A Polyconvex Anisotropic Strain-Energy Function for Soft Collagenous Tissues

Model Structure

Polyconvexity of a strain energy function is a very important mathematical condition, especially in the context of a boundary-value problem. In the paper presented here, the authors Itskov, Ehret and Mavrilas propose an exponential polyconvex anisotropic strain energy function. It is represented by a series with an arbitrary number of terms and associated material constants. Each term of this series a priori satisfies the condition of the energy- and stress free natural state so that no additional restrictions have to be imposed. The proposed strain energy function has an exponential form and is, therefore, very suitable for the application to soft biological tissues. Thus, a good agreement with experimental data on different types of tissues is achieved.

Histologically, soft biological tissues consist of various cell types and the extracellular matrix. The latter is composed of proteins such as fibrous collagen, elastin and of the ground substance. The mechanical behavior of soft tissues under quasi-static loading is dominated by the performance of its fibrous components, primarily collagen and elastin fibers.

Collagen is the main load-carrying element of tissue. The most common form being the fiber-forming collagen I. The variety of types, the amount and the structural organization of collagen fibrils and fibers among different collagenous tissues are responsible for their strong anisotropy and influence their mechanical properties accordingly.

Elastin fibers are thin strands of a rubbery consistence. The ensemble consisting of collagen and elastin leads to the characteristic exponential or "J-shaped" (Holzapfel 2001) stress strain response of soft biological tissues in tension tests. At smaller strains, the collagen fibers remain unstretched, wavy and crimped so that the mechanical response of the tissue is controlled by the soft and almost isotropic elastin. With increasing load, the collagen fibers gradually straighten and tend to align in the direction of loading, which causes a strong increase in the stiffness of the material.

At quasi-static loading, this response can be described by a hyperelastic constitutive model with an exponential strain energy function (Fung et al. 1979).

Soft collagenous tissues can be described more or less accurately using a model known as the Fung-elastic model (a strain energy function model). The Fung-Model although used to model soft-tissues, is not that accurate as it is generally not elliptic and for this reason not polyconvex and thus exhibits non-physical behaviour. Polyconvexity represents a very important mathematical condition, especially in the context of a boundary-value problem. In contrast, the polyconvex strain energy function that already exists, exhibits non-physical behaviour.

To avoid such non-physical behavior, Itskov et al. have proposed an exponential polyconvex anisotropic strain energy function that allows soft biological tissues to be more accurately modelled.

The model was implemented in a manner that could be used for peforming finite element model simulations on the CMISS software program developed at the Bioengineering Institute, University of Auckland.

For additional information on implementation of cellML files in CMISS, please refer to the following Link.

The complete original paper reference is cited below:

A Polyconvex anisotropic strain-energy function for soft collagenous tissues, M. Itskov, A.E. Ehret and D. Mavrilas, 2006. Biomechanics and Modeling in Mechanobiology , 5(1), 17-26. PubMed ID: Unknown