A numerical analysis of a model of growth tumor

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1 Applied Mathematics and Computation 67 (2005) A numerical analysis of a model of growth tumor Andrés Barrea a, *, Cristina Turner b a CIEM, Universidad Nacional de Córdoba, Argentina b Universidad Nacional de Córdoba, CIEM-CONICET, FaMAF, Medina Allende sn, Cordoba 5000, Argentina Abstract In this paper we study a free boundary problem modeling the growth of tumors. The model uses the conventional ideas of nutrient diffusion and consumption by the cells. We consider the radially symmetric case of this free boundary problem. We apply a spectral numerical method to the system of equations. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Growth tumor Spectral method. Introduction and preliminaries A variety of PDE for tumor growth have been developed in the last decades. Some models are based on reaction diffusion equations [4,5]. Other models include hyperbolic equations, we refer to [ 3,6].In[3] the authors have studied a particular model with three cells populations: proliferating cells, quiescent cells and dead cells. This model includes densities P, Q, and D of proliferating, * Corresponding author. addresses: abarrea@mate.uncor.edu (A. Barrea), turner@mate.uncor.edu (C. Turner) /$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:0.06/.amc

2 346 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) quiescent and dead (necrotic) cell respectively, and concentration C of nutrients (generally oxygen). These densities satisfy a system of nonlinear first order hyperbolic equations in the tumor, with tumor surface as a free boundary. The cells in different states are assumed to be mixed within the tumor, and to have the same size. They assumed that the tumor is uniformly packed with cells, so that P þ Q þ D ¼ const N: Due to proliferation of cells and to removal of necrotic cells, there is a continuous movement of cells within the tumor and that the field velocity ~v satisfies the DarcyÕs law. They treat the tumor tissue as a porous medium, that is ~v ¼rr r pressure: In this model they assume that K Q ðcþ is increasing in C (rate of change from proliferating state to quiescent state). K P ðcþ is decreasing in C (rate of change from quiescent state to proliferating state). K D ðcþ is decreasing in C (quiescent cells become necrotic at a rate K D ðcþ). K A ðcþ is decreasing in C (the death rate by apoptosis). K B ðcþ is increasing in C (the proliferation rate). K B ðcþ > K A ðcþ. K R is a positive constant (the rate of removal of dead cells). The concentration C satisfies the next diffusion equation r 2 C kc ¼ 0enXðtÞ ðk > 0Þ C ¼ C 0 sobre oxðtþ where X is the tumor region at time t. The densities P, Q and D satisfies the following system: op ot þ divðpvþ ¼½K BðCÞ K Q ðcþ K A ðcþšp þ K P ðcþq oq ot þ divðqvþ ¼K QðCÞP ½K P ðcþþk D ðcþšq od ot þ divðdvþ ¼K AðCÞP þ K D ðcþq K R D: If we add the last equations then we obtain an equation for the pressure r Nr 2 r ¼ K B ðcþp K R D:

3 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) Now, we set c ¼ C=C 0 p ¼ P=N q ¼ Q=N we arrive at the following system of equations: r 2 c kc ¼ 0 in XðtÞ c ¼ on oxðtþ op ot þ divðprrþ ¼½K BðcÞ K Q ðcþ K A ðcþšp þ K P ðcþq in XðtÞ oq ot þ divðqrrþ ¼K QðcÞp ½K P ðcþ K D ðcþšq in XðtÞ r 2 r ¼ K R þ½k B ðcþþk R Šp þ K R q in XðtÞ where K i ðcþ ¼K i ðc 0 cþ for i ¼ A B D P Q: The pressure r on the surface of the tumor is equal to the surface tension, that is r ¼ c on oxðtþ ðc > 0Þ where is the mean curvature. The motion of the free boundary is given by the continuity equation or ~v ~n ¼ V n or o~n ¼ V n on oxðtþ where ~n is the outward normal and V n is the velocity of the free boundary of the free boundary in the outward normal direction. Given initial conditions Xð0Þ pðx 0Þ qðx 0Þ we would like to determine the family of domains X(t) and the functions p(x,t), q(x,t), c(x,t) and r(x,t) satisfying the last system. 2. The radial model and its properties We note that tumors grown in vitro are typically of spherical shape, which makes the study of radially symmetric solutions quite relevant. The radially symmetric case of the the general model is given by the following equations system: r 2 o or oc ðr2 Þ¼kc ð0 < r < RðtÞ t > 0Þ or oc ð0 tþ ¼0 cðrðtþ tþ ¼ ðt > 0Þ or

4 348 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) op op þ u ot or ¼½K BðcÞ K Q ðcþ K A ðcþšp þ K P ðcþq ½ðK B ðc þ K R Þp þ K R q K R Šp ð0 6 r 6 RðtÞ t > 0Þ oq oq þ u ot or ¼ K QðcÞp ½K P ðcþþk D ðcþšq ½ðK B ðcþþk R Þp þ K R q K R Šq ð0 6 r 6 RðtÞ t > 0Þ o r 2 or ðr2 uþ ¼½K B ðcþþk R Šp þ K R q K R uð0 tþ ¼0 ðt > 0Þ drðtþ ¼ uðrðtþ tþ ðt > 0Þ dt with initial data Rð0Þ pðr 0Þ qðr 0Þ: To transform the above free boundary problem in a problem with fixed boundary we first note that, for R(t) given cðr tþ is given by pffiffi RðtÞ sinhð k rþ r cðr tþ ¼ pffiffiffi ¼ c r sinhð k RðtÞÞ RðtÞ RðtÞ where op ot þ v op or ¼½K BðcÞ K Q ðcþ K A ðcþšp þ K P ðcþq ½ðK B ðcþþk R Þp þ K R q K R Šp ð0 6 r 6 t > 0Þ oq ot þ v oq or ¼ K QðcÞp ½K P ðcþþk D ðcþšq ½ðK B ðcþþk R Þp þ K R q K R Šq ð0 6 r 6 t > 0Þ vðr tþ ¼uðr tþ ruð tþ ð0 6 r 6 t > 0Þ o r 2 or ðr2 uþ¼½k B ðcþþk R Šp þ K R q K R ð0 6 r 6 t > 0Þ uð0 tþ ¼0 ðt > 0Þ drðtþ dt ¼ RðtÞuð tþ ðt > 0Þ

5 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) with initial data Rð0Þ ¼R 0 pðr 0Þ ¼p 0 ðrþ qðr 0Þ ¼q 0 ðrþ where R 0 > 0 p 0 ðrþ P 0 q 0 ðrþ P 0 p 0 ðrþþq 0 ðrþ 6 ð0 6 r 6 Þ: In [3] the authors show that the last system has a unique solution for 0 6 r 6, 0 6 t >, and it has the following properties:. p(r,t) P 0, q(r,t) P 0, p(r,t) +q(r,t) 6, 2. R 0 e 3 KRt 6 RðtÞ 6 R 0 e 3 KBðÞt, 3. lim t! R(t) = if K R =0, 4. d 0 6 R(t) 6 M for all t P 0if0<K R <. 3. The numerical method Spectral methods are based in simple ideas of interpolation [7], we use these ideas for the numerical method. We give a description of our numerical scheme for the problem with fixed boundary. Let r i ¼ i for 0 < i 6 N + be a partition of the interval [0,] into subintervals I i =[r i,r i+ ], of length h ¼. First, Nþ Nþ we consider the following matrix: D r i ¼ Nþ Y a i ðr i r k Þ¼ ði 6¼ Þ a a k¼ ðr i r Þ and where D r ii ¼ XNþ ðr r k Þ k¼ a ¼ YNþ ðr i r Þ: k¼ We use the following notation P ¼ðP ðtþþ ¼ðpðr tþþ Q ¼ðQ ðtþþ ¼ðqðr tþþ U ¼ðU ðtþþ ¼ðuðr tþþ V ðtþ ¼V ðtþ ¼U ðtþ U Nþ ðtþ

6 350 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) where (Æ) is a vector with components. We approximate D r P op or ðr tþ D r Q oq or ðr tþ : Now, we consider the following approximations op ot ðr tþþvðr tþ op or ðr tþ dp dt þ diagðv ÞD r P where [Æ] denote the -component of the vector (Æ). oq ot ðr tþþvðr tþ oq or ðr tþ dq dt þ diagðv ÞD r Q r 2 o or ðr2 uþðr tþ diag r 2 k D r r 2 U We remark that K i (c) =K i (r,r) for i = A,B,P,Q,D. We define the following matrices: M ðrþ ¼diagðK B ðr RÞ K Q ðr RÞ K A ðr RÞÞ M 2 ðrþ ¼diagðK P ðr RÞÞ M 3 ðrþ ¼diagðK B ðr RÞþK R Þ M 4 ðrþ ¼diagðK Q ðr RÞÞ M 5 ðrþ ¼diagðK P ðr RÞþK D ðr RÞÞ and the following vectors: P 2 ðtþ ¼ðP 2 ðtþþ Q 2 ðtþ ¼ðQ 2 ðtþþ P:QðtÞ ¼ðP ðtþq ðtþþ r 2 UðtÞ ¼ðr 2 U ðtþþ ¼ðÞ : We have the following approximations dp dt þ diagðv ÞD r P ¼ M P þ M 2 Q M 3 P 2 K R PQ þ K R P dq dt þ diagðv ÞD r Q ¼ M 4 P M 5 Q M 3 PQ K R Q 2 K R Q :

7 diag D r r 2 U ¼ M r 2 3 P þ K R Q K R k! U ¼ diag ðd r Þ diagðr 2 r 2 k ÞðM 3P þ K R Q K R Þ: From the last equation, we can obtain the vector U as function of the P, Q and R. Then we can express diag(v ) as function of P, Q, and R we obtain the following system of ordinary differential equations dp dt ¼ F ðr P QÞ dq dt ¼ F 2ðR P QÞ dr dt ¼ F 3ðR P QÞ with initial conditions A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) Pð0Þ ¼ðpðr 0ÞÞ ¼ðp 0 ðr ÞÞ Qð0Þ ¼ðqðr 0ÞÞ ¼ðq 0 ðr ÞÞ Rð0Þ ¼R 0 : If we define W ¼½P Q RŠ t then we can use the following notation dw ¼ FðWÞ dt Wð0Þ ¼W 0 : This system can be resolved by means the standard numerical methods for the ordinary differential equations. 4. Numerical examples From [6], in this section we set the following values for the parameters of the problem:. R(0) = 3, 2. k = 0.05, 3. K A (c) =0, 4. K P (c) = 0.05c, 5. K Q (c) = 0.05(c + ), 6. K D (c) = c +, 7. K B (c) =c.

8 352 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) The parameters K R and the functions p 0 (r) andq 0 (r) will be different in the examples. We set 0 points for the discretization of the interval [0, ]. A Runge Kutta method is used by integrate the ordinary differential equations. Example. In this example, we consider several values for parameter K R,we remark that lim RðtÞ ¼ if K R ¼ 0 t! we set p 0 (r) = 0.3 and q 0 (r) = 0.6. From Fig. we can see that the solution is increasing with respect to K R. Example 2. Now p 0 (r) = sin(px/2(n )) where N = 0 the number of points in [0, ] for the discretization. In Figs. 2 and 3 we show the behavior of the densities for different times. Example 3. In the last example, the parameters are the same in Example, with K R =. We remember that R 0 e 3 KRt 6 RðtÞ 6 R 0 e 3 KBðÞt Fig. 4 shows that the analytic bounds are preserved by the numerical solution. Fig.. R(t) for several values of K R.

9 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) P(r,t) t r Fig. 2. P(r,t) for different times Q(r,t) t r Fig. 3. Q(r,t) for different times.

10 354 A. Barrea, C. Turner / Appl. Math. Comput. 67 (2005) R(0)exp( 0.33K R t) solution R(0)exp(0.33K B ()t) R(t) Conclusion t Fig. 4. R(t), lower and upper bounds. In this paper we have investigated the numerical solution of a model of tumor growth. An special approach of transforming a PDE system into a ODE system has been adopted, this approach is based in simple ideas of interpolation. The analytic properties of the solution are preserved for the numerical method. References [] S. Cui, A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math Biosci. 64 (2000) [2] S. Cui, A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (200) [3] S. Cui, A. Friedman, A hyperbolic free boundary problem modeling tumor growth, Interf. Free Bound. 5 (2003) [4] J.A. Adam, A simplified mathematical model of tumor growth, Math. Biosci. 8 (986) [5] H. Byrne, M. Chaplain, Free boundary value problems associated with growth and development of multicellular spheroids, Eur. J. Appl. Math. 8 (997) [6] G. Pettet, C. Please, M. Tindall, D. McElwain, The migration of cells in multicell tumor spheroids, Bull. Math. Biol. 63 (200) [7] L. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000.

Universidad Nacional de Córdoba CIEM-CONICET, Argentina

Selçuk J. Appl. Math. Vol. 10. No. 1. pp. 147-155, 2009 Selçuk Journal of Applied Mathematics A Simple Discrete Model for the Growth Tumor Andrés Barrea, Cristina Turner Universidad Nacional de Córdoba