Tissue Mechanics

The myocardium is a non-linear, hyperelastic material, meaning its deformation is not directly proportional to the applied force, and it can stretch and return to its original shape. Additionally, it is nearly incompressible, meaning its tissue volume doesn't change significantly under pressure. The myocardium is orthotropic, which means its mechanical properties vary along three distinct directions that are aligned with the structure of the heart. These directions representing a layered organization of myocytes and fibres are characterized by a right-handed orthonormal set of basis vectors. These basis vectors consist of the fiber axis ${\mathbf{f}}_0\left(\mathbf{x}\right)$, which coincides with the prevailing orientation of the myocytes at location $\mathbf{x}$, the sheet axis ${\mathbf{s}}_0\left(\mathbf{x}\right)$, and the sheet-normal axis ${\mathbf{n}}_0\left(\mathbf{x}\right)$. The mechanical deformation of the tissue is described by Cauchy equation of motion.

Equation 1: Cauchy's Equation of Motion

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

Constitutive models

Constitutive models are mathematical descriptions used to represent the mechanical behavior of materials under various loading conditions. In the case of the myocardium it exhibits complex mechanical properties as listed above that need to be accurately captured by these models. To model the mechanical behavior of the myocardium, constitutive models often use a strain energy function ψ to describe how the tissue stores energy as it deforms. For nearly incompressible materials like the myocardium, this strain energy function is typically divided into two parts: a volumetric part which accounts for changes in volume and an isochoric part which accounts for changes in shape without altering volume.

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.