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

When Cardiac Biophysics Meets Groupwise Statistics: Complementary Modelling Approaches for Patient-Specific Medicine

This habilitation manuscript contains research on biophysical and statistical modeling of the heart, as well as interactions between these two approaches

When Cardiac Biophysics Meets Groupwise Statistics: Complementary Modelling Approaches for Patient-Specific Medicine

This habilitation manuscript contains research on biophysical and statistical modeling of the heart, as well as interactions between these two approaches

Integrated Heart - Coupling multiscale and multiphysics models for the simulation of the cardiac function

Mathematical modelling of the human heart and its function can expand our understanding of various cardiac diseases, which remain the most common cause of death in the developed world. Like other physiological systems, the heart can be understood as a complex multiscale system involving interacting phenomena at the molecular, cellular, tissue, and organ levels. This article addresses the numerical modelling of many aspects of heart function, including the interaction of the cardiac electrophysiology system with contractile muscle tissue, the sub-cellular activation-contraction mechanisms, as well as the hemodynamics inside the heart chambers. Resolution of each of these sub-systems requires separate mathematical analysis and specially developed numerical algorithms, which we review in detail. By using specific sub-systems as examples, we also look at systemic stability, and explain for example how physiological concepts such as microscopic force generation in cardiac muscle cells, translate to coupled systems of differential equations, and how their stability properties influence the choice of numerical coupling algorithms. Several numerical examples illustrate three fundamental challenges of developing multiphysics and multiscale numerical models for simulating heart function, namely: (i) the correct upscaling from single-cell models to the entire cardiac muscle, (ii) the proper coupling of electrophysiology and tissue mechanics to simulate electromechanical feedback, and (iii) the stable simulation of ventricular hemodynamics during rapid valve opening and closure

Personalized noninvasive imaging of volumetric cardiac electrophysiology

Three-dimensionally distributed electrical functioning is the trigger of mechanical contraction of the heart. Disturbance of this electrical flow is known to predispose to mechanical catastrophe but, due to its amenability to certain intervention techniques, a detailed understanding of subject-specific cardiac electrophysiological conditions is of great medical interest. In current clinical practice, body surface potential recording is the standard tool for diagnosing cardiac electrical dysfunctions. However, successful treatments normally require invasive catheter mapping for a more detailed observation of these dysfunctions. In this dissertation, we take a system approach to pursue personalized noninvasive imaging of volumetric cardiac electrophysiology. Under the guidance of existing scientific knowledge of the cardiac electrophysiological system, we extract the subject specific cardiac electrical information from noninvasive body surface potential mapping and tomographic imaging data of individual subjects. In this way, a priori knowledge of system physiology leads the physiologically meaningful interpretation of personal data; at the same time, subject-specific information contained in the data identifies parameters in individual systems that differ from prior knowledge. Based on this perspective, we develop a physiological model-constrained statistical framework for the quantitative reconstruction of the electrical dynamics and inherent electrophysiological property of each individual cardiac system. To accomplish this, we first develop a coupled meshfree-BE (boundary element) modeling approach to represent existing physiological knowledge of the cardiac electrophysiological system on personalized heart-torso structures. Through a state space system approach and sequential data assimilation techniques, we then develop statistical model-data coupling algorithms for quantitative reconstruction of volumetric transmembrane potential dynamics and tissue property of 3D myocardium from body surface potential recoding of individual subjects. We also introduce a data integration component to build personalized cardiac electrophysiology by fusing tomographic image and BSP sequence of the same subject. In addition, we develop a computational reduction strategy that improves the efficiency and stability of the framework. Phantom experiments and real-data human studies are performed for validating each of the framework’s major components. These experiments demonstrate the potential of our framework in providing quantitative understanding of volumetric cardiac electrophysiology for individual subjects and in identifying latent threats in individual’s heart. This may aid in personalized diagnose, treatment planning, and fundamentally, prevention of fatal cardiac arrhythmia