Analysis of travelling waves associated with the modelling of aerosolised skin grafts

A previous model developed by the authors investigates the growth patterns of keratinocyte cell colonies after they have been applied to a burn site using a spray technique. In this paper, we investigate a simplified one-dimensional version of the model. This model yields travelling wave solutions and we analyse the behaviour of the travelling waves. Approximations for the rate of healing and maximum values for both the active healing and the healed cell densities are obtained

A mathematical model of dynamic glioma-host interactions: receptor-mediated invasion and local proteolysis

We present a mathematical model of glioma spread based on cellular movement by receptor-mediated haptotaxis, local proteolysis of healthy tissue components by glioma-derived proteinases, malignant proliferative enhancement and host up-regulation of specific key extracellular matrix (ECM) components in response to the invading glioma. We subsequently consider the nature of glioma–host interactions as predicted by our model in order to test the hypothesis given in (Knott et al. (1998) that production of adhesive ECM components by the brain in response to the invading glioma may have the counter-intuitive effect of enhancing glioma invasion by assisting haptotactic migration. We suggest that host production of certain adhesive ECM chemicals can have a profound effect on both glioma invasion speed and the character of the glioma–host interface. In particular, we conclude that up-regulation of host ECM production in the vicinity of the glioma may produce a less diffuse glioma, providing clearer demarcation between glioma and healthy tissue, and thus improving the possibility of surgical resection within reasonable bounds

Avascular tumour dynamics and necrosis

We consider the dynamic growth of a tumour, concentrating on the possible development of a necrotic region and examine some simple tumour geometries in detail. The growth and death rates of the cells in the viable rim of the tumour are taken to be determined by the local oxygen concentration. Crucially the cell motion is determined by the forces generated by cell affinity, by cell interaction and by the need to get the waste products of cell death, primarily water, out of the tumour and products for cell growth, again primarily water, into the tumour. A consolidation type model with surface tension on the cells, slow viscous flow of the cells and porous media flow of the extracellular water is derived. The dynamic behaviour of this model is examined. Considering the very simple case where resistance to extracellular water flow dominates the problem, the model accounts naturally for the formation of a necrotic region. In regions where the extracellular water pressure gets too large, the cells are assumed to be ripped from the extracellular matrix and die. This model contrasts significantly from previous models which typically assume a necrotic region exists and that its behaviour is primarily governed directly by oxygen concentration. Here, the stress determines the necrotic region behaviour and this is affected by the oxygen only indirectly through the cell growth and death rates. The predicted time-dependent growth of one-dimensional, and spherical tumours are illustrated by numerical calculations

The migration of cells in multicell tumour spheroids

A mathematical model is proposed to explain the observed internalization of microspheres and3H-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

Novel solutions for a model of wound healing angiogenesis

We prove the existence of novel, shock-fronted travelling wave solutions to a model of wound healing angiogenesis studied in Pettet et al (2000 IMA J. Math. App. Med. 17 395–413) assuming two conjectures hold. In the previous work, the authors showed that for certain parameter values, a heteroclinic orbit in the phase plane representing a smooth travelling wave solution exists. However, upon varying one of the parameters, the heteroclinic orbit was destroyed, or rather cut-off, by a wall of singularities in the phase plane. As a result, they concluded that under this parameter regime no travelling wave solutions existed. Using techniques from geometric singular perturbation theory and canard theory, we show that a travelling wave solution actually still exists for this parameter regime. We construct a heteroclinic orbit passing through the wall of singularities via a folded saddle canard point onto a repelling slow manifold. The orbit leaves this manifold via the fast dynamics and lands on the attracting slow manifold, finally connecting to its end state. This new travelling wave is no longer smooth but exhibits a sharp front or shock. Finally, we identify regions in parameter space where we expect that similar solutions exist. Moreover, we discuss the possibility of more exotic solutions

Modelling the effects of bone fragment contact in fracture healing

The fracture healing process is modulated by the mechanical environment created by imposed loads and motion between the bone fragments. Contact between the fragments obviously results in a significantly different stress and strain environment to a uniform fracture gap containing only soft tissue (e.g. haematoma). The assumption of the latter in existing computational models of the healing process will hence exaggerate the inter-fragmentary strain in many clinically-relevant cases. To address this issue, we introduce the concept of a contact zone that represents a variable degree of contact between cortices by the relative proportions of bone and soft tissue present. This is introduced as an initial condition in a two-dimensional iterative finite element model of a healing tibial fracture, in which material properties are defined by the volume fractions of each tissue present. The algorithm governing the formation of cartilage and bone in the fracture callus uses fuzzy logic rules based on strain energy density resulting from axial compression. The model predicts that increasing the degree of initial bone contact reduces the amount of callus formed (periosteal callus thickness 3.1mm without contact, down to 0.5mm with 10% bone in contact zone). This is consistent with the greater effective stiffness in the contact zone and hence, a smaller inter-fragmentary strain. These results demonstrate that the contact zone strategy reasonably simulates the differences in the healing sequence resulting from the closeness of reduction