D.L.S. McElwain

A mathematical model of tumour-induced capillary growth.

The corneal limbal vessels of an animal host respond to the presence of a source of Tumour Angiogenesis Factor (TAF) implanted in the cornea by the formation of new capillaries which grow towards the source. This neovasculature can be easily seen and studied and this paper describes a mathematical model of some of the important features of the growth. The model includes the diffusion of TAF, the formation of sprouts from pre-existing vessels and models the movement of these sprouts to form new capillaries as a chemotactic response to the presence of TAF. Numerical results are produced for various values of the parameters which characterize the model and it is suggested that the model might form the framework for further theoretical work on related phenomena such as wound healing or to develop strategies for the investigation of anti-angiogenesis.

A Two-Phase Model of Solid Tumour Growth

Many solid tumour growth models are formulated as systems of parabolic and/or hyperbolic equations. Here an alternative, two-phase theory is developed to describe solid tumour growth. Versions of earlier models are recovered when suitable limits of the new model are taken. We contend that the multiphase approach represents a more general, and natural, modelling framework for studying solid tumour growth than existing theories.

A model of wound-healing angiogenesis in soft tissue

Angiogenesis, or blood vessel growth, is a critical step in the wound-healing process, involving the chemotactic response of blood vessel endothelial cells to macrophage-derived factors produced in the wound space. In this article, we formulate a system of partial differential equations that model the evolution of the capillary-tip endothelial cells, macrophage-derived chemoattractants, and the new blood vessels during the tissue repair process. Chemotaxis is incorporated as a dominant feature of the model, driving the wave-like ingrowth of the wound-healing unit. The resulting model admits traveling wave solutions that exhibit many of the features characteristic of wound healing in soft tissue. The steady propagation of the healing unit through the wound space, the development of a dense band of fine, tipped capillaries near the leading edge of the wound-healing unit (the brush-border effect), and an elevated vessel density associated with newly healed wounds, prior to vascular remodeling, are all discernible from numerical simulations of the full model. Numerical simulations mimic not only the normal progression of wound healing but also the potential for some wounds to fail to heal. Through the development and analysis of a simplified model, insight is gained into how the balance between chemotaxis, tip proliferation, and tip death affects the structure and speed of propagation of the healing unit. Further, expressions defining the healed vessel density and the wavespeed in terms of known parameters lead naturally to the identification of a maximum wavespeed for the wound-healing process and to bounds on the healed vessel density. The implications of these results for wound-healing management are also discussed.

ON THE ROLE OF ANGIOGENESIS IN WOUND HEALING

Angiogenesis, the formation of blood vessels, may be described as a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then develop and organize themselves into a dendritic structure. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours. In this paper we present a mathematical model which describes the rôle of angiogenesis as observed during (soft-tissue) wound healing. We focus attention on certain principal players involved in this complex process, namely capillary tips, capillary sprouts, fibroblasts, macrophage-derived chemical attractants, oxygen and extracellular matrix. The model consists of a system of nonlinear partial differential equations describing the interactions in space and time of the above substances. Numerical simulations are presented which are in very good qualitative agreement with experimental observations.

A re-examination of oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics

Abstract Oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is re-examined and the results of a recent paper by Lin corrected. An extension of this model to include external diffusion is made and its effect is shown to be significant. A model which attempts to model the nucleus as a central sphere which does not consume any oxygen is also investigated. Finally, a perturbation solution for a small Michaelis constant is developed.

Travelling waves in a wound healing assay

Several authors have predicted that cell propagation in a number of biological contexts, for example, wound healing, tumour cell invasion, angiogenesis etc., occurs due to a constant speed travelling wave of invasion. The analyses of these models to arrive at this prediction is, in many cases, essentially an extension of the classical analysis of Fishers equation. Here, we show that a very simple wound healing assay does indeed give rise to constant speed travelling waves. To our knowledge, this is the first verification of Fishers equation in a medical context.

A model for the growth of a solid tumor with non-uniform oxygen consumption

Abstract A model for the growth of a solid in vitro tumor including the effects of non-uniformity in oxygen consumption and proliferation rate is developed. The appropriate equations which describe the diffusion of oxygen are set up and then solved. The growth equation is formulated and a method of solution described. The results of this new model are compared with the predictions of an earlier model which assumes that the consumption of nutrient by live cells is uniform.

A NEW APPROACH TO MODELLING THE FORMATION OF NECROTIC REGIONS IN TUMOURS

We present a mathematical model of the growth of tumours. The cells in the tumour are taken to proliferate and die at rates determined by the concentration of oxygen which diffuses into the tumour across its surface. Tumour cells are assumed to be composed primarily of water while the extracellular water is taken to move through the tumour as a porous media flow between the cells. Exchange of water between the two is governed by cell proliferation. We model the mass of cells as an inviscid fluid with the pressure in the fluid, keeping the cells loosely packed together. Cells move in response to pressure in both fluids until the extracellular water pressure exceeds the cell pressure, resulting in the rupture of the tumour cells as they are ripped from one another. The resulting model is one of porous media flow with distributed sources and sinks determined by the oxygen concentration. The boundary conditions change type depending on whether the tumour surface is retreating or advancing. Retreating interfaces leave ruptured cells creating necrotic regions. An example of the model behaviour in one dimension is presented.

Cell migration in multicell spheroids: Swimming against the tide

Multicell spheroids, small spherical clusters of cancer cells, have become an important in vitro model for studying tumour development given the diffusion limited geometry associated with many solid tumour growths. Spheroids expand until they reach a dormant state where they exhibit a grossly static three-layered structure. However, at a cellular level, the spheroid is demonstrably dynamic with constituent cells migrating from the outer well-nourished region of the spheroid toward the necrotic central core. The mechanism that drives the migrating cells in the spheroid is not well understood. In this paper we demonstrate that recent experiments on internationalization can be adequately described by implicating pressure gradients caused by differential cell proliferation and cell death as the primary mechanism. Although chemotaxis plays a role in cell movement, we argue that it acts against the passive movement caused by pressure differences.

Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth

Abstract A model for the growth of spheroids including a recently discovered cell loss mechanism is formulated. This model reproduces the observed growth patterns and predicts that, under certain conditions, a dormant state can exist without the formation of a central coagulative necrotic region, in agreement with recent experiment evidence. Growth patterns are presented for several parameter sets.