Radial basis function-generated finite difference scheme for simulating the brain cancer growth model under...

We extend the original mathematical model, i.e., Swanson's reaction-diffusion equation to the surfaces with no boundary, and we find a new numerical method based on a meshless approach for solving numerically Swanson's reaction-diffusion model in the square and on the sphere.

To solve numerically the Swanson's reaction-diffusion model and its extension version, a collocation meshless technique, namely radial basis function-generated finite difference (RBF-FD) scheme is employed for approximating the spatial variables in the square domain and on the sphere, respectively. Also, to approximate the time variable of the studied models, a first-order semi-implicit backward Euler scheme is used. The resulting fully discrete scheme is a linear system of algebraic equations per time step that is solved via the biconjugate gradient stabilized (BiCGSTAB) iterative algorithm with a zero-fill incomplete lower-upper (ILU) preconditioner.

The numerical simulations show the growth of untreated and treated brain tumors with radiotherapy using estimated and clinical data (given from magnetic resonance imaging (MRI) scans of patients). Moreover, the results reported here can be used for improving the treatment strategies of the invasive brain tumor.

Using the developed numerical scheme in this paper, we can simulate the behavior of the invasive form of brain tumor response to radiotherapy. Also, we can see the effects of radiation response on the brain tumor cell concentration of individual patients. The proposed meshless technique, which is applied for solving numerically the studied model, does not depend on any background mesh or triangulation for approximation in comparison with mesh-dependent methods. Moreover, we apply this technique to the sphere via any set of distributed points easily.