A REACTION-DIFFUSION MODEL OF LEUKEMIA TREATMENT BY CHEMOTHERAPY

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume, Number 3, Fall 3 A REACTION-DIFFUSION MODEL OF LEUKEMIA TREATMENT BY CHEMOTHERAPY WENXIANG LIU AND H. I. FREEDMAN ABSTRACT. A model of leukemia treatment by chemotherapy techniques is proposed utilizing a system of reaction diffusion equations representing the change in densities of normal cells, competing cancer cells, and chemotherapy in a given organ or area. We view the interactions between normal and cancer cells as being competitive for available resources, and we think of the chemotherapy agent as a predator on both normal and leukemia cells. The existence, uniqueness, and boundedness of the solutions are established by means of a comparison principle and a monotonicity method. We analyze the constant solutions and their stabilities. The main method used in studying the stability is the spectral analysis of the linearized operators. Persistence criteria for the normal cells and cancer cells are also derived. The analysis is carried out both analytically and numerically. Introduction Leukemia can be described as the disorganization of the hematopoietic system in which a malignant clone of cells acts to impede the growth of normal hemopoietic tissue. When leukemia develops, the body produces large numbers of abnormal blood cells. In most types of leukemias, the abnormal cells are white blood cells, and they usually look different from normal blood cells, and do not function properly. Leukemia is either acute or chronic. In acute leukemia, the abnormal blood cells are blasts that remain very immature and cannot carry out their normal functions. The number of blasts increases rapidly, and the disease gets worse quickly. In chronic leukemia, some blast cells are present, but in general, these cells are more mature and can carry out some of their normal functions. Also, the number of blasts increases less rapidly than in acute leukemia. As a result, chronic leukemia gets Keywords: solutions. Competition models, chemotherapy, persistence, lower and upper Copyright c Applied Mathematics Institute, University of Alberta. 49

2 5 WENXIANG LIU AND H. I. FREEDMAN worse gradually. However, without treatment, even the chronic disorders may be fatal [, ]. It has been postulated [3 6] that the origins of acute leukemia can be found in pluripotent stem cells. Therefore, it is suggestive that in the acute leukemia state a pluripotent stem cell in the bone marrow becomes malignant, proliferates and displaces normal cells in the marrow. These abnormal cells fill the blood and the marrow and produce a malfunction of the body s immune system. Also, leukemia cells inhibit the colonyforming capabilities of the normal proliferative cells. Consequently, the body s hematopoietic system becomes disorganized. In cancer treatment today, four types of treatment are most commonly used in efforts to obtain long-term periods of disease-free remission. These include surgery, radiotherapy, chemotherapy and immunotherapy. Cancer chemotherapy has demonstrated a definite capacity for controlling disseminated metastatic cancer and is therefore widely used Dorr and Von Hoff [7], Frei [8], Liotta [9], Perry []). In cancer chemotherapy, anti-neoplastic drugs are designed to selectively destroy or inhibit the proliferative activity of cancer cells while the normal cells are affected to a lesser extent Dorr and Von Hoff [7], Frei [8]). The ultimate role of mathematical modelling in cancer chemotherapy is to provide a more rational basis for experimental design of the anti-cancer drugs and to make qualitative predictions with regard to the dynamic evolution of the disease based on the cytokinetic parameters of the patient and the drug parametric configuration. The fact that it is reasonable to view the interaction between normal cells and cancer cells as competitive is justified in Freedman [] and Nani and Freedman [, 3]. In [], a model of cancer treatment by chemotherapy was presented and conditions for the boundedness of solutions were analyzed. The equilibria and their stabilities, and conditions for the existence of small amplitude periodic solutions were also discussed. Persistence and extinction criteria of the normal cells and cancer cells were also derived. It is well known that the distributions of populations in general, being heterogeneous, depend not only on time, but also on the spatial positions in the habitat. So it is natural and more precise to study the corresponding P.D.E. problem as suggested by the authors in []. For a detailed explanation of the ecological background of the problem, the reader is referred to []. Motivated by the conception of persistence in [4, 5], we introduce it into this paper and establish the existence, uniqueness, boundedness and persistence of the solutions for the Neumman boundary condition problem by means of a comparison principle and a monotonicity method, see e.g., [7, 8]). The main method used

3 A REACTION-DIFFUSION MODEL 5 in studying the stability of constant solutions is the spectral analysis of the linearized operators. The idea of modelling cancer interactions with healthy tissue as a competition process was first proposed by Gatenby []. However, his paper did not consider treatment. The first paper to incorporate chemotherapy treatment was Nani and Freedman []. An extension to modelling cancer at several sites metastasis) with chemotherapy treatment was carried out in [3]. This paper utilizes a special case logistic) of the model developed in [] see [] for the derivation of the model), and extends it to the diffusion case to model the spread of cancer within a site such as leukemia in the bone marrow). The organization of this paper is as follows. In Section we describe the model. Section 3 deals with the no diffusion case, whereas in Section 4, we analyze the diffusion case. We conclude with a final section which contains numerical examples to illustrate our results and a short discussion. The model We take as our model of leukemia treatment by chemotherapy a system of reaction diffusion equations where u x, t) represents the density of normal cells, u x, t) the density of leukemia cells, and vx, t) the density of chemotherapy agents in the affected region at time t. We view u x, t) and u x, t) as competing for nutrient, oxygen, etc. and we think of vx, t) as a predator capable of destroying both u x, t) and u x, t), but selectively is more lethal to u x, t). The model then takes the form ) u t u t = D u x + α u = D u x + α u v t = D v 3 x + with initial conditions u K u K ) q u u p u v, a + u ) q u u p u v, a + u [ ξ + c u a + u + c u a + u ] v, u x, ) = u x), u x, ) = u x), vx, ) = v x) in Ω, and Neumann homogeneous boundary conditions u n = u n = v = on Ω, n

4 5 WENXIANG LIU AND H. I. FREEDMAN where Ω is an open boundary region with smooth boundary Ω, / n denotes the derivative along the outward normal, u x), u x), v x), are smooth functions on Ω. The constants in system) may be interpreted as follows: α i, i =,, are the specific birth rates of the normal and cancer cells for small densities. K i, i =,, are the respective carrying capacities. q i, i =,, are the competition coefficients between u x, t) and u x, t). p i, i =,, are the predation coefficients of vx, t) on u i x, t). a i, i =,, determine the speeds at which u i x, t), in the absence of competition and predation, reaches carrying capacity. is the infusion rate of the chemotherapy to the specific region. ξ is the loss of poison for the chemotherapy agent within this region. c i, i =, are the combination rates of the chemotherapy agent with the cells. Hence they are proportional to p i, i =,. All constants are positive. To make this model more realistic, we impose certain inequalities among the parameters. It is well known that leukemia cells grow at a much faster rate than normal cells. Further, if no treatment is offered, most of time leukemia cells out-compete the normal cells independent of initial conditions. Furthermore, the chemotherapy agent must be considerably more effective in killing leukemia than in killing normal cells in order for the treatment to be effective. These lead to the following sets of inequalities: α > α, p >> p. In addition, there are other inequalities which we will list in the next section since they depend on homogeneous steady states values. 3 The no diffusion case When there is no spatial variation we obtain a system of ordinary differential equations of the form. = α u u ) q u u p u v ) du dt du dt K = α u u dv dt = K, a + u ) q u u p u v, a + u [ ξ + c u a + u + c u a + u ] v,

5 A REACTION-DIFFUSION MODEL 53 with initial conditions u ) = u >, u ) = u, v) = v >. At this point we will establish two important properties of solutions to system ).. All solutions with positive initial values remain positive. Proof. By uniqueness of solutions, since u is a solution of the first equation of ), no solution with u t) > at any time t can become zero in finite time. Similarly, the same is true for u t). Since v) = >, no solution vt) of ) with vt) > can become zero.. System ) is dissipative. Proof. Since the initial conditions are nonnegative, then so are the solutions. From ), we have du t) dt α u t) u t) K ), du t) dt α u t) u t) K ). From standard comparison theory, we obtain lim sup u t) K, t lim sup u t) K. t Similarly, dvt)/dt ξvt) gives lim sup vt) ξ. t Hence, the region R = {u, u, v) R 3 + : u K, u K, v ξ } is an attracting invariant region proving the property. Definition. In analytical terms persistence means that lim inf t xt) > for each population xt); in geometric terms, that each trajectory of the modelling system of differential equations is eventually bounded away from the coordinate planes.

6 54 WENXIANG LIU AND H. I. FREEDMAN 3. The homogeneous steady states These equilibria are: E,, ξ ), E û,, ˆv), E, ũ, ṽ), E 3 u, u, v ). Note that the trivial steady state E,, ξ ) always exists, and E û,, ˆv), E, ũ, ṽ), E 3 u, u, v ) may or may not exist. In particular E û,, ˆv) represents the steady state where leukemia is eliminated, which is the most desirable state. E, ũ, ṽ) represents the case where the leukemia has completely taken over the site. E 3 u, u, v ) means coexistence which is only desirable for small numbers of leukemia cells. The equilibrium E û,, ˆv) exists provided that the algebraic system 3) α u ) p v =, K a + u ξ + c ) u v = a + u has a positive solution û, ˆv). This system has a unique positive solution provided 4) P < α a ξ. Necessary and sufficient conditions for 3) to have two positive solutions are ξa < K ξ + c ), 5) ξa < p < α [a ξ K ξ + c )]. α 4K ξ + c ) 6) Analogously, E, ũ, ṽ) exists provided that the algebraic system has a positive solution ũ, ṽ). α u ) p v =, K a + u ξ + c ) u v = a + u

7 A REACTION-DIFFUSION MODEL 55 Similar to the analysis of E, system 6) has a unique positive solution provided 7) P < α a ξ, and exactly two positive solutions if ξa < K ξ + c ), 8) ξa < p < α [a ξ K ξ + c )]. α 4K ξ + c ) As a result, we have the following theorems. Theorem. If 4) holds, then E û,, ˆv) exists uniquely. If 5) holds, then there exists two distinct equilibria of type E û,, ˆv). Theorem. If 7) holds, then E, ũ, ṽ) exists uniquely. If 8) holds, there exists two distinct equilibria of type E, ũ, ṽ). In the next sections, we will analyze both cases when model ) has only one or more than one solution of type E û,, ˆv), E, ũ, ṽ). Here we have their coordinates. In the case of only one solution of each type, we consider just the first solution with the following coordinates. û = α [K ξ + c ) a ξ] α ξ + c ) ± {α [a ξ K ξ + c )] 4α K ξ + c )p α a ξ)} α ξ + c ) ˆv = α + û ) ξa + ξ + c )û,, ũ = α [K ξ + c ) a ξ] α ξ + c ) ± {α [a ξ K ξ + c )] 4α K ξ + c )p α a ξ)} α ξ + c ) ṽ = α + ũ ) ξa + ξ + c )ũ.,

8 56 WENXIANG LIU AND H. I. FREEDMAN 3. Local stability In order to compute the stability of the various equilibria of system ), we let M be the Jacobian matrix about the point u, u, v). Then α α u K p u q u a q p v u a + u a + u ) α u α M = K p u q u q u a p v a + u. a + u ) a ξ + c u c v a c v a + u a + u ) a + u ) + c ) u a + u Computing M at E,, ξ ), we get and the eigenvalues are α p a ξ M = c a ξ α p a ξ, ξ c a ξ λ = α p a ξ, λ = α p a ξ, λ 3 = ξ. As a result, we have: Theorem 3. If P < α a ξ or P < α a ξ, then E,, ξ ) is a hyperbolic saddle point. If P > α a ξ and P > α a ξ, then E,, ξ ) is locally asymptotically stable.

9 A REACTION-DIFFUSION MODEL 57 It is easy to see that one or both of E û,, ˆv), E, ũ, ṽ) will exist if E,, ξ ) is a hyperbolic saddle point, and neither of them will exist if E,, ξ ) is stable. Computing M at E û,, ˆv), we obtain M = α û ) K p û a q p ˆv û a + û a + û ) α q û p ˆv a c a + û ) c ˆv a ξ + c ) û a + û. Hence the eigenvalues are λ = α q û a p ˆv, σa) = {λ i λ Tr A)λ + deta) =, i =, 3}, where α û ) K A = a c ˆv a + û ) a p ˆv a + û ) p û a + û ξ + c û a + û ). By the Routh-Hurwitz criteria, if Tr A) < and deta) >, then the eigenvalues of A have negative real parts. If û > K /, then Tr A) = α α û a p ˆv K a + û ) ξ + c ) û < a + û

10 58 WENXIANG LIU AND H. I. FREEDMAN and deta) = [α û ) a ] p ˆv K a + û ) ξ + c ) û a c p û ˆv a + û a + û ) = α û ) ξ + c ) û p α ξ + K a + û a + û )[a ξ + ξ + a )û ] >. As a result, we have the following theorem. Theorem 4. Suppose that û > K / and α q û + a p ˆv. If α > q û + a p ˆv, then E û,, ˆv) is a hyperbolic saddle point. If α < q û + a p ˆv, then E û,, ˆv) is locally asymptotically stable. 6 Trajectories of Populations in the Case of No Diffusion. u t) u t) vt) time t FIGURE : Solutions for model ) with α =.5, α =., K = 46., K =., q =.75, q =.5, p =.8, p =.8, a =., a =., c =.4, c =.6, =., ξ = ; u ) = 8., u ) =., v) = 9.. Here the boundary equilibrium E 46.,,.) is locally stable.

11 A REACTION-DIFFUSION MODEL 59 Computing M at E, ũ, ṽ), one obtains α q ũ p ṽ a M = q ũ α α ũ a p ṽ K a + ũ ) p ũ. a + ũ c ṽ a c ṽ a a + ũ ) ξ + c ) ũ a + ũ Hence the eigenvalues are λ = α q ũ a p ṽ, σb) = {λ i λ Tr B)λ + detb) =, i =, 3}, where α ũ ) K B = a c ṽ a + ũ ) a p ṽ a + û ) p ũ a + ũ ξ + c ũ a + ũ ). Similar to the analysis of E, we have the following lemma. Lemma If ũ > K /, then the real parts of eigenvalues λ, λ 3 are negative. Based on Lemma, we have the following theorem. Theorem 5. Suppose that ũ > K / and α q ũ + a p ṽ. If α > q ũ + a p ṽ, then E, ũ, ṽ) is a hyperbolic saddle point. If α < q ũ + a p ṽ, then E, ũ, ṽ) is locally asymptotically stable. Now we wish to examine criteria for there to be no limit cycles in the u v plane, u v plane, and the u u plane. In the u v plane: du dt = α u u ) p u v K a + u dv dt = ξ + c ) u v. a + u

12 6 WENXIANG LIU AND H. I. FREEDMAN 5 Trajectories of Populations in the Case of No Diffusion. u t) u t) vt) time t FIGURE : Solutions for model ) approach E,, 97.6) if the treatment intensity is weak small values of a p and/or ξ ) with a large initial u ) =.. Other parameters and initial conditions are the same as in Figure except p =.4. Using Dulac s negative criterion, we define Du, v) = [ a + u α u u ) p )] u v u p u K a + u + v [ a + u p u ξ + c )) ] u v a + u = α K a ) p K α u p K ξa + u ) p u c p. Clearly, Du, v) < for u, v > if K < a. Therefore, if K < a, then there are no periodic solutions in the u v plane for Du, v). A similar statement holds for the corresponding system in the u v plane, that is, if K < a, then there are no periodic solutions in the u v plane.

13 A REACTION-DIFFUSION MODEL 6 Trajectories of Populations in the Case of No Diffusion. u t) u t) vt) time t FIGURE 3: A solution for model ) with α = 3.9, α = 37., K = 5., K = 67., q =.8, q =.8, p =.5, p = 8., a =., a = 8., c =., c = 36., = 5., ξ = 5; u ) =., u ) = 6., v) =.. Here system ) is uniformly persistent and the interior equilibrium E 39., 4., 96.) is locally stable. Since there is no equilibrium in the u u plane, there are no periodic solutions in this plane. Based on the above results, we may address the question of an interior equilibrium in u u v space by using the techniques in Freedman and Waltman [5, 6], and the results in Butler et al. [7], and obtain the following theorem. Theorem 6. Suppose that K min{a, û } and K min{a, ũ }. If α > q ũ + a p ṽ and α > q û + a p ˆv, then system ) is uniformly persistent, and hence E 3 u, u, v ) exists. 4 The diffusion case To study the effects of spatial variations, we first note that the non-uniform diffusive steady state produces equations that can not be solved in closed form. We therefore consider the effects of small space - time perturbations of the uniform steady states, E, E,

14 6 WENXIANG LIU AND H. I. FREEDMAN E, E 3. Let us assume that x = s for the case of one spatial dimension with s a, where a is a constant. This spatial dimension could be measured over a section of the bone marrow or over the total space of distribution where leukemia activity may be significant. Before we study the stabilities of these steady states, we first establish the existence and uniqueness of solutions of system ). 4. Preliminaries In this section we introduce the concept of upper and lower solutions as well as an existence-comparison theorem, which will be very useful to us in establishing the existence, uniqueness, and boundedness, and even in studying the asymptotic behavior in some sense) of the solutions. We first consider the more general system 9) with boundary condition u t L u = f u, u, u 3 ), u t L u = f u, u, u 3 ), u 3 t L 3u 3 = f 3 u, u, u 3 ), B i [u i ] = α i x)u i + β i x) u i n = h ix), i =,, 3 on Ω R +, and initial condition u i x, ) = u i x), i =,, 3 in Ω, where L i is a uniformly elliptic operator in Ω, i =,, 3. We assume that α i, β i and u i are smooth nonnegative functions with u i, α i + β i > and that f i is continuously differentiable with respect to its variables for u k, i, k =,, 3. In addition, we require that f = f, f, f 3 ) is a quasi-monotone function, i.e.: f u, f u, f 3 u, f u 3, f u 3, f 3 u

15 A REACTION-DIFFUSION MODEL 63 for u i, i =,, 3. Now, we give the definition of upper and lower solutions. Definition 4. Ordered smooth functions ū = ū, ū, ū 3 ) and u = u, u, u 3 ) in Ω T are called upper and lower solutions of 9) respectively, if they satisfy the following inequalities in Ω T. ū ) t L ū f ū, u, u 3 ) u ) t L u f u, ū, ū 3 ), ū ) t L ū f u, ū, u 3 ) u ) t L u f ū, u, ū 3 ), ū 3 ) t L 3 ū 3 f 3 u, u, ū 3 ) u 3 ) t L 3 u 3 f 3 ū, ū, u 3 ) B i [ū i ] h i x) B i [u i ], i =,, 3 on S T, ū i x, ) u i x) u i x, ), i =,, 3 on Ω, where Ω T = Ω, T ], S T = Ω, T ], and T < but can be arbitrarily large. Suppose ū and u exist. Denote Σ = {u, u, u 3 ) R 3 : ρ i u i ρ i, i =,, 3}, M i = sup P { } f i u i, i =,, 3, where ρ i = inf x,t) ΩT u i x, t), ρ i = sup x,t) ΩT ū i x, t), i =,, 3. We construct the sequences {ū k) } and {u k) } with ū ) = ū and u ) = u as follows: ū k) ) t L ū k) + M ū k) = M ū k ) + f ū k ), u k ), u k ) 3 ), ū k) ) t L ū k) + M ū k) = M ū k ) + f u k ), ū k ), u k ) 3 ), ū k) 3 ) t L 3 ū k) 3 + M 3 ū k) 3 = M 3 ū k ) 3 + f 3 u k ), u k ), ū k ) 3 ), u k) ) t L u k) + M u k) = M u k ) + f u k ), ū k ), ū k ) 3 ), u k) ) t L u k) + M u k) = M u k ) + f ū k ), u k ), ū k ) 3 ), u k) 3 ) t L 3 u k) 3 + M 3 u k) 3 = M 3 u k ) 3 + f 3 ū k ), ū k ), u k ) 3 ),

16 64 WENXIANG LIU AND H. I. FREEDMAN and B i [ū k) i ] = h i x) = B i [u k) i ], i =,, 3, x, t) S T, ū k) i x, ) = u i x) = u k) i x, ), i =,, 3, x Ω. By using standard techniques e.g., C. V. Pao [8]), we can establish the following existence-comparison theorem. Theorem 7. Suppose that f = f, f, f 3 ) is a quasi-monotone function and there exists a pair of upper and lower solutions ū = ū, ū, ū 3 ) and u = u, u, u 3 ) satisfying u i ū i, i =,, 3. Then the sequences {ū k) } and {u k) } obtained as above converge monotonically from above and below, respectively, to a unique solution u = u, u, u 3 ) of 9) such that u i x, t) u i x, t) ū i x, t), i =,, 3, x, t) Ω T. In view of Theorem 7, to obtain the existence and uniqueness of solutions of ), we need only to find a pair of upper and lower solutions of ). We do this as follows by using an appropriate O.D.E. problem to find upper and lower solutions. For an upper solution, we study the O.D.E. system du dt du dt = α u u ), K = α u u ), K dv dt = ξv, with initial conditions u i ) = ũ i sup u i x) >, i =,, 3, Ω v) = ṽ sup v x) >. Ω

17 A REACTION-DIFFUSION MODEL 65 Then we have [ ū t) = K + K ] ũ e αt, ũ [ ū t) = K + K ] ũ e αt, ũ vt) = ξ + ṽ ξ )e tξ. Clearly,,, ) and ū t), ū t), vt)) are a pair of lower and upper solutions of ). Hence we can use Theorem 7 for any T > and obtain: Theorem 8. There exists a unique solution u x, t), u x, t), vx, t)) to system ) satisfying u i x, t) ū i t), i =,, vx, t) vt). We have established the global existence and uniqueness of the solutions of ). Now, we will prove the global boundedness of these solutions. From the above, it is easy to see that u x, t) ū t) max{k, ũ }, u x, t) ū t) max{k, ũ }, vx, t) vt) max{ ξ, ṽ}. Hence, all solutions of ) are uniformly bounded for x, t) Ω R +. Next, we analyze the asymptotic behavior of the three populations. Theorem 9. Suppose that α q K a ξ p >, α q K a ξ p >. Then system ) is persistent. Proof. We are trying to find a pair of upper and lower solutions with

18 66 WENXIANG LIU AND H. I. FREEDMAN the property: dū t) dt dū t) dt ) = α ū ū q ū u K, ) = α ū ū q u K ū, and du t) dt du t) dt d vt) dt = vξ, = α u u ) q K u K a p u v, = α u u ) q K u K a p u v, dvt) dt with initial conditions and = ξ + a c K + a c K ) v, ū i ) = ũ i sup u i x) >, i =,, Ω v) = ṽ sup v x) >, Ω u i ) = u i inf Ω u i x) >, i =,, v) = v inf Ω v x) >. Here, we take K i = max{k i, ũ i }, i =, for convenience. Obviously, ū, ū, v) and u, u, v) are a pair of lower and upper solutions of ); moreover, we have vt) = ξ + ṽ ξ )e tξ, vt) = ξ + a c K + a c K ) + [ṽ ξ + a c K a c K ) ]e tξ+a ck+a ck).

19 A REACTION-DIFFUSION MODEL 67 Therefore, lim inf t lim inf t Again, vt) = ξ + a c K + a c K ) >, vx, t) lim inf t vt) = ξ + a c K + a c K ) >. u t) = α q K a p v)ce α qk a p v)t + α K ceα qk a where c is a constant. Hence lim inf t u x, t) lim inf t Similarly, we have lim inf t u x, t) lim inf t The proof is complete. p v)t, u x, t) = α q K a ξ p α K >. u x, t) = α q K a ξ p α K >. Now we will analyze the stability of all possible equilibria of system ), E,, ξ ), E û,, ˆv), E, ũ, ṽ), E 3 u, u, v ). 4. The analysis of E,, ξ ) To examine the stability of the uniform steady state E,, ξ ) to perturbations, we write ) u x, t) = + ε x, t), u x, t) = + ε x, t), vx, t) = ξ + ηx, t).

20 68 WENXIANG LIU AND H. I. FREEDMAN By substituting ) into ), and linearizing the equations, we obtain ε t = D ε x + α p a ξ )ε + ε + η, ε t = D ε x + ε + α p a ξ )ε + η, ) η t = D η 3 x c a ξ ε c a ξ ε ξη, ε x ε x = ε x= x = η x= x =, x= = ε x=a x = η x=a x =. x=a For an examination of linear stability, it is sufficient to assume solutions of ) are in the form ε e tλ cos kx, ε e tλ cos kx, η e tλ cos kx, where λ and k are the frequency and wave number respectively. The eigenvalue equation then reads λ + D k +a ξ p α λ + D k +a =. ξ p α a ξ c a ξc λ + D 3 k + ξ Hence λ = α a ξ p D k, λ = α a ξ p D k, λ 3 = ξ D 3 k.

21 A REACTION-DIFFUSION MODEL 69 Note that the diffusion has no effect on the stability of E,, ξ ) in this case. Now by applying the boundary conditions ) we obtain k = nπ/a,where n is an integral constant. If a is very small, then D i k becomes very large. Within this frame work, it can be realized that the steady state would be stable to small space-time perturbations for all time, even though it is unstable without diffusion, which means leukemia can not win the competition within the small space-time perturbations and this is not true in the case where there is no diffusion when P i < α i a i ξ, i =,. Therefore, the steady state E,, ξ ) is unstable in the homogeneous case, as is shown by the analysis of system ), but is stable in the presence of spatial variations. 4.3 The analysis of E û,, ˆv) Again, to examine the stability of the uniform steady state E to perturbations, we write ) u x, t) = û + ε x, t), u x, t) = + ε x, t), vx, t) = ˆv + ηx, t). By substituting ) into ) and linearizing the equations, we obtain ε t = D ε x + a ε + a ε + a 3 η, 3) ε t = D ε x + a ε + a ε + a 3 η, η t = D η 3 x + a 3ε + a 3 ε + a 33 η, and 4) ε x ε x = ε x= x = η x= x =, x= = ε x=a x = η x=a x =, x=a

22 7 WENXIANG LIU AND H. I. FREEDMAN where a = α û K a =, ) a p ˆv a + û ), a = q û, a 3 = p û a + û, a = α q û a p ˆv, a 3 =, a 3 = a c ˆv a + û ), a 3 = a Let c ˆv, a 33 = ε e tλ cos kx, ε e tλ cos kx η e tλ cos kx, ξ + c û a + û where λ and k are the frequency and wave-number respectively. The eigenvalue equation then reads λ + D k a a a 3 λ + D k a =. a 3 a 3 λ + D 3 k a 33 Hence where λ = a D k, σa) = {λ i λ Tr A)λ + deta) =, i =, 3}, A = [ ] a D k a 3 a 3 a 33 D 3 k. The condition k = corresponds to neglecting diffusion and the constant solution E û,, ˆv) is unstable with α > q û + a p ˆv, û > K /. Now we will discuss the stability with the case k. Case : û > K /. Tr A) = a + a 33 D + D 3 )k <, deta) = D D 3 k 4 a D 3 + a 33 D )k + a a 33 a 3 a 3 ) >, and so, a diffusion-driven instability can be immediately excluded with the case, α < q û + a p ˆv. ).

23 A REACTION-DIFFUSION MODEL 7 Evolution of Normal Cells in the Space and Time u x,t) Time t Distance x.8 FIGURE 4: A solution for model ) with the same parameters and initial conditions as in Figure along with, D =, D =, D 3 = 3. Here the uniform steady sate u x, t) = û is locally stable and diffusion has no effects on the stability. Theorem. Suppose that û > K /, and α q û +a p ˆv. Then the diffusion has little effect on the stability of the uniform steady state compared with the case where there is no diffusion. That is, if α > q û + a p ˆv + D k, then E û,, ˆv) is unstable. If α < q û + a p ˆv + D k, then E û,, ˆv) is locally asymptotically stable. From the boundary conditions 4), we can see that k = nπ/a. If a is sufficiently small, then α > q û + a p ˆv + D k could be easily violated, and then leukemia will eventually be driven to extinction. If a is large enough, i.e. the diffusion region is large enough, the chemotherapy agent is not effective anymore and leukemia will eventually kill the normal cells and will win the competition. Let δ = α q û a p ˆv >, a s = nπ D /δ. Then we obtain the stable region, [, a s ]. Within this region, the leukemia will be eventually excluded and the normal cells will win the competition.

24 7 WENXIANG LIU AND H. I. FREEDMAN Evolution of Abnormal Cells in the Space and Time...8 u x,t) Time t Distance x.8 FIGURE 5: A solution for model ) with the same parameters and initial conditions as in Figure 4. Here cancer cells decay with time and diffusion has no effects on the stability. 6 Solution Prolifes at a Selection of Distances. u x) u x) vx) Time t FIGURE 6: A collection of solution profiles for model ) with a selection of distances in both Figure 4 and 5, which demonstrates that the uniform steady steady E is locally stable in the diffusion case.

25 A REACTION-DIFFUSION MODEL 73 Case : µ < û < µ < K /). In this case, Tr A) = a + a 33 D + D 3 )k, deta) = D D 3 k 4 a D 3 + a 33 D )k + a a 33 a 3 a 3 ). By the Routh-Hurwitz criterion, if Tr A) < and deta) >, then the eigenvalues of A have negative real parts. Let k m = max { a + a 33 D + D, a D 3 + a 33 D + {a D 3 + a 33 D ) + 4D D 3 a 3 a 3 a a 33 )} / D D 3 Then, Tr A) <, deta) > if k > k m. Theorem. Suppose µ < û < µ < K /) and α q û +a p ˆv. If α > q û + a p ˆv + D k, then E û,, ˆv) is unstable. If α < q û + a p ˆv + D k and k > k m, then E û,, ˆv) is locally asymptotically stable. By applying the boundary conditions, we obtain the stable region [, nπ/k m ]. Within this region, the leukemia will be eventually driven to extinction. The condition α > q û +a p ˆv +D k, which guarantees that leukemia wins could be easily violated again when a is sufficiently small and the other parameters are fixed. That is, when the diffusion region is small and falls into [, nπ/k m ], the chemotherapy agent is very effective and will kill all leukemia eventually. However, when the diffusion region grows out of the stable region, the chemotherapy agent is not effective enough to kill the leukemia and the normal cells finally lose the competition. 4.4 The analysis of E, ũ, ṽ) To examine the stability of the uniform steady state E to perturbations, we write }. 5) u x, t) = + ε x, t), u x, t) = ũ + ε x, t), vx, t) = ṽ + ηx, t).

26 74 WENXIANG LIU AND H. I. FREEDMAN By substituting 5) into ) and linearizing the equations, we obtain ε t = D ε x + b ε + b ε + b 3 η, 6) and 7) ε t = D ε x + b ε + b ε + b 3 η, η t = D η 3 x + b 3ε + b 3 ε + b 33 η, ε x ε x = ε x= x = η x= x =, x= = ε x=a x = η x=a x =, x=a where b = α q ũ a p ṽ, b =, b 3 =, b = q ũ, b = α ũ ) a p ṽ K a + ũ ), b 3 = p ũ, a + ũ b 3 = a c ṽ, b 3 = a c ṽ a + ũ ), b 33 = ξ + c ) ũ. a + ũ Letting ε e tλ cos kx, ε e tλ cos kx, η e tλ cos kx, where λ and k are the frequency and wave-number respectively; thus, we get the eigenvalue equation λ + D k b b λ + D k b b 3 =. b 3 b 3 λ + D 3 k b 33

27 A REACTION-DIFFUSION MODEL 75 Hence where λ = b D k, σb) = {λ i λ Tr B)λ + detb) =, i =, 3}, 8) B = [ b D k b 3 b 3 b 33 D 3 k The condition k = corresponds to neglecting diffusion and the constant solution E, ũ, ṽ) is unstable with α > q ũ + a p ṽ, ũ > K /. Again, we will try to work on the stability with the case k. Case : ũ > K /. Here we have Tr B) = b + b 33 D + D 3 )k <, detb) = D D 3 k 4 b D 3 + b 33 D )k + b b 33 b 3 b 3 ) >. Theorem. Suppose that ũ > K /, and α q ũ +a p ṽ+d k. If α > q ũ + a p ṽ + D k, then E, ũ, ṽ) is unstable. If α < q ũ + a p ṽ + D k, then E, ũ, ṽ) is locally asymptotically stable. From the boundary conditions 3), we can see that k = nπ/a. If a is sufficiently small, then α > q ũ + a p ṽ + D k could be easily violated, and then the leukemia will eventually be eliminated. If a is large enough, i.e. the diffusion region is large enough, the chemotherapy agent is not effective anymore and leukemia will eventually kill the normal cells and will win the competition. Let σ = α q ũ a p ṽ >, a u = nπ D /σ. Then we obtain the unstable region, [a u, ). Within this region, the leukemia would be eventually eliminated and the normal cells win the competition. Case : τ < ũ < τ < K /). Then Tr B) = b + b 33 D + D 3 )k, detb) = D D 3 k 4 b D 3 + b 33 D )k + b b 33 b 3 b 3 ). ].

28 76 WENXIANG LIU AND H. I. FREEDMAN By the Routh-Hurwitz criterion, if Tr B) < and detb) >, then the eigenvalues of B have negative real parts. Let { km b + b 33 = max, D + D 3 b D 3 + b 33 D + {b D 3 + b 33 D ) + 4D D 3 b 3 b 3 b b 33 )} / D D 3 Then, Tr B) <, detb) > if k > k m. As a result, we have the following theorem. Theorem 3. Suppose τ < ũ < τ < K /) and α q ũ +a p ṽ+ D k. If α > q ũ + a p ṽ + D k, then E, ũ, ṽ) is unstable. If α < q ũ + a p ṽ + D k and k > k m, then E, ũ, ṽ) is locally asymptotically stable. By applying the boundary conditions, we obtain the stable region [, nπ/k m ]. Within this region, normal cells will be eventually extinct. }. 5 Numerical results and discussion In order to perform the numerical simulations of system ) and ), we impose some conditions based both on the analytical results and on some physiological arguments: a) α > α cancer cells grow faster than normal cells). b) K K the carrying capacity of cancer cells may be greater than that of normal cells). c) p >> p the drug is more potent against the cancer cells than against the normal cells). d) c >> c a consequence of the above item). e) the hypotheses of Theorems and which guarantee the existence of E and E. Also the initial condition is such that: u > u in general). The three most important steady states of our model from a physiological point of view are E, E and E 3. E represents a cancer free state, and of course the most desirable result would be to have E globally asymptotically stable. E represents a state which is exclusively

29 A REACTION-DIFFUSION MODEL 77 cancerous and is a lethal state for the individual. E 3 represents a state where normal and cancer cells exist simultaneously. We are able to obtain analytic criteria for E to be locally asymptotic stable, but not globally stable. Numerically, we can show that if the initial cancer value is sufficiently small, solutions may approach E Figure ). However, if u is large, solutions may approach E if the treatment intensity is weak small values of a p and/or ξ ), and may approach E 3 Figure 3) if the treatment intensity is strong higher values of a p and/or ξ ). This demonstrates an equi-asymptotical stability in the large for E 3. Also the region of stability of E is found numerically to increase with higher treatment levels. On the other hand, we show that it is possible to choose parameters and initial values so that solutions of system ) approach a positive physiologically) state E 3 with the cancer level small. In these parameters we note that ξ is significantly higher than in the previous cases. In the small space region, we find numerically that diffusion has no effect on the stability of constant solutions if the treatment is strong enough and solutions of system ) approach the uniform steady state E Figures 4, 5 and 6). However, for a large initial cancer value u, solutions would not approach the uniform steady state E under a reasonable weak treatment intensity, but they eventually approach the uniform steady state E 3 exhibiting coexistence of three populations with a small cancer lever.figures 7, 8 and 9). When we increase the cancer initial value a little more under the same treatment condition, solutions still approach the uniform interior steady sate but with a large cancer level Figures ). This implies that there may be accumulations of cells at certain sites and depletions at other sites and through such processes leukemic cells may occupy sites of normal cells as the propagation of spatial heterogeneities occur. As a result, it may be suggested that the positions occupied by the leukemic cells as they expand, may be very fertile areas that are rich in nutrients needed for hematopoiesis. This is because those positions used to be occupied by the displaced normal cells. Thus, leukemic cell numbers may increase very rapidly. Also, upon introduction of more leukemic cells, existing normal cell colonies go through a process of shrinkage Figure )as their positions are invaded by emerging colonies of abnormal cells. The resulting leukemic dominance may cause damage to and disturb the colony-forming capabilities of the normal cells. Proceeding from the facts above, it is appropriate to suggest that through certain diffusive processes and mechanisms, the normal cells are displaced from their positions by colonies of leukemic cells over a wide region of space and are eventually driven to extinction. Essentially the

30 78 WENXIANG LIU AND H. I. FREEDMAN leukemic colonies display a tendency to invade the spaces designated for normal cell growth. Also, the rapid increase in the leukemic population over a period of time may result in a high leukemic cell density. This could lead to a migration of leukemic cells, possibly through a diffusive process as we described in this article to regions of lower cell density and nutrient availability. This could account for the reasons why other organs of the body become clogged with masses of abnormal cells, as is noted in [9]. It is important to mention that the predictions of the model may hold for some acute leukemias but not for the chronic leukemias in which there is a more gradual procession towards leukemic dominance []. Finally, we point out how our results may be useful to leukemia treatment. For those situations in which our models may apply, the theorems identify which combination of coefficients, i.e., sufficiently high or low values would lead to low levels of cancer, or elimination altogether. It is known [] that different leukemias can be treated with various successes or failures), and our results may be used to help in improving the successes i.e., the rate of success or length of remission time). Evolution of Normal Cells in the Space and Time. 4 3 u x,t) Time t Distance x.8 FIGURE 7: A solution for model ) with the same parameters and initial conditions as in Figure 4 except p =.4a weak intensive treatment, i.e. small a p) and a small initial cancer number u =.. Here the normal cells eventually approach a uniform interior steady state.

31 A REACTION-DIFFUSION MODEL 79 Evolution of Abnormal Cells in the Space and Time..5 u x,t) Time t Distance x.8 FIGURE 8: A solution for model ) with the same parameters and initial conditions as in Figure 7. Here the abnormal population survives the treatment and eventually approaches an interior steady state u =.5 under the weak treatment intensity. 4 Evolution of Solution Profiles at a Selection of Distance. u x) u x) vx) Time t FIGURE 9: A collection of solution profiles for model ) with a selection of distances in both Figure 7 and 8, which shows that solutions of model ) approach an interior uniform steady state with a small cancer lever under a weak treatment intensity.

32 8 WENXIANG LIU AND H. I. FREEDMAN Evolution of Normal Cells in the Space and Time. 8 u x,t) Time t Distance x.8 FIGURE : A solution for model ) with the same parameters and initial conditions as in Figure 7 except a large initial cancer number u =.6. Here the normal cells begin to go through a process of shrinkage. Evolution of Abnormal Cells in the Space and Time. 4 u x,t) Time t Distance x.8 FIGURE : A solution for model ) with the same parameters and initial conditions as in Figure. Here the abnormal population survives the treatment and eventually dominates the populations.

33 A REACTION-DIFFUSION MODEL 8 5 Evolution of Solution Profiles at a Selection of Distances. u x) u x) vx) Time t FIGURE : A collection of solution profiles for model ) with a selection of distances in both Figure and, which shows that solutions of model ) approach an interior uniform steady state with a high cancer level under a weak treatment intensity. REFERENCES. S. L. Schrier, The leukemia and the myeloproliferative disorders, Sci. Am. Med. VIII 994),.. S. L. Schrier, Hematopoiesis and red blood cell function, Sci. Med. I 998),. 3. E. A. McCulloch and J.E. Till, Blast cells in acute myeloblastic leukemia: A model, Blood cells 7 98), E. A. McCulloch, Stem cells in normal and leukemia hemopoiesis, Blood 6) 983),. 5. E. A. McCulloch, The blast cells of acute myeloblastic leukemia, Clin. Haematol ), E. A. McCulloch, A. F. Howatson, R. N. Buick, M. Minden and C. A. Izaguirre, Acute myeloblastic leukemia considered as a clonal hemoathy, Blood Cells 5 979), R. T. Dorr and D. D. Von Hoff, Cancer Chemotherapy Handbook, Appleton and Lange, N ORWALK, Connecticut, E. Frei, III, Curative caner chemotherapy, Cancer Res ), L. A. Liotta, Cancer cell invasion and metastasis, Scientific American, February 99), M. C. Perry ed), The Chemotherapy Sourcebook, Williams and Wilkins, Baltimore, 99.. H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 98.

34 8 WENXIANG LIU AND H. I. FREEDMAN. F. K. Nani and H. I. Freedman, A mathematical model of cancer treatment by chemotherapy, to appear. 3. S. T. R. Pinho, H. I. Freedman and F. K. Nani, A chemotherapy model for the treatment of cancer with metastasis, Math. Comp. Model. 36 ), H. I. Freedman and P. Moson, Persistence definitions and their connections, Proc. Amer. Math. Soc. 9 99), H. I. Freedman and P. Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci ), G. J. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Aer. Math. Soc. 963) 986), C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 8798), J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, E. S. Henderson, Acute leukemia in adults, in: Hematologic Malignancies B. Hoogsstraten, ed.), Springer, Berlin, p. 7, W. Liu, H. I. Freedman, Dynamic processes leading to blast cell dominance in chronic myelocytic leukemia, to appear.. R. A. Gatenby, Models of tumor-host interaction as competing populations: implications for tumor biology and treatment, J. Theor. Biol ), Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G G address: [email protected] address: [email protected]