Mathematical modelling of metastatic processes on a network

Singh, Khimeer
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This study investigates metastasis - the process characterized by cancer spreading away from the primary site and moving to secondary locations. Metastatic cancer is notoriously difficult to treat and typically results in death. Some of this difficulty stems from the mystery surrounding metastasis. The model constructed in this dissertation aims to elucidate the process with mathematical methods. Given some primary site, the model intends to predict the secondary metastatic sites. Furthermore, the model explores the relationship between metastasis and blood flow and between metastasis and the diffusive behaviour of cancer. To model cancer, various techniques across different mathematical disciplines were used. Continuum techniques model the growth and local spread of cancer. A graph network represents the organs and the blood vessels between them. Differential equations and simulation procedures govern blood flow between organs and through the vessels. An analysis of the model predictions showed a good correlation with clinical data for some cancer types. Particularly for cancers originating in the gut. The model also predicted an inverse relationship between blood velocity and the concentration of cancer cells in secondary organs. Finally, for particular diffusive behaviour, a slowing effect was observed. The investigation yields some valuable results for clinical practitioners and researchers – as it clarifies some aspects of cancer that have hitherto been difficult to study. The model provides a good framework for studying cancer progression using patient and cancer-specific information when simulating metastasis
A dissertation submitted in fulfilment of the requirements for the degree of Master of Science to the Faculty of Science, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, 2022