress in the LV (top) and RV (bottom) for different choice of fiber angle. The fiber angles were defined symmetrically across the wall with a negative angle on the epicardium and a positive angle on the endocardium ranging from 30 with increments of 10 . On the x axis we plot t he normalized time with respect to end - diastole (ED) and end - systole (ES). Horizontal dotted lines indicate timings of opening of the aortic and mitral valve. 126 F igure 6.6 . Comparison of fiber stress and cardiac contraction using the active strain and activ e stress approach using 60 fiber angle. To the left, average traces of the active stress (top) parameter in Eq. (6.8), and the active strain (bottom) parameter in 3. The active stress parameter, with unit kPa, is plotted on a logarithmic scale for ea sier comparison with the active strain parameter. To the right, estimated Cauchy fiber stress using the active stress (top) and active strain formulation (bottom). On the x axis we plot the normalized time with respect to end - diastole (ED) and end - systole (ES). Horizontal dotted lines indicate timings of opening and closing of the aortic valve. Table 6.1 . Average LV and RV fiber stress (kPa) at end - diastole and end - systole Case ID LV (ED) RV(ED) LV(ES) RV(ES) Case1 5.76 4.87 48.3 19.2 Case2 4.04 3.57 54.4 22.7 Case3 9.94 8.34 40.3 18.2 Average ± std 6.58 ± 2.48 5.59 ± 2.01 47.65 ± 5.74 20.03 ± 1.91 127 Figure 6.7 . Snap shots of the end - diastolic and end - systolic configuration and the estimated fiber stresses shown as color - map 6.4 Discussion In this study, we have presented a novel and highly efficient method for non - invasive perso nalization of an image - based bi ventricular mechanics model based on regional measurements of circumferential strain, as well as global measurements of volumes and press ures in the LV and RV. A gradient based optimization method was used to minimize the model - data mismatch by solving a PDE - constrained optimization problem for each measurement point in order to calibrate model parameters. Passive material parameters and an unloaded (zero - pressure) geometry were estimated using the bi ventricular geometry that was reconstructed from MR images acquired at late diastole. Time variation of an active contraction parameter was estimated throughout the cardiac cycle starting from ED. This framework was applied using measurements from three 128 normal subjects to extract estimates of regional fiber stress as well as indices of myocardial contractility. Sensitivity analysis of the model outputs with respect to the choice of fiber angle d istribution and active formulation were also conducted. The described framework is effective and efficient in capturing cardiac mechanical behavior throughout the cardiac cycle, and gave low patient - to - patient variability in the extracted mechanical featur es. As such, it has potential clinical utility in the quantification of contractile function and myocardial stress in vivo and the potential differentiation of pathological states. 6.4.1 Data compatibility and multi - objective optimization The objective fu nctional in problem Eq. ( 6. 13) consists of a weighted sum of different objective functionals. Such problems are referred to as multi - o bjective optimization problems [252] . While it is possible to perfectly match the strain or volume data individually with the chosen control parameters (data not shown), there is expected to be a trade - off when fitting bo th the volume and strain data in a combined objective functional. In such cases, a single, unique optimum does not always exist, but rather a family of so called Pareto optimal solutions can be found [252] . The particular solution found will depend on the chosen weights of each objective. In a previous study [226] , the weights in the total functional given in Eq. ( 6. 14) were determined by performing an exhaustive search, testing several combinations of weights of the strain and volume functionals, and choosing the corner - point of str ain versus volume error curve. However, the weights will depend on the data source, and in our case ch oosing the weights proposed in [226] , resulted in an excellent fit of the volume, while a relatively poor fi t of the strain, and hence a higher weight was chosen for the strain. Nevertheless, neither 129 was captured exactly, and which data source is more important for model utility remains to be determined. In addition, other general methods for solving multi - objec tive optimization problems [252] may be superior to the weighted sum method used here, and will be considered in future studies. 6. 4.2 Contractility Higher value of the active strain parameter or the active stress parameter indicates that the myocardium is contracting more forcefully and our results also suggest that the LV generates a higher contractile force per myocardial volume than the RV. One of the underlying mechanisms that modulate the contract ile forces is calcium dynamics [237] , [235] , and while and cannot be directly compared to the calcium concentration because of the difference in units, their time traces have similar shapes when compared to the calcium transient. Further investigation o f these estimates is needed, but we hypothesize that these measurements may provide useful biomarkers. Due to the observed consistent results in normal patients, even for wide ranging PV loops, these estimates of contractility could therefore potentially s erve as important diagnostic estimates in cases where disease alters myocardial contractility. 6. 4.3 Fiber stress As there is no direct way to measure myocardial fiber stresses, we have compared our results with other patient specific modeling studies. Th e range of reported values for humans are broad, and are mostly confined to the LV. For example, [211] reported fiber stress computed at ED (2.21 ± 0.58kPa) and ES (16.64 ± 4.73 kPa) in normal humans, whereas [253] conducted a stress analysis on healthy bi ventricles and found that wall stress at ES was 65.7 ± 12.3 kPa in the LV and 23.6 ± 14.2 in the RV. 130 Our estimated average fiber stresses ( Table 6.1 ) are well wit hin the range of values reported in these studies. Fiber stress distributions at ED and ES ( Figure 6.7 ) are also consistent in the three subjects investigated here. Furthermore, our results also show small variations in fiber stress with respect to the cho ice of fiber field. Fiber stresses obtained from active strain and stress formulations are also comparable during late diastole and early systole. A large difference in the fiber stress between these two formulations, however, can be found at late systole and during the isovolumic relaxation when the ventricles are in their most compressed state. In particular, the active stress formulation produces elevated stresses during this time interval. We found that the elevated stresses are always accompanied by ve ry high hydrostatic pressure , suggesting that the enforcement of incompressibility, which does not hold in vivo due to blood perfusion in the myocardial wall, is causing this artifact. Future studies are needed to examine the effect of compressibility to our results. 6.4.4 Fiber angle sensitivity In this work, we applied a rule - based algorithm [4] t o assign myocardial fiber orientations to the bi ventricular geometries, and investigated the sensitivity of the data matching and the extracted mechanical features to the choice of fiber field. Different fiber fields, in which the fiber angle varies linear ly across the myocardium wall from at the at the epicardium were tested, for the range 30 . Our results show that , in order to fit the PV loop and circumferential strain measurements simulta neously. The validation study (s ection 6. 3.1), where we compared our model results with the longitudinal strain, confirms this finding. 131 This highlights a major challenge in building accurate mechanics models of the myocardium. T he choice of fiber field is very important, as it con trols the amount of longitudinal and radial shortening during contraction. Unless the correct fiber field is used, strain measurements cannot be reproduced simultaneously in the model with the measured p ressure volume relationships. Accurate measurements of the underlying ventricular microstructure are lacking, however, and therefore, rule - based methods ( [4] ) are often the only alternative to prescribe muscle fiber field in subject - specific modeling of cardiac mechanics. Our fundamental knowledge of the myocardial architecture is based largely on early histo logica l studies [234] , which found that the muscle fiber orientation varies linearly across the myocardial wall with an angle = 60 at the endocardium at the epicardium. This fiber field is often prescribed in ventricular models without questioning. On the other h and, diffusion tensor MRI (DT - MRI) is now becoming an important tool to measure fiber orientations and could potentially do so in vivo [254] . M ore recent histological studies [255] on the canine left ventricle have shown that the muscle fibers are more longitudinally oriented at the sub - endocardium and sub - epicardium than those obtained using DT - MRI, and such fiber orientation can better reproduce the longitudinal motions ob served in the experiments [256] . Our results support this finding. A few hypotheses on the basis of cardiac muscle fiber orientation in the ventricles have been put forward. For example, it has been hypothesized that myocardial fiber orientations adapts to achieve a minima l fiber - cross fiber shear s train during the cardiac cycle [257] . While our results showed that the active strain parameter varied a little with 132 respect to the different fiber angle prescribed in the model, they also show that the peak active strain is lowest when lies between 60 and 70 in all 3 cases. This finding suggests that the tight range of found here is optimal in the sense that the active shortening necessary to produce the same stroke volume is at its minimum. 6. 5 Limitations In this work, we clearly see a variability of model - data fit with respect to choice of the fiber angle, suggesting that the fiber field can be calibrated to better fit the data. Here, we have only prescribed a linear transmural fiber angle variation that has opposite signs at the endo - and epicardium in both LV and RV. Dissection studies, however, generally found that the fibers are more circumferentially oriented at the sub - epicardium and more longi tudinally orientated at the sub - endocardium in the RV [258] . This suggests that one should also consider non - symmetric fiber fields across the wall as well as different fiber field in the LV and RV. We seek to investigate these issues in future studies, possibly together with in vivo measures of fiber angles. As noted above, the constants that balance the terms of the cost fu nctional Eq. ( 6. 14) were adjusted based on a previous study [226] of a single LV, where an L - curve type analysis was performed to estimate an optimal set of weights. Since the present study also considers the R V and the current cost functional form differs from the LV case, it is likely that these differences may affect the optimal choice of weight parameters. Hence, although the chosen parameter values gave good results for our applications here, there may exis t even better choices. 133 While we were able to obtain stress measures across a small cohort of healthy subjects that were both internally consistent as well as in broad agreement with other published studies, the effect of our modelling assumptions remains to be determined. Here we used an incompressible model of the myocardium, even though it is well known that the myocardium is compressible due perfusion of blood. Future studies should investigate the role of compressibility, and in particular how fiber s tress is altered when the material is allowed to compress. We clearly see stress effects related to the hydrostatic term in our model, and this will be investigated more closely in future studies. Residual stresses are hypothesized to be important in stre ss estima tion in soft biological tissue [259] , and can be incorporated into the finite element model [260] , [245] . Because it has been shown in a previous study that resi dual stresses have little e ffects on ventricular function [261] , they were not considered in the present study. Nevertheless, the effect of including residual stresses in the estimation of fiber stress and contractility remains to be investigated in future studies . The late diastolic pressure - volume curve is fitted by estimating one material parameter, while fixing the remaining parameters to previously reported values [220] . This is a limitation in our study, and future studies will be geared towards reducing the need for fixing these model parameters by incorporating more clini cal data or by revising the constitutive model. In particular, the incorporation of diastolic strains may be useful in more clearly defining material properties. The simple estimation of unloaded configuration using only one iteration of the backward displacement method can be used with different initial material parameters and still recapitulate the end - diastolic volumes with different optimized material parameter s. 134 Several studies have jointly esti mated the unloaded left - ventricular geometry and material parameters [220], [247], [262] , but estimation of the unloaded co nfiguration with bi - ventricular geometries is not a well - posed problem, since buckling of the RV free wall might occur. More work on formulating well - posed algorithms for determining the unloaded configuration should be considered in future studies. Fina lly, in this study we only considered three normal subjects, and in the future we would like to apply this framework to more individuals and use it to study larger cohorts as well as patients with cardiac pathology. 135 CHAPTER 7 PATIENT DATA ASSIMILATION ANALYSIS SHOWS AN INVERSE LINEAR RELATIONSHIP BETWEEN RV CONTRACTION AND REMODELING IN PULMONARY ARTERIAL HYPERTENSION 136 Abstract Pulmonary arterial hypertension (PAH) causes an increase in the mechanical loading imposed on the right ventricle (RV) that results in progressive changes to its mechanics and function. Here, we quantify the mechanical changes associated with PAH by assimilating clinical data consisting of reconstructed 3D geometry, pressure and volume waveforms as well as regional strains mea sured in PAH patients (n= 12) and controls (n= 6) within a computational modeling framework of the ventricles. Modeling parameters reflecting regional passive stiffness and load - independent contractility as indexed by the tissue active tension were optimized so that simulation results matched the measurements The optimized parameters were compared with clinical metrics to find usable indicators associated with the underlying mechanical changes. Peak contractility of the RV free wall (RVF W) was found to be strongly corr elated, and had an inverse rela tionship with the RV and left ventricle (LV) end - diastolic volume ratio (i.e., RVEDV/LVEDV) ( = - 0.13 (RVEDV/LVEDV) + 0.44, R 2 = 0.77). Correlation with RV eje ction fraction (R 2 = 0.50) and end - diastolic volume index ( R 2 =0.40) were comparatively weaker. Patients with RVEDV/LVEDV 5 had 18% higher (P = 0.09) t han that of the control whereas those with RVEDV/LVEDV <1. 5 had 25% lower (P<0.05). On average, RVFW pas sive stiffness increased progressively with the degree of remod eling as indexed by RVEDV/LVEDV and RVFW myofiber stress was increased by 49% on = 0.14). These results provide the mechanical basis of using RVEDV/LVEDV as a clinical index for delineating disease severity and estimating RVFW contractility in PAH patients. 137 7.1 Introduction Pulmonary arterial hypertension (PAH) is a complex disorder caused by an increased vascular resis tance in the pulmonary arterial circulatory system. As a consequence, pressure becomes elevated in the pulmonary artery and the right ventricle (RV). Unlike systemic hypertension, PAH is difficult to detect in routine clinica l examination; the current gold standard for diagnosing PAH is through invasive right heart catheterization [263] . Because of the difficulty in detecting PAH, the current estimated prevalence of this disease (15 to 50 per million) is most likely under estimated [264] [266] . Perhaps also because of this difficulty, PAH is arguably less well - studied compared to systemi c hypertension, and there are currently no effective treatments for PAH. Existing treatments of PAH have so far been confined to relieving the symptoms and attenua ting the disease progression [267] . As the disease progresses, the RV remodels both structurally and geometrically, becomes dysfunctional as a result, and eventually this progression leads to decompensated heart failure and death. A significant part of our current understanding of the ventricular mechanical alterations due to PAH has been developed usin g animal models [59], [60], [268] [270] . Animal models, however, may not fully reproduce the pathologies found in humans [271] . Human studies have so far been largely confined to measuring ventricular function of PAH p atients at the global level [62] . Specifically, RV contractility and wall stresses are typically quantified globally using the maximal elastance (Emax) [272], [273] (maximum value of the ratio of ventricular pressure to volume during the cardiac cycle), and [274] inaccurate when applied to the RV bec aus e of its irregular geometry [274] ; similarly, the 138 accuracy of Emax estimated using single beat methods (because manipulation of venous return is difficu lt in practice [272] ) has been questioned [275], [276] . On the other hand, while magnetic resonance (MR) and echo imaging can quantify regional (including RV) myocardial strain or motion in vivo [187], [277] , [278] , strain is a load - dependent quantity and not truly a measure of myocardial contractility. As described by Reichek [279] , the popular notion of equating myocardial contractility with strain measures (that are load off the mark [and] if contractility means anything, it is an expression of the ability of a given piece of myocardium to generate tension and shortening under any loading conditions Mathematical modeling offers an opportunity to overcome this limitation through direct quan tification of both passive and active mechanical interactions under vary ing loading conditions. Such modeling has been heavily used to assess left ventricular dynamics, and a smal l number of com putational modeling studies has been conducted to investigate alterations of the RV mechanics in PAH [188], [280], [281] . These studies are, however, limited either by the use of non - human (rat) PAH data [281] , the lack of consideration of p at ient - specific RV geometry [280] , or the failure to take into account of the vari ability across PAH patients [188] . Consequently, there exists a gap in our current understanding on the changes in RV mechanics during the progression of PAH in humans. Here, we seek to close this gap by using a recently developed gradient - based data assimilation method [282] to evaluate patient - s pecific regional myocardial contractility and ventricular wall stresses from clinical PAH data. Specifically, we seek to answer the following three questions: First, is my ocardial contractility, as indexed by the load - independent active tension generated b y the tissue, altered regionally in PAH? Second, 139 are ventricular wall stresses, particularly in the RV, altered regionally in PAH? Third, how are myocardial contractility and ventricular wall stresses associated with the progression of remodeling in PAH? W e believe that answering these questions will help us to better understand the mechanical mechanisms of PAH so that existing treatment strategies can be optimized and new therapies can be developed. 140 Table 7.1 . Demographics of the PAH patient and control groups 141 7.2 Methods 7.2.1 Patient c ohort and data processing Twelve PAH patients were recruited in the study and underwent both cardiac magnetic resonance (CMR) scans and right heart catheterization (RHC) that were performed at rest using standard techniques. Pulmonary arterial hypertension was defined as havi ng a mean pulmonary artery pres sure (mPAP) greater than or equal to 25 mmHg). Six human subjects who had no known cardiovascular disease or other co - morbidities also underwent CMR scans and served as control in this s tudy. Invasive hemodynamics mea surements were not acquired in the control group. Demographics of the two stud y groups are summarized in Table 7. 1 . The protocol was approved by the Local Institutional Review Board, and informed consents were obtained from all subjects. Cardiac magnetic resonance imaging was performed using steady - state free precession (SSFP) cine gradient echo sequences. All subjects were imaged in a 3.0T Philips scanner (Philips - Adrglam; Philips Healthcare, Netherlands) with a 12 - element body matrix coil. Steady state free precession end - expiratory breath - hold cine images were acquired in multi - p lanar short - and long - axis views. The typical imaging parameters were: TR/TE 34/1 ms, flip angle 45 , slice thickness 8 mm, echo time 1.404 ms, repetition time 2.808 ms, pixel bandwidth 1796 Hz, percent phase field of view 100 mm, temporal resolution 35 ms , in - plane spatial resolution 1.6 x 1.6 - 1.8 x 1.8 mm, 40 frame/cardiac cycle for the control group and 30 for the PAH group. Pressure - volume (PV) loops of the LV and RV of the PAH pat ients were reconstructed by syn chronizing the pressure waveforms measu red from RHC and volume 142 waveforms measured from the cardiac magnetic resonance image s as described in Xi et al. [188] . Similar to our previous approach [188] , the volume waveforms of the control subjects were synchronized with surrogated normal human pressure waveforms from measurements in previous studies as RHC was not performed on these subjects. Specifically , we applied the mean normal RV pressure - waveform acquired from healthy human s ubjects in a previous study [231] to all the control subjects. On the other hand, we applied a normal LV pressure - waveform in which the e nd - systolic pressure was scaled to be equal to 0.9 of a previous empirical study [232] . Regional circumferential strain and longitudinal strain were estimated from the cine CMR images using a hyperelastic warping method as pr eviously described [40], [181], [186], [283] . Briefly, the biventricular unit was reconstructed from the cine CMR images and partitioned into 3 regions consisting of the left ventricular free wall (LVFW), right ventric ular free wall (RVFW) and inter ventr icular septum (SEPT) [282] . A hyperelastic warping method was then applied to deform the reconstructed biventricular geometry from the template image int o alignment with the correspond ing obj ect in the target image [181], [233] . Normal strains in the circumferential and longitudin al directions at the LVFW, SEPT and RVFW regions were computed from the displacement field using end - diastole as the reference configuration. The circumferential and longitudinal directions in the biventricular unit were prescribed using a Laplace - Dir ichlet - Rule - Based algorithm [4] . 143 7.2.2 Construction of personalized m odels Personalized computational models o f biventricular mechanics that fit the pressure, volume, and regional strain data were created using p reviously described methods (See details in s ection 6.2 ) . Briefly, the computational models were formulated based on class ical lar ge - deformation solid me chanics, and active contraction of the ventricular wall was incorpo rated by a multiplicative decom position of the deformation gradien t [235] : (7.1) Here, is the total deformation gradient computed from the displacement field , is associated with an inelastic deformation resulting from the actively contracting muscle fibers, and is associated with the elastic deformation that preserves (kinematic) compatibility of the tissue under load. We employ the following form of ; (7.2) w here is the unit fiber direction in the reference configuration, and is a parameter that repre sents the relative active shortening strain along the muscle fibers i.e., a measure of the local load independent active tension gene rated by the tissue. Meanwhile, passive mechanics was modeled using a purely incompressible transversely isotropic hyperelastic mate rial low [16] with an isochoric strain energy density given by (7.3) w here , , , are material constants and , are invariants that are defined as , (7.4) 144 w ith being the volume preserving contribution of the elastic component to right Cauchy Green strain tensor and the elastic volumetric deformation. The ventricular base was fixed in the longitudinal direction and the biventricular geometry was anchored by constraining the epicardial surface using a Robin - type boundary condition with a linear spring of stiffness k= 0.5 kPa/cm [188] . Measured cavity pressure in the LV ( ) and RV ( ) were applied as a Neumann condition at the endocardial surfaces. The force - balance equations were solved using the finite element m ethod implemented in FEniCS . A mixed formulation was used to enforce incompressibility of the elastic deformation ( =1 ) [16] . In order to separately determine the active and passive material properties of the computational model for each patient, the clinical measurements were divided into a passive phase and an active phase. Models were then optimized using the active and passive material parameters as controls variables to minimize a weighted cost functio n based on the differences between model predictions and measurements of pressure, vo lume and circumferential strain ( ) , data as described previously [247], [282] . Longitudinal strains were not included in the optimization due to the limited number of control parameters in the simulations, but served as a n independent measure of the final model fit as in our previous study [282] . To solve the minimization problem, we applied a sequential quadratic programming algorithm (SQP) [251] , a gradient - based optimization algorithm where the functional gradient of the cost function with respect to all the control variables was estimated efficiently by solving an automatically derived adjoint eq uation using dolfin - adjoint [230] . 145 I n the passive phase of the optimization, we allowed the isotropic parameter in Eq. ( 7. 3) to var y while keeping all other parameters fixed. To capture regional differences of this parameter, we allow associated with the LV (LVFW + SEPT) ( ) and the RVFW ( ) to vary independently from each other. Similar to Finsberg et al. [282] , we first apply one iteration of the b ackward displacement method [244] to estimate an unloaded, stress - free configuration, thus n eglecting residual stresses [245] . The passive parameter was then determined by assimilating the passive phase measurements, starting from an initial guess of = 1.291kPa in both regions. After fitting the model to the passive phase of the PV - curve and strain data, the parameters and were held fixed and the relative active fibe r shortening strain in Eq. ( 7. 2) was chosen as the control variable for the active phase. We allow associated with the LVFW ( ), SEPT ( ) and RVFW ( ) to vary independently from each other. For each time point, we e stimated , , and , to obtain their variation with time over a cardiac cycle. The spatially resolved isotropic parameters and - waveforms were estimated with different linear transmural variation of the myofiber helix angles varyin g from at the epicardium to + at the endocardium, with ranging from 30 to 80 [282] . The set of parameter s yielding the lowest mean square error between the predicted and measured strain and PV data was taken to be the optimal one and used to post - process regional biventricular myofiber wall stresses . 146 7.3 Results 7.3.1 Patient data and regional strains The control and PAH groups have comparable demographic characteristics, with the majority of the patients in the latter group (8/12) classified in NYHA functional class II. In terms of hemodynamics, the PAH group had an mPAP of 39 ± 9 mmHg with a pulmonary capillary wedge pressure (PCWP) 11 ± 3 mmHg. No differences in the systemic hemodynamics measurements (i.e., blood pressure) were detected between the 2 groups. Evaluation of cardiac magnetic resonance images revealed that the PAH group had sig nificantly (P<0.05) reduced right ventricular ejection fraction (RVEF) (37±14% vs. 54 ±11%), increased RVFW thickness at ED (3.83 ± 0.61mm vs. 1.75 ± 0.17mm) and ES (6.22 ±1.88mm vs. 2.27 ± 0.24mm) compared to the control group. Left ventricular ejection f raction (LVEF) and stroke volume (LVSV) were also significantly reduced in the PAH group compared to the control group (LVEF: 58 ± 12% vs. 73 ± 7%; LVSV: 49 ± 14 ml vs. 68 ± 7 ml). Right ventricular end - diastolic volume (RVEDV) and right ventricular end systolic volume (RVESV) were larger, but not statistically significant, in the PAH group. Absolute peak circumferential strain in the PAH group was significantly lower (one - tailed: P <0.05) than the control group at the LVFW (13 ±4% vs. 17 ± 2%) and RVFW (8±4% vs.11± 2%) ( Fig ure 7. 1 ). Similarly, peak longitudinal strain in the PAH group was also significantly lower at the LVFW (13± 5% vs. 17 ± 3%) and RVFW (10 ± 4% vs. 15 ± 3%) compared to the control group. While both and at the septum were lower in the PAH group than the control group, these reductions were not statistically significant (one - tailed P = 0.11 for and 0.05 for ). 147 Figure 7.1. Comparison of peak regional (a) circumferential and (b) longitudinal strains between control (unstriped) and PAH (striped) groups with (*) denoting that the latter value is statistically lower (P<0.05) than the former. Time traces of RVFW longitudinal strain of the (c) PAH patients and (d) control subjects. Thick line denot es average and error bar denotes standard deviation. 148 7.3.2 Model results; contractility, stiffness, and stress Figure 7.2 shows an example of the patient specific simulations, assimilated with the corresponding measured PV loops as well as estimated reg ional , from the control and PAH groups. The overall fit of the LV and RV volumes in the patient - specific computational models is very good, with the simulation results agreeing closely with the measurem ents in the cardiac cycle ( Figure 7. 3 ). The overall root mean square error (RMSE) of the fit is 3.89 mL for the RV and 6.6 mL for the LV. Compared to the volumes, the fit of the regional strains in the cardiac cycle shows significantly more scatter, especially at lower LV systolic strains. The RMSEs of the regional circumferential strain fit are 4.7% for the LVFW, 1.9% for the SEPT, and 2.6% for the RVFW. Meanwhile, serving as an independent measure of model fit, longitudinal strains, not included in the optimization, had an RMSE of 6.2% for the LVFW , 6.2% for the SEPT, and 5.2% for the RVFW . Regional contractility for the biventricular units was estimated using fitted values of the cor responding regional active strains du ring systole. Peak RVFW contrac tility is not uniformly decreased in the PAH gro up, but instead presented a pat tern with respect to the level of remodeling ( Fig ure 7. 4 ). Using the ratio of RVEDV to LVEDV (i.e., RVEDV/LVEDV) as an indicator of remodeling, we fo und that RVEDV/LVEDV varies sub stantially in the PAH group (1.55 ± 0.50) but little in the control group (1.11 ± 0.12). Using RVEDV/LVEDV = 1.5 (that is 3 SD from the mean value of the control group) as a threshold to delineate between PAH patients with mild RV remodeling (RVEDV/LVEDV<1.5) and those with severe RV remodeling (RVED , we 14 9 found that in PAH patients with mild RV remodeling, their peak RVFW contractility is larger than that in the control group (P= 0. 09). We note that the threshold RVEDV/LVEDV = 1.5 Figure 7.2. Examples of simulations results from a control (left column) and an extensively remodeled (RVDEV/LVEDV = 1.75) PAH patient (right column). A) PV loop data (open markers) and simulations (solid lines) for the LV (red) and RV (blue). B) Calculated fiber stresses at end diastole, and C) Calculated fiber stresses at end systole. 150 is approximately the midpoint for the range (1.27 RVEDV/LVEDV 1.69) proposed RVEDV/LVEDV, however, decreases linearly s o that in PAH patients with (severely remodeled PAH) is sig nificantly less than that in the control group. The inverse linear relationship between and RVEDV/LVEDV is strong and has a coefficient of determination R 2 = 0.77. By comparison, has a weaker linear relationship with RVEF (R 2 = 0.50) and RVEDV index (RVEDVi) (R 2 = 0 . 40). Figure 7.3. Overall data assimilation errors in the control (circle) and PAH patient (diamond) populations. Comparison between measured and simulated (a) volumes for both the RV (blue) and LV (red); (b) for the LVFW (red), SEPT (green), and RVFW (blue) at all cardiac time points. A y = x line is also plotted to show the zero error reference. 151 Figure 7.4. Analysis of peak RVFW contractility for control and PAH patient groups. (a) Graphical depiction of for both controls (diamonds) and patients (black dots) as a function of the ratio of RVEDV/LVEDV. Dashed line shows a l inear fit to the PAH patient group ( = - 0.13(RVEDV/LVEDV) +0.44, R 2 = 0.77). (b) Average for controls patients with RVEDV/LVEDV < 1.5 (similar to controls) and patients with RVEDVi ( = - 0.001(RVEDVi) +0.32, R 2 = 0.40) (d) Linear fit of with RVEF ( = - 0.39(RVEF) +0.09, R 2 = 0.50). 152 Peak contractility in the LVFW , unlike , did not exhibit any relationship with the level of remodeling as measured by RVEDV/LVEDV ( Fig ure 7. 5 ). Specifically, we found that peak contractility in the LVFW is significantly smaller in the mildly remodeled PAH group with RVEDV/LVEDV<1.5 (0.33±0.02) compared t o the control (0.38±0.03) (P <0.02). On the other hand, while the average in the severely remodeled PAH group (0.33±0.07) is decreased compared to the control group, that decrease is not statistically significant due to the large standard deviation (P= 0.1 4). Figure 7.5. Analysis of peak LVFW contractility for control and PAH patient groups. (a) Graphical depiction of peak LVFW contractility for both controls (diamonds) and patients (black dots) as a function of the ration of RVEDV/LVEDV. (b) Average for controls, patients with RVEDV/LVEDV 1.5. 153 Fitted values of the regional material isotropic parameters and are measures of the tissue passive stiffness in the LVFW+SEPT and RVFW of the bivent ricular unit, respectively. Sep arating the fitted values in the PAH group based on the degree of remodeling (i.e., RVEDV/LVEDV) revealed a progressive increase in the mean value of and with remodeling ( Figure 7.6 ). The value of one patient in the severely remodeled PAH group disregarded as it appears to be an outlier (Z score >2, a = 36.78kPa). In the severely remodeled PAH group, the mean value of (4.3± 3.5 kPa) is 2.4 times higher than that of the control group (1.8± 0.6 kPa) but that increase was not si gnificant due to the large standard deviation in the former group. On the other hand, the mean value of in the severely remodeled PAH group (3.00 ± 2.5 kPa) was significantly higher (P<0.05) than that of the control (0.48 ± 0.12 kPa). Figure 7.6. Co mparison of passive tissue stiffness parameter between control and the PAH groups in the (a) LVFW + SEPT and (b) RVFW regions 154 Using the data - assimilated computational models, we computed the peak regional wall stress in the myofiber direction ( ) (i. e., maximum myofiber load) ( Figure 7.7 ). We found that at the RVFW is, on average, the same between the control (36.4 ± 5.7 kPa) and the mildly remodeled PAH (36.5 ± 12 kPa) groups. In the severely remodeled PAH group, however, at the RVFW i s on average 1.5 times larger (54.27 ± 25.1 kPa) than these 2 groups but that difference is not significant (P= 0.14) due to its large standard deviation. In comparison, at the LVFW is significantly reduced in the mildly remodeled PAH group (55.4 ± 7.4 kPa ) compared to the control group (84± 17.6 kPa). In the severely remodeled PAH group, at the LVFW of two patients are disregarded as they appeared to be outliers with substantially large values (Z score>2, = 544.1 , 1659.9 kPa), which is due likely to the pre sence of local stress - concentration in the model. Peak myofiber stress at the LVFW in this group (71.22 ± 7.8 kPa) is significantly larger than the mildly remodeled PAH group but lower (not statistically significant) than the control. Figure 7.7. Comparison of peak myofiber wall stress between control and the PAH groups in the (a) LVFW and (b) RVFW regions. 155 7.4 Discussion We have used a previously established data assimilation technique [247], [282] to quantify changes in regional myocardial properti es and stresses in PAH based on measurements of PV loops and my ocardial strains from patients and a cohort of healthy subje cts serving as control. The major finding of this study is a strong inverse linear relationship between RVFW load - independent contractility as indexed by the fitted , shown previously to be biomark ers of ventricular failure [ 247] , and the degree of RV remodeling as indexed by the RVEDV/LVEDV in PAH patients. The relationship between RVEDV/LVEDV (R 2 = 0.77) is stronger than that between and RVEF (R 2 = 0.50) or RVEDVi (R 2 = 0.40) in PAH patients. We also found that RVFW contr actility is increased by about 20% in the PAH patients when little remodeling is present (i.e., RVEDV/LVEDV<1.5), but decreases linearly with increasing RVEDV/LVEDV. The strong RVEDV/LVEDV relationship also provides a mechanistic explanation for recent clinical findings that RVEDV/LVEDV is a better metric (with a higher sensitivity) than RVEDVi for identifying PAH patients b ased on all - cause mortality [284] as well as for detecting RV enlargement [285] . Besides PAH, this metric is also used for assessing RV dilation and indicating pulmonary valve replacement in patients who h ave tetralogy of Fallot [286], [287] . There are, however, no other clear mechanistic basis for applying RVEDV/LVEDV to delineate the severity in PAH or electing surgery other than a statistical association of this met ric with clinical endpoints [284] or from clinical experience [287] . Our finding therefore prov ides such basis and suggests that the underlying reason why RVEDV/LVEDV is a better metric in determining PAH severity is 156 because of its close association with RVFW contractility in PAH. Based on this relationship ( Figure 7. 4 ), the threshold of RVEDV/LVEDV 2 for distinguishing PAH patients with severe RV dilation as well as electing patients for pulmonary valve replacement is associated with a reduction of RVFW contractility by about 30% from normal . Fundamentally, our study also provides an insight into the changes in regional contractility d uring the progression of PAH. Our result suggests there is an increase in the RV contractility in early stages of PAH (RVEDV/LVEDV<1.5), perhaps as a compensatory mechanism to main tain RVEF in response to the increas ed RV pressure, before decreasing as the disease progresses. The similarity in RVEF in PAH patients with RVEDV/LVEDV<1.5 (49±5%) compared to normal (54± 11%) ( Figure 7.4 ) support this theory. Because wall thickness is already accounted for geometrically in the computational models, the increase in RVFW contractility indicates that RV cardiomyocytes becomes hypercontractile in an attempt to normalize RVEF in the presence of increased pulmonary afterload at early stages of PAH. This finding is supported by a r ecent study, which shows that the maximal tension of skinned myocytes of idiopathic PAH patients is 28% higher (than normal) in early stages of disease with RVEF at 46 ± 7% [288] , close to that found in PAH patients with RVEDV/LVEDV <1.5 in this study. Other than affecting the RV, we also found that LV contractility in PAH patients is reduced. This reduction is observed even in PAH patients exhibiting little RV remodeling suggesting that there is some early influence on LV function in this disease, a result broadly consistent with findings of reduced myocyte contractility in the LV of PAH patients [289] . 157 Besides cont ractility, we also found that there is, on average, an increase in the LV and RV passive stiffness (reflected by the increase in values of and and that control the general passive isotropic tissue stiffness) with RVEDV/LVEDV in the PAH groups. While this increase is largely not statistically significant due to the large variance of the fitted parameter values, this result is c onsistent with experimental [290] and clinical [288] findings that PAH is associated with cardiac fibrosis and myocyte passive stiffening. Computation of the peak RV fiber stress using the fitted parameters revealed that it is increased only in PAH patients (P = 0.14) with RVEDV/LVEDV 1. 5 although that increase is not statistically significant ( Figure 7.7 ). Because myocardial wall stress is directly correlated with myocardial oxygen consumption (MVO2), this result suggests that MVO2 is increased in the RV of PAH patients with RVEDV/LVEDV 1.5 but less so in patients without substantial RV remodeling. When taken into account with our finding that RVFW contractility is reduced in PAH patients with RVEDV/LVEDV 1.5, this result further suggests that coronary flow may not be sufficient to me et the increase in MVO2 due to a higher workload in the RV, and as a result, ischemia sets in this cohort of patients producing a lower contractility. Our finding t hat RVEDV/LVEDV 1.5 may represent the threshold at which RV may become ischemic (with reduced contractility) is also consistent with clinical observations that RV ischemia may play a role in later stages in PAH patients [291], [292] . Compared to previous patient data assimilation techniques that have so far been only applied to the LV [42], [43], [247], [293] with a relatively small number of patients (n 6), we have shown that the semi - automatic data assimilation pipeline when applied to the biventricular unit is robust in regards to a sizable patient cohort ( n= 12) that has 158 features reflecting those found in the general heterogeneous PAH population. Specifically, PAH patients recruited in this study had mPAP = 39 ± 9 mmHg and PCWP = 11 ± 3 mmHg, which falls within the clinical definition of this disease (mPAP 25 mmHg and PCWP <15 mmHg) [294] . In terms of geometry and function, the PAH patients had thicker RV wall, higher mean RVEDV, lower RV and LV EF as well as significantly reduced longitudinal and circumferential strains at the RVFW and LVFW that are all consistent with previous clinical observations of this disease [274] , [295] , [196], [200], [296] [298] . The data assimilation process produces relatively few data outliers for both the control and patient groups, and is able to fit the patient - specific ventricular volumes and regional strains well, especially considering the fact that only a small number of control variables is used in fitting the data. 7.5 Limitations Some limitations and challenges still remain in regards to this study. First, the models were fitted using only a small number of control variables so that we do not run into the risk of having non - unique solutions and over - fitting. While we are able to fit the PV loops and regional circumferential strains relatively well in this study, it will be useful to further explore the potential impact of the choice of the computational model and the control variables used to fit the patient data. Second, the cohort size (n = 12) is small although it is still comparatively larger than many other patient - specific computational modeling studies. Nevertheless, we are able to provide a mechanistic explanation as to why some clinical indices are better at characterizing PAH severity as well as features found during the progression of PAH, which are also supported by other clinical studies. A larger cohort 159 will be considered i n future studies. Third, we have applied surrogate pressure waveforms to the control subjects as RHC was not performed on them. Future studies can consider estimating pressures in control healthy subjects using D oppler echocardiography or including human s ubjects who have had false - positive diagnosis of PAH after undergoing RHC. 7.6 Conclusion In conclusion, we have show n that RVEDV/LVEDV is strongly associated with the RVFW load - independent contractility estimated from assimilating a computational model of active biventricular mechanics with clinical imaging and hemodynamics data acquired from PAH patients and control sub jects. Our study therefore provides a mechanistic basis for using RVEDV/LVEDV as a non - invasive metric for assessing PAH severity as well as a non - invasive approach of estimating RV contractility from measurements of RVEDV/LVEDV. 160 CHAPTER 8 CONCLUSIONS 161 The aim of this research was to develop microstructure - based and patient - specific computational models to understand progressive changes in ventricular mechanics associated with HFpEF and PAH. First, w e developed and validated a microstructure - based constitutive model of the passive my ocardium in a three - di mensional FE modeling framework. To the best of our knowledge, this is the first FE implementation of a microstructural constitutive model to simulate passive filling of a LV in a representative 3D geometry with experimentally measure d transmural muscle fiber orientation. Our results have shown that the LV filling function is sensitive to the collagen ultrastructure and the load taken up by the tissue constituents varies depending on the LV transmural location. These findings underscor e the importance of quantifying changes in the collagen fiber network ultrastructure during remodeling, particularly in diseases where fibrosis is a key feature such as HFpEF . Second, we developed patient - specific computational models to investigate effe cts of PAH on ventricular mechanics and function PAH patients, and compare them to those found in age - and gender - matched norma l controls. Our results showed that RV longitudinal, circumferential and radial strain were depressed in PAH patients compared with controls; RV passive stiffness increased progressively with the degree of remodeling as indexed by the RV and LV end - diastolic volume ratio (RVEDV/LVEDV); Peak contractility of the RV was found to be strongly correlated, and had an inverse relationship with RVEDV/LVEDV. These results provide the mechanical basis of using RVEDV/LVEDV as a clinical index for delineating disease severity and est imating RVFW contractility in PAH patients. 162 Overall, the presented research benefits the biomechanical community by addressing multiple gaps in the literature and promote the understanding of the effects of HFpEF and PAH on progressive changes in ventricul ar mechanics through computational modeling. 163 APPENDICES 164 APPENDIX A: Plane stress formulation In the 6 canine specimens found in the experiments, the loading directions ( coincided with the local material coordinate system i.e., . Since the top and bottom of the specimens are traction - free, the normal stress in the tissue sheet normal direction can be assumed to be zero. The norma l second Piola - Kirchhoff stresses in the muscle fiber cross - fiber and out - of - plane directions can therefore be derived, respectively, as: (A1) (A2) (A3) In Eqs. (A1 - 3), the stress contribution of the non - fibrous ground matrix is given by: (A4) On the other hand, the stress contribution of the muscle fibers can be derived from Eq. (5) in the main text, and is given as: (A5) where . Finally, the stress contribution of the collagen fiber network can be derived from Eq. (14) in the main text and is given by: (A6) (A7) 165 Substitution of Eq. (A3) into (A1 - 2) and invoking the incompressibility constraint (i.e., ) enable one to express and as a function of the stretch in the muscle fiber directi on and stretch in the cross - fiber direction To compare with the experiments, we transformed the second Piola - Kirchhoff stress components to the Cauchy stress components by: (A8) 166 APPENDIX B: Results of biaxial tests for other specimens Specimen 0119 Fig ure B1 . (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0119. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial f luid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxia l test of specimen 0119 167 Specimen 0124 Figure B2. (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxia l and constant = 1.1, 1.15 and 1.2 tests of specimen 01 24 . (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bott om) directions in the equi - biaxial test of specimen 0124 168 Specimen 0217 Figure B3. (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0217 . (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) directions in the equi - biaxia l test of specimen 0217 169 Specimen 0303 Figure B4. (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibiaxial and constant = 1.1, 1.15 and 1.2 tests of specimen 0 303. (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial f luid matrix to the total stress in the fiber direction (top) and the cross - fiber (bottom) dire ctions in the equi - biaxial test of specimen 0303 170 Specimen 0330 Figure B5. (a) Comparison of the fitted and experimental Cauchy stress - stretch data for equibia xial and constant = 1.1, 1.15 and 1.2 tests of specimen 0 330 . (b) Contribution of the collagen fiber network, muscle fibers, non - fibrous ground matrix and interstitial fluid matrix to the total stress in the fiber direction (top) and the cross - fiber (bo ttom) directions in the equi - biaxia l test of specimen 0330 171 APPENDIX C: M uscle fiber stress and stiffness modulus From Eq. ( 3. 4) in the main text, the Cauchy stress of the muscle fiber with stiffness coefficients of and at the given engineering strain is given by (C1) The corresponding stiffness modulus for the muscle fiber is given by: (C2) 172 APPENDIX D: Strains as a function of volume change Figure D1. Normal strain components vs. LV cavity volume change at the sub - endocardial, midwall and sub - epicardial regions. Colored solid lines: model predictions using best - fit parameters and their mean values (Table 3.2 of the main text). Experimental data [2] are plotted as means ± SD. Note: sub - epicardial, midwall and sub - endocardial regions are reported as 2.0 ± 0.3%, 40.6 ± 5.5%, 79.1 ± 10.6% of the total wall thickness wit h respect to the epicardium in the experiments, which in our case, roughly corresponds to layer 1 (0 - 14%), layer 3 (29 - 43%) and layer 6 (71 - 86%) of the LV FE model shown in Figure 3.2 in the main text. 173 Figure D2. Shear strain components vs. LV cavity volume change at the sub - endocardial, midwall and sub - epicardial regions. Colored solid lines: model predictions using the best - fit parameters and their mean values in Table 3.2 of the main text. Experimental data [2] are plotted as means ± SD. Refer to Figure D1 for details of the transmural locations. 174 APPENDIX E: M esh sensitivity and computational efficiency Figure E1 shows the passive pressure vs. volume change curves using mean values of the fitted model parameters for the LV FE mesh with the number of finit e elements ranging from 3565 to 8780. Our results showed that the difference in the change of LV volume at a pressure of 20 mmHg was about 3% between meshes with 4734 and 6435 elements, suggesting that a FE mesh with around 5000 elements should yield suffi ciently accurate passive pressure volume relations. Figure E1. Effects of number of elements on the passive pressure volume relation using the mean fitted model parameters from biaxial tests. 175 Figure E2 shows the model - predicted transmural distribution of fiber stress using the mean fitted model parameters for the LV FE mesh with the number of finite elements ranging from 6035 to 38985. Our results showed that the difference in transmural fiber stress was less than 2% at each transmural laye r between meshes with 17600 and 38985 elements, suggesting that the LV FE mesh with 17600 elements is sufficient to capture the deformation and transmural stress field. Figure E2. Effects of mesh density (number of cell elements) on the estimated transmural fiber stress at LV pressure of 8mmHg or 1.07kpa using the mean fitted model parameters (Table 3.2) from the biaxial tests. 176 The FEniCS library uses domain decomposition method that is inherently parallelizable (i.e., partitio ning FE problem into sub - problems that can be solved in each processor). Our results showed an approximately linear parallel scalability of the code ( Table E1) . Table E1 . Running time of passive inflation simulation for a FE mesh of 3565 elements with di fferent number of processors (Intel ® Xeon (R) CPU E5 - 1660 v3@ 3.00GHz x 16) Number of Processors Running Time 1 14 h 22 min 2 8 h 12 min 4 5h 30 min 8 3h 6 min 16 1h 37 min Estimation of running time for simulating active mechanics: Assuming that we set a time step of 1ms to simulate a cardiac cycle of 1000ms, then we need 1000 time steps, which should take about 39 hours to run for a mesh of 17600 elements with 16 processors (Intel ® Xeon (R) CPU E5 - 1660 v3@ 3.00GHz). Because of the linear sc alability exhibited by the code, the computing time can be reduced substantially if we use more processors. 177 APPENDIX F: Parameter sensitivity analysis A sensitivity analysis on key parameters associated with the RV function ( , a nd ) was performed on the nor mal case by perturbing the parameters from their fitted values. The results of this analysis ( Table A1 and Figure A1 ) show the changes in predicted quantities (e. g., stroke volume and peak pres sure) arising from perturbation of the parameter values. Specifically, the results show t hat these perturbations can pro duce a change in peak systolic pressure of 4 mmHg, which is the standard deviation of the normal RVP found in the measurements [20]. The results also show that the perturbation is substantially smaller than the fitted parameter values found in the PAH case, suggesting that differences in fitted parameter values between the PAH and normal case cannot be att ributed to the uncertainty asso ciated with the measur ements of RVP (4 mmHg). To confirm this result, we have also rerun the normal case with parameters found from the PAH case, and the resulting PV loops are substantially different ( Figure A2 ). Table A1 . 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