A multiscale model for vascular cancer growth
The modelling of cancer provides an enormous mathematical challenge because of its inherent multi-scale nature. For example, in vascular tumours, nutrient is transported by the vascular system, which operates on a tissue level. However, it affects processes occurring on a molecular level. Molecular and intra-cellular events in turn affect the vascular network and therefore the nutrient dynamics. Our modelling approach is to model, using partial differential equations, processes on the tissue level and couple these to the intercellular events (modelled by ordinary differential equations) via cells modelled as automaton units. Thusfar, within this framework we have modelled structural adaptation at the vessel level and have shown that environmental heterogeneity, which a number of models have ignored, can significantly affect the ability of cancer tissue to grow. See publications 161, 165.
Simulations showing the evolution of the spatial distribution of cells for growth in inhomogeneous (top panels) and homogeneous (bottom panels) environments. The white spaces are occupied by cancer cells, the black spaces are either empty or occupied by the (hexagonal) array of blood vessels (a highly idealised vascular network). The graphs show the time evolution of the number of cancer cells for each case. The yellow lines denote total number of cancer cells (proliferating + quiescient) while the blue lines denotes number of quiescient cells. Note the order of magnitude difference in scale. In each case the normal cells tend to zero.
Simulations for vascular networks of different sparsity. First row corresponds to time evolution of the cancer colony, where each iteration represents 15 hours; second row shows the stationary distribution of oxygen concentration, and the third row shows the stationary distribution of cells (see above). [Reproduced with permission from publication 165]
** Work carried out in collaboration with T. Alarcon and H.M. Byrne