Curtin, Lee
(2019)
Mathematical modelling of brain tumour growth and therapy.
PhD thesis, University of Nottingham.
Abstract
Glioblastoma multiforme (GBM) is the most aggressive primary brain tumour with an expected median survival of just 15 months post-diagnosis in spite of treatment. GBMs are highly invasive, such that their cells can be found in low densities far away from the apparent tumour edge indicated by magnetic resonance imaging (MRI). It is this invasiveness that can limit the efficacy of surgery. However, surgery acts as a unique opportunity for further local drug delivery which is currently not exploited as a standard of care.
A local drug delivery polymer is in development at the Children's Brain Tumour Research Centre at the University of Nottingham; this biodegradable polymer can be loaded with chemotherapy drugs and pasted onto the cavity wall immediately following the surgical removal of the tumour bulk, to treat the remaining GBM cells that could not be removed. The drug delivery dynamics through the polymer are not fully understood. We have developed in vitro experiments and an analogous mathematical model to characterise the dynamics of diffusion through the polymer. We noticed that the polymer constructs swell during in vitro experiments, and our mathematical model fits suggest a temporally decreasing diffusion coefficient of drug travelling through the polymer. Together, these suggest that the polymer swelling inhibits drug diffusion over time.
In further collaboration with the Hounsfield Facility at the University of Nottingham, we have gathered state-of-the-art microCT data of unswollen polymer constructs. We have applied homogenisation of diffusion theory to diffusive drug transport in the mixture of polymer particles and liquid medium. The microCT data has been used for input geometries of the resulting cell problems, whose solutions define the homogenised tensor of diffusion coefficients. We have found these solutions using a finite element method in FEniCS. This theory allows us to quantify the rates of effective diffusion and the effect of the internal geometries on this rate. The homogenised coefficients suggest that both channels of polymer and gaps of liquid medium are present in the polymer constructs, such that the liquid channels are the primary mechanism by which drug would diffuse through the polymer in these constructs. We see a significantly lower average diffusion rate in the direction that the polymer constructs were packed into their moulds, highlighting the importance of the method of application of this treatment and its subsequent impact on the diffusion rate of chemotherapy drugs.
With the Precision Neurotherapeutics Innovation Program at the Mayo Clinic, we have extended the Proliferation Invasion Hypoxia Necrosis Angiogenesis (PIHNA) model, a PDE model that recapitulates the dynamics between angiogenesis and GBM growth at the tissue level. We show that the original term that denotes the efficiency of the vasculature to provide nutrients to the tumour can produce counter-intuitive behaviour in some cases, and hence include a more realistic term. Unlike previous PIHNA publications, we have allowed for different hypoxic and normoxic cell diffusion rates, as literature suggests the hypoxic cell rate is faster than its normoxic counterpart. We linearised the PIHNA model to find an analytical expression to predict the minimum possible wavespeed, and have shown that this minimal wavespeed is reached if the diffusivity of normoxic cells is greater than or equal to that of their hypoxic counterpart. However, when the hypoxic diffusion rate is markedly faster than the normoxic diffusion rate, we see a faster numerical wavespeed, suggesting that the hypoxic cells can drive the tumour growth. We then apply this model to perioperative ischemia, a reduction in functional blood vessels that can occur as a surgical complication, which has been seen clinically to increase the chance of patients having a distally recurring tumour. A distally recurring tumour returns with a different progression pattern such that it appears far away from the original site (distant recurrence) or more diffuse (diffuse recurrence) on MRI. We see in the PIHNA model that if a tumour becomes hypoxic and necrotic at a fast enough rate due to the decrease in nutrient availability caused by ischemia, and the ischemia is severe enough, the tumour can reappear on simulated MRI far away from the expected local recurrence location if it has a high enough migration rate and low enough proliferation rate - giving a distant recurrence. This work therefore supports the hypothesis that hypoxia can act as a driver for distantly recurring GBM, and suggests conditions by which this can occur.
Ultimately, we plan to bring together the two themes of this thesis with a tumour growth model similar to PIHNA that incorporates the local drug delivery polymer to test the potential efficacy of this treatment for brain tumour patients.
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