Predicting the effect of a combination drug therapy on the prostate tumor growth via an improvement of a direct radial basis function partition of unity technique for a diffuse-interface model.

Chemotherapy is usually applied to treat advanced prostate cancer that cancer cells spread outside the prostate gland. The treatment uses cytotoxic drugs to target cells that grow and divide quickly. On the other hand, the growth of such cancerous tumors depends on angiogenesis. In this paper, we numerically study a diffuse-interface model in a two-dimensional space related to the therapies of prostate cancer. The proposed model describes the tumor growth driven by a generic nutrient and producing the prostate-specific antigen. More precisely, the effect of cytotoxic chemotherapy in the model is evaluated by considering a time-dependent function in the tumor dynamics. Also, another function related to the antiangiogenic therapy is considered to show the reducing intratumoral nutrient supply in the nutrient dynamics. Here, a meshless approximation, i.e., a generalized form of the direct radial basis function partition of unity (D-RBF-PU) method is presented for finding the numerical simulations of this model utilizing in medical oncology. The method uses the lower number of trial points in each patch than the original D-RBF-PU scheme for approximating the trial function per test point. Hence, the time complexity of the method is less than the D-RBF-PU technique. Besides, a semi-implicit time discretization of order 1 has been used to deal with the time variable. Consequently, a linear system of algebraic equations could be solved iteratively per time step by the use of the biconjugate gradient stabilized method with zero-fill incomplete lower-upper preconditioner. Finally, the obtained results without using any adaptive algorithm demonstrate the response of the prostate tumor growth to the chemotherapy, antiangiogenic therapy and a combined therapy.

Radial basis function-generated finite difference scheme for simulating the brain cancer growth model under radiotherapy in various types of computational domains.

BACKGROUND AND OBJECTIVES: 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. METHODS: 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. RESULTS: 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. CONCLUSIONS: 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.