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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 726837, 7 pages
A Spherically Symmetric Model for the Tumor Growth
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
Received 25 May 2013; Accepted 19 December 2013; Published 19 January 2014
Academic Editor: Renat Zhdanov
Copyright © 2014 Saeed M. Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The tumor growth has been usually modeled as a reaction-diffusion process in the literature. Jones et al.  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.  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  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.  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 , 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  which is based upon finding Lie point symmetries of the PDEs that leave them invariant. In order to find the Lie symmetry generators of (4) and obtain closed-form solutions for all , we consider the one-parameter Lie point transformation that leaves (4) invariant. These transformation formulae [7, 9] are given as follows: where defines the symmetry generator  associated with (2) given by Requiring invariance of (4) with respect to the prolonged symmetry generator, with 0 representing and 1 representing , and the coefficients and , of the derivatives with respect to dependent variables in (8) are to be evaluated using the expressions: Using (9) into (8) and comparing terms involving derivatives of the dependent variable , we obtain the following system of differential equations: To determine the unknown functions , , and , we solve the above coupled system of differential equations by first considering (11). Differentiating this equation twice with respect to leads to the following expression: Using (10) into (14) reduces to We proceed from the above equation to obtain a complete classification of both and as shown in the next section.
In order to perform a complete classification of solution of (4), we notice that the following three cases arise from (15)(I),(II),(III).For obtaining a complete classification, we consider all the three cases one by one. Since procedure of classification in all the three cases is similar, we present a complete analysis in the first case and briefly state the results in the remaining cases. To begin the classification, we proceed as follows.
3.1. Case I
3.1.1. Subcase (a)
But depend only on , so that two possibilities arise from expression of :(a.1) and ,(a.2) and .
Subsubcase (a.1). This subcase leads to the fact that and are arbitrary and the infinitesimals are Only one symmetry generator is associated with above infinitesimals which is .
Subsubcase (a.2). As a result of this subcase, and are arbitrary and the infinitesimals are In this subcase, we have two generators which are
3.1.2. Subcase (b)
Using these conditions arising in this case into (13), it becomes Differentiating (25) with respect to , we get The above equation is a Cauchy Euler differential equation, its solution given by To require consistency of found above, we use (27) into (25). This suggests that (25) is satisfied when the following differential constraint is met: Differentiating (28) four times with respect to , we get and hence . Therefore, Using (29) into (17), we conclude that From (16) and (30), we infer that and consequently the general expressions of , , , , and are The two symmetry generators associated with above infinitesimals are given by The commutation relation for each of the above symmetry generators is listed in Table 1.
3.1.3. Subcase (c)
Following the procedure adopted in earlier cases, we obtain the same symmetry generators in cases (a) and (b).
3.2. Case II
Solving equation for instantly yields where is a constant. Using (33) into (11), we obtain Using (34) and (33) into (13), we obtain a differential relation in the given by Differentiating (35) with respect to , we obtain From (36) two cases arise: (a),(b).
3.2.1. Subcase (a)
This subcase gives that Using (37) into (35) yields Differentiating (38) with respect to 4 times, we conclude that and hence . Therefore, As a result of (39), we obtain Using (39) and (40) into (12), we obtain Differentiating (41) with respect to yields The above equation holds if , and substituting this value into (41) gives Differentiating (43) with respect to The equality in (44) holds if where is constant.
Subsubcase (a.1) ( and ). Using these conditions arising in this case into (47), we obtain and hence the infinitesimals are determined The two symmetry generators associated with the above infinitesimals are given by The commutation relation for these generators is given in Table 2.
Subsubcase (a.2) ( and ). Similarly, using these conditions arising in this case into (47), we obtain and hence the infinitesimals are determined The three symmetry generators associated with the above infinitesimals are given by The commutation relation for these generators is given in Table 3.
Subsubcase (a.3) ( and ). using these conditions arising in this case into (47), we obtain and hence the infinitesimals are The two symmetry generators associated with the above infinitesimals are given by The commutation relation for these generators is given in Table 4.
3.2.2. Subcase (b)
In accordance with this subcase, (35) becomes Using (33) into (35) gives and hence Using (56) into (12) yields Differentiating (57) with respect to The equality in (58) holds if In accordance with (59), we have Using (60) into (57) yields Differentiating (61) with respect to leads to From (62) three cases arise: (b.1), and ,(b.2) and ,(b.3) and .
Subsubcase (b.1) ( and ). In accordance with this subsubcase and (61), we obtain and hence the infinitesimals are determined as The generators associated with this subsubcase are The commutation relation for these generators is given in Table 5.
Subsubcase (b.2) ( and ). This subsubcase with (61) yields to and hence the infinitesimals are The generators associated with this subsubcase are The commutation relation for these generators is given in Table 6.
Subsubcase (b.3) ( and ). In accordance with this subsubcase and (61), we obtain and hence the infinitesimals are The generators corresponding to this subsubcase are
Remark 1. Case (III) represents the particular case of (I) and (II). So by similar manipulation as in the previous cases, we obtain the same symmetry generators.
4. Some Reduction
In this section, we present solutions of (4) via reductions. These reductions are obtained by the similarity variables obtained through symmetry generators.
Case 1 ( and are arbitrary). In this case, we have only one generator; that is, . Thus, the characteristic equation corresponding to this generator is Solving the above equation it is straightforward  to find that it yields the similarity variables and . Replacing in (4) in terms of new variables it becomes
Case 2 ( and are arbitrary). We consider the generator . Thus, the characteristic equation associated with this generator is The similarity variables corresponding to the above equation become and . These new variables reduce (4) to a ODE of the form Choosing in the above equation, then the solution of the resulting equation is Recasting the above equation in its original coordinates, the exact solution of (4) becomes The graph of this solution is plotted in Figure 1.
Case 3 ( and ). We take the generator and its characteristic equation gives the similarity variables and . In accordance of these similarities, (4) is transformed to the ODE in the form Choosing in the above equation, then its solution becomes Recasting the above equation in its original coordinates, the exact solution of (4) becomes The graph of this solution is plotted in Figure 2.
Case 4 ( and ). In this case, we consider the generator The characteristic equation corresponding to this generator is solving the above characteristic equation gives with . These variables can be used to recast (4) to an ODE, Choosing in (82), then its solution becomes Recasting the above equation in its original coordinates, the exact solution of (4) becomes The graph of this solution is plotted in Figure 3.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to acknowledge the support provided by the Deanship of Research at King Fahd University of Petroleum and Minerals (KFUPM) for funding this work through Project IN111008.
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