ANALYSIS OF A DELAYED FREE BOUNDARY PROBLEM FOR TUMOR GROWTH

. In this paper we study a delayed free boundary problem for the growth of tumors. The establishment of the model is based on the diﬀusion of nutrient and mass conservation for the two process proliferation and apopto-sis(cell death due to aging). It is assumed the process of proliferation is delayed compared to apoptosis. By L p theory of parabolic equations and the Banach ﬁxed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the asymptotic behavior of the solution, and prove that in the case c is suﬃciently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a dormant state as t → ∞ .


1.
Introduction. Over the last thirty years, a variety of partial differential equation models for tumor growth or therapy have been developed, cf. [2-4, 9, 13-14, 17-18] and references therein. Most of those models are based on the reaction diffusion equations and mass conservation law. Analysis of such free boundary problems has drawn great interest, and many interesting results have been established, cf. [1, 5-8, 10-12, 15-17, 19-23] and references therein.
In this paper we study the following problem: c ∂σ ∂t = ∆ r σ(r, t) − λσ(r, t), 0 < r < R(t), t > 0, ∂σ ∂r (0, t) = 0, σ(R(t), t) = σ ∞ , t > 0, (2) d dt σ(r, t) = ψ(r, t), 0 ≤ r ≤ R(t), −τ ≤ t ≤ 0, where r is the radial variable scaled by the tumor-cell radius, t is the time variable scaled by the tumor-cell doubling time, the variable σ(r, t) represents the scaled nutrient concentration at radius r and time t and the variable R(t) represents the scaled radius of the tumor at time t, ∆ r = 1 r 2 ∂ ∂r (r 2 ∂. ∂r ). The term λσ in (1.1) is the scaled consumption rate of nutrient in a unit volume within a unite time interval; µσ is the scaled proliferation rate of tumor cells in a unit volume(=the number new-born cells in a unite volume within a unite time interval); σ ∞ reflects scaled constant supply of nutrient that the tumor receives from its surface, ψ and ϕ are given initial date of σ and R. c is a constant and c ≪ 1(cf [12]).
The study of the effects of time delays in the growth of tumors by using the methods of mathematical models was initiated by [2]. Experiments suggest that changes in the proliferation rate can trigger changes in apoptotic cell loss and that these changes do not occur instantaneously: they are mediated by growth factors expressed by the tumor cells (see [2]). In [2], the author considered two ways of modifying the standard model of avascular tumor growth by incorporating into the net proliferation rate a time-delayed factor. In the first case, the delay represents the time taken for cells to undergo mitosis. In the second case, the delay represents the time for changes in the proliferation rate to stimulate compensatory changes in apoptotic cell loss. The models presented in [2] are quasi-stationary version (i.e. c=0) of free boundary problem with a necrotic core inside the tumor and the nutrient is consumed by tumor cells with constant rate. Numerical and asymptotic techniques were used to show how a tumor's growth dynamics are affected by including such delay terms. In particular for the first case, the author showed that the size of the delay does not affect the limiting behavior of the tumor. Recently, Foryś and Bodnar [10] made a rigorous analysis of the model with time delay presented in Byrne [2] of the first case in the frame work of delay differential equations. In [10], they only considered a simpler model without a necrotic core inside the tumor.They mainly considered a tumor's growth dynamics and showed that how the time delay affect the tumor growth. They proved existence of periodic solutions for some parameter and global stability of steady-state solutions for other ones.
The idea of the model studied in this paper come from the paper Byrne [2], Foryś and Bodnar [10] and Cui and Xu [8]. The model is established by modifying the model A. Friedman [12] by considering the time delay effect as the first case in Byrne [2](i.e. the time delay represents the time taken for cells to undergo mitosis). Obviously, this is a fully non-stationary version a free boundary problem for tumor growth with a time delay. The limiting case where c = 0 (i.e. the quasi-stationary version) is studied by [8]. In [8] rigorous analysis of the limiting case of the model is given. Final mathematical formulations to the limiting case is a retarded differential equation of the form By using a comparison method, the authors discussed the dynamical behavior of solutions to the model. They proved that the dynamical behavior of solutions of the problem is similar to that of solutions for corresponding problem without time delay. In the limiting case where c = 0 Eq.(1), (2) can be solved exactly, and the exact expression of the evolution equation for R can be obtained. This is clearly not the case for present model and the method used in [8] can not be used to present model. Using Banach fixed point theorem, a compare method and some mathematical techniques, we mainly prove the existence and uniqueness of the global solution to the problem and asymptotic behavior of the solutions to the problem. The results show that in the case c is sufficiently small and σ ∞ >σ, the volume of the tumor cannot expand unlimitedly. It will evolve to a dormant state as t → ∞ which is similar to that of the corresponding problem without time delay (see [12]). We also show that in the case c is sufficiently small and σ ∞ <σ, the volume of the tumor also cannot expand unlimitedly. It will disappear. The paper is arranged as follows: In Section 2 we prove the existence and uniqueness of the global solution to the system (1)- (5). Section 3 is devoted to the asymptotic behavior of the solutions to the system (1)- (5). In the last Section we give some conclusions.

2.
Global existence and uniqueness. We shall prove a global existence and uniqueness theorem for the problem (1)-(5) under the following assumptions: For a given number T > 0 and a given positive function R ∈ C[0, T ] we introduce the following notations: denotes the ball in R 3 centered at origin with radius R 0 . Lemma 2.1. (see [20] Lemma 1 or [7] Lemma 2.1) Let c, T be given positive numbers. Let R(t) ∈ C 1 [0, T ], R(t) > 0 for all 0 ≤ t ≤ T, and R(0) = R 0 . Let ψ(0, x) ∈ D p,σ∞ for some 5 2 < p < ∞, and F ∈ C(Q R T ). Then the following initial value problem: σ(x, 0) = ψ(x, 0), |x| ≤ R 0 (9) has a unique solution σ in the sense that satisfies the following three conditions: (2) σ satisfies the equation (7) a.e. in Q R T ; (3) σ satisfies the conditions (8) and (9). Moreover, the following assertions hold: (i) If ψ(x, 0) and F (x, t) are spherically symmetric in x then σ is also spherically symmetric in x.
(ii) There exists a positive constant C depending on σ ∞ , c and Our main results of this section are as follows.
Next, we prove the mapping F is a contraction mapping for small T. Let (σ i , R i ) ∈ S T , (i = 1, 2) and denote F (σ i , R i ) = (σ i ,R i ), (i = 1, 2). From (15), we have for any 0 ≤ t ≤ T, Substituting this estimate to (22) we have Noticing that Then h satisfies the problem Let h 1 = h 1 (r, t)(r > 0, 0 ≤ r ≤ T ) be the solution to the following initial value problem: Letting Then by maximum principle we have

It follows that max
Then max r≥0,0≤t≤T For m(t) ≤ r ≤ M (t) and 0 ≤ t ≤ T and p > 5, by Lemma 2.1 (ii) and the embedding theorem, we have Noticing that max r≥0,−τ ≤t≤0 we have max r≥0,−τ ≤t≤T By (25) and (34), we have Therefore, F is a contraction mapping for small T. 3. Asymptotic behavior of the solutions to the system (1)- (5). In this section, we study asymptotic behavior of the solutions to (1)-(5). First we consider the case σ ∞ <σ.
From the left inequality above we can get Set ω(t) = R 3 (t), by the right inequality of (35) we have Set c is the unique real value root of the equation z = −µσ + µσ ∞ e −τ z . By σ ∞ <σ we readily have c < 0. Consider the following initial value problem: The solution to the above problem is x(t) = C 3 e ct . Since when −τ ≤ t ≤ 0, Next, we consider the case σ ∞ >σ. By Theorem 2.1 [12], the system (1)-(5) has a unique stationary solution (σ s (r), R s ), R s > 0 in this case. In the following, we prove that (σ s (r), R s ) is asymptotically stable provided c is small.
. Then the following assertions hold: Proof. The proof of (1) can be found in [12] and the proof of (3),(4) can be found in [8]. Next we prove (2), from Lemma 3.3 [6] we know that and xp ′′ (x) p ′ (x) is strictly monotone decreasing for all x > 0. By simple computation, it follows that From [23] we know that lim x→0 p ′ (x) = 1, then we have −2 < xp ′′ (x) p ′ (x) < 1. This completes the proof. Lemma 3.4. Let (σ(r, t), R(t)) is the solution to (1)- (5). Assume that the conditions (A 1 ) -(A 3 ) are satisfied. If σ ∞ >σ, assume for some ε > 0, ε ≤ R(t) ≤ 1 ε for −τ ≤ t ≤ 0. Then there exists a positive constant c 0 independent of c, R(t) for −τ ≤ t ≤ 0 such that Proof. From (35) and the assumption ε ≤ R(t) ≤ 1 ε for −τ ≤ t ≤ 0, we have Assume that (44) is not valid for some t. It follows that there exists T > τ such that for 0 ≤ t < T, By (35) and the fact that for 0 ≤ t < T, we have R ′ (t) ≤ L 0 , L 0 is a positive constant independent of c and T . Obviously for arbitrary 0 ≤ r ≤ R(t), 0 ≤ t < T and 0 < c ≤ c 0 . Then we have for t > τ It follows that for T > τ where we have used the fact x 3 p(x) for x > 0 is monotone increasing (Lemma 3.3 (3)). From Lemma 3.3 (1) we known function p(x) is monotone decreasing for any x > 0, noticing R(T ) > R s we have 3µσ ∞ p( √ λR(T )) − λσ < 0, then if c 0 is sufficiently small and 0 < c ≤ c 0 it follows that R ′ (T ) < 0 which contracts to the fact R ′ (T ) ≥ 0.
If R(T ) = 1 2 min(ε, R s ) similar arguments can prove the desired assertion. This completes the proof.
Lemma 3.6. Assume that the conditions (A 1 ) -(A 3 ) are satisfied. Assume further that for all t > −τ, where C * and C * are two constants independent of c and α. If σ ∞ >σ, Then there exist positive constants c 0 , θ, T 0 and C independent of c such that the following assertions holds: If 0 < c ≤ c 0 , for any α ∈ (0, α 0 ], if the inequalities hold for all 0 ≤ r ≤ R(t), t ≥ −τ and |R ′ (t)| ≤ α holds for all 0 ≤ r ≤ R(t), t ≥ 0, then also the inequalities ,(52) and |R ′ (t)| ≤ α holds for all 0 ≤ r ≤ R(t), t ≥ 0, we can get for t > τ Noticing for t ≥ 2τ, e − λ(t−τ ) c ≤ e − λτ c < c λτ , it follows that for t > 2τ where C 2 is a positive constant independent of α and c. Here and hereafter for easy of notation we use the same notation to denote various different positive constants independent of c and α. Consider initial value problems By Lemma 3.5 we know that there exist positive constants α 0 , c 0 such that for α ∈ (0, α 0 ], c ∈ (0, c 0 ] the equations G(x, x) ± Cαc = 0 has respectively unique solutions R ± s , and the corresponding solutions of the equations to the initial problem above which we respectively denote as R ± (t), converge respectively to R ± s as t → ∞. By the fact p(x) is monotone decreasing for all x > 0, we can get Actually, since R ± s respectively satisfies the equations p( √ λR ± s )−σ 3σ∞ = ∓Cαc and R s satisfies the equation p( √ λR s ) −σ 3σ∞ = 0, by the fact (51) and p(x) is monotone decreasing for all x > 0, we readily have | R ± s − R s |≤ Cαc. Then there exists M * > 0 and T > 0 such that for all t > T, R ± (t) ≤ M * . If we choose M > M * , comparison principle (cf. [8] Lemma 3.1) implies for all t ≥ T R − (t) ≤ R(t) ≤ R + (t).