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This article investigates heterogeneous proliferation within a seeded three-dimensional scaffold structure with the purpose of improving protocols for engineered tissue growth. A simple mathematical model is developed to examine the very strong interaction between evolving oxygen profiles and cell distributions within cartilaginous constructs. A comparison between predictions based on the model and experimental evidence is given for both spatial and temporal evolution of the oxygen tension and cell number density, showing that behaviour for the first 14 days can be explained well by the mathematical model. The dependency of the cellular proliferation rate on the oxygen tension is examined and shown to be similar in size to previous work but linear in form. The results show that cell-scaffold constructs that rely solely on diffusion for their supply of nutrients will inevitably produce proliferation-dominated regions near the outer edge of the scaffold in situations when the cell number density and oxygen consumption rate exceed a critical level. Possible strategies for reducing such non-uniform proliferation, including the conventional methods of enhancing oxygen transport, are outlined based on the model predictions.

During the development of vertebrate embryos, cell migrations occur on an underlying tissue domain in response to some factor, such as nutrient. Over the time scale of days in which this cell migration occurs, the underlying tissue is itself growing. Consequently cell migration and colonization is strongly affected by the tissue domain growth. Numerical solutions for a mathematical model of chemotactic migration with no domain growth can lead to travelling waves of cells with constant velocity; the addition of domain growth can lead to travelling waves with nonconstant velocity. These observations suggest a mathematical approximation to the full system equations, allowing the method of characteristics to be applied to a simplified chemotactic migration model. The evolution of the leading front of the migrating cell wave is analysed. Linear, exponential and logistic uniform domain growths are considered. Successful colonization of a growing domain depends on the competition between cell migration velocity and the velocity and form of the domain growth, as well as the initial penetration distance of the cells. In some instances the cells will never successfully colonize the growing domain. These models provide an insight into cell migration during embryonic growth, and its dependence upon the form and timing of the domain growth.

A mathematical model is proposed to explain the observed internalization of microspheres and 3H-thymidine labelled cells in steady-state multicellular spheroids. The model uses the conventional ideas of nutrient diffusion and consumption by the cells. In addition, a very simple model of the progress of the cells through the cell cycle is considered. Cells are divided into two classes, those proliferating (being in G1, S, G2 or M phases) and those that are quiescent (being in G0). Furthermore, the two categories are presumed to have different chemotactic responses to the nutrient gradient. The model accounts for the spatial and temporal variations in the cell categories together with mitosis, conversion between categories and cell death. Numerical solutions demonstrate that the model predicts the behavior similar to existing models but has some novel effects. It allows for spheroids to approach a steady-state size in a non-monotonic manner, it predicts self-sorting of the cell classes to produce a thin layer of rapidly proliferating cells near the outer surface and significant numbers of cells within the spheroid stalled in a proliferating state. The model predicts that overall tumor growth is not only determined by proliferation rates but also by the ability of cells to convert readily between the classes. Moreover, the steady-state structure of the spheroid indicates that if the outer layers are removed then the tumor grows quickly by recruiting cells stalled in a proliferating state. Questions are raised about the chemotactic response of cells in differing phases and to the dependency of cell cycle rates to nutrient levels.

Tactically driven cell movement modelled by coupled advection-reactiondiffusion (ARD) equations typically exhibit smooth travelling waves, and less frequently sharp interfaces in the wave form. We study the existence of travelling waves with smooth and sharp interfaces in coupled ARD models by using geometric singular perturbation techniques. In particular, we show that a travelling wave analysis under an appropriate Liénard transformation reveals a generic fold condition to observe shock-like interfaces in the wave form. This geometric approach further explains automatically well-known jump and entropy conditions for shocks in hyperbolic PDE theory (Rankine-Hugoniot and Lax conditions). Our analysis also shows that canards, a special class of solutions within singular perturbation problems, play an important role in the construction of travelling waves with smooth and sharp interfaces.