A Spherically Symmetric Model for the Tumor Growth

The nonlinear tumor equation in spherical coordinates assuming that both the diffusivity and the killing rate are functions of concentration of tumor cell is studied. A complete classification with regard to the diffusivity and net killing rate is obtained using Lie symmetry analysis. The reduction of the nonlinear governing equation is carried out in some interesting cases and exact solutions are obtained. 1. Introduction The tumor growth has been usually modeled as a reaction-diffusion process in the literature. Jones et al. [1] have given a simple tumor model based upon this idea. A model describing the growth of the tumor in brain taking into account diffusion or motility as well as proliferation of tumor cells has been developed in a series of papers [2, 3]. In continuation of this approach, Tracqui et al. [4] suggest a model which takes into account treatment and thus killing rate of tumor cells along with the above factors. The governing equation in this case is where is the concentration of tumor cells, is the diffusion coefficient, is the proliferation rate, and is the killing rate. Assuming complete radial summery, Moyo and Leach [3] have studied this model with being variable. The resulting governing equation reduces to the simple form where . They have performed Lie symmetry analysis and presented some exact solutions based upon this approach. Consequently, Bokhari et al. [5] used Lie symmetry analysis to obtain a number of invariant reductions and exact solutions in the case of killing rate being function of . The present study is based upon the fact that the diffusivity is not necessarily a constant and may depend upon the concentration of tumor cells. Moreover, the net killing rate is also taken to be -dependent. This introduces nonlinearity in the governing equation. Keeping these assumptions in mind (1) becomes which in spherical coordinates and with radial symmetry assumption becomes where is the diffusivity of the medium and is the net killing rate. We present a classification of the functions and using Lie symmetry analysis. The Lie symmetry approach, first proposed by Lie [6], has been used to classify nonlinear differential equations, find appropriate similarity transformations, and find exact solutions. One may refer to [7–10] for a good account of this method. Some recent studies in nonlinear diffusion equations using this approach can be found in [1, 6]. 2. Symmetry Analysis of the Tumor Equation In this section, we perform the symmetry analysis of (4). To this end, we use the Lie symmetry method [10] which is based upon finding