This thesis describes the development of a framework for computational modelling of electromechanics in the mouse, with the purpose of being able to integrate cellular and tissue scale observations in the mouse and investigate physiological hypotheses. Specifically, the framework is applied to interpret electromechanical coupling mechanisms and the progression of heart failure in genetically modified mice. Chapter 1 introduces the field of computational biology and provides context for the topics to be investigated. Chapter 2 reviews the biological background and mathematical bases for electromechanical models, as well as their limitations. In Chapter 3, a set of efficient computational methods for coupled cardiac electromechanics was developed. Among these are a modified Newton method combined with a solution predictor which achieves a 98% reduction in computational time for mechanics problems. In Chapter 4, this computational framework is extended to a multiscale electromechanical model of the mouse. This electromechanical model includes our novel cardiac cellular contraction model for mice, which is able to reproduce murine contraction dynamics at body temperature and high pacing frequencies, and provides a novel explanation for the biphasic force-calcium relation seen in cardiac myocytes. Furthermore, our electromechanical model of the left ventricle of the mouse makes novel predictions on the importance of strong velocity-dependent coupling mechanisms in generating a plateau phase of ventricular pressure transients during ejection. In Chapter 5, the framework was applied to investigate the progression of heart failure in genetically modified 'Serca2 knockout' mice, which have a major disruption in mechanisms governing calcium regulation in cardiac myocytes. Our modelling framework was instrumental in showing for the first time the incompatibility between previously measured cellular calcium transients and ventricular ejection. We were then able to integrate new experimental data collected in response to these observations to show the importance of beta-adrenergic stimulation in the progression of heart failure in these knockout mice. Chapter 6 presents the conclusions and discusses possibilities for future work.

Time splitting is a numerical procedure used in solution of partial differential equations whose solutions allow multiple time scales. Numerical schemes are split for handling the stiffness in equations, i.e. when there are multiple time scales with a few time scales being smaller than the others. When there are such terms with smaller time scales, due to the Courant number restriction, the computational cost becomes high if these terms are treated explicitly. In the present work a nonlinear advective equation is solved numerically using different techniques based on a generalised framework for splitting methods. The nonlinear advective equation was chosen because it has an analytical solution making comparisons with numerical schemes amenable and also because its nonlinearity mimics the equations encountered in atmospheric modelling. Using the nonlinear advective equation as a test bed, an analysis of the splitting methods and their influence on the split solutions has been made. An understanding of influence of splitting schemes requires knowledge of behaviour of unsplit schemes beforehand. Hence a study on unsplit methods has also been made. In the present work, using the nonlinear advective equation, it shown that the three time level schemes have high phase errors and underestimate energy (even though they have a higher order of accuracy in time). It is also found that the leap-frog method, which is used widely in atmospheric modelling, is the worst among examined unsplit methods. The semi implicit method, again a popular splitting method with atmospheric modellers is the worst among examined split methods. Three time-level schemes also need explicit filtering to remove the computational mode. This filtering can have a significant impact on the obtained numerical solutions, and hence three-time level schemes appear to be unattractive in the context of the nonlinear convective equation. Based on this experience, splitting methods for the two-time level schemes is proposed. These schemes realistically capture the phase and energy of the nonlinear advective equation.