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Harsh Jain : A delay differential equation model of cancer chemotherapy, with applications to ovarian carcinoma treatment

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The use of delay differential equations (DDEs) to study biological phenomena has a long history, when the rate of change of model variables depends their previous history. Today, DDEs occupy a central place in models of infectious disease dynamics, epidemiology, ecology and tumor growth. In this talk, I will present a delayed partial differential equation (PDE) model of tumor growth and treatment. The model accounts for cell cycle arrest and cell death induced by chemotherapy, and explicitly includes intracellular signaling pathways relevant to drug action. The model is simplified to give a 1D hyperbolic PDE, which is further reduced to a nonlinear, non-autonomous DDE by projecting along characteristics. Necessary and sufficient conditions for the global stability of the cancer-free equilibrium are derived and conditions under which the system evolves to periodic solutions are determined. This has clinical implications since it leads to a lower bound for the amount of therapy required to affect a cure. Finally, I will present a clinical application of the model, by applying it to the treatment of ovarian cancers. Two types of drugs are considered – platinum-based chemotherapeutic agents that are the current standard of care for most solid tumors, and small molecule cell death inducers that are currently under development. The model is calibrated versus in vitro experimental results, and is then used to predict optimal doses and administration time scheduling for the treatment of a tumor growing in vivo.

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