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Most of the existing mathematical models for tumour growth and tumour-induced angiogenesis neglect blood flow. This is an important factor on which both nutrient and metabolite supply depend. In this paper we aim to address this shortcoming by developing a mathematical model which shows how blood flow and red blood cell heterogeneity influence the growth of systems of normal and cancerous cells. The model is developed in two stages. First we determine the distribution of oxygen in a native vascular network, incorporating into our model features of blood flow and vascular dynamics such as structural adaptation, complex rheology and red blood cell circulation. Once we have calculated the oxygen distribution, we then study the dynamics of a colony of normal and cancerous cells, placed in such a heterogeneous environment. During this second stage, we assume that the vascular network does not evolve and is independent of the dynamics of the surrounding tissue. The cells are considered as elements of a cellular automaton, whose evolution rules are inspired by the different behaviour of normal and cancer cells. Our aim is to show that blood flow and red blood cell heterogeneity play major roles in the development of such colonies, even when the red blood cells are flowing through the vasculature of normal, healthy tissue.

This Timeline article charts progress in mathematical modelling of cancer over the past 50 years, highlighting the different theoretical approaches that have been used to dissect the disease and the insights that have arisen. Although most of this research was conducted with little involvement from experimentalists or clinicians, there are signs that the tide is turning and that increasing numbers of those involved in cancer research and mathematical modellers are recognizing that by working together they might more rapidly advance our understanding of cancer and improve its treatment.

In this paper the theory of mixtures is used to develop a two-phase model of an avascular tumour, which comprises a solid, cellular, phase and a liquid phase. Mass and momentum balances which are used to derive the governing equations are supplemented by constitutive laws that distinguish the two phases and enable the stresses within the tumour to be calculated. Novel features of the model include the dependence of the cell proliferation rate on the cellular stress and the incorporation of mass exchange between the two phases. A combination of numerical and analytical techniques is used to investigate the sensitivity of equilibrium tumour configurations to changes in the model parameters. Variation of parameters such as the maximum cell proliferation rate and the rate of natural cell death yield results which are consistent with analyses performed on simpler tumour growth models and indicate that the two-phase formulation is a natural extension of the earlier models. New predictions relate to the impact of mechanical effects on the tumour's equilibrium size which decreases under increasing stress and/or external loading. In particular, as a parameter which measures the reduction in cell proliferation due to cell stress is increased a critical value is reached, above which the tumour is eliminated.

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In this paper we compare two alternative theoretical approaches for simulating the growth of cell aggregates in vitro: individual cell (agent)-based models and continuum models. We show by a quantitative analysis of both a biophysical agent-based and a continuum mechanical model that for densely packed aggregates the expansion of the cell population is dominated by cell proliferation controlled by mechanical stress. The biophysical agent-based model introduced earlier (Drasdo and Hoehme in Phys Biol 2:133-147, 2005) approximates each cell as an isotropic, homogeneous, elastic, spherical object parameterised by measurable biophysical and cell-biological quantities and has been shown by comparison to experimental findings to explain the growth patterns of dense monolayers and multicellular spheroids. Both models exhibit the same growth kinetics, with initial exponential growth of the population size and aggregate diameter followed by linear growth of the diameter and power-law growth of the cell population size. Very sparse monolayers can be explained by a very small or absent cell-cell adhesion and large random cell migration. In this case the expansion speed is not controlled by mechanical stress but by random cell migration and can be modelled by the Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) reaction-diffusion equation. The growth kinetics differs from that of densely packed aggregates in that the initial spread, as quantified by the radius of gyration, is diffusive. Since simulations of the lattice-free agent-based model in the case of very large random migration are too long to be practical, lattice-based cellular automaton (CA) models have to be used for a quantitative analysis of sparse monolayers. Analysis of these dense monolayers leads to the identification of a critical parameter of the CA model so that eventually a hierarchy of three model types (a detailed biophysical lattice-free model, a rule-based cellular automaton and a continuum approach) emerge which yield the same growth pattern for dense and sparse cell aggregates.

Objectives : The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation. Methods : At the subcellular level, deterministic models characterise molecular networks, such as cell-cycle control and Wnt signalling. The output of these models determines the behaviour of each epithelial cell in response to intra-, inter-and extracellular cues. The modular nature of the model enables us to easily modify individual assumptions and analyse their effects on the system as a whole. Results : We perform virtual microdissection and labelling-index experiments, evaluate the impact of various model extensions, obtain new insight into clonal expansion in the crypt, and compare our predictions with recent mitochondrial DNA mutation data. Conclusions : We demonstrate that relaxing the assumption that stem-cell positions are fixed enables clonal expansion and niche succession to occur. We also predict that the presence of extracellular factors near the base of the crypt alone suffices to explain the observed spatial variation in nuclear beta-catenin levels along the crypt axis.

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.