DEVELOPMENT OF FINITE ELEMENT MODELING FRAMEWORK TO INVESTIGATE CARDIAC HYPERTROPHY IN HEART DISEASES By Joy Mojumder A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering – Doctor of Philosophy 2022 ABSTRACT Heart is a prime organ in the human body, and continuously adapting and evolving through growth and remodeling processes to maintain a balance between the demand and supply of blood and oxygen during physiological/developmental (i.e., from birth to adult) and pathological (i.e., various heart diseases) conditions. The exact mechanism of the progression of disease and the growth and remodeling processes are, however, unclear. While numerous experimental studies have been performed on animal models to investigate the mechanism of heart diseases, they are associated with some limitations. To address these limitations, computational frameworks based on idealized, and patient specific heart have been developed. Considering the short history of computational cardiac mechanics compared to experimental studies, many improvements are necessary to advance computational cardiac models. Here we developed both patient and animal specific computational models to investigate the mechanics found in 3 different heart diseases. First, we developed a computational growth framework based on human biventricular geometry to investigate the growth and remodeling processes associated with mechanical dyssynchrony, a disease caused by the asynchronous contraction of the left ventricle (LV). Cardiac mechanics was described using an active stress formulation and growth model was formulated based on volumetric growth framework. Through prescribing myofiber stretch as growth stimulus, our model can quantitatively reproduce the thickening and thinning of ventricular wall at the late and early activated regions, respectively, for two activation sites, namely, interventricular septum and LV free wall. The model is also able to reproduce global LV dilation found in mechanical dyssynchrony, which is consistent with reported experimental studies. Second, we developed a computational-experimental approach based on swine model of pressure overload to investigate the correlation between local growth as indexed by changes in regional thickness and local mechanical quantities. The LV pressure and volume data were acquired from 4 aortic constriction swine models to calibrate the model. From the analysis using the Pearson correlation coefficient, we found a strong correlation between local growth and local myofiber stress induced by an instant rise in peak systolic pressure due to aortic constriction. Third, we developed a computational framework based on idealized LV model to investigate how pathological features, such as a reduction in global longitudinal strain (GLS), myofiber disarray and hypertrophy, affects LV mechanics in hypertrophic cardiomyopathy (HCM), a genetic heart disease. In this modeling framework, LV mechanics was described using an active stress formulation and myofiber disarray was described using a structural tensor in the constitutive models. Both the LV function indexed by ejection fraction and stroke volume and mechanics indexed by circumferential and longitudinal strain were reduced with increasing myofiber disarray. Last, we developed patient specific computational models of LV using clinical measurements of 2 female HCM patients based on two different phenotypes (obstructive and non- obstructive) and a control subject. After calibrating our models with clinical data, the results showed that without consideration of myofiber disarray, peak myofiber tension was lowest in the obstructive HCM subject (60kPa), followed by the non-obstructive subject (242kPa) and the control subject (375kPa). With increasing myofiber disarray, peak tension has to increase in the HCM models to match with the clinical measurements. The computational modeling workflow proposed here can be used in future studies with more clinical and experimental data. Copyright by JOY MOJUMDER 2022 This dissertation is dedicated to my parents and my friend, Dr. Robert N. Coffey Jr. Thank you for always motivating me. v TABLE OF CONTENTS LIST OF ABBREVIATIONS………………………………………………………................viii CHAPTER 1 GENERAL BACKGROUND………………………………………………….....1 1.1 Anatomy of heart…………………………………………………………………………2 1.2 Growth and remodeling of heart………………………………………………………...3 1.3 Review on computational modeling of cardiac hypertrophy…………………………..5 1.4 Background of this dissertation………………………………………………………….6 1.5 Objectives of this dissertation…………………………………………………………..17 CHAPTER 2 BIVENTRICULAR MODEL ON LEFT BUNDLE BRUNCH BLOCK …….19 2.1 Introduction……………………………………………………………………………..20 2.2 Methods………………………………………………………………………………….21 2.3 Results & Discussion ……………………………………………………………………30 2.4 Conclusion ………………………………………………………………………………34 CHAPTER 3 ANIMAL SPECIFIC LEFT VENTRICULAR MODEL ON PRESSURE OVERLOAD ……………………………………………………………………………………35 3.1 Introduction……………………………………………………………………………..36 3.2 Methods………………………………………………………………………………….36 3.3 Results & Discussion ……………………………………………………………………40 3.4 Conclusion ………………………………………………………………………………47 CHAPTER 4 IDEALIZED LEFT VENTRICULAR MODEL ON HYPERTROPHIC CARDIOMYOPATHY…………………………………………………………………………49 4.1 Introduction……………………………………………………………………………..50 4.2 Methods………………………………………………………………………………….51 4.3 Results …………………………………………………………………………………...60 4.4 Discussion………………………………………………………………………………..67 4.5 Conclusion ………………………………………………………………………………68 CHAPTER 5 PATIENT SPECIFIC LEFT VENTRICULAR MODEL ON HYPERTROPHIC CARDIOMYOPATHY…………………………………………………..69 5.1 Introduction……………………………………………………………………………..70 5.2 Methods………………………………………………………………………………….72 5.3 Results …………………………………………………………………………………...79 5.4 Discussion ……………………………………………………………………………….87 5.5 Conclusion……………………………………………………………………………….91 CHAPTER 6 LIMITAIONS AND FUTURE SCOPE………………………………………...93 6.1 Biventricular model on Left Bundle Brunch Block…………………………………...94 6.2 Animal specific LV model on pressure overload………………………………………94 6.3 Idealized LV model on Hypertrophic Cardiomyopathy………………………………95 6.4 Patient specific LV model on Hypertrophic Cardiomyopathy……………………….95 6.5 Conclusion ………………………………………………………………………………95 vi BIBLIOGRAPHY………………………………………………………………………………96 APPENDIX A: MODEL PARAMETERS WITHOUT DISARRAY……………………….113 APPENDIX B: MODEL PARAMETERS WITH DISARRAY……………………………..115 vii LIST OF ABBREVIATIONS Cardiac magnetic resonance imaging, CMR Cardiac resynchronization therapy, CRT Ejection fraction, EF End-diastolic volume, EDV Endothelin-1, ET-1 End-systolic volume, ESV Extracellular volume, ECV Finite element, FE Fractional anisotropy, FA Global longitudinal strain, GLS Growth and remodeling, G&R Hypertrophic cardiomyopathy, HCM Induced pluripotent stem cells, iPSC Late gadolinium enhancement, LGE Left atrium, LA Left bundle brunch block, LBBB Left ventricle, LV Left ventricular assist device, LVAD Left ventricular free wall, LVFW Left ventricular outflow tract, LVOT Magnetic resonance imaging, MR Mechanical dyssynchrony, MD viii Non-sustained ventricular tachycardia, NSVT Right atrium, RA Right ventricle, RV Right ventricular free wall, RVFW Root mean square error, RMSE Sudden cardiac death, SCD Three dimensional, 3D ix CHAPTER 1 GENERAL BACKGROUND 1 1.1 Anatomy of heart The heart is a critical component of the cardiovascular system, which ensures that adequate blood flow is delivered to the body organs to facilitate the exchange of gases, fluid, electrolytes, large molecules and heat between the cells and outside environment [1]. The heart consists of four chambers, namely, the left atrium (LA), left ventricle (LV), right atrium (RA) and right ventricle (RV) (Fig 1.1). At the tissue level, the heart wall consists of myofibers that are oriented helically Figure 1.1: Basic anatomy of heart (left). Pressure-volume relationship in LV during a cardiac cycle (right). During systole, isovolumic contraction (b) and ejection (c) occurs, while during diastole, isovolumic relaxation (d) and ventricular filling (a) occurs. The figures are adapted from internet. with their orientation varying transmurally from the endocardium (inner periphery of the heart) to the epicardium (outer periphery of the heart). When operating in vivo, the heart undergoes a sequence of mechanical events that are associated with different phases in the cardiac cycle. Specifically, the cardiac cycle is divided into 2 general phases, namely systole and diastole. The systole phase refers to events associated with ventricular contraction and ejection, whereas the diastole phase refers to the rest of cycle that includes ventricular relaxation and filling (Figure 2 1.1). The heart cyclically contracts over a cardiac cycle to generate a pressure gradient to perfuse all body organs including itself. At a smaller tissue scale on the other hand, the myocardium in the heart wall operates as a system where its function depends on the highly complex and tightly orchestrated collective interactions between cells and sub-constituents [2]. Figure 1.2: Cardiac hypertrophy geometries. The figure is adapted from Maillet M. et al [3]. 1.2 Growth and remodeling of heart In response to electrical, mechanical, chemical and neurohormonal cues, the myocardium can also undergo long term adaptive (i.e., favoring myocyte survival) or maladaptive (i.e., promoting apoptosis) processes that are commonly referred to as “growth and remodeling” (G&R). These processes can lead to geometrical and functional changes of the heart. As shown in Figure 1.2, the nature of G&R can be pathological (e.g., in heart diseases) or physiological (e.g., during 3 growth and development, exercise, pregnancy, aging) [3]. Growth and remodeling of the heart can be broadly classified into two types, namely, eccentric hypertrophy and concentric hypertrophy. In eccentric hypertrophy, the LV wall becomes thinner via serial sarcomerogenesis (i.e., addition of sarcomere) with significant increase in the chamber volume. In concentric hypertrophy, the LV wall thickens via parallel sarcomerogenesis with little or no change in the cavity volume [4–6]. Pathological conditions such as hypertension, aortic stenosis and mitral regurgitation are associated with concentric, eccentric or a mixture of both types of hypertrophy. Besides geometrical changes, hypertrophy also produce changes to the local mechanical quantities of the LV, specifically, myocardial wall stresses and stretches [7]. It is believed that a change in tissue mechanics is one of the major driving forces of growth at both the cellular and organ levels [8]. Based on in vivo studies at the organ level, volume overload increases the passive stretches or stresses of the muscle cells during ventricular filling in diastole that is associated with the chronic dilation of the heart chamber [9], whereas, an increase in afterload in pressure overload not only produces an increase in stresses of the muscle cells during systole [10–12], but it may also affect the stretches of the cells that are associated with chronic ventricular wall thickening. According to the systolic stress-correction hypothesis proposed by Grossman et al. [13], the increase in wall thickness in concentric hypertrophy helps normalize wall stress to the baseline homeostatic levels. Besides sarcomerogenesis, ventricular remodeling are also associated with cardiac fibrosis, which is characterized by the net accumulation of extracellular matrix in the myocardium [6, 14–16]. Remodeling associated with progressive fibrosis can lead to the development of diastolic heart failure in elderly patients. On the other hand, during pressure overload, extensive cardiac fibrosis is associated with ventricular dilation and combined diastolic and systolic heart failure. Cardiac fibrosis, during volume overload, characterized by disproportionately large 4 amounts of non-collagenous matrix, may lead to chamber dilation and the development of systolic dysfunction. In addition to sarcomerogenesis and fibrosis, myofiber disarray (e.g., in hypertrophic cardiomyopathy, (HCM) [17]), loss of myocyte function (e.g., contractile dysfunction in pressure overload and HCM [18, 19]), alteration of molecular pathways and genetic mutations [20] are also observed during cardiac remodeling. A reversal of remodeling may also occur in some heart failure treatments (e.g., left ventricular assist device (LVAD), cardiac resynchronization therapy (CRT) [21]), which is widely considered to be a sign of recovery for the patient. Overall, cardiac G&R has very significant clinical implications and is widely considered to be an important determinant of the clinical course of heart failure. 1.3 Review on computational modeling of cardiac hypertrophy Despite the clinical significance of cardiac G&R, the exact mechanisms of myocardial G&R are, however, not known [22]. For example, the type of mechanical cues that myocytes sense and the way they respond to those mechanical cues have not been fully elucidated. An in-depth understanding of the various mechanisms of G&R can provide key insights to develop effective heart failure therapies. Given the complexity of the multitude of G&R pathways and their interactions, computational modeling integrated with experiments have been extensively used to predict and understand pathological and physiological behaviors of the heart across multiple scales [22–26]. Several computational modeling frameworks have been developed to predict long-term changes associated with cardiac G&R [25]. Specifically, cardiac growth constitutive models have been formulated based on the volumetric growth framework in which the deformation gradient tensor is multiplicatively decomposed into an elastic and a growth component to describe local changes in shape and size of the myocytes in response to local alterations of cardiac mechanics 5 (i.e., stresses) and/or kinematics (i.e., strains) [27, 28]. These constitutive models are usually coupled with a computational cardiac mechanics model to simulate how geometrical changes of the myocytes collectively affect ventricular geometry when the loading conditions are altered [22, 29–31]. For example, Goktepe et al. [30, 32] and Rausch et al. [33] both proposed a stress-driven growth constitutive model to describe ventricular wall thickening associated with pressure overload in the heart. On the other hand, Kerckhoffs et al. proposed a unified strain-driven growth law that is able to reproduce features found in concentric hypertrophy associated with aortic stenosis and eccentric hypertrophy associated with mitral valve regurgitation [31]. Based on this unified strain driven growth law but with different homeostatic set points for growth, Yoshida et al. showed that the model is able to predict forward growth with pressure overload, but is unable to predict reverse growth with the removal of pressure overload [34]. While these phenomenological G&R models can capture the global features and/or some features of either pressure overload or volume overloaded heart [23], there are still questions to be answered and issues to be tackled with computational modeling of cardiac growth. Also, since most of these models are based on idealized LV ellipsoidal geometry, they cannot be applied directly to individual patients because the outcome is a rough estimate and based on averages [35]. Hence, it is necessary to develop a patient-specific modeling framework to tailor treatment and optimize an individual’s therapy. 1.4 Background of this dissertation 1.4.1 Mechanical dyssynchrony Left ventricular mechanical dyssynchrony (MD) is a disease associated with mechanical contraction or relaxation occurring asynchronously between different segments of the LV. During a cardiac cycle, MD can affect the systolic phase by decreasing the efficiency of contraction and 6 the diastolic phase by decreasing the efficiency of LV filling. It can also affect both systolic and diastolic phases. Besides being associated with alteration of the acute electro-mechanical behaviors such as a prolonged QRS duration, a reduction in wall motion and changes in blood flow etc. [36], MD can also lead to long-term ventricular remodeling [37]. Sometimes MD and electrical dyssynchrony (defined by the inhomogenous LV activation of activation delay between ventricles [38]) are both found in patients with left bundle branch block (LBBB). Patients with LBBB have showed an increased risk of developing cardiac diseases such as hypertension, congestive heart failure [39]. The mortality rate is also higher in these patients with MD if not treated [39–41]. Several experimental studies have been performed on animal models to investigate the effects of asynchronous electrical activation and contraction pattern in the ventricles induced by ventricular pacing at different sites of the LV. The activation timing and pacing locations in these animal models altered the ventricular mechanics and pump function of the heart. More specifically, ventricular pacing at different locations of the canine heart (i.e., the RA, the LV free wall, the LV apex or the RV outflow tract) resulted in a reduction of myofiber shortening, contractile work, myocardial blood flow, and oxygen consumption in early activated region. These quantities, however, are increased in the late activated region [36, 42]. Besides these acute changes, ventricular enlargement (represented by increased LV cavity volume), increased wall mass and asymmetrical LV wall hypertrophy were found with long term asynchronous electrical activation [43, 44]. The asymmetrical LV wall hypertrophy is associated with the thickening and thinning of the late- and early- activated regions, respectively. While all these experimental studies have contributed to our understanding on the alteration of LV mechanics with MD, it is difficult to determine the possible mechanism(s) of hypertrophy associated with mechanical cues solely from these experiments. Hence, it is necessary to develop 7 a mathematical framework describing the G&R associated with MD to increase our understanding of its mechanism. To address this limitation, we developed a finite element framework seeking the stimuli associated with chronic G&R in MD, as described in Chapter 2. 1.4.2 Pressure overload Ventricular afterload is an important determinant of cardiac function and chronic G&R under physiological and pathological conditions. Afterload is often indexed by the pressure of the LV during ejection e.g. peak LV pressure or end systolic pressure. Based on Laplace’s law, 𝑃𝑅 𝜎= (1.1) 2𝑡 an increase in afterload contributes to an increase in total wall stress [45]. In the above equation, 𝜎, 𝑃, 𝑅 and 𝑡 denote the ventricular wall stress, end systolic pressure, end-systolic radius and wall thickness, respectively. An increase in afterload is associated with an increase in left ventricular output impedance and consequently, is associated with an increase in ventricular pressure during systole as seen in various pathological conditions such as aortic stenosis, hypertension, increased total peripheral resistance, HCM etc. [45, 46]. An increase in LV systolic pressure develops higher wall stress, which leads to ventricular remodeling where wall thickness is increased (initially) as a compensatory mechanism. Pressure overload hypertrophy occurs as a result. Additionally, coronary blood blow may be affected when a new balance between oxygen supply and the increased demand is reached with the increase in wall tension [45]. An increase in afterload can lead to the development of heart failure [47] (see a brief review on G&R induced by pressure overload in section 1.2). The progression of hypertrophy in heart diseases associated with pressure overload is still under investigation. Several surgical techniques performed on animal models have been developed to mimic the nature of mechanical cues related to pressure overload and investigate how the cells and heart response to these cues over a long 8 period time [48]. For example, animal models of ascending or transverse aortic constriction mimic aortic stenosis while abdominal aortic constriction or renal warping mimic cues related to hypertension [47]. One of the most frequently used surgical technique to induce pressure overload is ascending aortic constriction (AAC), where a stricture is placed around ascending aorta. Another common model is transverse aortic constriction (TAC) associated with constricting the aorta between the brachiocephalic trunk and the left common carotid artery. These surgical models have both advantages and disadvantages. For example, while the quantification of pressure gradient across the aortic stenosis and stratification of LV hypertrophy are easier with TAC, the higher mortality rate in rats at early state of TAC due to acute cardiac insufficiency makes the application of this technique limited to certain types of animals [49]. On the other hand, AAC is less complicated and time-consuming. It also has high intra- and inter-surgeon reproducibility, low postoperative mortality and reproducible HF phenotypes [50]. The progression and frequency of development of HF induced by these surgical models depend on various factors including banding severity, location, rodent strain, animal type and time course etc. Overall, the consideration of advantages and disadvantages of an animal model along with the purpose and method of experiment will play vital role on the success of these surgical experiments. While animal models are widely being used to recreate the features associated with pressure overload, the intrinsic mechanism of the disease progression associated with pressure overloaded can be investigated by computational modeling. Computational models have been widely used to investigate change in those mechanical properties such as stress or perfusion, which are not easy to measure experimentally or clinically in the deep layer of the myocardium. Also, several computational models have been developed to investigate G&R due to pressure overload. A brief review of existing models is given in section 1.3 and 3.1. However, to our best knowledge, 9 the change in LV mechanics due to the change in instant pressure induced by experimental condition have not been investigated yet. To address this limitation, we develop animal specific model to investigate the mechanics and how its changes is correlated to growth during pressure overload in Chapter 3. 1.4.3 Hypertrophic cardiomyopathy HCM is a genetic heart disease resulting from sarcomeric protein mutations in 60% of patients [51–57]. It has a prevalence of 1 per 500 and a mortality rate that is 4-fold higher in young adults than the general US population [58–63]. This disease is associated with sudden cardiac death (SCD). The annual incidence due to SCD is approximately 1% and far higher in asymptomatic young adults and pediatric patients, respectively [64–66]. Clinical risk factors of SCD include a family history of SCD, unexplained syncope, non-sustained ventricular tachycardia (NSVT), maximum left ventricular wall thickness, and an abnormal blood pressure response during exercise [67]. In symptomatic HCM patients, typical symptoms include dyspnea, chest pain, exercise intolerance, palpitations, and syncope [57]. Treatments widely vary in HCM patients depending on the severity of symptoms and risk factors. Most treatments (e.g., septal myectomy and pharmacological treatments) of HCM are designed to alleviate symptoms and decrease the risk of SCD [68]. Recently, the drug Mavacamtem has showed promising results as a treatment for HCM patients [69–71], especially in obstructive HCM patients where it showed an attenuation in cardiac remodeling [70]. The scope of Mavacamtem on non-obstructive hypertrophy, however, is still under investigation [72]. 10 Figure 1.3: Morphologic subtypes of HCM demonstrated by echocardiography and magnetic resonance imaging. A)Reverse curvature, B)Sigmoid septum, C)Apical HCM, D)Mid- ventricular septum and E)Neutral septum. In each subtype, end diastolic (left) and end-systolic (right) echocardiography images of heart are shown at upper images. In lower images of each subtype, left and middle columns show heart in a 3-chamber orientation in end-diastole and end- systole, respectively, whereas, right column shows myocardial delayed enhancement (MDE) images. This image is adapted from Syed et al. [73]. There are several phenotypes of HCM with different features. Two of the most widely considered phenotypes of this disease are, namely, non-obstructive HCM (30 %) and obstructive HCM (70 %). These 2 phenotypes of HCM are distinguished based on whether left ventricular outflow tract (LVOT) obstruction, as defined by a maximal left ventricular gradient greater than or equal to 30 mm Hg at rest or with provocation, is present [74–76]. In addition, HCM can also 11 be classified based on the variation of hypertrophy distribution that can be generalized into four types. Type I HCM is associated with hypertrophy at the basal septum, type II HCM is associated with hypertrophy involving the whole septum, type III HCM is associated with hypertrophy involving the septum, anterior, and anterolateral walls and type IV HCM is associated with LV apical hypertrophy [77]. In addition to these 4 types of HCM, five major anatomic subsets have been suggested based on the extent of hypertrophy and septal contour, namely, reverse septum curvature, sigmoidal septum, apical form, mid-ventricular form, and neutral contour (Figure 1.3) [73]. Three functional phenotypes of HCM, namely sub-aortic obstruction, mid-ventricular obstruction and cavity obliteration, were also suggested [78]. Several techniques have been developed to diagnose HCM. Among these techniques, echocardiography has played a vital role in the diagnosis and monitoring of HCM patients. In echocardiography, it is recommended to measure the thickness of LV segments from base to apex for all patients. Additional assessment of the apical segments are required to measure the hypertrophy at the LV apex in patients with apical hypertrophy [79]. The use of contrast agents for optimal LV opacification or better imaging techniques such as cardiac magnetic resonance imaging (CMR) is also preferred to adequately visualize the LV segment. At the tissue and organ level, HCM is characterized by myofiber disarray [17, 80–83], disorganized myocardial architecture [84–89], abnormal septal hypertrophy compared to the left ventricular free wall (LVFW), changes in the myocardial contractility, and interstitial and replacement fibrosis [81–89]. These features have been associated with changes in the LV function seen in HCM patients, such as a reduction in (global and segmental) longitudinal and circumferential strains [63, 90–92], an increase in relative ATP consumption during tension generation [93], and a reduction in myocardial work [94]. Additionally, microvascular 12 dysfunction, diffused myocardial ischemia and myocardial cell death are also reported in HCM patients [95]. Figure 1.4: Histological phenotypes of HCM. A thin myocardial section showing A. organized myocardial architecture in normal patient. B. disorganized myocardial architecture in HCM patient. C. myocyte disarray at higher magnification in HCM patient. D. interstitial fibrosis at blue region in a thin myocardial section stained with Masson trichrome. Figure is adapted from Marian et al [96]. Myocardial disarray (Figure 1.4) is an archetypal feature of HCM. This pathological feature is independent of LV wall thickness and may be present in both normal and hypertrophied regions [97]. Although it does not exhibit significant variations between the various regions in the heart of HCM patients, myofiber disarray appears slightly more frequently in the interventricular septum [96]. The exact stimuli inducing myofiber disarray in HCM heart is still unknown. In an in vitro study, using induced pluripotent stem cells (iPSCs)-derived cardiomyocytes, a group of researchers found that Endothelin (ET)-1 peptide enhanced the incidence of myofibrillar disarray in the HCM iPSC‐derived cardiomyocytes. Using mouse HCM model, they also confirmed that myofibrillar disarray was induced by ET-1 [98]. However, due to the differences in nature between adult cardiomyocytes and iPSC‐derived cardiomyocytes, the underlying mechanism causing 13 myofiber disarray is still under investigation. While the genetic backgrounds causing myofiber disarray in HCM heart is still under investigation, with the advancement of imaging techniques such as Diffusion Tensor – CMR, in vivo visualization of normal and HCM myocardial structure have provided substantial insights on myofiber disarray (shown in Figure 1.5) in HCM patients. Along with these techniques is the introduction of a new marker, fractional anisotropy, to describe the degree of myofiber disarray in the cardiac wall quantitatively [17]. Figure 1.5: Disarray and fibrosis depicted by Fractional anisotropy and late gadolinium enhancement, respectively, using diffusion tensor-CMR in HCM and control patients. This figure is adapted from Ariga et al [99]. Table 1.1: Clinical data of circumferential strain (%) LV Segment Young et al 1993[100] Sun et al 2009[91] Piella et al 2010[92] Normal HCM Normal HCM Normal HCM Septal 19.67±2.67 15±5.67 23.97±5.47 16.17±7.17 14.28±1.58 9.38±3.24 Lateral 21.3±2.33 19.67±4 16.87±5.89 15.77±6.63 13.4±1.54 9.86±3.32 Inferior - - 20.67±6.5 17.5±5.57 13.73±2.07 8.6±2.63 14 Table 1.1 (cont’d) Anterior 21.67±2.33 18.67±6 20.23±5.2 15.47±7.6 12.5±1 8.63±3.32 Posterior 19.67±3 17.3±4.67 17.1±6.2 17.7±6.43 - - Anterior - - 23.7±5.03 15.9±8.2 - - septal Table 1.2: Clinical data of longitudinal strain (%) LV Segment Young et al 1993[100] Normal HCM Septal 16±2.33 9.67±6.3 Lateral 17.67±3.33 12.3±5 Inferior - - Anterior 16.67±2.33 10.67±5 Posterior 17.67±3.33 10.3±5.67 Anterior septal - - Reduction in global longitudinal strain (GLS) is a feature of HCM at early stages and before the development of hypertrophy in relatives of HCM patients [101, 102]. GLS is not only a sensitive indicator of global left ventricular function, but is also a prognostic marker to predict mortality and cardiac events in other cardiac diseases [103–106]. A significant association between worse LV-GLS and increased composite cardiac outcomes has been showed in a systematic review over the prognostic value of GLS in HCM. Based on a 3-year follow-up period, patients with GLS > −16% had a significantly high risk for sustained ventricular tachycardia/fibrillation, heart failure, cardiac transplantation, and all-cause death compared to patients with GLS < − 16% [90, 107]. Patients with GLS > −10% had four times higher risk of events compared to patients with GLS value ≤ −16% [107]. Besides global reduction of strain, regional variation of strain has also been found in this disease. Specifically, circumferential strain was reduced significantly (~5% 15 minimum[92], 7.8% maximum [91]) at septal regions in HCM patients compared to healthy normal humans (Table-1.1, Table-1.2). Compared to the septal region, the reduction in circumferential strain in the lateral regions (LVFW) was lower (~1% minimum [91], 3.5% maximum [92]). In the study by Sun et al [91], however, they reported a 0.6% increase in circumferential strain at the LV posterior region. On the other hand, the decrease in longitudinal strain was larger in the septum than lateral regions (6.3% vs 5.4% decrease) in HCM patients compared to healthy humans. This heterogeneity in strain distribution could be due to the regional distribution of myocardial disarray and fibrosis. The exact mechanism of how strain is affected by this disease and how it predicts outcome, however, remains unclear. Hypertrophy of the LV is a key feature of HCM. Specific to this disease, hypertrophy is largely asymmetric with heterogeneous wall thickening [108]. Left ventricular wall thickness is typically analyzed in HCM patients with echocardiography based mostly on the short-axis view images acquired at multiple levels at end diastole [109], [79]. A well-known cut-off value of LV wall thickness for defining hypertrophy in adults, relatives and pediatric patients are ≥15 mm, >12–15 mm and ≥2 Standard Deviation greater than the Body-Surface-related normal values, respectively [110, 111]. The presence of asymmetric septal hypertrophy in HCM patients is defined by a septal-to-posterior diastolic wall thickness ratio ≥ 1.3 (or ≥1.5 in hypertensive patients), with or without subaortic obstruction [109]. Myocardial fibrosis is a key feature and a marker to predict mortality rate, SCD and progression toward heart failure in HCM patients [112, 113], [114]. This feature can be evaluated using magnetic resonance imaging with late gadolinium enhancement (LGE). The distribution of fibrosis vary greatly between various regions of the LV wall, including septum, LV free wall, lateral wall, apex, and RV insertion point in HCM patients [115, 116]. Extracellular volume (ECV) 16 estimated from CMR imaging with LGE has also been found to correlate with the hypertrophied region in HCM patients [117]. About one third of HCM patients with LVOT obstruction have systolic anterior motion of the mitral valve due to severe interventricular septum hypertrophy, mitral leaflet abnormalities, papillary muscle hypertrophy, and displacement. In about one third of these patients, latent LVOT obstruction is provoked due to changes in preload and/or afterload, or altered LV contractility [76]. Motivated by the diverse nature of HCM and lack of computational models on the mechanics of HCM, we have developed two finite element frameworks based on an idealized LV geometry and patient specific geometries to investigate the effects of remodeling features on the altered mechanics of HCM. These models are briefly explained at Chapter 4 & 5. 1.5 Objectives of this dissertation The overall objectives of this dissertation are to develop computational framework to evaluate and describe 1) volumetric changes in the heart wall (hypertrophy/atrophy) in heart diseases and 2) tissue microstructure (myofiber disarray) changes in heart diseases. More specifically, the objectives explained in following chapters are as follows. Chapter 2: A coupled electromechanics-growth model was developed to simulate the long- term effects during MD. Using myofiber stretch as stimulus, this model can quantitatively reproduce asymmetrical hypertrophy by wall thinning of early activated region and wall thickening of late activated region. Chapter 3: A computational-experimental approach was developed to identify the mechanical stimuli during pressure overload. The computational framework was calibrated against experimental measurements from 4 aortic constriction porcine models, and the results showed a strong correlation between myofiber stress and growth. 17 Chapter 4: A computational framework describing the effects of myofiber disarray in the LV of HCM patients was formulated and developed. The computational framework was developed based on an idealized LV geometry and calibrated using published data associated with healthy humans and HCM patients. The effects of geometry and myofiber disarray globally was investigated using the model. The simulated results showed that the mechanics of left ventricle got impaired by varying myofiber disarray. Chapter 5: A patient specific computational model was developed to investigate the ventricular mechanics associated with obstructive and non-obstructive HCM patients. The model was validated using patient-specific clinical measurements of the HCM patients. The effects of varying degree of myofiber disarray was investigated using the model. Using this model we found that the contractile force generated by the cell to reproduce clinical measurements is increased with an increase in global myofiber disarray. 18 CHAPTER 2 BIVENTRICULAR MODEL ON LEFT BUNDLE BRUNCH BLOCK 19 2.1 Introduction Mechanical dyssynchrony [36, 39–41] is a disease associated with asynchronous contraction or relaxation of the RV and LV[118]. Experimental studies using animal models have shown that ventricular pacing produces ventricular dilation and asymmetrical hypertrophy [43, 44]. While existing computational cardiac growth models mentioned earlier (in Section 1.3) have largely focused on describing pathologies associated with the global alterations in loading conditions, such as pressure and volume overload that produce concentric and eccentric hypertrophy, respectively [30, 31, 119, 120], little work has been done to simulate long-term changes associated with alterations of the electrical conduction pattern in the heart except for study [121]. In order to simulate chronic changes associated with MD, it is necessary to prescribe the appropriate stimulus driving G&R. While the exact stimulus driving growth is still unknown, insights provided by an experimental study on cardiomyocyte growth suggest that longitudinal stretch can produce both longitudinal and transverse growth by series and parallel addition of sarcomeres, respectively [122]. Motivated by these experimental observations, we seek here to investigate, i) if prescribing myofiber stretch as a single stimulus that controls growth in the myofiber and transverse directions (with different sensitivity) can quantitatively reproduce long term changes in ventricular geometry associated with MD ii) if it is possible to find different forward and reverse growth rates in the longitudinal and transverse directions that can simultaneously and quantitatively reproduce global and local asymmetrical changes in biventricular geometry. 20 2.2 Methods Figure 2.1: Top: Simulated chronic pacing timelines are shown. Pacing locations are indicated in the geometry using a red star. Bottom: Myofiber architecture and lumped circulation model. On the basis of single-cell experiments [122] and an existing computational modeling framework [29, 123], we developed an anisotropic G&R constitutive model in which the changes in lengths of the tissue in 3 orthogonal material directions are driven locally by the deviation of maximum elastic myofiber stretch (over a cardiac cycle) from its corresponding homeostatic set point value. This model was coupled to an electromechanics modeling framework [21, 124–127] to simulate the long-term effects of asynchronous activation associated with LVFW pacing (Figure 2.1). After appropriate calibration of parameters, the model predictions were compared with local 21 and global measurements in canine experiments with similar chronic LVFW pacing protocol, as shown in Table 2.1. Table 2.1: Comparison with experimental data (adapted from [128] ) Parameter Month LVFW LVFW Pacing LBBB LVFW Pacing pacing Experiment Experiment Experiment simulation [129] [118] [44] LVEDV (% 0 100 100 ± 27.8 100 ± 29.8 100 of Normal) 2 102.3 − 117.5 ± 12.8 − 𝑅 4 104.6 − 129.8 ± 50.9 − 𝑀 6 109 107.4 ± 29.9 − − 8 113.7 − − − LVEF (%) 0 48.7 35.3 ± 7.0 43 ± 4.0 100 2 48.3 − − − 4 48.6 − 33 ± 6𝑅 − 𝑀 6 47.8 39.6 ± 8.9 − − 8 48.3 − − − LVESV (% 0 100 100 − 100 Change) 2 104.9 − − − 4 108.7 − − − 6 118.5 − − − 8 126.4 − − − Early 0 100𝐿 100𝐿 100𝐶 100𝐿 activated 2 90.3𝐿 90.7 ± 8.7𝐿 − 88.9 ± 6.8𝐿,𝑅 region 4 83.9𝐿 87.0 ± 7.2𝐿 − 79.7 ± 8.0𝐿,𝑅,∗ thickness 6 81.3 𝐿 86.5 ± 16.7 𝐿,𝑅 − − (% change) 8 79.6 𝐿 − − − Late 𝐶 𝐶 𝐿 0 100 100 100 100𝐶 activated 2 105.2 𝐶 108.4 ± 11.3 𝐶 − 96.8𝐶,𝑀 region 4 113.7𝐶 110.5 ± 16.8𝐶 − 103.0 ± 7.5𝐶,𝑀,∗ thickness 6 119.6𝐶 122.5 ± 11.3𝐶,𝑅 − − (% change) 8 127.5 𝐶 − − − RV 0 100 100 100 100 thickness 2 91.4 − − − (% 4 91.1 − − − Change) 6 88.6 − − − 8 88.7 − − − 𝑀 𝑅 denotes no significant change over time ; denotes significant change over time (p<0.05) 𝐿 denotes LVFW thickness ; 𝐶 denotes septum thickness ∗ denotes 3 months ; − denotes not reported or measured 22 2.2.1 Growth constitutive model Figure 2.2: (a) The left ventricle and a cutout from the wall; (b) the structure through the thickness from the epicardium to the endocardium; (c) Transmural variation of fibers at five longitudinal–circumferential sections at regular intervals from 10 to 90 per cent of the wall thickness; (d) the layered organization of myocytes and the collagen fibres between the sheets referred to a right-handed orthonormal coordinate system with fibre axis 𝒇𝟎 , sheet axis 𝒔𝟎 and sheet-normal axis 𝒏𝟎 ; and (e) a cube of layered tissue showing local material coordinates (𝒇𝟎 , 𝒔𝟎 , 𝒏𝟎 ). The figure is adapted from [130]. Let, 𝜒𝜅0 (𝑿, 𝑡) describes the mapping from an unloaded reference configuration 𝜅0 with position X to a current configuration 𝜅 with the corresponding material position 𝒙 = 𝜒𝜅0 (𝑿, 𝑡). The displacement field is given by u = x − X and the deformation gradient tensor is defined 23 as 𝑭 = 𝜕𝒙⁄𝜕𝑿 . In the volumetric growth framework, the deformation gradient tensor, 𝑭, is multiplicatively decomposed into an elastic and a growth tensor as follows 𝑭 = 𝑭𝒆 𝑭𝒈 , (2.1) Here Fe and Fg are the elastic and growth deformation gradients, respectively. The growth deformation gradient Fg was described by 𝑭𝒈 = 𝜃𝑓 𝒇𝟎 ⊗ 𝒇𝟎 + 𝜃𝑠 𝒔𝟎 ⊗ 𝒔𝟎 + 𝜃𝑛 𝒏𝟎 ⊗ 𝒏𝟎 , (2.2) where, 𝒇𝟎 , 𝒔𝟎 and 𝒏𝟎 are the local myofiber, sheet, and sheet-normal directions in the reference configuration, respectively (Figure 2.2). The evolution of the growth multipliers associated with the deviations of a prescribed stimulant 𝑠𝑖 from its homeostatic value 𝑠𝑖,ℎ is given by 𝜃𝑖̇ = 𝑘𝑖 (𝜃𝑖 , 𝑠𝑖 )𝑔𝑖 (𝑠𝑖 , 𝑠𝑖,ℎ ), (2.3) where 𝜃𝑖̇ is the derivative of growth multipliers with respect to time t. Based on the local stimulus, the function 𝑔𝑖 (𝑠𝑖 , 𝑠𝑖,ℎ ) is prescribed as 𝑔𝑖 (𝑠𝑖 , 𝑠𝑖,ℎ ) = 𝑠𝑖 − 𝑠𝑖,ℎ . The rate limiting function, which restricts forward and reverse growth rates, is defined as follows 𝛾𝑔,𝑖 1 𝜃𝑚𝑎𝑥,𝑖 − 𝜃𝑖 ( ) if 𝑔𝑖 (𝑠𝑖 , 𝑠𝑖,ℎ ) ≥ 0 𝜏𝑔,𝑖 𝜃𝑚𝑎𝑥,𝑖 − 𝜃𝑚𝑖𝑛,𝑖 𝑘𝑖 (𝜃𝑖 , 𝑠𝑖 ) = { 𝛾𝑟𝑔,𝑖 , (2.4) 1 𝜃𝑖 − 𝜃𝑚𝑖𝑛,𝑖 ( ) if g i (si , si,h ) < 0 𝜏𝑟𝑔,𝑖 𝜃𝑚𝑎𝑥,𝑖 − 𝜃𝑚𝑖𝑛,𝑖 where the subscript 𝑖 ∈ 𝑓, 𝑠, 𝑛 denote the association with the myofiber, sheet, and sheet-normal directions. The growth constitutive model parameters are 𝜏𝑔,𝑖 , 𝛾𝑔,𝑖 , 𝜏𝑟𝑔,𝑖 and 𝛾𝑟𝑔,𝑖 . The application of rate-limiting function are two folds. One, it restricts the evolution of the growth multipliers 𝜃𝑖 within some prescribed limits. Second, prescribing different value of 𝑘𝑖 in each 𝑖 direction enables a broad spectrum of anisotropic growth deformation. 24 Figure 2.3: Schematic of the anisotropic growth evolution in response to local stimulus. Maximum elastic myofiber stretch was prescribed as the growth stimuli in all 3 material directions. The myofiber stretch is defined as 𝜆𝑓 = √𝒇𝟎 ∙ 𝑪 ∙ 𝒇𝟎 , (2.5) where C denotes the right Cauchy-Green deformation tensor with respect to the end-diastolic configuration. Positive deviations from the homeostatic point will results towards the evolution of maximum growth multiplier, whereas negative deviations from the homeostatic point will drag the evolution towards the minimum as depicted in Figure 2.3. 2.2.2 Electrophysiology model Based on modified Fitzhugh-Nagumo model, cardiac electrical activity and its propagation was modeled. Specifically, the spatio-temporal evolution of cardiac action potential 𝜑 is described in the reference configurations by 𝜑̇ = 𝑑𝑖𝑣(𝑫𝑔𝑟𝑎𝑑𝜑) + 𝑓𝜑 (𝜑, 𝑟) + 𝐼𝑠 , (2.6) 25 𝑟̇ = 𝑓𝑟 (𝜑, 𝑟) , (2.7) where 𝑫 is the anisotropic electrical conductivity tensor, Is is the constant electrical stimulus for prescribing local excitation initiation during pacing, and r is a dimensionless recovery variable. The excitation properties of cardiac tissue are defined by 𝑓𝜑 = 𝑐𝜑 (𝜑 − 𝑎)(1 − 𝜑) − 𝑟𝜑 , (2.8) 𝜇1 𝑓𝑟 = (𝛾 + 𝑟 𝜇 ) (−𝑟𝑐𝜑(𝜑 − 𝑏 − 1)), (2.9) 2 +𝜑 Here 𝑐, 𝛼, 𝑏, 𝛾, 𝜇1 and 𝜇2 are the Fitzhugh-Nagumo model parameters. 2.2.3 Mechanics model Mechanical behavior of the cardiac tissue was described by an active stress formulation. In this formulation, mechanical behavior is additively decomposed into a passive component and an active component. More specifically, the second Piola-Kirchhoff or PK2 stress tensor 𝑺 has a passive component, 𝑺𝒑 and an active component, 𝑺𝒂 , i.e. 𝑺 = 𝑺𝑝 + 𝑺𝑎 , (2.10) Passive mechanical properties is described using the following Fung-type strain energy function 𝐶 𝑤(𝑬) = (𝑒 𝑄 − 1.0), (2.11a) 2 2 2 2 2 2 2 2 2 2 ), 𝑄 = 𝑏𝑓 𝐸𝑓𝑓 + 𝑏𝑓𝑠 (𝐸𝑓𝑠 + 𝐸𝑠𝑓 + 𝐸𝑓𝑛 + 𝐸𝑛𝑓 ) + 𝑏𝑥𝑥 (𝐸𝑠𝑠 + 𝐸𝑛𝑛 + 𝐸𝑛𝑠 + 𝐸𝑠𝑛 (2.11b) In the above equations, C, 𝑏𝑓 , 𝑏𝑓𝑠 , 𝑏𝑥𝑥 are material parameters. And 𝐸𝑖𝑗 with (𝑖, 𝑗) ∈ (𝑓, 𝑠, 𝑛) 1 denote components of the elastic Green-Lagrange strain tensor, 𝑬 = (𝑭𝑻𝒆 𝑭𝒆 − 1). The passive 2 stress is determined from the strain energy function by 𝜕𝑤(𝑬𝒆 ) 𝑺𝒑 = , (2.11c) 𝜕𝑬𝒆 Based on a phenomenological active contraction model, active mechanical behavior of the cardiac tissue is described by an active stress tensor directed in the myofiber direction, i.e., 26 𝐶𝑎02 1−cos (𝜔(𝑡,𝑡𝑖𝑛𝑖𝑡 ,𝐸𝑓𝑓 )) 𝑺𝑎 = 𝑇𝑚𝑎𝑥 2 (𝐸 ) 𝒇𝟎 ⊗ 𝒇𝟎 , (2.12a) 1+ 𝐸𝐶𝑎50 𝑓𝑓 2 (𝐶𝑎0 )𝑚𝑎𝑥 𝐸𝐶𝑎50 = , (2.12b) √exp(𝐵(𝑙− 𝑙0 ))−1 Here, 𝑇𝑚𝑎𝑥 , 𝐶𝑎0 , and 𝐸𝐶𝑎50 are the scaling factor associated with the tissue contractility, the peak intracellular calcium concentration and the length-dependent calcium sensitivity, respectively. Also, (𝐶𝑎0 )𝑚𝑎𝑥 , B, 𝑙0 are the maximum peak intracellular calcium concentration, a material constant, and the sarcomere length at which no active tension develops, respectively. The instantaneous sarcomere length is defined as 𝑙 = 𝑙𝑠0 √𝒇𝟎 ∙ 𝑪 ∙ 𝒇𝟎 with the prescribed initial length of sarcomere, 𝑙𝑠0 . To incorporate a spatially heterogeneous activation initiation time, 𝑡𝑖𝑛𝑖𝑡 (𝑿), the active contraction model is modified in the function ω i.e., 𝑡𝑠𝑎 𝜋 if 0 ≤ 𝑡𝑠𝑎 < 𝑡0 , 𝑡0 𝜔= 𝑡𝑠𝑎 − 𝑡0 + 𝑡𝑟 (2.12c) 𝜋 if 𝑡0 ≤ 𝑡𝑠𝑎 < 𝑡0 + 𝑡𝑟 , 𝑡𝑟 { 0 if 𝑡0 ≤ 𝑡𝑠𝑎 < 𝑡0 + 𝑡𝑟 , Here, 𝑡0 is the prescribed time to maximum active tension and 𝑡𝑟 is the sarcomere length- dependent active tension relaxation time that is given by 𝑡𝑟 = 𝑚𝑙 + 𝑏 with parameters m and b. Time since activation 𝑡𝑠𝑎 (𝑿) = 𝑡𝑐𝑢𝑟𝑟𝑒𝑛𝑡 − 𝑡𝑖𝑛𝑖𝑡 (𝑿) couples cardiac electrophysiology and mechanics, where 𝑡𝑐𝑢𝑟𝑟𝑒𝑛𝑡 denotes the current time in the cardiac cycle and 𝑡𝑖𝑛𝑖𝑡 (𝑿) defines the local initiation time that is given as 𝑡𝑖𝑛𝑖𝑡 (𝑿) = 𝑖𝑛𝑓 {𝑡(𝑿)|𝜑(𝑿, 𝑡) ≥ 0.9} , (2.13) 2.2.4 Computational approximation Finite element formulation of the BiV mechanics problem was obtained by minimizing the following Lagrangian functional [128, 131] 27 ℒ = ∫𝛺 𝛹𝑇 (𝒖) 𝑑𝑉 – ∫𝛺 𝑝(𝐽 − 1)𝑑𝑉 – 𝑃𝐿𝑉 (𝑉𝐿𝑉,cav (𝒖) − 𝑉𝐿𝑉 ) − 𝑃𝑅𝑉 (𝑉𝑅𝑉,𝑐𝑎𝑣 (𝒖) − 𝑉𝑅𝑉 ) – 1 1 ∫ 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 (𝒖. 𝒖)𝑑𝑆 − 2 ∫𝑑𝛺 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 (𝒖. 𝒖)𝑑𝑆 – 𝒄1 ∫Ω 𝒖𝑑𝑉 – 𝒄2 ∫Ω 𝑿 × 𝒖𝑑𝑉, (2.14) 2 𝑑Ω𝑒𝑝𝑖 𝑏 In Eq. (2.14), 𝛹𝑇 is the total strain energy of the myocardium, 𝒖 ∈ 𝑯𝟏 (𝛺0 ) is the displacement field. On the other hand, (𝑃𝐿𝑉 , 𝑃𝑅𝑉 ) ∈ 𝑅, 𝑝 ∈ 𝐿2 (𝛺 ), 𝒄1 ∈ 𝑅 3 and 𝒄2 ∈ 𝑅 3 are the Lagrange multipliers for, respectively, constraining the cavity volume 𝑉𝐿𝑉,𝑐𝑎𝑣 (𝒖) and 𝑉𝑅𝑉,𝑐𝑎𝑣 (𝒖) to the prescribed value 𝑉𝐿𝑉 and 𝑉𝑅𝑉 , respectively, enforcing incompressibility in which the Jacobian of the deformation gradient tensor 𝐽 = 1, enforcing zero mean translation and enforcing zero mean rotation, respectively. Spring (robin-type) boundary conditions with spring constant 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 and 𝑘𝑠𝑝𝑟𝑖𝑛𝑔2 were also imposed on the epicardial surface 𝑑Ω𝑒𝑝𝑖 and base 𝑑Ω𝑏 , respectively. The approximate solution of the weak formulation of the acute electromechanics problem are obtained from solving Euler-Lagrange problem by finding 𝒖 ∈ 𝐻1 (Ω ), 𝑝 ∈ 𝐿2 (Ω ), 𝑃𝐿𝑉 ∈ ℝ, 𝑃𝑅𝑉 ∈ ℝ, 𝒄1 ∈ ℝ3 , 𝒄2 ∈ ℝ3 , 𝜑 ∈ 𝐻1 (Ω ), 𝑟 ∈ 𝐻 0 (𝛺 ) that satisfies 𝛿ℒ = ∫Ω (𝑭𝑺 − 𝐽𝑭−𝑻 ): 𝛻𝛿𝑢 𝑑𝑉 − ∫Ω 𝛿𝑝(𝐽 − 1)𝑑𝑉 − 𝑃𝐿𝑉 ∫Ω 𝐽𝑭−𝑻 : 𝛻𝛿𝑢 𝑑𝑉 − LV 𝑃𝑅𝑉 ∫Ω 𝐽𝑭−𝑻 : 𝛻𝛿𝑢 𝑑𝑉 − 𝛿𝑃𝐿𝑉 (𝑉𝐿𝑉,cav (𝒖) − 𝑉𝐿𝑉 ) − 𝛿𝑃𝑅𝑉 (𝑉𝑅𝑉,cav (𝒖) − 𝑉𝑅𝑉 ) − RV 𝛿𝒄𝟏 ∙ ∫Ω 𝒖 𝑑𝑉 − 𝛿𝒄𝟐 ∙ ∫Ω 𝑿 × 𝒖 𝑑𝑉 − 𝒄𝟏 ∙ ∫Ω 𝛿𝒖 𝑑𝑉 − 𝒄𝟐 ∙ ∫Ω 𝑿 × 𝛿𝒖 𝑑𝑉 − ∫𝛿𝛺 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 𝒖 ∙ 𝛿𝒖 𝑑𝑆 − ∫𝛿𝛺 𝑘𝑠𝑝𝑟𝑖𝑛𝑔2 𝒖 ∙ 𝛿𝒖 𝑑𝑆 = 0, (2.15a) 𝑒𝑝𝑖 𝑏 ∫Ω (𝜑 − 𝜑𝑛 )Δ𝑡 −1 𝛿𝜑 = ∫Ω 𝑫 𝑔𝑟𝑎𝑑𝜑. 𝑔𝑟𝑎𝑑𝛿𝜑 + ∫Ω (𝑓𝜑 + 𝐼𝑠 )𝛿𝜑, (2.15b) ∫𝛺 (𝑟 − 𝑟𝑛 )𝛥𝑡 −1 𝛿𝑟 = ∫𝛺 𝑓𝑟 𝛿𝜑, (2.15c) for all test functions 𝛿𝒖 ∈ 𝐻1 (Ω ), 𝛿𝑝 ∈ 𝐿2 (Ω ), 𝛿𝑃𝐿𝑉 ∈ ℝ, 𝛿𝑃𝑅𝑉 ∈ ℝ, 𝛿𝒄1 ∈ ℝ3 , 𝛿𝒄2 ∈ ℝ3 , δ𝜑 ∈ 𝐻1 (Ω ), 𝛿𝑟 ∈ 𝐻 0 (𝛺 ). In Eq. (2.15), 𝛿𝒖, 𝛿𝑝, 𝛿𝑃𝐿𝑉 , 𝛿𝑃𝑅𝑉 , 𝛿𝒄1 , 𝛿𝒄2 are the first variation of the displacement field, Lagrange multipliers for enforcing incompressibility (𝐽 = 1) 28 and volume constraint for LV and RV, zero mean translation and rotation, respectively. Besides, 𝛿𝜑, 𝛿𝑟 are the first variation of the action potential and recovery state variable, respectively. Spring constants 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 and 𝑘𝑠𝑝𝑟𝑖𝑛𝑔2 are associated with boundary conditions imposed at epicardium and basal surface, respectively. Similarly, approximate solution of the weak formulation of the G&R problem were obtained from solving Euler-Lagrange problem by finding 𝒖 ∈ 𝐻1 (Ω ), 𝑝 ∈ 𝐿2 (Ω ), that satisfies 𝛿ℒ𝐺 = ∫Ω (𝑭𝑺 − 𝐽𝑭−𝑻 ): 𝑔𝑟𝑎𝑑 𝛿𝒖 𝑑𝑉 − ∫𝛺 𝛿𝑝(𝐽 − 1)𝑑𝑉 − ∫𝛿Ω 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 𝒖 ∙ 𝛿𝒖 𝑑𝑆 = 0, (2.16) 𝑒𝑝𝑖 for all test functions 𝛿𝒖 ∈ 𝐻1 (Ω ), 𝛿𝑝 ∈ 𝐿2 (Ω ). Here, 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 is associated with boundary conditions imposed at epicardium. 2.2.5 Simulation scheme Two cases differing in terms of the prescribed activation initiation location were simulated. These cases are, namely, • Normal: activation was initiated at the septum near the base, • Pacing: activation was initiated at the LVFW near the base. Table 2.2: Growth Parameters Direction 𝝉𝒈 𝝉𝒓𝒈 γ 𝜽𝒎𝒊𝒏 𝜽𝟎 𝜽𝒎𝒂𝒙 Days Days Days (no units) (no units) (no units) 𝜃𝑓 3.8 9.6 1.0 0.5 1.0 2.0 𝜃𝑠 9.6 3.8 1.0 0.5 1.0 2.0 𝜃𝑛 9.6 3.8 1.0 0.5 1.0 2.0 A schematic of the simulation timeline and pacing location is shown in Figure 2.1. The homeostatic set point for the maximum elastic myofiber stretch in the growth constitutive model was prescribed using the local values obtained from the Normal case with septal activation. 29 Deviations of the maximum elastic myofiber stretch in the Pacing case from the homeostatic values were used as growth stimuli. The calibrated growth parameters are mentioned in Table2.2. 2.3 Results & Discussion Figure 2.4: (a) Propagation of the depolarization isochrones in the Normal (top) and Pacing (bottom) cases. (b)Long term changes in RV (left) and LV (right) PV loops in the Pacing case; M0–8 denote results at 0–8 month. Refer to (c) for line color. (c) Myofiber stretch, 𝜆𝑓 , as a function of time over a cardiac cycle at 0–8 month. Normal case is in black. Our simulations showed that due to the presence of electromechanics alterations induced by LVFW pacing, a pre-stretch occurs at the late activated regions (Septum + RVFW) in the beginning of systole (Figure 2.4) that produced a higher maximum elastic myofiber stretch 30 compared to the homeostatic set point value (during normal activation) found in those regions. On the other hand, the early activated region (LVFW) has a lower maximum elastic myofiber stretch when compared to its corresponding homeostatic set point value. These results are consistent with observations in animal models of asynchronous activation [132, 133] and LBBB patients [21], where abnormal stretching of the tissue at the beginning of systole (i.e., pre-systolic stretching) was found at the late activated regions. Consequently, myofiber stretch in the septum + RVFW and LVFW of the Pacing case deviated positively and negatively from the homeostatic value in the Normal case. This heterogeneity in myofiber stretch 𝜆𝑓 resulted in the evolution of growth scalars 𝜃𝑖 ’s towards 𝜃𝑖,𝑚𝑎𝑥 in the late activated septum/RV, and 𝜃𝑖,𝑚𝑎𝑥 in the early-activated LVFW, leading to long-term asymmetrical geometrical changes. Using the alteration of elastic myofiber stretch as a stimulant for G&R in all 3 material directions, the model predictions, after appropriate calibration of parameters, were compared with local and global measurements in experiments where a similar chronic LVFW pacing protocol was applied to the canine model [43, 44]. In terms of long-term hemodynamic changes in the LV and RV (Figure 2.3c), there was no immediate substantial reduction in the pump function in the Pacing case (0 month). Changes were, however, noticeable at 2 months with the onset of progressive LV dilation. Specifically, in the span of 8 months, LV end-diastolic volume (EDV) increased from 104.3 ml to 118.6 ml whereas end-systolic volume (ESV) increased from 55.65 ml to 70.3 ml. This led to a rightward shift in the LV PV loop that was accompanied by a slight reduction in ejection fraction (EF) from 48.7% to 48.3% at 8 months. On the other hand, the simulations also show long-term changes of the RV PV loops arising largely from the thickening of septum. RV EDV was slightly decreased from 104.9 ml to 103.7 ml whereas RV ESV increased from 53.6 ml to 57.1 ml at 8 months. 31 Figure 2.5: Long-term changes in biventricular geometry (blue) are superimposed on the original (red outline). Left: short-axis view; Right: long-axis view. Long-term changes in the geometry is highly asymmetrical in the biventricular unit in the Pacing case with radial wall thickening occurring at the late-activated septum and wall thinning occurring at the early-activated LVFW (Figure 2.5). In terms of local geometrical changes (Figure 2.6), the model predicted an increase in septum wall thickness by 18.5% (cf. 23 ± 12% in the experiments [43]) and a decrease in LVFW thickness by 19.7% (cf. 17 ± 17% in the experiments [43]) after 6 months of pacing. In terms of global geometrical changes, the model predicted an increase in LV EDV by 9% (cf. 7.4 ± 29% in the experiments) and LVFW + Septum wall volume 32 by 9.5% (cf. 15 ± 17% in the experiments) for the same duration. The chronic features of LVFW pacing predicted by the model are also found in LBBB [118], which produces MD via an opposite activation pattern (i.e., septum is activated first followed by the LVFW). Figure 2.6: Long-term local geometrical changes. Left: wall thickness; Middle: wall volume; Right: Cavity volume. Figure 2.7: Effect of varying G&R parameters. Changes in Left: τg from 40.0 to 1.0 for 𝜆𝑓 − 𝜆𝑓,ℎ = 0.06; Middle: 𝜏𝑔 from 40.0 to 1.0 for 𝜆𝑓 − 𝜆𝑓,ℎ = −0.06; Right: 𝜆𝑓 − 𝜆𝑓,ℎ from 0.01 to 0.1 for 𝜏𝑔 = 9.6. Red: Parameter values in Table 2.2. Calibration of G&R parameters showed that in order to reproduce asymmetrical hypertrophy, it is necessary to impose different forward growth rate 𝜏𝑔 and reverse growth rate 𝜏𝑟𝑔 33 not only because of the heterogeneity in myofiber stretch in heart, but also due to the high sensitivity of the G&R parameters to the deviation (Figure 2.7). 2.4 Conclusion With appropriate calibration, we showed that the prescription of a single growth stimuli based only on the elastic myofiber stretch can quantitatively reproduce the largely local G&R features found with MD (Table 2.1), which reinforce the theory that transverse growth maybe controlled, at least to some extent, by elastic myofiber stretch. 34 CHAPTER 3 ANIMAL SPECIFIC LEFT VENTRICULAR MODEL ON PRESSURE OVERLOAD 35 3.1 Introduction LV afterload as reflected by an elevated systolic pressure [45] can lead to acute changes in cardiac mechanics and produce chronic G&R. An increase in afterload caused by pathological conditions (e.g., aortic stenosis and hypertension) can impair longitudinal myocardial deformation and produce local changes in myocardial stresses and stretches, which in turn, can trigger the development of concentric hypertrophy. Several computational models [30, 32] focusing mostly on global G&R features during pressure overload have been developed. These models, however, do not consider local/regional G&R features and how they are correlate with changes in the local stresses or strains, which can provide insights into which mechanical quantity is driving G&R. To address this issue, we used a combination of computational modeling and experiments to investigate whether normal stresses or strains along 3 orthogonal material directions can better correlate with regional measurements of growth in swine models during aortic banding. While changes in stretch in the pressure overloaded hearts can, in principle, be measured experimentally, the combination of a complex ventricular wall structure with highly nonlinear mechanical behavior and the limitations of current available techniques, however, do not allow for stresses in the muscle fibers to be quantified directly through experiment [11]. As such, we developed animal-specific finite element (FE) models of the LV to simulate the acute effects of pressure overload and estimate the regional changes in normal stresses or strains, which were then correlated with the corresponding regional growth. 3.2 Methods 3.2.1 LV geometry model Left ventricular geometries segmented from the 3D echocardiography (3D echo) images were discretized using (~4000) tetrahedral elements (Figure 3.1), which is sufficient for 36 convergence based on a previous study using a biventricular mesh [134]. Myofiber direction 𝒇𝒐 , was prescribed based on a linear transmural variation of the helix angle from 60° at the endocardium to −60° at the epicardium [135] across the wall using a Laplace-Dirichlet rule-based algorithm [136]. Figure 3.1: Construction of animal-specific LV FE model. (a) Segmentation of LV surfaces from 3D echo images, (b) Meshing of geometry to construct a LV FE model that is connected to a 3-element Windkessel model [137] (c) Transmural distribution of myofiber angle from +600 at endocardium to -600 at epicardium is prescribed in the LV FE model. 3.2.2 Experimental measurements Measurements were acquired in vivo from the animals before aortic constriction (baseline) and 2 weeks after aortic constriction (growth). Specifically, aortic and LV pressure waveforms were measured using catheterization while 3D echo (EPIQ-C system, Philips Healthcare, Andover, MA, USA) was performed on the animals from which the LV geometry and volume waveforms were acquired. The LV pressure and volume waveforms were synchronized to obtain pressure- volume (PV) loops in each animal at baseline and after growth. Based on the LV geometry segmented from 3D echo, we also computed the regional thickness by measuring the local shortest distance between the endocardium and epicardium. The local regional wall thickness was projected 37 on the endocardium and regional growth was indexed by the difference between thickness at baseline and after growth. 3.2.3 Mechanics model Left ventricular mechanics was described using an active stress formulation, where total stress tensor was additively decomposed into an active and a passive component. The passive mechanics model is briefly described in section 2.2.3 by Eq. 2.11. Active mechanics was described using an active contraction model modified from that of Guccione et al. [138, 139] with the active stress tensor given as 𝐶𝑎2 2 ) 𝐶𝑡 𝒇𝟎 ⊗ 𝒇𝟎 , (3.1a) 0 𝑺𝒂 = 𝑇𝑚𝑎𝑥 (𝐶𝑎2 +𝐸𝐶𝑎 0 50 1 𝐶𝑡 = (1 − 𝑐𝑜𝑠𝜔), (3.1b) 2 𝑡 𝜋 𝑡 ; 0 ≤ 𝑡 < 𝑡0 0 𝜔 = 𝜋 𝑡−𝑡0+𝑡𝑟 ; 𝑡0 ≤ 𝑡 < 𝑡0 + 𝑡𝑟 , (3.1c) 𝑡𝑟 { 0; 𝑡0 + 𝑡𝑟 ≤ 𝑡 𝑡𝑟 = 𝑚𝑙 + 𝑏, (3.1d) In Eq. (3.1), 𝑇𝑚𝑎𝑥 is the isometric tension achieved at the longest sarcomere length, 𝐶𝑎0 denotes the peak intracellular calcium concentration, 𝐸𝐶𝑎50 is the length dependent calcium sensitivity. The parameter t0 is the prescribed time to maximum active tension, whereas 𝑡𝑟 denotes the duration of relaxation that varies linearly with the instantaneous sarcomere length governed by parameters 𝑚 and 𝑏. 3.2.4 Finite element formulation Finite element formulation of the LV mechanics problem was obtained by minimizing the following Lagrangian functional [128, 131] 38 1 ℒ(𝒖, 𝑃, 𝑐1 , 𝑐2 ) = ∫𝛺 𝛹𝑇 (𝒖)𝑑𝑉 – ∫𝛺 𝑝(𝐽 − 1)𝑑𝑉 – 𝑃(𝑉𝑐𝑎𝑣 (𝒖) − 𝑉𝑝 ) – ∫𝑑Ω 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 (𝒖. 𝒖)𝑑𝑆 – 0 0 2 0,𝑒𝑝𝑖 𝒄1 ∫Ω 𝒖𝑑𝑉 – 𝒄2 ∫Ω 𝑿 × 𝒖𝑑𝑉, (3.2) 0 0 In Eq. (3.2), 𝛹𝑇 is the total strain energy of the myocardium, 𝒖 ∈ 𝑯𝟏 (𝛺0 ) is the displacement field. On the other hand, 𝑃 ∈ 𝑅, 𝑝 ∈ 𝐿2 (𝛺0 ), 𝒄1 ∈ 𝑅 3 and 𝒄2 ∈ 𝑅 3 are the Lagrange multipliers for, respectively, constraining the cavity volume 𝑉𝑐𝑎𝑣 (𝒖) to the prescribed value 𝑉𝑝 , enforcing incompressibility in which the Jacobian of the deformation gradient tensor 𝐽 = 1, enforcing zero mean translation and enforcing zero mean rotation, respectively. A spring (robin-type) boundary condition with spring constant 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 was also imposed on the epicardial surface 𝑑Ω0,𝑒𝑝𝑖 to describe the loading by the pericardial fluid. The weak formulation was then obtained by taking the first variation of the Lagrangian functional as follows: 𝛿ℒ(𝒖, p, P, 𝑐1 , 𝑐2 ) = ∫𝛺 𝑭𝑺: 𝑔𝑟𝑎𝑑𝛿𝒖 𝑑𝑉 − ∫𝛺 (𝑝𝐽𝑭−𝑻 : 𝑔𝑟𝑎𝑑𝛿𝒖 + 𝛿𝑝(𝐽 − 1))𝑑𝑉 − 0 0 𝛿𝑃(𝑉𝑐𝑎𝑣 (𝒖) − 𝑉𝑝 ) − ∫𝛺 𝑃𝐽𝑭−𝑻 : 𝑔𝑟𝑎𝑑𝛿𝒖 𝑑𝑉 − ∫𝑑𝛺 𝑘𝑠𝑝𝑟𝑖𝑛𝑔 (𝒖. 𝛿𝒖)𝑑𝑆 − 𝑐𝑎𝑣 0,𝑒𝑝𝑖 ∫𝛺0(𝑐1 𝛿𝒖 + 𝒖𝛿𝑐1 )𝑑𝑉 − ∫𝛺0{𝛿𝑐2 (𝑿 × 𝒖) + 𝑐𝟐 (𝑿 × 𝛿𝒖}𝑑𝑉 = 0, (3.3) In the above equation, 𝛿𝒖 ∈ 𝑯1 (Ω0 ), 𝛿𝑝 ∈ 𝐿2 (Ω0 ), 𝛿𝑃 ∈ 𝑅, 𝛿𝒄𝟏 ∈ 𝑅 3 , 𝛿𝒄𝟐 ∈ 𝑅 3 are the test functions corresponding to 𝒖, 𝑝, 𝑃, 𝒄𝟏 and 𝒄𝟐 , respectively. Displacement at the LV base was constrained from moving out of plane i.e., 𝐮. 𝐧𝐛𝐚𝐬𝐞 = 0, (3.4) The displacement field 𝒖(𝑿) and Lagrange multiplier 𝑝 were interpolated using quadratic and linear tetrahedral elements, respectively. An implicit backward Euler scheme was used for 39 numerical time-integration with a fixed time step. The modeling framework was implemented using the open-source FE library FEniCS [140]. 3.2.5 Simulation cases We considered two simulation cases for each of the 4 swine models: • Baseline: Before aortic banding was performed on the animals • Acute overload: Acute effects of aortic banding To simulate aortic constriction in the acute (immediate) pressure overload cases, only parameters of the Windkessel model were adjusted to match the elevated peak systolic pressure measured at 2nd week after aortic banding in the animals. As the pressure was not measured immediately after banding, we assumed that the elevated pressure associated with aortic banding was sustained for the 2 weeks. Also, we assumed that end-diastolic volume was not changed immediately after aortic banding based on a previous canine study by Crozatier et al. [141], which found no increase in acute end-diastolic diameter after aortic stenosis in most of the animals. Other model parameters as well as myofiber orientation distribution in the acute overload case for each swine were prescribed to be the same as those in the corresponding baseline case. 3.3 Results & Discussion Experimental measurements of four swine models before and after 2 weeks of aortic banding are tabulated in Table 3.1. A significant increase in the mean peak systolic pressure (~ 43%) was found in the post aortic banding animals. Mean EDV was increased (~ 10%) while mean EF was decreased (~5% absolute) after 2 weeks of banding. Mean aortic pressure was also increased (~53%) whereas average maximum and minimum thickness remained relatively unchanged. The mean septum thickness was decreased (~10%), however, while the mean free-wall thickness was increased (~8%). The experimental results are consistent with previous studies of 40 pressure overload using aortic constriction large animal (porcine and sheep) model, which reported an increase in systolic pressure (~27% vs. ~42% here), elevated aortic pressure gradient (~40 mmHg vs. ~31 mmHg here), increase in LV diameter and EDV [142–144]. The increase in LVFW thickness found here is also consistent with a study, which reported an increase in posterior wall thickness (~31% vs. ~8% here) and a reduction in EF (~12% vs. ~10% here) after 2 weeks of aortic constriction in a mouse model [145]. We note, however, that some experimental studies have reported a decrease in EDV (~31%) (over a longer time period of 4 weeks) in a swine model of severe aortic stenosis [146] and preserved ejection fraction [147]. Table 3.1: Experimental measurements Parameters 0th Week 2nd Week End Diastolic Volume, EDV (ml) 72 ± 14.38 79 ± 19.18 End Diastolic Pressure, EDP (mm Hg) 14.4 ± 5.23 28.48 ± 16.3 End Systolic Volume, ESV (ml) 39 ± 8.5 46.27 ± 10.04 Stroke volume, SV (ml) 33 ± 7 32.35 ± 10 Ejection Fraction, EF (%) 46 ± 3 41 ± 3 Peak Systolic pressure (mm Hg) 79 ± 8.41* 112.36 ± 16* Aortic Pressure (mm Hg) 58 ± 3 89 ± 19 Maximum Thickness (mm) 13 ± 4 13 ± 2.06 Minimum Thickness (mm) 5±1 5±1 Septum Thickness (mm) 11 ± 3 9.49 ± 2.02 LV Free wall Thickness (mm) 9±2 9.4 ± 1.22 Wall volume (ml) 63 ± 23 66 ± 13 Data are expressed as mean ± SD. *𝛼 < 0.05 Regional wall thickness and growth as indexed by the change in the LV wall thickness before and after banding are shown in Figure 3.2. In 3 swine models (1–3), the septum became thinner, and the LV free wall became thicker after banding. In swine model 4, both septum and free wall are thickened. 41 Figure 3.2: Regional measured wall thickness and growth of 4 swine models based on 17 AHA segmentation [148]. Unit in mm. 42 Figure 3.3: Comparison between model prediction and experimental measurements. (a) RMSE (expressed as mean ± SD) for pressure, volume waveforms and end systolic pressure over a cardiac cycle. (b) Pressure waveforms, (c) PV loop and (d) volume waveforms from one representative animal (Swine 2). Baseline simulation case (black line); experimental measurement (black dots); acute overload simulation case (red line). Note that there are no corresponding measurements for the acute overload simulation case. Model predictions of the baseline cases are in good agreement with the corresponding pressure and volume measurements. Specifically, the normalized root mean square error (RMSE) between model prediction and experimental measurement is 11.47 ± 5% for the pressure waveform and 29.6 ± 15.4% for the volume waveform (Figure 3.3a). Differences in stroke volume and EF 43 between the baseline model prediction and experimental measurements are 0.5% and 0.85%, respectively. Representative pressure waveform (Figure 3.3b), volume waveform (Figure 3.3c) as well as PV loop (Figure 3.3d) are also presented for one swine. For the acute overload LV FE models, which were calibrated to match peak systolic pressure measured at 2 weeks after aortic constriction, the RMSE between model prediction and measurements of the pressure is 6 ± 6.1%. Figure 3.4: Comparison of normal stress and stretch in the myofiber, sheet and sheet-normal directions between the baseline and acute overload cases. Spatially averaged waveforms of the baseline (blue) and acute overload (red) for: (a) myofiber stretch (b) sheet stretch (c) sheet- normal stretch (d) myofiber stress (in Pa) (e) sheet stress (in Pa) and (g) sheet-normal stress (in Pa). Stretch was computed with end-diastolic configuration as reference. Average maximum absolute deviation of (g) stretch and (h) stress (in kPa) in four swines. 44 A comparison of the spatially averaged normal stress and stretch in the myofiber, sheet and sheet-normal directions between the baseline and acute overload cases reveals that the amount of normal stretch was reduced in all directions in the latter (Figure 3.4a-c). This corresponds to a reduction in ESV in the acute overload cases (Figure 3.4c). On the other hand, spatially averaged normal stress in all directions was increased in the acute overload cases (Figure 3.4d-f). Among the 3 stretch components, spatially averaged normal stretch in the sheet direction has the largest change (0.47 ± 0.194) followed by normal stretch in the myofiber (0.1 ± 0.041) and sheet-normal (0.086 ± 0.04) directions (Figure 3.4g) associated with acute overload. Conversely, the spatially averaged myofiber stress has the largest change (10.66 ± 4.68 kPa) associated with acute overload followed by the sheet-normal (1.29 ± 0.82 kPa) and sheet stresses (0.48 ± 0.2 kPa) (Figure 3.4h). Figure 3.5: (a) Pearson correlation coefficients of growth with changes in maximum, minimum and mean stress and stretch over a cardiac cycle. (b) Regional growth measured experimentally. (c) – (e): Regional changes in maximum, minimum and mean myofiber stress, myofiber stretch and sheet-normal stress, respectively. Quantities are averaged over 4 swines. 45 By performing a correlation analysis of regional growth with the regional changes in mechanics of 4 aortic banding swine models, we found that the changes in maximum and mean myofiber stress exhibits the strongest (positive) correlation with growth (Figure 3.5a), where regions that has the largest (smallest) changes in maximum and mean myofiber stress correspond to regions that has the largest (smallest) increase in wall thickness (Figure 3.5b). Pearson and Spearman rank correlation coefficients were computed to quantify the degree of correlation of growth with the change in maximum, mean and minimum in the 6 mechanical quantities over a cardiac cycle (18 coefficients in total) for each swine. Averaging the coefficients over the 4 swine models reveals that the changes in maximum myofiber stress (0.5471) has the strongest correlation with growth, followed by the changes in the mean sheet-normal stress (0.5266) based on the Pearson correlation coefficient (Figure 3.5a). Based on the Spearman rank correlation coefficient, the changes in mean sheet normal stress (0.5256) and mean myofiber stress (0.5204) show the strongest correlation with growth, followed by changes in the maximum myofiber stress (0.5111) (Figure 3.5b). On the other hand, none of the stretch components has a good correlation with growth, with changes in the mean sheet-normal stretch showing the worst correlation with growth (Pearson = 0.02066, Spearman rank = 0.04267). Averaging the change in maximum myofiber stress over the 4 swine models reveals that the largest increase occurs in the LV free wall, which also shows the greatest increase in wall thickness (Figure 3.5c). These results support the “systolic stress-correction hypothesis” that had been applied in some growth constitutive model. Scatter plots of the local changes in maximum myofiber, mean sheet-normal and mean myofiber stresses (that have the best correlation) with local growth in the LV are shown in Figure 3.6 for a representative case. In this case, the changes in the maximum myofiber stress (Pearson: 0.5471, Spearman: 0.5111) showed the strongest correlation with growth whereas changes in the 46 minimum sheet-normal stretch (Pearson: 0.02066, Spearman: 0.04267) showed the worst correlation. Figure 3.6: Scatter plots of the local growth with changes in stimuli in the LV of a representative case (swine 1). These three stimuli show best correlation with growth (see Figure 3.5a). Interestingly, our result also shows that regional (acute) changes in myocardial stretches are not correlated with the regional changes in LV wall thickness. This is despite our findings showing that the myocardial stretches change globally in response to pressure overload in a manner that is consistent with clinical studies of aortic stenosis patients [149–151] and acute experimental studies of pressure overload in dogs [141] (where the amount of shortening in both the major and minor axes and the amount of thickening are reduced). Correspondingly, these results suggest that while using the changes in myocardial stretches as growth stimuli may be sufficient to describe changes in global features of remodeling, it may not be sufficient to reproduce regional changes in LV wall thickness associated with pressure overload. 3.4 Conclusion Based on our study, local myofiber stress is strongly correlated local growth compared to local myofiber stretch. This result suggests that prescribing local myofiber stresses as the local stimuli in the growth constitutive law will better capture regional geometrical changes in the LV thickness 47 associated with pressure overload than prescribing local myofiber stretch as the local growth stimuli. 48 CHAPTER 4 IDEALIZED LEFT VENTRICULAR MODEL ON HYPERTROPHIC CARDIOMYOPATHY 49 4.1 Introduction HCM (as discussed in Chapter 1) is characterized by myofiber disarray, fibrosis, asymmetrical hypertrophy and a reduction in both longitudinal and circumferential strains. Mathematical modeling incorporating these features can be helpful in investigating the adaptive or maladaptive changes in LV mechanics associated with this disease. Computational models based on idealized ellipsoidal geometry have been developed to investigate the effects of remodeling features in both normal and diseased LV. Specifically, Usyk et al [152] developed a mathematical model based on an idealized LV geometry to investigate the mechanism of regional dysfunction caused by myofiber dispersion in mice heart. By increasing the myofiber angular dispersion and reducing sarcomere length in their model, they showed that focal changes in the microstructural properties of the disarrayed myocardium are directly responsible for the patterns of regional dysfunction in the LV. In another study, Deng et al [153] developed an idealized LV model for healthy, subaortic obstructive and midventricular obstructive phenotypes of HCM to investigate the genesis of apical aneurysm and reported that higher myofiber stress at the apex might initiate the formation process of aneurysm. Recently, a few directions and limitations of developing multiscale models in HCM have also been reviewed by Campbell et al. [154]. None of these studies, however, have investigated the change in LV mechanics due to myofiber disarray in the HCM heart. To address these limitations, we developed computational frameworks based on an idealized LV model to investigate the changes in ventricular mechanics associated with myofiber disarray in HCM heart. The finite element framework coupled with closed loop circulatory model was applied on two different geometry, namely normal LV and HCM LV. The strain along longitudinal and circumferential directions for normal LV without disarray were validated using 50 published data. To investigate the effects of myofiber disarray on LV mechanics, different degrees of myofiber disarray were applied globally (in both models) and regionally (in normal LV model). 4.2 Methods 4.2.1 Geometry reconstruction Figure 4.1: Construction of LV FE model. (a) Variation in septum (𝑡𝑠 ) and LVFW (𝑡𝑓𝑤 ) thickness on Baseline and HCM model. Here 𝑅 and 𝐿 are the inner radius along short axis and outer radius along long axis, respectively, of the ellipsoid. (b) Schematic representation of LV mesh coupled with closed loop Windkessel model. An idealized half prolate geometry was used to represent a normal and HCM LV. The normal LV has a wall thickness that is axisymmetric whereas the HCM LV has a wall thickness that is asymmetrical about the long axis (Figure 4.1a). Geometrical parameters were prescribed based on clinical measurements. The ratio of septum vs. LVFW thickness for HCM model was prescribed to be 1.63, which is within the range found in Tanaka et al [155]. The geometries were discretized with 4353 quadratic tetrahedral elements. Mean myofiber direction was prescribed based on a linear transmural variation of the helix angle from +60° at the endocardium to −60° at 51 the epicardium across the LV wall using a Laplace-Dirichlet rule-based algorithm. 4.2.2 Incorporation of myofiber disarray Figure 4.2: (a) Myofiber dispersion following a probability distribution density function. Solid red arrow represents the initial myofiber direction, where dashed line represents one of many possible orientations of respective myofiber. (b) The two dimensional representation of distribution density function for varying angle (𝜃) and disarray (𝜅). Figure (b) is adapted from Gasser et al [156]. Figure 4.3: Three-dimensional graphical representation of the orientation for the fibers based on transversely isotropic density function. The figure is adapted from Gasser et al [156]. Based on the assumption of axisymmetric fiber distribution, myofiber disarray was incorporated through a structure tensor 𝑯 [156] describing a conical dispersion of myofibers about 52 a mean myofiber direction, 𝒆𝒇𝟎 . The structure tensor is given by 𝑯 = 𝜅𝑰 + (1 − 3𝜅)𝒆𝒇𝟎 ⊗ 𝒆𝒇𝟎 , (4.1) where 𝑰 is the identity tensor and κ represents the fiber distribution in an integral sense that is defined as, 1 𝜋 𝜅= ∫ 𝜌(𝜃)𝑠𝑖𝑛3 𝜃𝑑𝜃. 4 0 (4.2) Here, 𝜌(𝜃) is the probability density function representing the fiber dispersion. At the lower limit of the disarray parameter (κ = 0), myofibers are perfectly aligned along the 𝒆𝒇𝟎 direction (i.e., the structure tensor reduces to 𝒆𝒇𝟎 ⊗ 𝒆𝒇𝟎 ). At the upper limit of the disarray parameter (κ = 1/3), the structure tensor reduces to 𝑰, representing a distribution of myofibers that produces an isotropic material response (i.e., a complete myofiber disarray). Hence, the structure tensor, 𝐇, depends on a single dispersion parameter, κ, which represents the fiber distribution in an integral sense and describes its “degree of anisotropy”. The von-mises distribution is depicted in Figure 4.2 for different degree of κ varying between 0 to 1/3. The distribution changes from a bone like structure when κ = 0 to a sphere in three dimension when κ = 1/3 is graphically (Figure 4.3). 4.2.3 Constitutive law for LV model An active stress formulation was used to describe the mechanical behavior of the ventricular geometry in the cardiac cycle. In this formulation, the stress tensor 𝑆 can be decomposed additively into a passive component 𝑺 p and an active component 𝑺 a (i.e., 𝑺 = 𝑺a + 𝑺p ). The passive stress tensor was defined based on the strain energy function of a Fung-type transversely-isotropic hyperelastic material [157], 1 (4.3a) 𝑊= 𝐶(𝑒 𝑄 − 1), 2 where 53 2 2 2 2 2 ) 2 2 2 2 𝑄 = 𝑏𝑓𝑓 𝐸𝑓𝑓 + 𝑏𝑥𝑥 (𝐸𝑠𝑠 + 𝐸𝑛𝑛 + 𝐸𝑠𝑛 + 𝐸𝑛𝑠 + 𝑏𝑓𝑥 (𝐸𝑓𝑛 + 𝐸𝑛𝑓 + 𝐸𝑓𝑠 + 𝐸𝑠𝑓 ). (4.3b) In Eq. (4.3b), Eij with (i, j) ∈ (f, s, n) are components of the Green-Lagrange strain tensor with f, s, n denoting the myocardial fiber, sheet and sheet normal directions, respectively. Material parameters of the passive constitutive model are denoted by C, bff, bxx and bfx. Based on a previously developed active contraction model [139, 158, 159], the active stress (𝑺a ) directed in the local myofiber direction was calculated as 𝐶𝑎02 (4.4a) 𝑺𝒂 = 𝑇𝑚𝑎𝑥 ( 2 ) 𝐶𝑡 𝑯 , 𝐶𝑎02 + 𝐸𝐶𝑎50 In the above equation, 𝑇𝑚𝑎𝑥 is the isometric tension achieved at the longest sarcomere length and Ca0 denotes the peak intracellular calcium concentration. The length dependent calcium sensitivity 𝐸𝐶𝑎50 and the variable Ct are given by, (𝐶𝑎0 )𝑚𝑎𝑥 (4.4b) 𝐸𝐶𝑎50 = , √exp(𝐵(𝑙 − 𝑙0 )) − 1 1 (1 − cos (𝜋𝑡⁄𝑡 )) ; 0 ≤ 𝑡 < 𝑡𝑟 2 0 𝐶𝑡 = , (4.4c) 1 𝜋𝑡𝑟 − (𝑡−𝑡𝑟 )⁄ {2 (1 − 𝑐𝑜𝑠 ( ⁄ 𝑡0 )) 𝑒 𝜏; 𝑡 ≥ 𝑡 𝑟 In Eq. (4.4b), B is a constant, (𝐶𝑎0 )𝑚𝑎𝑥 is the maximum peak intracellular calcium concentration and 𝑙0 is the sarcomere length at which no active tension develops. In Eq. (4.4c), 𝑡0 , 𝑡𝑟 and 𝜏 are the time taken to reach peak tension, the duration of relaxation and the relaxation time constant, respectively. The sarcomere length 𝑙 is calculated from the myofiber stretch 𝜆LV by 𝜆𝐿𝑉 = √𝑡𝑟(𝑯𝑪) , (4.5a) 𝑙 = 𝜆𝐿𝑉 𝑙𝑟 , (4.5b) 54 In Eq. (4.5), 𝑪 = 𝑭𝑻 𝑭 is the right Cauchy-Green deformation tensor and 𝑙𝑟 is the relaxed sarcomere length. 4.2.4 Closed-loop circulatory model The LV FE model was coupled to a closed-loop lumped parameter modeling framework that describes the circulatory system (Figure 4.1b). The ventricular model consists of five compartments (namely LA, LV, venous, peripheral, and distal artery) yielding five volume states (𝑉𝐿𝐴 , 𝑉𝐿𝑉 , 𝑉𝑣𝑒𝑛 , 𝑉𝑎,𝑝 , 𝑉𝑎,𝑑 ). Based on mass conservation, the rate of volume change in each storage compartment of the circulatory system depends on the variation in flow rates, both in and out, (𝑞𝑚𝑣 , 𝑞𝑎𝑜 , 𝑞𝑎,𝑝 , 𝑞𝑎,𝑑 , 𝑞𝑣𝑒𝑛 ) at different segments, 𝑑𝑉𝐿𝐴 (𝑡) (4.6a) = 𝑞𝑣𝑒𝑛 (𝑡) − 𝑞𝑚𝑣 (𝑡), 𝑑𝑡 𝑑𝑉𝐿𝑉 (𝑡) (4.6b) = 𝑞𝑚𝑣 (𝑡) − 𝑞𝑎𝑜 (𝑡), 𝑑𝑡 𝑑𝑉𝑎,𝑝 (𝑡) (4.6c) = 𝑞𝑎𝑜 (𝑡) − 𝑞𝑎,𝑝 (𝑡), 𝑑𝑡 𝑑𝑉𝑎,𝑑 (𝑡) (4.6d) = 𝑞𝑎,𝑝 (𝑡) − 𝑞𝑎,𝑑 (𝑡), 𝑑𝑡 𝑑𝑉𝑣𝑒𝑛 (𝑡) (4.6e) = 𝑞𝑎,𝑑 (𝑡) − 𝑞𝑣𝑒𝑛 (𝑡), 𝑑𝑡 Flowrate at different segments of the circulatory model depends on their resistance to flow (𝑅𝑚𝑣 , 𝑅𝑎𝑜 , 𝑅𝑎,𝑝 , 𝑅𝑎,𝑑 ,𝑅𝑣𝑒𝑛 ) and the pressure difference between the connecting storage compartments (i.e., pressure gradient). The flow rates are given by, 𝑃𝐿𝑉 (𝑡) − 𝑃𝑎𝑜 (𝑡) (4.7a) ; 𝑖𝑓𝑃𝐿𝑉 > 𝑃𝑎𝑜 𝑞𝑎𝑜 (𝑡) = { 𝑅𝑎𝑜 , 0; 𝑖𝑓𝑃𝐿𝑉 < 𝑃𝑎𝑜 𝑃𝑎𝑜 (𝑡)−𝑃𝑎,𝑑 (𝑡) (4.7b) 𝑞𝑎,𝑝 (𝑡) = , 𝑅𝑎,𝑝 55 𝑃𝑎,𝑑 (𝑡)−𝑃𝑣𝑒𝑛 (𝑡) (4.7c) 𝑞𝑎,𝑑 (𝑡) = , 𝑅𝑎,𝑑 𝑃𝑣𝑒𝑛 (𝑡)−𝑃𝐿𝐴 (𝑡) (4.7d) 𝑞𝑣𝑒𝑛 (𝑡) = , 𝑅𝑣𝑒𝑛 𝑃𝐿𝐴 (𝑡)−𝑃𝐿𝑉 (𝑡) (4.7e) ; 𝑖𝑓𝑃𝐿𝐴 > 𝑃𝐿𝑉 𝑞𝑚𝑣 (𝑡) = { 𝑅𝑚𝑣 , 0; 𝑖𝑓𝑃𝐿𝐴 < 𝑃𝐿𝑉 A time varying elastance function was used to describe the contraction of LA [131]. Specifically, pressure in the LA 𝑃𝐿𝐴 (𝑡) was prescribed to be a function of its volume 𝑉𝐿𝐴 (𝑡) by the following equations that describe its contraction using a time-varying elastance function 𝑒(𝑡): 𝑃𝐿𝐴 (𝑡) = 𝑒(𝑡)𝑃𝑒𝑠,𝐿𝐴 (𝑉𝐿𝐴 (𝑡)) + (1 − 𝑒(𝑡))𝑃𝑒𝑑,𝐿𝐴 (𝑉𝐿𝐴 (𝑡)) , (4.8a) 𝑃𝑒𝑠,𝐿𝐴 (𝑉𝐿𝐴 (𝑡)) = 𝐸𝑒𝑠,𝐿𝐴 (𝑉𝐿𝐴 (𝑡) − 𝑉0,𝐿𝐴 ), (4.8b) 𝑃𝑒𝑑,𝐿𝐴 (𝑉𝐿𝐴 (𝑡)) = 𝐴𝐿𝐴 (𝑒 𝐵𝐿𝐴(𝑉𝐿𝐴 (𝑡)−𝑉0,𝐿𝐴 ) − 1), (4.8c) 1 𝜋 𝜋 3 (4.8d) (𝑠𝑖𝑛 [( ) 𝑡 − ] + 1) ; 0 < 𝑡 ≤ 𝑡𝑚𝑎𝑥 2 𝑡𝑚𝑎𝑥 2 2 𝑒(𝑡) = , 1 −(𝑡−3 𝑡𝑚𝑎𝑥 )⁄𝜏 3 { 𝑒 2 ; 𝑡 > 𝑡𝑚𝑎𝑥 2 2 In Eq. (4.8a-d), 𝐸𝑒𝑠,𝐿𝐴 is the end-systolic elastance of the LA, 𝑉0,𝐿𝐴 is the volume-intercept of the end-systolic pressure volume relationship (ESPVR), and both 𝐴𝐿𝐴 and 𝐵𝐿𝐴 are parameters of the end-diastolic pressure volume relationship (EDPVR) of the LA. The driving function 𝑒(𝑡) is given in Eq. (4.8d) in which 𝑡𝑚𝑎𝑥 is the point of maximal chamber elastance and 𝜏 is the time constant of relaxation. Pressure in each vessel (arteries and veins) in both systemic and pulmonary circulation was calculated by a simplified pressure volume relationship 𝑉𝑣𝑒𝑛 (𝑡) − 𝑉𝑣𝑒𝑛,0 (4.9a) 𝑃𝑣𝑒𝑛 (𝑡) = , 𝐶𝑣𝑒𝑛 𝑉𝑎,𝑝 (𝑡)−𝑉𝑎𝑝,0 (4.9b) 𝑃𝑎,𝑝 (𝑡) = , 𝐶𝑎,𝑝 56 𝑉𝑎,𝑑 (𝑡)−𝑉𝑎𝑑,0 (4.9c) 𝑃𝑎,𝑑 (𝑡) = , 𝐶𝑎,𝑑 where, 𝑉𝑣𝑒𝑛,0 , 𝑉𝑎𝑝,0 , 𝑉𝑎𝑑,0 are constants representing the resting volumes and 𝐶𝑣𝑒𝑛 , 𝐶𝑎,𝑝 , 𝐶𝑎,𝑑 are the total compliance of the veins, proximal and distal arteries, respectively. Finally, pressure in the LV depends on their corresponding volume through a non-closed form function, 𝑃𝐿𝑉,𝑐𝑎𝑣 (𝑡) = 𝑓 𝐹𝐸 (𝑉𝐿𝑉 (𝑡)). (4.10) The functional relationship between pressure and volume in the LV was obtained using the FE method as described in the next section. 4.2.5 Ventricular FE model The Lagrangian functional for the left ventricular FE formulation is given by, ℒ(𝒖, 𝑝, 𝑃𝐿𝑉 , 𝒄1 , 𝒄2 ) = ∫ 𝑊(𝒖)𝑑𝑉 − ∫ 𝑝(𝐽 − 1)𝑑𝑉 𝛺0 𝛺0 (4.11) − 𝑃𝐿𝑉 (𝑉𝐿𝑉,𝑐𝑎𝑣 (𝒖) − 𝑉𝐿𝑉 ) − 𝒄1 ∙ ∫ 𝒖 𝑑𝑉 𝛺0 − 𝒄2 ∙ ∫ 𝒙 × 𝒖 𝑑𝑉 , 𝛺0 In the above equation, 𝒖 is the displacement field, 𝑝 is a Lagrange multiplier to enforce incompressibility of the tissue (i.e., Jacobian of the deformation gradient tensor, 𝐽 = 1), 𝑃𝐿𝑉 is the Lagrange multiplier to constrain the LV cavity volume 𝑉𝐿𝑉,cav (𝒖) to a prescribed value 𝑉𝐿𝑉 [160], and both 𝒄1 and 𝒄2 are Lagrange multipliers to constrain rigid body translation (i.e., zero mean translation) and rotation (i.e., zero mean rotation) [161]. The functional relationship between the cavity volumes of the LV and RV to the displacement field is given by, 1 (4.12) 𝑉𝐿𝑉,cav (𝒖) = ∫ 𝑑𝑣 = − ∫ 𝒙. 𝒏 𝑑𝑎 , 3 Ω𝑘,𝑖𝑛𝑛𝑒𝑟 Γ𝑘,𝑖𝑛𝑛𝑒𝑟 57 where Ω𝑘,𝑖𝑛𝑛𝑒𝑟 is the volume enclosed by the inner surface Γ𝑘,𝑖𝑛𝑛𝑒𝑟 and the basal surface at z = 0, and 𝒏 is the outward unit normal vector. The first variation of the Lagrangian functional in Eq. (4.11) leads to the following expression: 𝛿ℒ(𝒖, 𝑝, 𝑃𝐿𝑉 , 𝑃𝑅𝑉 , 𝒄1 , 𝒄2 ) = ∫Ω (𝑷 − 𝑝𝑭−𝑇 ): 𝛻𝛿𝒖 𝑑𝑉 − (4.13) 0 ∫Ω0 𝛿𝑝(𝐽 − 1)𝑑𝑉 − 𝑃𝐿𝑉,cav ∫Ω0 𝑐𝑜𝑓(𝑭): 𝛻𝛿𝒖 𝑑𝑉 − 𝛿𝑃𝐿𝑉,cav (𝑉𝐿𝑉,cav (𝒖) − 𝑉𝐿𝑉 ) − 𝛿𝒄1 ∙ ∫Ω 𝒖 𝑑𝑉 − 0 𝛿𝒄2 ∙ ∫Ω 𝑿 × 𝒖 𝑑𝑉 − 𝒄1 ∙ ∫Ω 𝛿𝒖 𝑑𝑉 − 0 0 𝒄2 ∙ ∫Ω 𝑿 × 𝛿𝒖 𝑑𝑉 , 0 In Eq. (4.13), 𝑷 is the first Piola Kirchhoff stress tensor, 𝑭 is the deformation gradient tensor, 𝛿𝒖, 𝛿𝑝, 𝛿𝑃𝐿𝑉,cav, 𝛿𝒄1 , 𝛿𝒄2 are the variation of the displacement field, Lagrange multipliers for enforcing incompressibility and volume constraint, zero mean translation and rotation, respectively. The Euler-Lagrange problem then becomes finding 𝒖 ∈ 𝐻1 (Ω0 ), 𝑝 ∈ 𝐿2 (Ω0 ), 𝑃𝐿𝑉,cav ∈ ℝ, , 𝒄1 ∈ ℝ3 , 𝒄2 ∈ ℝ3 that satisfies, 𝛿ℒ(𝒖, 𝑝, 𝑃𝐿𝑉,cav , 𝒄1, , 𝒄2 ) = 0 , (4.14) and 𝒖. 𝒏|𝑏𝑎𝑠𝑒 = 0 (for constraining the basal deformation to be in-plane) ∀ 𝛿𝒖 ∈ 𝐻1 (Ω0 ), 𝛿𝑝 ∈ 𝐿2 (Ω0 ), 𝛿𝑃𝐿𝑉,cav ∈ ℝ, 𝛿𝒄1 ∈ ℝ3 , 𝛿𝒄2 ∈ ℝ3 . An explicit time integration scheme was used to solve the five ODEs in Eq. (4.6). The compartment volumes ( 𝑉𝐿𝐴 , 𝑉𝐿𝑉 , 𝑉𝑎,𝑝 , 𝑉𝑎,𝑑 , 𝑉𝑣𝑒𝑛 ) at each timestep 𝑡𝑖 were determined from their respective values and the segmental flow rates (𝑞𝑚𝑣 , 𝑞𝑎𝑜 , 𝑞𝑎,𝑝 , 𝑞𝑎,𝑑 , 𝑞𝑣𝑒𝑛 ) were determined using Eq. (4.7) at previous timestep 𝑡𝑖−1 . The computed compartment volumes at 𝑡𝑖 were used to update the corresponding pressures (𝑃𝐿𝐴 , 𝑃𝐿𝑉 , 𝑃𝑎,𝑝 , 𝑃𝑎,𝑑 , 𝑃𝑣𝑒𝑛 ). Pressures in LA (𝑃𝐿𝐴 ) and vessels (𝑃𝑎,𝑝 , 𝑃𝑎,𝑑 , 𝑃𝑣𝑒𝑛 ) were computed from Eq. (4.8a) and (4.9a-c), respectively. On the other hand, 58 pressures in the LV (𝑃𝐿𝑉,𝑐𝑎𝑣 ) was computed from the FE solutions of Eq. (4.10) with the volumes (𝑉𝐿𝑉 ) at timestep 𝑡𝑖 as input. We note here that (𝑃𝐿𝑉,𝑐𝑎𝑣 ) are scalar Lagrange multipliers in the FE formulation for constraining the cavity volumes to the prescribed values (𝑉𝐿𝑉 , 𝑉𝑎𝑟𝑡 ). The computed pressures at timestep 𝑡𝑖 were then used to update the segmental flow rates in Eq. (4.7) that will be used to compute the compartment volumes at timestep 𝑡𝑖+1 in the next iteration. 4.2.6 Model parameterization Parameters of LV FE model associated with the normal geometry without disarray was manually adjusted so that its predictions agree well with the previously reported clinical studies (Figure 4.4). The predicted longitudinal strain showed a good match with the clinical data (Root mean squared error (RMSE) ~25% compared with Gorcsan et al [162], ~21% compared with Smiseth et al. [103]). The circumferential strain also shows a good match with the clinical data [162, 163]. To calibrate the model, preload was adjusted by changing the venous return (𝑉𝑣𝑒𝑛,0 ) in the model, whereas the afterload was adjusted by changing the peripheral resistance (𝑅𝑎,𝑝 ). The same parameters were used in subsequent simulations using the normal and HCM LV ellipsoidal geometries, where the disarray index (𝜅) was varied from 0 to 1/3 in both cases. Later the LV model was segmented into 4 sections and the disarray parameter was varied regionally between the septum and LVFW. Specifically, the simulation cases are: • Normal LV: Uniform wall thickness of 1.13 cm with 𝜅 varying from 0 to 1/3 globally. • HCM LV: Septal thickness (1.63 cm) is higher than LVFW (1.13 cm) with 𝜅 varying from 0 to 1/3 globally. • Normal LV with regional myofiber disarray: Uniform wall thickness of 1.13 cm with 𝜅 varying from 0 to 1/3 regionally in the septum and 𝜅 = 0 in the LVFW. For each case, the simulation was performed over several cardiac cycles at a heart rate of 75bpm 59 until the pressure-volume loop reached a steady state. Figure 4.4: Strain comparison between experiments and simulated results for normal LV model without disarray. (a) Longitudinal strain, (b) Circumferential strain for 𝜅 = 0. 4.3 Results 4.3.1 Pressure-volume loop The steady state pressure-volume loops of the LV for normal LV and HCM LV cases were obtained from the FE model (Figure 4.5 c, f). Hemodynamics of the LV was greatly altered with increasing myofiber disarray with increasing 𝜅 (from 0 to 1/3). In the normal LV case, a significant reduction in peak systolic pressure (~55% decrease), stroke volume (~65% decrease) and EF (~ 68% decrease) were observed in the results for complete myofiber disarray compared to the case without myofiber disarray. Similar trend was also observed in the HCM LV case. 60 Figure 4.5: Volume-time plot (a,d), pressure-time plot(b,e), PV loop (c,f). The top and bottom row denoted normal LV geometry and HCM LV geometry, respectively. 61 4.3.2 Myocardial strains Figure 4.6: Myocardial strain-time profile for normal (a,b) and HCM model (c,d) from endocardium to epicardium. (e, f) Peak global strain comparison with varying disarray parameter. The global strain results are shown with solid lines. 62 Myocardial strains at the endocardium and epicardium over a cardiac cycle with varying myofiber disarray for both normal LV and HCM LV cases are shown in Figure 4.6 a-d. With an increase in myofiber disarray, the strain is reduced significantly in both cases. With the increase 1 in 𝜅 from 0 to 3, the reduction in global peak circumferential strain between endocardium and 1 epicardium was significant in both cases (~15.8% when 𝜅 = 0 vs. ~1.6% when 𝜅 = 3 in the normal 1 LV case; ~20.75% when 𝜅 = 0 vs. ~2.2% when 𝜅 = 3 in the HCM LV case). Also, the reduction in peak longitudinal strain between the endocardium and epicardium was similar (~2.8% when 1 1 𝜅 = 0 vs. ~0.5% when 𝜅 = 3 in the normal LV case; ~3.5% when 𝜅 = 0 vs. ~1% when 𝜅 = 3 in the HCM LV case). The average circumferential and longitudinal strain in both the normal LV and HCM LV cases (shown by the solid line in the strain profile) were reduced with an increase in 𝜅 1 from 0 to 3. Although the peak longitudinal strain was higher in the normal LV case compared to the HCM LV case (~21% at normal vs ~18.8% at HCM) at 𝜅 = 0, the difference was mostly diminished with complete myofiber disarray. This lower value of longitudinal strain in the HCM LV case when 𝜅 = 0 could be due the increase in septum thickness of the HCM LV geometry. While peak circumferential strain is reduced with increasing myofiber disarray, it did not show any substantial difference between the normal LV and HCM LV cases. Regional strain variation over the cardiac cycle for both normal LV and HCM LV cases is shown in Figure 4.7. While there was no significant variation in strain distribution between the septum and LVFW with myofiber disarray in the normal LV case, a slight decrease in peak strain at the septum (compared to LVFW) was observed in the HCM LV case. This decrease could be due to the increase in thickness at the septum in the LV geometry of the HCM LV case. At 𝜅 = 0 and 1/9, a slight reduction in peak circumferential and longitudinal strains at the thicker septum 63 region in the HCM LV case was also found compared to the thinner LVFW region. This difference was diminished at higher value of 𝜅. Figure 4.7: Myocardial strain-time profile for normal (a,b) and HCM model (c,d) from septum to LVFW. (e, f) Regional peak strain comparison with varying disarray parameter. The global strain results are shown with solid lines. 64 4.3.3 Spatial variation of myofiber disarray Figure 4.8: Results of variation of myofiber disarray in septum. Segmentation of LV (a-left) and distribution of disarray parameter of 1/3 at septum and 0 at rest of LV (a-right). pressure- time plot(b), Volume-time plot (c), PV loop (d) of Normal LV model. The value of disarray parameter was increased from 0 to 1/3 at septum while kept 0 at other regions. The disarray parameter 𝜅 was distributed regionally and was prescribed different values at the septum and LVFW (Figure 4.8). The value of 𝜅 = 0 was kept constant at the rest of the LV while at septum region, 𝜅 was varied from 0 to 1/3. With the increase of 𝜅, the performance of LV was reduced, although not as significantly as seen previously in Figure 4.5, when 𝜅 was varied globally. With increasing 𝜅 from 0 to 1/3 at septum region, the peak pressure (~22% reduction), stroke volume (~26% reduction) and EF (~27% reduction) were all reduced with complete myofiber disarray at the septum compared to without disarray. 65 Figure 4.9: Strain profile for spatial variation of disarray parameter in normal LV model. Strain distribution from septum to LVFW: Circumferential strain (a), Longitudinal strain(b). Strain distribution from endocardium to epicardium: Circumferential strain (c), Longitudinal strain(d). The strain distribution over a cardiac cycle with varying degree of myofiber disarray is shown in Figure 4.9. With an increase in septal myofiber disarray, a reduction in peak global 1 circumferential (~26% when 𝜅 = 0 vs. ~17.7% when = 3 ) and longitudinal (~21% when 𝜅 = 0 1 vs. ~17% when 𝜅 = 3 ) strains was found. Myocardial strains in the septum region were also 1 reduced in the circumferential (~26% when 𝜅 = 0 vs. ~6.7% when = 3 ) and longitudinal (~21% 1 when 𝜅 = 0 vs. ~14.1% when 𝜅 = 3 ) directions with increasing septal myofiber disarray. In the LVFW region, however, longitudinal strain was only slightly reduced (~21% when 𝜅 = 0 vs. ~19% 66 1 when = ), whereas circumferential strain, was slightly increased (~25.4% when 𝜅 = 0 vs. 3 1 ~27.7% when 𝜅 = 3 ) with increasing septal myofiber disarray. A pre-systolic strain up to ~5% (i.e., positive strain values at the onset of contraction) was also observed at the septum with increasing septal myofiber disarray. Both circumferential and longitudinal strains were also reduced at the endocardium and epicardium with the increase in septal disarray. 4.4 Discussion We developed an LV model based on an idealized ellipsoidal geometry with different septum wall thickness based on those published in the literature to investigate the effects of global and regional myofiber disarray. The model showed that with increasing disarray, LV mechanics gets impaired. More specifically, our results showing reduced circumferential and longitudinal strains with increasing disarray both globally and regionally at septum are consistent with the clinical results (Section 1.4.3). The pump function also reduced as the ejection fraction and SV decreased with increasing disarray. The ratio of septum wall thickness to the LVFW wall thickness in our HCM model was 1.6, which is over the threshold (≥ 1.3) prescribed in other studies [109]. The LV function represented by stroke volume, ejection fraction and peak systolic pressure was impaired with increasing disarray in both models. Besides, the reduction in global longitudinal strain and circumferential strain are consistent with previous studies [90–92, 100, 107]. The increased wall thickness at the septum also led to a reduction in longitudinal and circumferential strains (without the presence of myofiber disarray) in the HCM LV case compared to normal LV case (Figure 4.7e, f) suggesting that the heterogeneity in LV wall thickness induced by hypertrophy (observed clinically) are affecting the reduction in wall strain. At higher degree of disarray, however, there is no difference in strain between septum and LVFW. One reason could be the consideration of the 67 1 extreme disarray (𝜅 = ), representing the fractional anisotropy value of 0 (Figure 5.2), might not 3 be clinically feasible. Even if it is feasible, the associated G&R induced by the disarray is not simulated here. Also, LV function and mechanics was impaired globally by regional variation of disarray in normal LV case. Specifically, increase of regional septum strain, the strain was reduced more along septum than LVFW, suggesting the effect of heterogeneous disarray will impact differently on both global and local mechanics. The limitations and future scope related to idealized HCM model will be described in Chapter 6. 4.5 Conclusion Based on published thickness data for HCM patients, we developed a FE modeling framework to investigate the changes in global and regional mechanics due to different degree of myofiber disarray based on an idealized LV geometry. The results showed that both LV function and mechanics are impaired with increasing disarray, which are consistent with the clinical results. Further development of a patient specific model will help further investigate the intricate mechanism associated with development and progression of HCM. 68 CHAPTER 5 PATIENT SPECIFIC LEFT VENTRICULAR MODEL ON HYPERTROPHIC CARDIOMYOPATHY 69 5.1 Introduction Myocardial fiber disarray is a histopathological hallmark in both obstructive and non- obstructive HCM [164]. The disarray of myofibers is either confined to some particular region in the LV or is distributed throughout the entire LV. In obstructive HCM patients, a pressure gradient > 50𝑚𝑚 𝐻𝑔 across the LV outflow tract, either at resting or provoked condition, is also present [165, 166]. Besides myofiber disarray, HCM is also associated with other key histopathological features such as asymmetrical septal hypertrophy in the LV, changes in the myocardial contractility, and cardiac fibrosis [81–89]. These features have been associated with changes in the LV function seen in HCM patients, such as a reduction in (global and segmental) longitudinal and circumferential strains [63, 90–92], active tension [167], an increase in relative ATP consumption during tension generation [93], and a reduction in myocardial work (pressure-strain loop area) [94]. Given the multiple histopathological features present in HCM patients, how each of these features contributes to the changes in the LV function is not clear. Although clinical studies can help reveal abnormalities of myocardial structure (e.g., myofiber disarray) associated with HCM [99], the causal link of these features to LV function is difficult to ascertain from these studies . As such, the relative contribution of these remodeling features (i.e., asymmetrical hypertrophy, myofiber disarray) to the impairment of LV function in HCM patients remains unclear. Mathematical modeling can help resolve this issue by quantifying the causal effects of the remodeling features to changes in the LV function in HCM patients. In relation to HCM, a few computational models have been developed to investigate the effects of remodeling features on LV function [154, 168]. Specifically, mathematical models based on an idealized ellipsoidal LV geometry has been developed to investigate how regional strain is affected by myofiber disarray 70 [152] and sarcomeric mutation [154]. A study was also conducted on the effects of remodeling features associated with HCM by perturbing the heart geometry of a healthy volunteer [168]. Besides LV wall mechanics, other studies have investigated the contribution of diffuse fibrosis distribution in promoting arrhythmogenesis and ventricular arrhythmia in HCM patients [169], as well as the effect of abnormal morphological and functional aspect of the LV on the behavior of intraventricular blood flow dynamics [170]. In the latter study, they found a correlation between higher pressure gradient across the LV outflow tract due to obstruction and the HCM-induced thickening at basal portion of the septum, which further led to clinical indications useful for designing possible surgical treatment by septal myectomy. All these studies, however, do not consider the difference in LV wall mechanics between obstructive and non-obstructive HCM and patient-specific LV geometries that encapsulate the heterogeneous distribution of wall thickness associated with this disease. Other computational studies are focused only on obstructive HCM [153][171], but they did not consider the effects of myofiber disarray. To address these limitations, we developed patient-specific FE LV models based on clinical measurements from patients with 2 different types of HCM (obstructive and non-obstructive) and a control subject here. These models were constructed based on patient-specific LV geometries that were segmented from cardiac magnetic resonance images of these subjects. The models were coupled to a closed loop circulatory model and calibrated using patient-specific clinical measurements of the LV volume waveform, blood pressures and peak global longitudinal strain (GLS). Contractile function of the cardiac muscle fibers in the 3 subjects were determined by the calibration. The calibrated models were then applied to investigate the effects of different degrees of myofiber disarray on LV function in both the obstructive and non-obstructive HCM subjects. 71 5.2 Methods 5.2.1 Clinical data Table 5.1: Clinical measurements of each subject Parameters Control Obstructive Non-obstructive Age (years) 69 57 61 Weight (kg) 58.1 97 75 Heart rate (bpm) 60 51 66 End diastolic volume (ml) 63 114 82 End systolic volume (ml) 18 38 12 Ejection fraction (%) 70 66.8 85.3 Global longitudinal strain (%) -20 -13 -19 Body surface area (m2) 1.56 2.04 1.72 Blood Pressure (mm Hg) 126/65 151/80 133/66 Clinical data of 2 female HCM patients (obstructive and non-obstructive) along with a control female subject were acquired from the University of California San Francisco Medical Center. Specifically, the data consists of CMR images, blood pressure measurements and peak GLS estimated from 3D echocardiographic images. Left ventricular cavity volume waveform of each subject was estimated by segmenting the endocardial wall from the CMR images (Figure 5.1a) over the cardiac cycle with MeVisLab (MeVis Medical Solutions AG). The clinical data are listed in Table 5.1. In addition to the patient-specific data, we also used published pressure waveforms from HCM patients and healthy human subjects to reconstruct the pressure-volume (PV) loop of each subject [172]. 5.2.2 Reconstruction of LV FE model Left ventricular endocardial and epicardial surfaces were segmented from the MR images associated with end-diastole (ED) (Figure 5.1b). Patient-specific 3D LV geometries were then reconstructed from these surfaces and a FE mesh was generated for each geometry. The meshes 72 consist of approximately 13000 tetrahedral elements (Figure 5.1c). Mean myofiber direction (Figure 5.1d) 𝒆𝒇0 was prescribed based on a linear transmural variation of the helix angle from +70° at the endocardium to −70° at the epicardium across the LV wall using a Laplace-Dirichlet rule-based algorithm [173]. Figure 5.1: Construction of the patient specific LV FE model. a: MR image segmentation; b: Segmented endocardium and epicardium of the LV; c: FE model overlaid on the MR image in a long axis view; d: Transmural variation of mean myofiber direction across the LV wall; e: Schematic representation of LV FE model coupled with a closed loop circulatory model. A sample representation is shown for non-obstructive HCM patient. 73 5.2.3 Circulatory model The computational framework consists of the LV FE model, LA, the proximal (a,p) and distal (a,d) arterial and venous (ven) compartments that are connected in a closed-loop circulatory system (Figure 5.1 e)[174–176]. This framework is previously explained in section 4.2.4. 5.2.4 FE model formulation Finite element formulation of the LV model has been described previously [128, 174, 177, 178]. Briefly, denoting z as the apex-to-base axis and x, y are axes orthogonal to z, the functional relationship between pressure and volume of the LV was obtained based on the Lagrangian functional given by, ℒ(𝒖, 𝑝, 𝑃𝐿𝑉 , 𝑐𝑥 , 𝑐𝑦 , 𝑐𝑧 ) = ∫𝛺 𝑊(𝒖)𝑑𝑉 − ∫𝛺 𝑝(𝐽 − 1)𝑑𝑉 − 0 0 (5.1) 𝑃𝐿𝑉 (𝑉𝐿𝑉 (𝒖) − 𝑉𝐿𝑉 ) − 𝑐𝑥 ∙ ∫𝛺 𝑢𝑥 𝑑𝑉 − 0 𝑐𝑦 ∙ ∫𝛺 𝑢𝑦 𝑑𝑉 − 𝑐𝑧 ∙ ∫𝛺 𝒛 × 𝒖 𝑑𝑉. 0 0 In the above equation, 𝒖 is the displacement field, 𝑝 is a Lagrange multiplier to enforce incompressibility of the tissue (i.e., Jacobian of the deformation gradient tensor, 𝐽 = 1), 𝑃𝐿𝑉 is the Lagrange multiplier to constrain the LV cavity volume 𝑉𝐿𝑉,cav (𝒖) to a prescribed value 𝑉𝐿𝑉 [160]. Both 𝑐𝑥 and 𝑐𝑦 are Lagrange multipliers to constrain rigid body translation in x, y directions and 𝑐𝑧 is the Lagrange multiplier to constrain rigid body rotation [161]. The functional relationship between the cavity volumes of the LV to the displacement field is given by, 1 (5.2) 𝑉𝐿𝑉 (𝒖) = ∫ 𝑑𝑣 = − ∫ 𝒙. 𝒏 𝑑𝑎 , 3 Ω𝑘,𝑖𝑛𝑛𝑒𝑟 Γ𝑘,𝑖𝑛𝑛𝑒𝑟 where Ω𝑘,𝑖𝑛𝑛𝑒𝑟 is the volume enclosed by the inner surface Γ𝑘,𝑖𝑛𝑛𝑒𝑟 and the basal surface at 𝑧 = 0, and 𝒏 is the outward unit normal vector. 74 The first variation of the Lagrangian functional in Eq. (5.1) leads to the following expression: 𝛿ℒ(𝒖, 𝑝, 𝑃𝐿𝑉 , 𝑐𝑥 , 𝑐𝑦 , 𝑐𝑧 ) = ∫Ω (𝑷 − 𝑝𝑭−𝑇 ): 𝛻𝛿𝒖 𝑑𝑉 − 0 ∫Ω0 𝛿𝑝(𝐽 − 1)𝑑𝑉 − 𝑃𝐿𝑉,cav ∫Ω0 𝑐𝑜𝑓(𝑭): 𝛻𝛿𝒖 𝑑𝑉 − 𝛿𝑃𝐿𝑉 (𝑉𝐿𝑉 (𝒖) − 𝑉𝐿𝑉 ) − 𝛿𝑐𝑥 ∙ ∫Ω 𝑢𝑥 𝑑𝑉 − (5.3) 0 𝛿𝑐𝑦 ∙ ∫𝛺 𝑢𝑦 𝑑𝑉 − 𝑐𝑦 ∙ ∫𝛺 𝛿𝑢𝑦 𝑑𝑉 − 0 0 𝛿𝑐𝑧 ∙ ∫Ω 𝒛 × 𝒖 𝑑𝑉 − 𝑐𝑥 ∙ ∫Ω 𝛿𝑢𝑥 𝑑𝑉 − 0 0 𝑐𝑧 ∙ ∫Ω 𝒛 × 𝛿𝒖 𝑑𝑉. 0 In Eq. (5.3), 𝑷 is the first Piola Kirchhoff stress tensor, 𝑭 is the deformation gradient tensor, 𝛿𝒖, 𝛿𝑝, 𝛿𝑃𝐿𝑉,cav, 𝛿𝑐𝑥 , 𝛿𝑐𝑦 , 𝛿𝑐𝑧 are the variation of the displacement field, Lagrange multipliers for enforcing incompressibility and volume constraint, zero mean translation along x and y directions and zero mean rotation along z direction, respectively. The Euler-Lagrange problem then becomes finding 𝒖 ∈ 𝐻1 (Ω0 ), 𝑝 ∈ 𝐿2 (Ω0 ), 𝑃𝐿𝑉,cav ∈ ℝ, 𝑐𝑥 ∈ ℝ , 𝑐𝑦 ∈ ℝ , 𝑐𝑧 ∈ ℝ that satisfies, 𝛿ℒ(𝒖, 𝑝, 𝑃𝐿𝑉 , 𝑐𝑥 , 𝑐𝑦 , 𝑐𝑧 ) = 0 , (5.4) and 𝒖. 𝒏|𝑏𝑎𝑠𝑒 = 0 (for constraining the basal deformation to be in-plane) ∀ 𝛿𝒖 ∈ 𝐻1 (Ω0 ), 𝛿𝑝 ∈ 𝐿2 (Ω0 ), 𝛿𝑃𝐿𝑉 ∈ ℝ, 𝛿𝑐𝑥 ∈ ℝ , 𝛿𝑐𝑦 ∈ ℝ , 𝛿𝑐𝑧 ∈ ℝ . 5.2.5 Constitutive relation Mechanical behavior of the LV was described using an active stress formulation in which the first Piola Kirchhoff stress tensor 𝑺 was decomposed additively into a passive component 𝑺 p and an active component 𝑺 a (i.e. 𝑺 = 𝑺a + 𝑺p ). The passive stress tensor was defined based on the strain energy function of the Holzapfel-Ogden constitutive model [130, 179, 180] given as 𝑎 𝑏(𝐼 −3) 𝑎𝑖 [𝑏 (𝐼 −1)2] 𝑎𝑓𝑠 2 (5.5a) 𝑊= 𝑒 1 + ∑ [𝑒 𝑖 4𝑖 − 1] + [𝑒 (𝑏𝑓𝑠 𝐼8𝑓𝑠 ) − 1] , 2𝑏 2𝑏𝑖 2𝑏𝑓𝑠 𝑖=𝑓,𝑠 where 75 𝑪 = 𝑭𝑻 𝑭, 𝐼1 = 𝑡𝑟(𝑪), 𝐼4𝑓 = 𝑪: 𝑯, (5.5b) 𝐼4𝑖 = 𝒆𝒊𝟎 ∙ (𝑪𝒆𝒊𝟎 ), 𝐼8𝑓𝑠 = 𝒆𝒇𝟎 ∙ (𝑪𝒆𝒔𝟎 ). In Eq. (5.5b), 𝑪 is the right Cauchy-Green deformation tensor, F is deformation gradient, 𝑯 is the structure tensor, 𝐼1 , 𝐼4𝑖 , 𝐼8𝑓𝑠 are invariants and 𝒆𝒊𝟎 with i ∈ (s, n) is a unit vector in the myocardial fiber (f ), sheet (s ) and sheet normal (n ) directions. The effect of myofiber disarray is incorporated via the invariant 𝐼4𝑓 . Material parameters of the passive constitutive model are denoted by 𝑎, 𝑏, 𝑎𝑓 , 𝑏𝑓 , 𝑎𝑠 , 𝑏𝑠 , 𝑎𝑓𝑠 𝑎𝑛𝑑 𝑏𝑓𝑠 . The structure tensor, 𝑯, has been previously explained in section 4.2.2. Active stress calculated based on a previously developed active contraction model [138, 139, 176] was explained briefly in section 4.2.3 by Eq. (4.4) and (4.5). 5.2.6 Simulation cases and protocol For each subject-specific LV FE model, the following simulations were performed sequentially. 1) Estimating the unloaded geometry: First, the unloaded LV configuration was estimated from the LV geometry reconstructed from the CMR images at ED using a backward displacement method [181]. To do so, passive material parameters in the Holzapfel-Ogden model were calibrated manually so that the EDPVR of the LV FE model matches that derived from the single-beat estimation by Klotz et al.[182, 183], which is also applied for HCM subjects. 2) Simulation of a beating heart without myofiber disarray (𝜅 = 0): Following the estimation of unloaded geometry, the unloaded LV FE model was coupled to a closed-loop lumped parameter model of the circulatory system to predict cardiac hemodynamics and mechanics. Myofiber contractility parameter Tmax in the active contraction model, resistances and compliances in the circulatory model in each subject-specific model were calibrated without myofiber disarray (i.e., 𝜅 = 0) to match the corresponding measured volume waveforms, blood pressure and peak 76 GLS. The models were also calibrated to maintain a pressure gradient of ~ 60mmHg across the LVOT assumed for the obstructive HCM subject [184, 185]. Figure 5.2: Relationship between fractional anisotropy and myofiber disarray. 3) Simulation of a beating heart with disarray (𝜅 > 0): Thereafter, the relationship between myofiber disarray and myofiber contractility Tmax was investigated in the 2 HCM patients. To do so, different values of 𝜅 was imposed globally into the HCM LV FE models based on fractional anisotropy (FA) measured in HCM patients in previous studies [17, 186, 187]. The relationship between FA and myofiber disarray is shown in Figure 5.2, and was established by assuming the structure tensor 𝑯 to be equivalent to the diffusion tensor measured in the diffusion-tensor MR images (DTMRI). Following the formulation described in Mukherjee et al [188], the eigenvalues (𝜆1 , 𝜆2 , 𝜆3 ) of the structure tensor were used to compute the FA based on the following relationship: √(𝜆1 − 𝜆2 )2 + (𝜆2 − 𝜆3 )2 + (𝜆3 − 𝜆1 )2 𝐹𝐴 = ⁄ √2(𝜆21 + 𝜆22 + 𝜆23 ), (5.7) 77 Based on the reported FA, the range of myofiber disarray parameter 𝜅 considered here lies between 0.0 to 0.22. For each value of 𝜅, myofiber contractility Tmax in the active contraction model was adjusted to match the clinical data. We note that the venous resting volume was also adjusted in the obstructive HCM subject in order to keep the EDV at the same value as the measurement and to maintain a pressure gradient across the LVOT as prescribed in previous studies. 5.2.7 Post-processing of simulation The following quantities were obtained for each simulation of the 3 subjects. Specifically, total normal stress of the myofibers was described by 𝑆𝑓 = 𝑺: 𝑯 , (5.8) where 𝑺 is the second Piola-Kirchoff stress and H is the structure tensor. respectively. Normal Green-Lagrange strain 𝐸𝑓 of the myofibers was determined by 𝐸𝑓 = 𝑬: 𝑯 , (5.9a) 𝑬 = (𝑪 − 𝑰)⁄2, (5.9b) We note that in the limiting case 𝜅 = 0 (perfect alignment of myofibers), 𝐸𝑓 = 𝒆𝒇0 ∙ 𝑬 ∙ 𝒆𝒇0 and 𝑆𝑓 = 𝒆𝒇0 ∙ 𝑺 ∙ 𝒆𝒇0 . These stress and strain quantities are used to compute the work density of the myofiber over a cardiac cycle by (5.10) 𝑊𝑓 = ∫ 𝑆𝑓 𝑑𝐸𝑓 , 𝐶𝑎𝑟𝑑𝑖𝑎𝑐 𝑐𝑦𝑐𝑙𝑒 Global longitudinal strain was calculated from the right Cauchy-Green stretch tensor with end diastole as the reference configuration 𝑪𝑬𝑫 by [175] 78 1 (5.11) 𝑒 = (1 − )⁄2, 𝒆𝒍 ∙ 𝑪𝑬𝑫 ∙ 𝒆𝒍 5.2.8 Determination of difference between model prediction and measurements Relative difference between the model predicted EDPVR and the one based on the empirical relationship by Klotz et al.[182, 183] is defined as 𝑁 𝑁 (5.12) 2 2 𝑒𝑟𝑟𝑝𝑎𝑠𝑠𝑖𝑣𝑒 = ∑(𝑃𝑘𝑙𝑜𝑡𝑧 (𝑉𝑖 ) − 𝑃𝑚𝑜𝑑𝑒𝑙 (𝑉𝑖 )) ⁄∑(𝑃𝑚𝑜𝑑𝑒𝑙 (𝑉𝑖 )) , 𝑖=1 𝑖=1 where 𝑃𝑘𝑙𝑜𝑡𝑧 (𝑉) and 𝑃𝑚𝑜𝑑𝑒𝑙 (𝑉) are the pressure at the same volume 𝑉and 𝑁 is the number of equally-distributed volume data points in the EDPVR for calculation of the difference. On the other hand, the relative difference between the model predicted and clinical measurements of pressure and volume waveforms over a cardiac cycle is defined as 𝑒𝑟𝑟𝑐𝑎𝑟𝑑𝑖𝑎𝑐𝑐𝑦𝑐𝑙𝑒 = ∑𝑀 2 𝑀 2 𝑖=1(𝑦𝑐𝑙𝑖𝑛𝑖𝑐𝑎𝑙 (𝑡𝑖 ) − 𝑦𝑚𝑜𝑑𝑒𝑙 (𝑡𝑖 )) ⁄∑𝑖=1(𝑦𝑚𝑜𝑑𝑒𝑙 (𝑡𝑖 )) . (5.13) In Eq. (5.13), 𝑦𝑐𝑙𝑖𝑛𝑖𝑐𝑎𝑙 𝜖 {𝑃𝑐𝑙𝑖𝑛𝑖𝑐𝑎𝑙 , 𝑉𝑐𝑙𝑖𝑛𝑖𝑐𝑎𝑙 } and 𝑦𝑚𝑜𝑑𝑒𝑙 𝜖 {𝑃𝑚𝑜𝑑𝑒𝑙 , 𝑉𝑚𝑜𝑑𝑒𝑙 } are, respectively, clinical measurements and model predictions of LV pressure and volume at a particular time point 𝑡 in the cardiac cycle. Also, 𝑀 is the no of equally-distributed time steps over a cardiac cycle used to calculate the difference. Relative difference between clinical measurements and model prediction of peak GLS and blood pressure was also calculated for each subject. 5.3 Results 5.3.1 Clinical measurements End diastolic volume was higher in both HCM subjects (Non-obstructive: 82ml; Obstructive: 115ml) compared to the control subject (63.13ml). Ejection fraction was highest in the non-obstructive HCM subject (85%), and was comparable between obstructive 79 HCM subject (67%) and the control subject (70%). Absolute peak GLS was reduced substantially in the obstructive HCM subject (13%), but was comparable between the obstructive HCM subject (19%) and the control subject (20%). 5.3.2 LV geometry Figure 5.3: a. LV geometry of the 3 subjects. b. Regional distribution of wall thickness (in cm) based on AHA segmentation and c. Violin plot of the wall thickness. 80 Left ventricular geometries reconstructed from the CMR images as well as the regional wall thickness based on AHA segmentation for each subject are shown in Figure 5.3. Septum wall thickness of the obstructive HCM subject (1.43 ± 0.36 cm) was largest followed by that of the non-obstructive HCM subject (0.85 ± 0.24 cm) and the control subject (0.73 ± 0.14 cm). In each HCM subject, LV free wall thickness was smaller (cf. septum) but was larger when compared to the same region in the control subject (Obstructive HCM: 1.07 ± 0.18 cm; Non-Obstructive HCM: 0.73 ± 0.13 cm; Control: 0.5 ± 0.08 cm). The resultant global wall thickness was higher in the HCM subjects compared to the control (Obstructive HCM: 1.27 ± 0.33 cm; Non- Obstructive HCM: 0.79 ± 0.23 cm; Control: 0.58 ± 0.15 cm). 5.3.3 LV mechanics without consideration of myofiber disarray The calibrated models’ prediction of the EDPVR relationship is consistent with that obtained from the single-beat estimation based on the Klotz relationship (Figure 5.4a). The passive material properties (APPENDIX A) reflected an increased isotropic stiffness (Obstructive: 334.8%, Non-obstructive: 769.6%) and a decrease in stiffness along the fiber direction (over 99%) in both HCM patients when compared to control. The calibrated models’ predictions of LV volume waveform, blood pressure and peak GLS also agree with the corresponding patient-specific clinical measurements (Figure 5.4b - e). While LV pressure was not measured in these subjects, the pressure waveforms predicted by the model are also comparable with measurements from previous clinical studies of HCM patients. Differences between the measurements and the model predictions are within about 10%, with the highest difference occurring in the comparison between the EDPVR derived from the empirical Klotz relationship and the model (Figure 5.4f). 81 Figure 5.4: Calibration of model parameters for each subject without myofiber disarray. a. EDPVR b. Volume waveform (Solid line – Simulated results, Dotted line – Clinical results) c. Pressure waveform d. PV loop e. Peak GLS f. Difference between model prediction and measurements. 82 Figure 5.5: a. Isometric tension plot; regional distribution of b. peak total fiber stress (in kPa) and c. peak longitudinal strain (absolute value in %) for each subject. Peak (isometric) myofiber tension derived from the calibrated active stress model parameters was found to be substantially smaller in the HCM subjects when compared to the control subject (Figure 5.5a). The obstructive HCM subject has the smallest peak myofiber tension of 60kPa and the non-obstructive HCM subject has a peak myofiber tension of 242kPa, which were both lower compared to that of the control subject (375kPa). Peak myofiber stress averaged over the entire LV was smallest in the obstructive HCM subject (39 ± 8.85 kPa) followed by the 83 non-obstructive HCM subject (40.6 ± 10.3 kPa) and the control subject (66.9 ± 21 kPa) (Figure 5.5b). Peak myofiber stress was lower at the septum than the LVFW in both HCM subjects, with the lowest value found in the obstructive HCM subject. Peak GLS was lower in the entire LV of the obstructive HCM subject compared to the other 2 subjects (Figure 5.5c). Longitudinal strain was higher at the LVFW (−19.8% ) compared to the septum (−12.5%) in the obstructive HCM subject. In the other 2 subjects, however, the difference between longitudinal strain at the septum and LVFW was not prominent (Control: septum −19.5% vs. LVFW −18.8%; non-obstructive HCM: septum −21.8% vs LVFW −18.7%). Figure 5.6: Work densities in the HCM and control subjects without myofiber disarray. a. stress- strain loop along mean fiber direction, b. regional distribution of myofiber work density. 84 Total myofiber work density (indexed by the area of the stress-strain loop along material direction) was lowest in the obstructive HCM subject (9.0 𝑘𝐽/𝑚3), followed by the control subject (11.2 𝑘𝐽/𝑚3 ) and the non-obstructive HCM subject (11.9 𝑘𝐽/𝑚3 ) (Figure 5.6). In terms of its regional distribution, myofiber work density was higher at the LVFW (control: 14.2 𝑘𝐽/𝑚3 ; non- obstructive: 13.1 𝑘𝐽/𝑚3 ; obstructive: 10.8 𝑘𝐽/𝑚3 ) compared to the septum (control: 8.5 𝑘𝐽/𝑚3 ; non-obstructive: 10.1 𝑘𝐽/𝑚3 ; obstructive: 7.8 𝑘𝐽/𝑚3) for all subjects. 5.3.4 Effects of myofiber disarray on the LV mechanics of HCM subjects Figure 5.7: PV loop for a. the non-obstructive and b. the obstructive HCM subject. With an increase in myofiber disarray, it is necessary to increase the scaling parameter Tmax (associated with myofiber contractility) to match the clinical data of the HCM subjects (Figure 5.7). The model parameters are tabulated in APPENDIX B. The resultant peak myofiber tension was therefore increased as a result with increasing myofiber disarray (Figure 5.8). Specifically, peak myofiber tension associated with the largest degree of disarray was 507.9kPa (𝜅 = 0.18) and 100.5 kPa (𝜅 = 0.22) for the non-obstructive and obstructive HCM patients, respectively. Peak GLS did not change substantially (~3%) with increasing myofiber disarray in both HCM subjects. Regional distribution of peak longitudinal strain, peak stress of the myofibers also did not change with different degree of myofiber disarray. In the obstructive HCM subject, peak stress 85 of the myofibers was decreased in both LVFW and septum with increasing myofiber disarray (Figure 5.8d). Conversely in the non-obstructive HCM subject, peak stress of the myofibers was slightly increased with increasing myofiber disarray (Figure 5.8e). Figure 5.8: Effects of myofiber disarray. Isometric tension-time plot of a. non-obstructive and b. obstructive HCM subjects. c. Peak GLS for the 2 HCM subjects. Peak stress of the myofibers at the septum and LVFW for d. obstructive and e. non-obstructive HCM subjects. Myofiber work density was reduced with increasing myofiber disarray in both the non- obstructive HCM subject and obstructive HCM subject (Figure 5.9a, b). The reduction in myofiber work density was highest in the septum and lowest in the anterior in the non-obstructive HCM subject (Septum: -74% ; Anterior: -71% at 𝜅 = 0.18 cf. 𝜅 = 0.0). On the other hand, in obstructive HCM subject, posterior and LVFW regions have the highest and lowest reduction in myofiber work density, respectively (Posterior: -87% ; LVFW: -81% at 𝜅 = 0.22 cf. 𝜅 = 0.0). 86 Figure 5.9: Effects of myofiber disarray on myofiber work densities for a. the non-obstructive and b. the obstructive HCM subject. 5.4 Discussion We have developed a patient-specific computational framework of LV mechanics to investigate the effects of myofiber disarray using clinical data of 2 HCM subjects with different 87 phenotypes (obstructive vs non-obstructive) along with a control subject. The key finding of this study suggests that the contractile force generated by the cardiac muscle cell is reduced in the obstructive HCM subject compared to the control subject. In the non-obstructive HCM subject, the contractile force is reduced only if the degree of global myofiber disarray 𝜅 is less than 0.14. Specifically, the study found that the contractile force generated by the cell to reproduce the clinical measurements is increased with an increase in global myofiber disarray. An increase in myofiber disarray led to a reduction in myofiber work density in both HCM subjects. The reconstructed LV geometries of the HCM subjects are consistent with those reported in previous clinical studies. Specifically, the maximum LV wall thickness in the obstructive and non-obstructive HCM subjects are 17.4mm and 12.3 mm are consistent with previous studies [110, 111]. Besides, the ratio of maximum septum wall thickness to minimum posterior wall thickness for the non-obstructive (1.9) and obstructive (1.54) HCM subjects are also within the threshold (≥ 1.3) used to define asymmetric septal hypertrophy in HCM patients [109]. Both HCM subjects have higher EDV than the control subject (Figure 5.3), although ejection fraction is normal (67%) and supra-normal (85%) for the obstructive and non-obstructive HCM subjects, respectively. The supra-normal EF in the non-obstructive HCM patient is a result of its small ESV. Peak GLS is slightly smaller in the non-obstructive HCM subject (19%) compared to the control subject (20%), but is substantially smaller in the obstructive HCM subject (13%). The smaller peak GLS in the obstructive HCM subject is within the range of -9.65% to - 16% reported in previous studies [90, 107]. The results suggest that the reduction in peak GLS is associated with a reduction in myofiber contractility that is indexed by the peak muscle fiber tension. Without considering myofiber disarray, the models predicted that the peak tension to 88 reproduce the clinical measurements is, respectively, 84% (absolute) and 35% (absolute) smaller in the obstructive and non-obstructive HCM subject when compared to the control subject. By considering myofiber disarray based on the range found in DTMRI studies with 𝜅 having values between 0 to 0.22, we found that the peak muscle fiber tension has to increase to compensate for an increasing degree of myofiber disarray in order to reproduce the clinical measurements. Within this range of 𝜅, peak GLS varies by only +/- 2% (absolute) in the obstructive HCM subject and is still depressed compared to the control subject (Figure 5.7c). At the highest degree of myofiber disarray in the obstructive HCM subject, however, the peak muscle fiber tension is still about 72.8 % (absolute) lower than that in the control subject. For the non-obstructive HCM subject, we found that the peak muscle fiber tension is equivalent to the control subject at a disarray 𝜅 = 0.14. At that value of 𝜅, peak GLS is -18 % and lies within the ranges reported previously [189, 190]. These findings therefore suggest myocardial contractility is likely reduced in the HCM subjects, especially in the obstructive phenotype, which can explain the results of a previous CMR study on HCM patients showing that a reduction in FA is associated with a reduction in myocardial strain [186]. The finding that a reduced peak GLS is associated with a reduction in myocardial contractility even with a normal EF is consistent with a previous modeling study based an idealized LV geometry [174]. In that study, only a reduction in myocardial contractility can explain the simultaneous features (including a reduction in GLS) found in patients with HFpEF. Specific to HCM, a reduction in myocardial contractility has also been found in animal studies and is attributed to the mutation of sarcomeric protein [54] [191]. The lower peak tension found here is also consistent with the reduced myofibril density found in vitro studies of myocytes obtained from myocardial biopsies of HCM patients [192]. 89 Peak stress of the myofibers is heterogeneously distributed in the LV (Figure 5.5b). Compared to the control subject, peak myofiber stress is smaller in the HCM subjects, and is smallest in the obstructive HCM subject. This result is largely due to the increase in wall thickness in the HCM subject, and is consistent with previous studies of HCM patients [153, 193] . Peak myofiber wall stress is also lower in the septum (thicker region) than LVFW (thinner region) in all subjects. Between non-obstructive and obstructive HCM subjects, peak stress of the myofibers behaves differently with increasing myofiber disarray (Figure 5.5d, e). With an increase in myofiber disarray, peak stress of the myofibers increases in the non-obstructive HCM subject, whereas decreases in obstructive HCM subject. This result suggests that the effects of myofiber disarray on myofiber stress may be sensitive to geometry. Global myocardial work density, indexed by the pressure volume area, is linearly correlated to the cardiac metabolism and total myocardial oxygen consumption[194–196]. Local myofiber work density 𝑊𝑓 is determined from the area in the average myofiber stress-strain loop (Figure 5.6). Without consideration of myofiber disarray, our analysis shows that the non-obstructive HCM subject has the highest mean 𝑊𝑓 (11.9 𝑘𝐽/𝑚3 ), followed by the control subject (11.2 𝑘𝐽/𝑚3 ), and the obstructive HCM subject (9.00 𝑘𝐽/𝑚3 ). With disarray where cardiac muscles are oriented in other directions other than the mean myofiber direction, 𝑊𝑓 decreases with increasing degree of myofiber disarray in both HCM subjects (Figure 5.8). These results showing a lower 𝑊𝑓 in the obstructive HCM subject than the non-obstructive HCM subject (and the normal) is consistent with published results of myocardial work index (pressure-strain loop area) assessed noninvasively using echocardiography and blood pressure measurement in HCM patients [197, 198]. The findings that septal 𝑊𝑓 is lower than that in the LVFW is also consistent with these studies, especially when in HCM phenotypes with substantial septal hypertrophy. We note that 𝑊𝑓 is 90 defined differently from the myocardial work index measured in the clinic as the latter relies on a global index of stress (i.e., pressure) rather than the local stress of the myofibers. Nevertheless, both of these indices are metric of the total work of the myofiber over a cardiac cycle. Our finding suggests that the development of myofiber disarray further worsens the already lower myofiber work in the HCM subjects, further suggesting that this feature is a contributor to the lower myocardial work index found clinically in HCM patients. The lower work arises because myofibers are disoriented and not contributing efficiently to the overall contraction of the heart (e.g., myofibers oriented in the radial directions are not performing work when the wall thickens during contraction). Therefore, myofiber disarray is one of the key contributors to the worsening of myocardial work in HCM patients (in addition to other features such as mechanical dyssynchrony). 5.5 Conclusion We have developed patient-specific computational models based on clinical data acquired in 2 HCM (obstructive and non-obstructive) and a control subject to investigate LV mechanics and the relationship between myofiber disarray and myofiber contractility in this disease. Using these models, we show that myofiber contractility must increase to compensate for an increase in myofiber disarray associated with HCM in order to maintain same LV function. For the range of myofiber disarray measured in HCM patients, however, we found that the myofiber contractility in the obstructive HCM subject is still reduced compared to the control subject at the highest degree of myofiber disarray. Myofiber contractility of the non-obstructive HCM subject is close to that of the control subject only when myofiber disarray is substantial with a fractional anisotropy of 0.75. An increase in myofiber disarray also led to a reduction in myofiber work in the HCM subjects. These findings suggest that myofiber contractile stress generated in HCM patients is reduced and 91 is associated with an increase in wall thickness, and the reduction in myofiber work seen in HCM patients may be due in part to myofiber disarray. 92 CHAPTER 6 LIMITATIONS AND FUTURE SCOPE 93 6.1 Biventricular model on Left Bundle Brunch Block Finite element model based on the idealized biventricular geometry can be developed further. Based on experimental data, incorporation of lateral stretch and wall stress as growth stimuli associated with parallel sarcomere addition can be scope of future study to determine if they are able to reproduce asymmetrical features associated with asynchronous activation. Investigation of change in microstructure property due to remodeling can also be considered. Since preload and afterload can be altered due to G&R, incorporation of the evolution of these two properties over long term G&R modeling to investigate the effect on these in diseased heart and treatment may provide more realism. The model can be further developed by considering electrical conduction through the purkinje fibers. Last, the generalized framework can be further developed through integrating patient specific data. 6.2 Animal specific LV model on pressure overload Our animal specific computational model is based on two weeks results. Further development of a growth model to simulate G&R associated with pressure overload based on the findings that local changes in myofiber stress is correlated with changes in wall thickness will help better understand the mechanism behind the progression of this disease. The model can be developed by calibrating for mechanical strain found in experimental data related to pressure overload. Besides, future study can investigate any association of local fibrosis seen in pressure overload diseases with local G&R such as changes in wall thickness. As oxygen perfusion gets impaired due to increase in wall thickness, incorporation of a perfusion model will be beneficial, especially for investigating the effect of hypertrophy on regional perfusion. 94 6.3 Idealized LV model on Hypertrophic Cardiomyopathy Future studies using an ellipsoidal LV model can investigate the effect of disarray in various geometrical phenotypes of HCM, Also, since in HCM heart, the interaction between actin-myosin play a vital role on the mechanics, incorporation of crossbridge model will be helpful to investigate the interaction of mechanics from cell to organ level. Besides, LV mechanics altered by other features such as fibrosis, extreme outflow tract obstruction, hypertrophy around mid-ventricle and apex can be considered in future. 6.4 Patient specific LV model on Hypertrophic Cardiomyopathy This study can be extended in future to consider the broad range of disease pattern and variation of morphological phenotypes (such as apical hypertrophy) found in HCM patients. Diffused and regional myofiber disarray based on local DTMRI measurements of myofiber disarray in HCM patients can also be incorporated into the model in future studies. Also, based on LGE quantity, local or diffuse fibrosis can be applied into the model. MD can be considered in future studies using an electromechanics model [128, 177]. 6.5 Conclusion All of these studies suggest that the underlying mechanisms of cardiac hypertrophy are different in different pathological conditions. 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Table A.1: Material parameters Parameters Unit Control Non-obstructive Obstructive Holzapfel-Ogden model a 𝑃𝑎 46 400 200 b 12 5 4 𝑎𝑓 𝑃𝑎 7.51e03 37.5 15 𝑏𝑓 5.893 1.47325 22.1 𝑎𝑠 𝑃𝑎 492 492 492 𝑏𝑠 3.393 3.393 3.393 𝑎𝑓𝑠 𝑃𝑎 70 70 70 𝑏𝑓𝑠 3.929 3.929 3.929 Guccione model 𝑇𝑚𝑎𝑥 𝑘𝑃𝑎 620 400 99.75 𝜏 𝑚𝑠 20 35 35 𝑡𝑡𝑟𝑎𝑛𝑠 𝑚𝑠 385 430 420 B 4.75 4.75 4.75 𝑡0 𝑚𝑠 350 400 350 𝑙0 𝜇𝑚 1.55 1.55 1.55 𝐶𝑎0 𝜇𝑀 4.35 4.35 4.35 𝐶𝑎0𝑚𝑎𝑥 𝜇𝑀 4.35 4.35 4.35 𝑙𝑟 𝜇𝑚 1.85 1.85 1.85 BCL 𝑚𝑠 1000 910 1180 113 The model parameters prescribed in circulatory model and time varying elastance model are enlisted in Table A.2. Table A.2: Circulatory and left atrium model parameters Parameter Unit Control Non-obstructive Obstructive Circulatory model 𝐶𝑎,𝑝 𝑚𝑙 𝑃𝑎 0.00208 0.00544 0.0048 𝐶𝑎,𝑑 𝑚𝑙 𝑃𝑎 0.02145 0.0561 0.0495 𝐶𝑣𝑒𝑛 𝑚𝑙 𝑃𝑎 0.196 0.378 0.014 𝑉𝑎,𝑝,0 𝑚𝑙 144 144 306 𝑉𝑎,𝑑,0 𝑚𝑙 160 160 160 𝑉𝑣𝑒𝑛,0 𝑚𝑙 4500 3100 4525 𝑅𝑎𝑜 𝑃𝑎 𝑚𝑠 𝑚𝑙 −1 3000 3000 31500 𝑅𝑣𝑒𝑛 𝑃𝑎 𝑚𝑠 𝑚𝑙 −1 10 10 100 𝑅𝑎,𝑝 𝑃𝑎 𝑚𝑠 𝑚𝑙 −1 108000 90000 45000 𝑅𝑎,𝑑 𝑃𝑎 𝑚𝑠 𝑚𝑙 −1 127200 84800 159000 𝐶𝑎,𝑝 𝑚𝑙 𝑃𝑎 0.00208 0.00544 0.0048 Time varying elastance model 𝐸𝑒𝑠,𝑙𝑎 𝑃𝑎/𝑚𝑙 9 7 10 𝐴𝑙𝑎 𝑃𝑎 0.801 0.6675 4.005 𝐵𝑙𝑎 𝑚𝑙 −1 0.0152 0.00475 0.021 𝑉0,𝑙𝑎 𝑚𝑙 10 10 10 𝑇𝑚𝑎𝑥,𝑙𝑎 𝑚𝑠 120 120 150 𝜏𝑙𝑎 𝑚𝑠 25 25 25 𝑡𝑑𝑒𝑙𝑎𝑦,𝑙𝑎 𝑚𝑠 140 140 140 114 APPENDIX B: MODEL PARAMETERS WITH DISARRAY The model parameters calibrated to match with clinical volume waveform and blood pressure with varying degree of disarray for 2 HCM subjects are listed below. Noted, the rest of the model parameters are same as described in Appendix A. Table B.1: Model parameters with disarray Parameter Unit 𝜅 = 0.07 𝜅 = 0.1 𝜅 = 0.14 𝜅 = 0.18 𝜅 = 0.22 Obstructive HCM 𝑇𝑚𝑎𝑥 𝑘𝑃𝑎 106.8 109.25 123.5 137.75 166.25 𝑉𝑣𝑒𝑛,0 𝑚𝑙 4550 4640 4660 4660 4660 Non-obstructive HCM 𝑇𝑚𝑎𝑥 𝑘𝑃𝑎 420 440 500 840 115