Multiscale modelling and homogenisation of fibre-reinforced hydrogels for tissue engineering.

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic-poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

A mathematical model for cell infiltration and proliferation in a chondral defect.

We develop a mathematical model to describe the regeneration of a hydrogel inserted into an ex vivo osteochondral explant. Specifically we use partial differential equations to describe the evolution of two populations of cells that migrate from the tissue surrounding the defect, proliferate, and compete for space and resources within the hydrogel. The two cell populations are chondrocytes and cells that infiltrate from the subchondral bone. Model simulations are used to investigate how different seeding strategies and growth factor placement within the hydrogel affect the spatial distribution of both cell types. Since chondrocyte migration is extremely slow, we conclude that the hydrogel should be seeded with chondrocytes prior to culture in order to obtain zonal chondrocyte distributions typical of those associated with healthy cartilage.

On a poroviscoelastic model for cell crawling.

In this paper a minimal, one-dimensional, two-phase, viscoelastic, reactive, flow model for a crawling cell is presented. Two-phase models are used with a variety of constitutive assumptions in the literature to model cell motility. We use an upper-convected Maxwell model and demonstrate that even the simplest of two-phase, viscoelastic models displays features relevant to cell motility. We also show care must be exercised in choosing parameters for such models as a poor choice can lead to an ill-posed problem. A stability analysis reveals that the initially stationary, spatially uniform strip of cytoplasm starts to crawl in response to a perturbation which breaks the symmetry of the network volume fraction or network stress. We also demonstrate numerically that there is a steady travelling-wave solution in which the crawling velocity has a bell-shaped dependence on adhesion strength, in agreement with biological observation.

Multiple travelling-wave solutions in a minimal model for cell motility.

Two-phase flow models have been used previously to model cell motility. In order to reduce the complexity inherent with describing the many physical processes, we formulate a minimal model. Here we demonstrate that even the simplest 1D, two-phase, poroviscous, reactive flow model displays various types of behaviour relevant to cell crawling. We present stability analyses that show that an asymmetric perturbation is required to cause a spatially uniform, stationary strip of cytoplasm to move, which is relevant to cell polarization. Our numerical simulations identify qualitatively distinct families of travelling-wave solutions that coexist at certain parameter values. Within each family, the crawling speed of the strip has a bell-shaped dependence on the adhesion strength. The model captures the experimentally observed behaviour that cells crawl quickest at intermediate adhesion strengths, when the substrate is neither too sticky nor too slippy.