Physically motivated strain energy for an architecturally detailed muscle model

INTRODUCTION The deformation of the muscle-tendon complex in response to forces depends on architectural design, tissue properties and activation patterns. In this study we derive a mathematical model of muscle which is amenable to finite element simulation, and which allows for flexible prescription of tissue properties and architecture. Architectural design within this model is characterized by its volume, pennation angle, orientation of fascicle planes and tissue distribution (both with respect to its properties and its dimensions). The model design is intended to be general enough to support both dynamic and isometric studies of muscle. Biological soft tissues exhibit anisotropy, nearincompressible behaviour and support large deformations. The material anisotropy is consistent with a 'fibre-reinforced' framework, where the material is isotropic in a plane transverse to the local fibre direction (transverse isotropy). Key to the computation of such hyperelastic deformations is a careful description of the constitutive laws and strain energy for the muscle-tendon complex. There are many studies that have modeled muscle responses in different loading conditions. In many earlier works simpler elements (i.e. spring-damper) and formulations were used. In order to have a muscle model with detailed architecture and be able to replicate many loading conditions a finite element study seems to be more practical. Among the strain energy functions that have been used for modeling soft tissues (including muscle) there are two distinct approach. The first approach (i.e. Weiss et al. 1996) had its strain energy based on invariants of a Cauchy-Green deformation tensor (classic formulation). The benefit of this formulation is that it is much easier to solve for the elasticity equation, and avoids higher levels of nonlinearity when compared to formulations from the second approach (i.e. Criscione et al. 2001). However, the second approach defined physically-based invariants and allows a faster and more understandable way in defining material constants using experimentally measured material properties. In this study we introduce a new strain energy function, derived from both these previous approaches, to model muscle tissue elasticity.