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. (C) 2021 Elsevier B.V. All rights reserved.