A curvilinear isogeometric framework for the electromechanical activation of thin muscular tissues

We propose an isogeometric approximation of the equations describing the propagation of an electrophysiologic stimulus over a thin cardiac tissue with the subsequent muscle contraction. The underlying method relies on the monodomain model for the electrophysiological sub-problem. This requires the solution of a reaction–diffusion equation over a surface in the three-dimensional space. Exploiting the benefits of the high-order NURBS basis functions within a curvilinear framework, the method is found to reproduce complex excitation patterns with a limited number of degrees of freedom. Furthermore, the curvilinear description of the diffusion term provides a flexible and easy-to-implement approach for general surfaces. At the discrete level, two different approaches for integrating the ionic current are investigated in the isogeometric analysis framework. The electrophysiological stimulus is converted into a mechanical load employing the well-established active strain approach. The multiplicative decomposition of the deformation gradient tensor is grafted into a classical finite elasticity weak formulation, providing the necessary tensor expressions in curvilinear coordinates. The derived expressions provide what is needed to implement the active strain approach in standard finite-element solvers without resorting to dedicated formulations. Such a formulation is valid for general three-dimensional geometries and isotropic hyperelastic materials. The formulation is then restricted to Kirchhoff–Love shells by means of the static condensation of the material tensor. The purely elastic response of the structure is investigated with simple static test-cases of thin shells undergoing different active strain patterns. Eventually, various numerical tests performed with a staggered scheme illustrate that the coupled electromechanical model can capture the excitation–contraction mechanism over thin tissues and reproduce complex curvature variations.