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In this paper we present a simple mathematical model for angiogenesis in wound healing and then compare the results of theoretical predictions from computer simulations with actual experimental data. Numerical simulations of the model equations exhibit many of the characteristic features of wound healing in soft tissue. For example, the steady propagation of the wound healing unit through the wound space, the development of a dense band of capillaries near the leading edge of the unit, and the elevated vessel density associated with newly healed wounds, prior to vascular remodelling, are all discernible from the simulations. The qualitative accuracy of the initial model is assessed by comparing the numerical results with independent clinical measurements that show how the surface area of a range of wounds changes over time. The model is subsequently modified to include the effect of vascular remodelling and its impact on the spatio-temporal structure of the vascular network investigated. Predictions are made concerning the effect that changes in physical parameters have on the healing process and also regarding the manner in which remodelling is initiated.

Oxygen heterogeneity in solid tumors is recognized as a limiting factor for therapeutic efficacy. This heterogeneity arises from the abnormal vascular structure of the tumor, but the precise mechanisms linking abnormal structure and compromised oxygen transport are only partially understood. In this paper, we investigate the role that red blood cell (RBC) transport plays in establishing oxygen heterogeneity in tumor tissue. We focus on heterogeneity driven by network effects, which are challenging to observe experimentally due to the reduced fields of view typically considered. Motivated by our findings of abnormal vascular patterns linked to deviations from current RBC transport theory, we calculated average vessel lengths L¯ and diameters d¯ from tumor allografts of three cancer cell lines and observed a substantial reduction in the ratio λ=L¯/d¯ compared to physiological conditions. Mathematical modeling reveals that small values of the ratio λ (i.e., λ<6) can bias hematocrit distribution in tumor vascular networks and drive heterogeneous oxygenation of tumor tissue. Finally, we show an increase in the value of λ in tumor vascular networks following treatment with the antiangiogenic cancer agent DC101. Based on our findings, we propose λ as an effective way of monitoring the efficacy of antiangiogenic agents and as a proxy measure of perfusion and oxygenation in tumor tissue undergoing antiangiogenic treatment.

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In this paper we present a simple mathematical model to describe the initial phase of placental development during which trophoblast cells invade the uterine tissue as a continuous mass of cells. The key physical variables involved in this crucial stage of mammalian development are assumed to be the invading trophoblast cells, the uterine tissue, trophoblast-derived proteases that degrade the uterine tissue, and protease inhibitors that neutralise the action of the proteases. Numerical simulations presented here are in good qualitative agreement with experimental observations and show how changes in the system parameters influence the rate and degree of trophoblast invasion. In particular we suggest that chemotactic migration is a key feature of trophoblast invasion and that the rate at which proteases are produced is crucial to the successful implantation of the embryo. For example, both insufficient and excess production of the proteases may result in premature halting of the trophoblasts. Such behaviour may represent the pathological condition of failed trophoblast implantation and subsequent spontaneous abortion.

Over the past twenty-five years there has been an unparalleled increase in understanding of cancer biology. This transformation is exemplified by Hanahan and Weinberg's decision in 2011 to expand their original Hallmarks of Cancer from six traits to ten! At the same time, mathematical modelling has emerged as a natural tool for unravelling the complex processes that contribute to the initiation and progression of tumours, for testing hypotheses about experimental and clinical observations, and assisting with the development of new approaches for improving its treatment. This article starts by reviewing some of the earliest models of tumour growth and tumour responses to radiotherapy. Following Hanahan and Weinberg's lead, attention then focuses on how closer collaboration with cancer scientists and access to experimental data is stimulating the development of new and increasingly detailed models which account, for example, for tumour-immune interactions and immunotherapy. The article concludes by discussing the ways in which mathematical modelling is being integrated with experimental and clinical work and outlining how this could improve disease diagnosis and the delivery of effective personalised treatments to cancer patients. As such, the article serves as an introduction to mathematical modelling of cancer and its treatments, suitable for researchers seeking to enter the field.

Abstract. A multiscale model for vascular tumour growth is presented which includes systems of ordinary differential equations for the cell cycle and regulation of apoptosis in individual cells, coupled to partial differential equations for the spatio-temporal dynamics of nutrient and key signalling chemicals. Furthermore, these subcellular and tissue layers are incorporated into a cellular automaton framework for cancerous and normal tissue with an embedded vascular network. The model is the extension of previous work and includes novel features such as cell movement and contact inhibition. We present a detailed simulation study of the effects of these additions on the invasive behaviour of tumour cells and the tumour's response to chemotherapy. In particular, we find that cell movement alone increases the rate of tumour growth and expansion, but that increasing the tumour cell carrying capacity leads to the formation of less invasive dense hypoxic tumours containing fewer tumour cells. However, when an increased carrying capacity is combined with significant tumour cell movement, the tumour grows and spreads more rapidly, accompanied by large spatio-temporal fluctuations in hypoxia, and hence in the number of quiescent cells. Since, in the model, hypoxic/quiescent cells produce VEGF which stimulates vascular adaptation, such fluctuations can dramatically affect drug delivery and the degree of success of chemotherapy.

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