Modelling cell movement and the cell cycle in multicellular tumour spheroids
The work presented in this thesis is concerned with modelling the effects of cell movement on the growth and formation of cell cycle phase specific regions within solid tumours. A model is proposed in the context of multicellular tumour spheroids (MCTS) and includes a simple model of the cell cycle, where cells move between each cell cycle phase depending on the availability of extracellular nutrient, as well as cell movement via chemotaxis, which varies depending upon the respective cell cycle phase of the cell. Numerical and asymptotic solutions show the model re-produces the well known MCTS structure of an internal quiescent cell region surrounded by a rim of proliferating cells. A further, more interesting result, describes a tumour surrounded by a rim of quiescent cells, with an inner quiescent and an interim proliferating cell region. The resultant solutions are a result of the different cell velocity profiles along with the effects of the cell cycle kinetics in different regions of the tumour. The non-linear form of the conservation equations describing the movement of cells means that solutions with spatial discontinuities in the cell concentrations (shocks) are observed for specific parameter values. Analysis of the effects of the chemotactic response and the cell cycle kinetics, both spatial and temporal, provide insight in to the model's behaviour and shows an understanding of cell cycle kinetics, cell movement and the spatial structure of tumours is important in assisting therapeutic strategies. The effectiveness of apoptosis, as an anti-cancer strategy, is shown to be dependent upon the concentration and spatial organisation of proliferating cells within the respective tumour. Comparison with the experimentally verified model of tumour growth developed by Gompertz allows specific model parameters to be expressed in terms of experimentally known variables. Such analysis shows that Gompertz's model is good at predicting the growth of solid tumours with a proliferating rim, but other models are required to understand the growth of non-uniform, heterogeneous tumours. Experimental justification of the model is provided by considering the observed internalisation of H3 Thymidine labelled cells and inert microspheres within MCTS. Here experimental results show that following adherence to the spheroid edge, the microspheres were all advected towards the centre of the spheroids whilst the labelled cells were spread throughout the proliferating and quiescent outer regions. The cell cycle model which is developed is, unlike previous models, able to account for this observed behaviour. Various simulations are discussed in relation to the original experimental results. These results show the importance of cell movement in providing possible ways of assisting with drug delivery to the more therapeutically resistant regions of solid tumours. Finally the importance of necrosis formation is discussed by a simple extension to the model. Necrosis as a result of quiescent cell death leads to the commonly observed formation of a necrotic core in each case. However, using the model to consider the more recent hypothesis that apoptosis leads to the formation of necrotic regions provides interesting theoretical results.