Numerical reconstruction of brain tumours

What is it about?

We investigate the reconstruction of brain tumours backwards in time by formulating the problem as finding the initial state of the solution to well-established parabolic reaction-diffusion models for tumour growth, given the tumour at a final instance in time. We then propose the nonlinear Landweber type method for this inverse problem to obtain a stable solution. Some mathematical analysis was undertaken to find the adjoint operator needed and to motivate convergence. Moreover, we give full 3-dimensional simulations of the tumour source localization for two realistic brain models being the 3-dimensional Shepp-Logan phantom and an MRI T1-weighted brain scan.

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Photo by jesse orrico on Unsplash

Photo by jesse orrico on Unsplash

Why is it important?

An additional challenge was introduced in that the simulations was undertaken with the data generated with a different set of parameters (including different nonlinearities). The obtained results corroborated well with what could be expected theoretically, and indicate that stable numerical results can be obtained opening for further investigations of more advanced tumour growth models backwards in time.