Bridging Scales: A Three-Dimensional Electromechanical Finite Element Model of Skeletal Muscle

Abstract

This paper introduces a framework for skeletal muscles that couples outputs from a detailed biophysically based electrophysiological cell model to a three-dimensional continuum-based finite element model of muscle mechanics. Due to the unique manner in which a skeletal muscle is activated, specifically the fact that neighboring fibers are electrically isolated and can act independently of each other, a completely new and novel coupling framework has been created. Within this framework, the electrical activity within a fiber is modeled with a biophysically based cell model, which is itself an amalgamation of several existing cell models. From this amalgamated cell model, specific output parameters that describe the level of crossbridge activity are computed and stored within a lookup table. This lookup table is then used to map the appropriate level of activity to all fibers within the muscle. To link the level of activity to a three-dimensional finite flement model of a skeletal muscle, which is based on principles of continuum mechanics, an upscaling method is introduced to compensate for the fact that the finite element mesh does not attempt to separately represent each individual fiber. This upscaling method allows the stress equilibrium equations to be computed at each Gauss point based on different values of the cell model outputs in all the neighboring cells. Since adjacent fibers can operate independently, the cell model outputs used in the finite element solution of the finite elasticity equations are discontinuous. The behavior and performance of the entire coupling framework is carefully analyzed in some simple test cases analyzing the reduction of the discretization error with respect to a sequence of uniformly refined meshes and different activation patterns. The results show that the error-reduction factors obtained from the electromechanical framework using triquadratic Lagrange and tricubic Hermite basis functions in solving the Galerkin finite element stress equilibrium equations are very similar to those obtained from a mechanics-only continuum-based model. Following this, an example of this process applied to the lateral pterygoid muscle is presented. The proposed framework can be used, for example, to investigate the mechanical effects with respect to cellular changes or to analyze the effects of different neuromuscular activation patterns on the tissue response.