Continuous versus discrete (invariant) representations of fibrous structure for modeling non-linear anisotropic soft tissue behavior

The non-linear anisotropic mechanical response of soft tissue is largely dependent on the structure of the underlying collagen network. Collagen structure has been successfully quantified for various tissue types in terms of a locally defined fiber orientation distribution function. The continuous distribution function derived from structural data can be directly incorporated into an integral representation of the strain energy function for modeling tissue behavior. Alternatively, non-integral (often invariant-based) strain energy functions have been developed in which the collagen network structure is approximated using a discrete set of fiber classes. The advantage of such an approach is increased computational efficiency since the values of the strain energy and its derivatives (e.g. stress) can be evaluated without numerical integration. However, because of the structural simplifications such models are presumably unable to predict mechanical data as accurately as the models which incorporate a continuous orientation distribution function. In this work the ability of discrete versus continuous fiber models to capture the non-linear anisotropic response of soft tissue is critically analyzed. Both unimodal and bimodal fiber distributions are considered. A general formulation has been developed in terms of an arbitrary fiber strain energy function, such that the analysis can be performed for any suitable fiber material model. For tissue structures in which a discrete representation is suitable, techniques are presented for establishing the range of loading conditions in which model accuracy is not significantly compromised, thus justifying the use of an invariant-based modeling approach.