Nobile F, Quarteroni A, Ruiz-Baier R
MOX—Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Italy.
Int J Numer Method Biomed Eng. 2012 Jan;28(1):52-71. doi: 10.1002/cnm.1468.
We propose a finite element approximation of a system of partial differential equations describing the coupling between the propagation of electrical potential and large deformations of the cardiac tissue. The underlying mathematical model is based on the active strain assumption, in which it is assumed that there is a multiplicative decomposition of the deformation tensor into a passive and active part holds, the latter carrying the information of the electrical potential propagation and anisotropy of the cardiac tissue into the equations of either incompressible or compressible nonlinear elasticity, governing the mechanical response of the biological material. In addition, by changing from a Eulerian to a Lagrangian configuration, the bidomain or monodomain equations modeling the evolution of the electrical propagation exhibit a nonlinear diffusion term. Piecewise quadratic finite elements are employed to approximate the displacements field, whereas for pressure, electrical potentials and ionic variables are approximated by piecewise linear elements. Various numerical tests performed with a parallel finite element code illustrate that the proposed model can capture some important features of the electromechanical coupling and show that our numerical scheme is efficient and accurate.
我们提出了一个偏微分方程组的有限元近似，该方程组描述了电势传播与心脏组织大变形之间的耦合。基础数学模型基于主动应变假设，即假设变形张量可乘性分解为被动部分和主动部分，后者将电势传播信息和心脏组织各向异性纳入不可压缩或可压缩非线性弹性方程中，这些方程控制生物材料的力学响应。此外，通过从欧拉构型转换到拉格朗日构型，模拟电传播演化的双域或单域方程呈现出一个非线性扩散项。采用分段二次有限元来近似位移场，而对于压力、电势和离子变量则用分段线性元进行近似。使用并行有限元代码进行的各种数值测试表明，所提出的模型能够捕捉机电耦合的一些重要特征，并表明我们的数值格式是高效且准确的。
