Abstract
Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state.
Highlights
Cancer is a complex disease not restricted to a single biological scale but involving processes occurring over multiple spatial scales, ranging from the subcellular scale to the tissue scale, and temporal scales.Many models developed previously focused on a single aspect of tumour growth, for example, cell cycle (Alarcon et al, 2004), cell apoptosis and necrosis (Byrne and Chaplain, 1998), diffusion of angiogenic factors (Chaplain and Stuart, 1991) or tumour angiogenesis (Balding and McElwain, 1985)
The focus of this work has been to develop a continuum model based on the multiscale model given in Owen et al (2011) for growth of a vascular tumour tissue
Both the models account for evolution of normal and tumour cells and formation of vessels in response to surrounding conditions like availability of oxygen and secretion of VEGF
Summary
Cancer is a complex disease not restricted to a single biological scale but involving processes occurring over multiple spatial scales, ranging from the subcellular scale (for example, progression through the cell cycle) to the tissue scale (for example, angiogenesis, vascular remodelling), and temporal scales (for example, oxygen diffusion on a relevant tissue length-scale occurs in minutes while the doubling time for cells is usually in days/weeks). Many models developed previously focused on a single aspect of tumour growth, for example, cell cycle (Alarcon et al, 2004), cell apoptosis and necrosis (Byrne and Chaplain, 1998), diffusion of angiogenic factors (Chaplain and Stuart, 1991) or tumour angiogenesis (Balding and McElwain, 1985). These models provide a good insight into the underlying mechanisms of these processes, they fail to address issues of how different phenomena occurring at different scales couple together to influence tumour growth. Continuum models enable us to perform certain analyses, like determining the steady states of the system and investigating the effect of model parameters and initial conditions on the evolution of the model variables
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