## Cardiac Electromechanics

## Tissue Mechanics

The ventricular myocardium is modeled as a nonlinear, hyperelastic, nearly incompressible, and anisotropic material with a layered organization of myocytes and fibres that is characterized by a right-handed orthonormal set of basis vectors. These basis vectors consist of the fiber axis \(\vec{f}_0(\vx)\), which coincides with the prevailing orientation of the myocytes at location \(\vx\), the sheet axis \(\vec{s}_0(\vx)\), and the sheet-normal axis \(\vec{n}_0(\vx)\). The mechanical deformation of the tissue is described by Cauchy equation of motion.

Under stationary equilibrium assumptions we have the following quasi-static boundary value problem: For a given pressure \(p(t)\), find the unknown displacement \(\vu\) such that $$ \begin{alignat}{3} \label{equ:bvp} - \nabla \cdot \sigma(\vu,t) & = 0 &\quad& \text{ in } \Omega \\ \sigma(\vu,t) \cdot \vn & = - p(t) \, \vn &\quad& \text{ on } \Gamma_N \notag \\ \sigma(\vu,t) \cdot \vn & = 0 &\quad& \text{ on } \Gamma_H \notag \\ \vu & = 0 &\quad& \text{ on } \Gamma_D \notag \end{alignat} $$ holds for \(t \in [0, T] \). By \(\Omega \subset \mathrm{R}^3\), we denote the deformed geometry and by \(\Gamma = \partial \Omega\) we define its boundary with \(\Gamma = \overline{\Gamma_D} \cup \overline{\Gamma_H} \cup \overline{\Gamma_N}\) and \(|\Gamma_D| > 0\). The normal outward vector of \(\Gamma\) is denoted by \(\vn\). The total Cauchy stress tensor \(\sigma\) refers to the sum of a passive stress tensor \(\sigmapas\) and an active stress tensor \(\sigmaact\). That is, \(\sigma = \sigmapas + \sigmaact\) with $$ \begin{equation} \sigmapas = J^{-1} \tensor{F} \left( 2 \, \frac{\partial \Psi(\tensor{C})}{\partial \tensor{C}} \right) \tensor{F}^{\top} \end{equation} $$ where \(\tensor{F}\) is the deformation gradient, \(\Psi\) is the strain energy function, \(J=\operatorname{det} \tensor{F}\) is the Jacobian and \(\tensor{C}=\tensor{F}^\top\tensor{F}\) is the right Cauchy--Green strain tensor.

## Electromechanical Coupling

Electrophysiology drives mechanical deformation through a process referred to as excitation-contraction coupling [Bers, 2002], whereas deformation in turn influences the electrophysiological state of the heart through mechano-electric feedback mechanisms [Quinn et al., 2014]. Depending on mechanical boundary conditions and external loads imposed, the active forces generated by the myocardium drive mechanical contraction and relaxation of the walls, forcing blood in, through, and out of the heart's chambers.