Consistent formulation of the growth process at the kinematic and constitutive level for soft tissues composed of multiple constituents

Abstract

Previous studies have investigated the possibilities of modelling the change in volume and change in density of biomaterials. This can be modelled at the constitutive or the kinematic level. This work introduces a consistent formulation at the kinematic and constitutive level for growth processes. Most biomaterials consist of many constituents and can be approximated as being incompressible. These two conditions (many constituents and incompressibility) suggest a straightforward implementation in the context of the finite element (FE) method which could now be validated more easily against histological measurements. Its key characteristic variable is the normalised partial mass change. Using the concept of homeostatic equilibrium, we suggest two complementary growth laws in which the evolution of the normalised partial mass change is governed by an ordinary differential equation in terms of either the Piola–Kirchhoff stress or the Green–Lagrange strain. We combine this approach with the classical incompatibility condition and illustrate its algorithmic implementation within a fully nonlinear FE approach. This approach is first illustrated for a simple uniaxial tension and extension test for pure volume change and pure density change and is validated against previous numerical results. Finally, a physiologically based example of a two-phase model is presented which is a combination of volume and density changes. It can be concluded that the effect of hyper-restoration may be due to the systemic effect of degradation and adaptation of given constituents.

Keywords:

• volume growth
• density change
• incompressibility
• growth and remodelling
• adaptation
• homeostatic equilibrium

Acknowledgements

The authors would like to thank Alexander Gnech, Christian Krawzyk and Simo Schmidt from RWTH Aachen University for the support in validating the code; also we would like to acknowledge Prof. Peter Hunter from the Auckland Bioengineering Institute for making the source code of CMISS available. Furthermore, helpful discussions with Dr Paul Watton from Oxford University are gratefully acknowledged.

Notes

^1. Migration of cells within the tissue would be a possible contribution to the convective term, yet would also happen on a longer timescale.

^2. Programmed cell death by inflammatory molecules.

^3. We neglect the word infinitesimal in the sequel.

^4. Note that this fraction is not a derivative but the division of two infinitesimal volumes at different times. The infinitesimal d is upright, whereas the differential d is slanted.

^5. The first invariant was chosen for the sake of simplicity. Experimental guidance is necessary to further qualify the dependence on possibly other invariants and possible nonlinear dependencies.

^6. Himpel et al. (2005) used a range of values for the so-called stress stimulus attractor (Beaupre et al. 2005). In our approach, using the target strain quantity to be the homeostatic one, equates to setting the stress stimulus attractor to zero.