Mathematical Modelling of Nanoparticle Delivery from PODS Crystals

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Cancer is one of the most challenging diseases and it is difficult to treat. Therefore, there are many studies to find effective treatments with minimal side effects. In this study, we examine photothermal therapy as an exciting new method for treating cancer. In this treatment, the tumour is injected with metal nanoparticles (NPs) (specifically gold which is often used as it is highly conductive to heat). Then, when the system as a whole is externally heated, the tumour heats much faster than the surrounding tissues. The aim of this thesis is to have a mathematical model for a monolayer of cells triggering apoptosis with some nutrients in the region outside and the NPs would be on the nutrients. Then we want to move NPs from the nutrients into the tissue and we want to see how fast it happens and how long it takes to get into the cells. Firstly, we consider the case when the NPs are in the nutrients where we predict two situations can happen either cell absorption of NPs or adhesion of particles to the cell. We show that at the fixed point, the number of NPs inside the cell settles to the number of NPs attached to it. Secondly, we study a model when the NPs are initially within a PODS crystal and diffuse out into the culture medium. We describe how the wall of the PODS becomes leaky over time, and thus the NPs can move from inside the PODS into the culture medium. We determine the fixed point of this system on which the mass of NPs within the PODS is equal to the mass of NPs in the culture medium. Finally, we conclude by coupling the two models above together, hence we get a model for the delivery of NPs carried in the PODS to the culture medium, and then into the cell. As a result, the time taken to transfer the NPs from the PODS to the cell is calculated and the amount of NPs is determined.