This dissertation investigates the mechanics of semiflexible filament network and the effects of polydispersity, anisotropy, and labile crosslinks through numerical simulations.

This study starts with exploring the effects of filament length polydispersity on the mechanical properties of semiflexible network. Extending previous studies on monodisperse network, the responses of networks of bimodal and exponential length distributions are tested numerically. It is found that in polydisperse networks, mixtures of long and short filaments interact cooperatively to generally produce mechanical response closer to the affine prediction than comparable monodisperse networks of either long or short filaments. Overall, length polydispersity has the effect of sharpening and shifting the affine/nonaffine(A/NA) transition to lower network densities.

The effect of adding long, stiff filaments to a semiflexible network is studied next. It is shown that the addition of a small fraction of longer and stiffer filaments (microtubules) to a nonaffine network (actin filaments) leads to a significant increase in its overall elastic moduli, even though the long filaments do not form a stress bearing network by themselves. Moreover, there is a strong negative correlation between long filament density and local geometric measures of nonaffinity.

Anisotropic, monodisperse network is then investigated and like isotropic networks, it undergoes an A/NA transition controlled by the ratio of the filament length to the nonaffinity length $\lambda$. Deep in the nonaffine regime, however, these anisotropic networks exhibit a vanishing linear response regime for highly ordered networks and a dependence of the shear modulus on shear direction. These features can be understood in terms of a generalized floppy modes analysis of the nonaffine mechanics and a type of cooperative Euler buckling, which are discussed in this study, too.

Lastly, by allowing crosslinks to break and rebind, network's dynamic response is studied. It is shown that force-induced crosslink breakage leads to significant creeping and the network retains part of its elastic modulus even after significant plastic flow. The spatial correlation of crosslink breakage is also studied.

The heart is a complex organ found in all vertebrates. It circulates blood throughout the body by cyclically contracting and relaxing. In an unhealthy heart, the pumping efficiency decreases, causing poor blood flow. What changes occur in a sick heart and how they affect its functioning is a subject of research. Much has been already understood about the electrophysiological changes that occurs in a sick heart. However, it is vastly unknown how these changes relate to mechanical dysfunction of the heart. A coupled electrophysiology and mechanics study of the heart may help answer some of the questions related to this. The focus of this dissertation is twofold. First we investigate efficient numerical techniques for solving the governing equations and discuss the convergence and accuracy of finite element electrophysiology simulation using the monodomain model. Secondly, we show the effect of mechanical deformation on stability of spiral waves, using combined electrophysiology and mechanics simulation.

Cardiac arrhythmias are a leading cause of death worldwide. Notably, the electrophysiologiy and microstructural requirements for a fatal ventricular arrhythmia remain incompletely understood, thereby the treatment remains largely empirical. Standard antiarrhythmic drug therapy has failed to reduce, and in some instances has increased, the incidence of Sudden Cardiac Death (SCD). Hence, a more complete understanding of the mechanisms that foment a fatal arrhythmia is needed and computational models offer an excellent way to test hypotheses about various changes to cellular electrophysiology and myocardial microstructure in a manner not easily achieved in experiments. The understanding of associated deformation is also a longstanding research field; to some extent it provides the paradigm of a complex system, as it incorporates several mathematical issues such as geometric and material nonlinearity, complex geometrical and material data, fluid-structure interaction, that are already challenging by themselves without even mentioning the social relevance of the problem.

This thesis is concerned with the development of a unified formulation of cardiac electromechanics. The computational requirements for physiologically and numerically accurate computational analysis of the coupled equations of cardiac electrophysiology and finite-deformation contractile mechanics are carefully examined. The validation criterion which needs to be satisfied by any generic model are laid out.

The voltage evolution in the heart is obtained by solving the reaction diffusion monodomain equations. The convergence properties of finite-element procedures, employing various combinations of different shape functions, quadrature methods, and operator splitting strategies are studied which place the most stringent limitations on mesh resolution. Computational speedup is achieved preferential by row-sum lumping of the capacitance and mass matrices. However, selective lumping of these matrices can have noticeable effects on the convergence.

Finite element model of a rabbit ventricle is developed using Diffusion Tensor(DT) - MRI images. His-Bundle is included in the model to provide the correct activation sequence. Different geometries of the conduction system are analyzed comparing the obtained activation pattern and six lead electrocardiogram (ECG). The ventricular model is further validated by reproducing scroll wave break up.

Cardiac excitation coupling is modeled using an active deformation formulation based on Calcium dynamics. Analysis of the formulation of the active part of the deformation gradient and its dependence on Calcium concentration is studied. Effects of material models and inclusion of fiber anisotropy are also studied. Using a simplified ellipsoid model with assumed fiber orientation consistent with commonly used values in literature we successfully reproduce the twisting action and 60% volume reduction which is typically observed in experiments.