A continuum model for the development of tissue-engineered cartilage around a chondrocyte.

The limited ability of cartilage to repair when damaged has led to the investigation of tissue engineering as a method for reconstructing cartilage. We propose a continuum multispecies model for the development of cartilage around a single chondrocyte. As in healthy cartilage, the model predicts a balance between synthesis, transport, binding and decay of matrix components. Two mechanisms are investigated for the transport of soluble matrix components: diffusion and advection, caused by displacement of the scaffold medium. Numerical results indicate that a parameter defined by the ratio of the flux of soluble components out of the chondrocyte and its diffusive flux determines which of these mechanisms is dominant. We investigate the diffusion-dominated and advection-dominated limiting cases using perturbation analysis. Using parameter values from the literature, our modelling results suggest that both diffusion and advection are significant mechanisms in developing cartilage. Moreover, in this parameter regime, results are particularly sensitive to parameter values. These two observations could explain differences observed experimentally between various scaffold media. Modelling results are also used to predict the minimum chondrocyte seeding density required to produce functional cartilage.

A traveling wave model for invasion by precursor and differentiated cells.

We develop and investigate a continuum model for invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric nervous system, but is presented in general terms and is of broader applicability. Two cell populations are identified and modeled explicitly; a population of precursor cells that migrate and proliferate, and a population of differentiated cells derived from the precursors which have impaired migration and proliferation. The equation describing the precursor cells is based on Fisher's equation with the addition of a carrying-capacity limited differentiation term. Two variations of the proliferation term are considered and compared. For most parameter values, the model admits a traveling wave solution for each population, both traveling at the same speed. The traveling wave solutions are investigated using perturbation analysis, phase plane methods, and numerical techniques. Analytical and numerical results suggest the existence of two wavespeed selection regimes. Regions of the parameter space are characterized according to existence, shape, and speed of traveling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, our results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells.

Dispersal and settling of translocated populations: a general study and a New Zealand amphibian case study.

Translocations are widely used to reintroduce threatened species to areas where they have disappeared. A continuum multi-species model framework describing dispersal and settling of translocated animals is developed. A variety of different dispersal and settling mechanisms, which may depend on local population density and/or a pheromone produced by the population, are considered. Steady state solutions are obtained using numerical techniques for each combination of dispersal and settling mechanism and for both single and double translocations at the same location. Each combination results in a different steady state population distribution and the distinguishing features are identified. In addition, for the case of a single translocation, a relationship between the radius of the settled region and the population size is determined, in some cases analytically. Finally, the model is applied to a case study of a double translocation of the Maud Island frog, Leiopelma pakeka. The models suggest that settling occurs at a constant rate, with repulsion evidently playing a significant role. Mathematical modelling of translocations is useful in suggesting design and monitoring strategies for future translocations, and as an aid in understanding observed behaviour.