a numerical simulation of vascular brain tumor growth using adomian decomposition method.

Abstract

A tumor develops when a single normal cell transforms due to mutations in certain key genes. To continue growing, it requires new sources of nutrients, hence develops new blood vessels that continue feeding it from the blood leading to vascularization. Statistics from World Health Organization (WHO) records shows that incidences of brain tumors in the year 2014 were already at 1/12,500 persons. The purpose of this study was to develop a numerical simulation of vascular brain tumor that will help medical practitioners to predict the size of the tumor for prognosis purposes instead of exposing patients to radiations through multiple scanning. In this work numerical simulation was developed from partial differential equations models, whereby cell nutrients concentration(C) was the dependent variable x, y, z were spatial independent variables, t was a variable for different time schedules, P was the variable for cells proliferation, P n was the variable for non- proliferating cells while N was the variable for necrotic cells. Objectives of study were, to develop a numerical simulation of vascular brain tumor growth in one, two and three dimensions, to determine the viable rate of consumption of the nutrients in tumor growth and development, to present validated results in tabular and graphical form, to determine the period within which angiogenic inhibitors are viable. In attaining the objectives above results were generated by Adomian Decomposition Method (ADM) whereby equations are decomposed into a series of Adomian polynomials. The method generates a solution in the form of a series whose terms are determined by a recursive relationship. Results obtained from the simulation of growth and dynamics of malignant brain tumor (glioma) compares well with those from medical literature. In one dimensional model, radius of the tumor in different time schedules was obtained, for example where the rate of diffusion of the nutrients was 11mm/year, in 560 days, simulation radius was found to be 25.4mm compared to an experimental radius of 25.0 mm. In two dimensional models, cross section area of the tumor in different time schedules was obtained, whereby in 560 days, simulation area was found to be 19.02cm 2 , whereas analytical area was 19.64cm 2 . In three dimensional models, volume of the tumor in different time schedules was obtained, whereby in 560 days, simulation volume was found to be 65.77cm 3 , whereas analytical volume was 65.48cm 3 . Thus obtained results were found to be consistent with available experimental data, hence may be used to complement traditional tumor diagnostic. Considering idealized cases of tumors, ADM gave realistic simulations, which can provide clinical practitioners with valuable information on the potential effects of therapies in their exact schedules. However for tumors with multiple distinct clones, current model may not be reliable thus further studies are needed to address this shortcoming.