title: Mathematical modelling of cardiac function: constitutive law, fibre dispersion, growth and remodelling
Creators: name: Guan, Debao
Publisher: name: University of Glasgow
Year: 2021

description: 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&amp;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&amp;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&amp;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&amp;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&amp;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&amp;R mainly occurs in endocardium, in which the myocyte is the primary responder for the G&amp;R process. G&amp;R laws of collagen fibre have significant impacts on G&amp;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&amp;R is investigated by including fibre dispersion model in the G&amp;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&amp;R process of heart under different pathological conditions. These studies support to choose suitable constitutive laws and fibre architectures in G&amp;R model and illustrate the underlying mechanism of mechanical triggers in G&amp;R. It presents the potential for understanding the mechanics of heart failure and reveal hidden roles of different constituents in myocardium.