Mathematical modelling of cardiac function: constitutive law, fibre dispersion, growth and remodelling

The heart is an immensely complex living organ. Myocardium has continually been undergoing adaptive or maladaptive response to surrounding environments, in which the significant importance of growth and remodelling (G&R) has been valued. This PhD project intends to study mechanics modelling of myocardium towards predictive stress/strain-driven growth. Constitutive laws and fibre structures in myocardium work together to determine the mechanical clues which trigger the growth mechanically. Therefore, this project includes two parts: (1) constitutive characterization of myocardium, and (2) myocardial G&R. Constitutive laws and myofibre architectures hold the key to accurately model the biomechanical behaviours of the heart. In the first part of this thesis, we firstly perform an analysis using combinations of uniaxial tension, biaxial tension and simple shear from three different sets of myocardial experimental tissue studies to investigate the descriptive and predictive capabilities of a general invariant-based model that is developed by Holzapfel and Ogden, denoted the HO model. We aim to reduce the constitutive law using the Akaike information criterion to maintain its mechanical integrity whilst achieve minimal computational cost. Our study shows that single-mode tests are insufficient to determine the myocardium responses. It is also essential to consider the transmural fibre rotation within the myocardial samples. We conclude that a competent myocardial material model can be obtained from the general HO model using Akaike information criterion analysis and a suitable combination of tissue tests. Secondly, we develop a neonatal porcine bi-ventricle model with three different myofibre architectures for the left side of the heart. The most realistic one is derived from ex vivo diffusion tensor magnetic resonance image, and the other two simplifications are based on the rule-based methods. We show that the most realistic myofibre architecture model can achieve better cardiac pump functions compared to those of the rule-based models under the same pre/after loads. Our results also reveal that when the cross-fibre contraction is included, the active stress seems to play a dual role: the sheet-normal component enhances the ventricular contraction while the sheet component does the opposite. This study highlights the importance of including myofibre dispersion in cardiac modelling if rule-based methods are used, especially in personalized model. To further describe the detailed fibre distribution, discrete fibre dispersion method is employed to compute passive response because of its advantages in excluding compressed fibres. An additive active stress method that includes cross-fibre active stress is proposed according to the generalised structure tensor method. We find that end-systolic volumes of simulated heart models are much more sensitive to dispersion parameter than end-diastolic volumes. G&R is the focus in the second part of this thesis. An updated reference approach is employed to track the evolution of the reference configuration during G&R, in which the nodal positions and the fibre structure are updated at the beginning of each new growth cycle. Moreover, the homogenised constrained mixture theory is used to describe the G&R process of each constituent within myocardium, which are the ground matrix, collagen network and myofibres. Our models can reproduce the eccentric growth driven by fibre stretch at the diastole, concentric growth driven by fibre stress at the systole, and heterogeneous growth after acute myocardium infarction. Ventricular wall G&R mainly occurs in endocardium, in which the myocyte is the primary responder for the G&R process. G&R laws of collagen fibre have significant impacts on G&R of heart. For example, purely remodeled collagen network without new deposition causes increasingly softer heart wall, leading to excessive heart dilation. Finally, the effects of fibre dispersion on G&R is investigated by including fibre dispersion model in the G&R of infarction model. Highly dispersed fibre structure in the infarcted zone significantly reduces the pump function. This thesis has been focusing on mathematical modelling of biomechanical behaviours of myocardium, firstly on the nonlinear cardiac mechanics including constitutive laws and fibre structures, and then on the G&R process of heart under different pathological conditions. These studies support to choose suitable constitutive laws and fibre architectures in G&R model and illustrate the underlying mechanism of mechanical triggers in G&R. It presents the potential for understanding the mechanics of heart failure and reveal hidden roles of different constituents in myocardium

Effect of myofibre architecture on ventricular pump function by using a neonatal porcine heart model: from DT-MRI to rule-based methods

Myofibre architecture is one of the essential components when constructing personalized cardiac models. In this study, we develop a neonatal porcine bi-ventricle model with three different myofibre architectures for the left ventricle (LV). The most realistic one is derived from ex vivo diffusion tensor magnetic resonance imaging, and other two simplifications are based on rule-based methods (RBM): one is regionally dependent by dividing the LV into 17 segments, each with different myofibre angles, and the other is more simplified by assigning a set of myofibre angles across the whole ventricle. Results from different myofibre architectures are compared in terms of cardiac pump function. We show that the model with the most realistic myofibre architecture can produce larger cardiac output, higher ejection fraction and larger apical twist compared with those of the rule-based models under the same pre/after-loads. Our results also reveal that when the cross-fibre contraction is included, the active stress seems to play a dual role: its sheet-normal component enhances the ventricular contraction while its sheet component does the opposite. We further show that by including non-symmetric fibre dispersion using a general structural tensor, even the most simplified rule-based myofibre model can achieve similar pump function as the most realistic one, and cross-fibre contraction components can be determined from this non-symmetric dispersion approach. Thus, our study highlights the importance of including myofibre dispersion in cardiac modelling if RBM are used, especially in personalized models

Growth and Remodelling of Right Ventricle Under Pulmonary Arterial Hypertension

Pulmonary arterial hypertension (PAH) is a fatal and rapidly progressive disease, and right ventricular failure is the main cause of death in patients with PAH. This study aims to determine the mechanical cues that can trigger the growth and remodelling of the heart. To this end, two bi-ventricular models of a control rat heart and a diseased rat heart with PAH are reconstructed, respectively, and then the growth amount is estimated by warping the PAH heart to the control heart. Finally, correlation analysis between mechanical cues (stress and strain) and growth tensors demonstrates principal strain may be a triggering cue for myocardial growth under PAH

An updated Lagrangian constrained mixture model of pathological cardiac growth and remodelling

Progressive left ventricular (LV) growth and remodelling (G&R) is often induced by volume and pressure overload, characterized by structural and functional adaptation through myocyte hypertrophy and extracellular matrix remodelling, which are dynamically regulated by biomechanical factors, inflammation, neurohormonal pathways, etc. When prolonged, it can eventually lead to irreversible heart failure. In this study, we have developed a new framework for modelling pathological cardiac G&R based on constrained mixture theory using an updated reference configuration, which is triggered by altered biomechanical factors to restore biomechanical homeostasis. Eccentric and concentric growth, and their combination have been explored in a patient-specific human LV model under volume and pressure overload. Eccentric growth is triggered by overstretching of myofibres due to volume overload, i.e. mitral regurgitation, whilst concentric growth is driven by excessive contractile stress due to pressure overload, i.e. aortic stenosis. Different biological constituent’s adaptations under pathological conditions are integrated together, which are the ground matrix, myofibres and collagen network. We have shown that this constrained mixture-motivated G&R model can capture different phenotypes of maladaptive LV G&R, such as chamber dilation and wall thinning under volume overload, wall thickening under pressure overload, and more complex patterns under both pressure and volume overload. We have further demonstrated how collagen G&R would affect LV structural and functional adaption by providing mechanistic insight on anti-fibrotic interventions. This updated Lagrangian constrained mixture based myocardial G&R model has the potential to understand the turnover processes of myocytes and collagen due to altered local mechanical stimuli in heart diseases, and in providing mechanistic links between biomechanical factors and biological adaption at both the organ and cellular levels. Once calibrated with patient data, it can be used for assessing heart failure risk and designing optimal treatment therapies

On the AIC-based model reduction for the general Holzapfel–Ogden myocardial constitutive law

© 2019, The Author(s). Constitutive laws that describe the mechanical responses of cardiac tissue under loading hold the key to accurately model the biomechanical behaviour of the heart. There have been ample choices of phenomenological constitutive laws derived from experiments, some of which are quite sophisticated and include effects of microscopic fibre structures of the myocardium. A typical example is the strain-invariant-based Holzapfel–Ogden 2009 model that is excellently fitted to simple shear tests. It has been widely used and regarded as the state-of-the-art constitutive law for myocardium. However, there has been no analysis to show if it has both adequate descriptive and predictive capabilities for other tissue tests of myocardium. Indeed, such an analysis is important for any constitutive laws for clinically useful computational simulations. In this work, we perform such an analysis using combinations of tissue tests, uniaxial tension, biaxial tension and simple shear from three different sets of myocardial tissue studies. Starting from the general 14-parameter myocardial constitutive law developed by Holzapfel and Ogden, denoted as the general HO model, we show that this model has good descriptive and predictive capabilities for all the experimental tests. However, to reliably determine all 14 parameters of the model from experiments remains a great challenge. Our aim is to reduce the constitutive law using Akaike information criterion, to maintain its mechanical integrity whilst achieving minimal computational cost. A competent constitutive law should have descriptive and predictive capabilities for different tissue tests. By competent, we mean the model has least terms but is still able to describe and predict experimental data. We also investigate the optimal combinations of tissue testsfor a given constitutive model. For example, our results show thatusing one of the reduced HO models, one mayneed just one shear response (along normal-fibredirection) and one biaxial stretch (ratio of 1 mean fibre : 1 cross-fibre) to satisfactorily describe Sommer et al. human myocardial mechanical properties. Our studysuggests that single-state tests (i.e. simple shear or stretching only) are insufficient to determine the myocardium responses. We also foundit is important to consider transmural fibre rotations within eachmyocardial sampleof tests during the fitting process.This is done byexcluding un-stretched fibres usingan “effective fibre ratio”, which depends on the sample size, shape, local myofibre architecture and loading conditions. We conclude that a competent myocardium material model can be obtained from the general HO model using AIC analysis and a suitable combination of tissue tests

Using LDDMM and a kinematic cardiac growth model to quantify growth and remodelling in rat hearts under PAH

Pulmonary arterial hypertension (PAH) is a rapidly progressive and fatal disease, with right ventricular failure being the primary cause of death in patients with PAH. This study aims to determine the mechanical stimuli that may initiate heart growth and remodelling (G&R). To achieve this, two bi-ventricular models were constructed: one for a control rat heart and another for a rat heart with PAH. The growth of the diseased heart was estimated by warping it to the control heart using an improved large deformation diffeomorphic metric mapping (LDDMM) framework. Correlation analysis was then performed between mechanical cues (stress and strain) and growth tensors, which revealed that principal strains may serve as a triggering stimulus for myocardial growth and remodelling under PAH. The growth tensors, estimated from in vivo images, could explain 84.3% of the observed geometrical changes in the diseased heart with PAH by using a kinematic cardiac growth model. Our approach has the potential to quantify G&R using sparse in vivo images and to provide insights into the underlying mechanism of triggering right heart failure from a biomechanical perspective

Using LDDMM and a kinematic cardiac growth model to quantify growth and remodelling in rat hearts under PAH

Pulmonary arterial hypertension (PAH) is a rapidly progressive and fatal disease, with right ventricular failure being the primary cause of death in patients with PAH. This study aims to determine the mechanical stimuli that may initiate heart growth and remodelling (G&R). To achieve this, two bi-ventricular models were constructed: one for a control rat heart and another for a rat heart with PAH. The growth of the diseased heart was estimated by warping it to the control heart using an improved large deformation diffeomorphic metric mapping (LDDMM) framework. Correlation analysis was then performed between mechanical cues (stress and strain) and growth tensors, which revealed that principal strains may serve as a triggering stimulus for myocardial growth and remodelling under PAH. The growth tensors, estimated from in vivo images, could explain 84.3% of the observed geometrical changes in the diseased heart with PAH by using a kinematic cardiac growth model. Our approach has the potential to quantify G&R using sparse in vivo images and to provide insights into the underlying mechanism of triggering right heart failure from a biomechanical perspective

Modelling of fibre dispersion and its effects on cardiac mechanics from diastole to systole

Detailed fibre architecture plays a crucial role in myocardial mechanics both passively and actively. Strong interest has been attracted over decades in mathematical modelling of fibrous tissue (arterial wall, myocardium, etc.) by taking into account realistic fibre structures, i.e. from perfectly aligned one family of fibres, to two families of fibres, and to dispersed fibres described by probability distribution functions. It is widely accepted that the fibres, i.e. collage, cannot bear the load when compressed, thus it is necessary to exclude compressed fibres when computing the stress in fibrous tissue. In this study, we have focused on mathematical modelling of fibre dispersion in myocardial mechanics, and studied how different fibre dispersions affect cardiac pump function. The fibre dispersion in myocardium is characterized by a non-rotationally symmetric distribution using a π-periodic Von Mises distribution based on recent experimental studies. In order to exclude compressed fibres for passive response, we adopted the discrete fibre dispersion model for approximating a continuous fibre distribution with finite fibre bundles, and then the general structural tensor was employed for describing dispersed active tension. We first studied the numerical accuracy of the integration of fibre contributions using the discrete fibre dispersion approach, then compared different mechanical responses in a uniaxially stretched myocardial sample with varied fibre dispersions. We finally studied the cardiac pump functions from diastole to systole in two heart models, a rabbit bi-ventricle model and a human left ventricle model. Our results show that the discrete fibre model is preferred for excluding compressed fibres because of its high computational efficiency. Both the diastolic filling and the systolic contraction will be affected by dispersed fibres depending on the in-plane and out-of-plane dispersion degrees, especially in systolic contraction. The in-plane dispersion seems affecting myocardial mechanics more than the out-of-plane dispersion. Despite different effects in the rabbit and human models caused by the fibre dispersion, large differences in pump function exist when fibres are highly dispersed at in-plane and out-of-plane. Our results highlight the necessity of using dispersed fibre models when modelling myocardial mechanics, especially when fibres are largely dispersed under pathological conditions, such as fibrosis

Effects of dispersed fibres in myocardial mechanics, Part I: passive response

It is widely acknowledged that an imbalanced biomechanical environment can have significant effects on myocardial pathology, leading to adverse remodelling of cardiac function if it persists. Accurate stress prediction essentially depends on the strain energy function which should have competent descriptive and predictive capabilities. Previous studies have focused on myofibre dispersion, but not on fibres along other directions. In this study, we will investigate how fibre dispersion affects myocardial biomechanical behaviours by taking into account both the myofibre dispersion and the sheet fibre dispersion, with a focus on the sheet fibre dispersion. Fibre dispersion is incorporated into a widely-used myocardial strain energy function using the discrete fibre bundle approach. We first study how different dispersion affects the descriptive capability of the strain energy function when fitting to ex vivo experimental data, and then the predictive capability in a human left ventricle during diastole. Our results show that the chosen strain energy function can achieve the best goodness-of-fit to the experimental data by including both fibre dispersion. Furthermore, noticeable differences in stress can be found in the LV model. Our results may suggest that it is necessary to include both dispersion for myofibres and the sheet fibres for the improved descriptive capability to the ex vivo experimental data and potentially more accurate stress prediction in cardiac mechanics

Constitutive modelling of soft biological tissue from ex vivo to in vivo: myocardium as an example

Imbalance of stress/strain microenvironment can lead to adverse remodelling and pathogenesis in various soft tissues, tumour included. Therefore, there is a critical need for accurate quantification of the biomechanical homeostasis in soft tissue through mathematical modelling, which is critically dependent on constitutive models, the mathematical descriptions that approximate the mechanical behaviours of material under specific conditions by considering information from subcellular, cellular and tissue levels. In most soft biological tissue, collagen is the major component of the extracellular matrix, its architecture largely determines the material property (stiffness). In this work, we will use myocardium as an example to show how we can develop a constitutive law from various ex vivo experiments within the continuum mechanics framework, and demonstrate the applications to real patient data. We will further focus on parameter calibrations from ex/in vivo measurements. We believe this approach of constitutive modelling and calibration can be applied to various soft biological tissues and shed light on physiological and pathological mechanobiology