Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth

What is it about?

We survey well-established models of reaction-diffusion type for studying brain tumours growth by discussing biological background and mathematical properties. We also give a variational formulation that opens the possibility of alternative derivations and modifications of the models. Moreover, we perform full 3-dimensional simulations of the tumours in time on two types of set-ups, the 3d Shepp-Logan phantom and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository (IBSR), and for both low and high grade glioma.

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Photo by Robina Weermeijer on Unsplash

Photo by Robina Weermeijer on Unsplash

Why is it important?

The reaction term is such that we have logistic growth. These simulations are obtained using standard finite difference discretisation of the space and time-derivatives with novel calculations to increase speed and accuracy, generating a simplistic approach that performs well. In fact, simulations show how the tumour deforms when it reaches regions into which it cannot grow further.