Minimal models of invasion and clonal selection in cancer
Paterson, Chay Giles Blair
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One of the defining features of cancer is cell migration: the tendency of malignant cells to become motile and move significant distances through intervening tissue. This is a necessary precondition for metastasis, the ability of cancers to spread, which once underway permits more rapid growth and complicates effective treatment. In addition, the emergence and development of cancer is currently believed to be an evolutionary process, in which the emergence of cancerous cell lines and the subsequent appearance of resistant clones is driven by selection. In this thesis we develop minimal models of the relationship between motility, growth, and evolution of cancer cells. These should be simple enough to be easily understood and analysed, but remain realistic in their biologically relevant assumptions. We utilise simple simulations of a population of individual cells in space to examine how changes in mechanical properties of invasive cells and their surroundings can affect the speed of cell migration. We similarly examine how differences in the speed of migration can affect the growth of tumours. From this we conclude that cells with a higher elastic stiffness experience stronger resistance to their movement through tissue, but this resistance is limited by the elasticity of the surrounding tissue. We also find that the growth rate of large lesions depends weakly on the migration speed of escaping cells, and has stronger and more complex dependencies on the rates of other stochastic processes in the model, namely the rate at which cells transition to being motile and the reverse rate at which cells cease to be motile. To examine how the rates of growth and evolution of an ensemble of cancerous lesions depends on their geometry and underlying fitness landscape, we develop an analytical framework in which the spatial structure is coarse grained and the cancer treated as a continuously growing system with stochastic migration events. Both the fully stochastic realisations of the system and deterministic population transport approaches are studied. Both approaches conclude that the whole ensemble can undergo migration-driven exponential growth regardless of the dependence of size on time of individual lesions, and that the relationship between growth rate and rate of migration is determined by the geometrical constraints of individual lesions. We also find that linear fitness landscapes result in faster-than-exponential growth of the ensemble, and we can determine the expected number of driver mutations present in several important cases of the model. Finally, we study data from a clinical study of the effectiveness of a new low-dose combined chemotherapy. This enables us to test some important hypotheses about the growth rate of pancreatic cancers and the speed with which evolution occurs in reality. We test a moderately successful simple model of the observed growth curves, and use it to infer how frequently drug resistant mutants appear in this clinical trial. We conclude that the main shortcomings of the model are the difficulty of avoiding over-interpretation in the face of noise and small datasets. Despite this, we find that the frequency of resistant mutants is far too high to be explained without resorting to novel mechanisms of cross-resistance to multiple drugs. We outline some speculative explanations and attempt to provide possible experimental tests.