ANALYSIS OF A FREE BOUNDARY PROBLEM FOR TUMOR GROWTH WITH GIBBS-THOMSON RELATION AND TIME DELAYS

. In this paper we study a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. It is assumed that the pro- cess of proliferation is delayed compared with apoptosis. The delay represents the time taken for cells to undergo mitosis. By employing stability theory for functional diﬀerential equations, comparison principle and some meticulous mathematical analysis, we mainly study the asymptotic behavior of the solu- tion, and prove that in the case c (the ratio of the diﬀusion time scale to the tumor doubling time scale) is suﬃciently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to one of two dormant states as t → ∞ . The results show that dynamical behavior of solutions of the model are similar to that of solutions for corresponding nonretarded problems under some conditions.


1.
Introduction. The process of tumor growth is a complex process. To describe the process, in recent years, an increasing number of mathematical models in forms of free boundary problems of partial differential equations have been proposed and studied, cf. [3,4,5,6,15,16,19,21]. The process of tumor growth has several different stages, starting from the very early stage of solid tumor without necrotic core inside (see,e.g., [4,8,9,10,14]) to the process of necrotic core formation(see,e.g., [2,5,7,12]). 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]). Following this idea, the study of time delayed mathematical model for tumor growth has drawn attentions of some other researchers(see,e.g., [10,11,13,22] and references cited therein).

SHIHE XU, MENG BAI AND FANGWEI ZHANG
In this paper we study the following problem: ∂σ ∂r (0, t) = 0, σ(R(t), t) = G(t), 0 < r < R(t), t > 0, (2) d dt µσr 2 dr , t > 0, (3) σ(r, t) = ψ(r, t), 0 < r < R(t), −τ ≤ t ≤ 0, (4) R(t) = ϕ(t), −τ ≤ t ≤ 0 (5) where λ, µ,σ,σ, c and τ are positive constants. λ is the nutrient consumption rate; σ is a threshold value of nutrient concentration for apoptosis; µ is the proliferation rate of tumor cells; r is the radial variable; the variable σ(r, t) represents the nutrient concentration at radius r and time t; the variable R(t) represents the radius of the tumor at time t; τ is the time delay in cell proliferation, i.e., τ is the length of the period that a tumor cell undergoes a full process of mitosis. G(t) is a given function representing the external nutrient supply. ϕ and ψ are given nonnegative functions. The two terms on the right hand side of (3) are explained as follows: The first term is the total volume increase in a unit time interval induced by cell proliferation; µσ is the cell proliferation rate in unit volume. The second term is total volume shrinkage in a unit time interval caused by cell apoptosis, or cell death due to aging. c represents the time scale of diffusion of nutrient and inhibitor compare to the time scale of the tumor doubling within the tumor and c << 1 (i.e., the time scale of the tumor doubling is more larger compared to the time scale of diffusion of nutrient and inhibitor within the tumor).
The model we studied in this paper is established by modifying the model studied in Wu [19] by considering the time delay effect as that in Byrne [3], i.e., introducing time delay in proliferation to the model studied in [19], we have the model (1)-(5). In Wu [19], under some assumptions, the author studied the model (1)-(5) without time delay (i.e., τ = 0). In this paper, as [19], let where H(·) is a smooth function such that H(r) = 0 for r ≤ γ, H(r) = 1 for r ≥ 2γ and 0 ≤ H (r) ≤ 2/γ for r ≥ 0. G(t) is induced by Gibbs-Thomson relation, please see [18,19,20] for details. Many other special cases where τ = 0 have been studied. For example, G(t) is assumed to be constantσ in Byrne and Chaplain [4], Friedman and Reitich [14], Cui [9] and G(t) is assumed to be a periodic function in Xu [21].
Motivated by [8,10,14,11], by employing stability theory for functional differential equations, comparison principle and some meticulous mathematical analysis, we mainly study the asymptotic behavior of the solution, and prove that in the case c (the ratio of the diffusion time scale to the tumor doubling time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to one of two dormant states as t → ∞.

A FREE BOUNDARY PROBLEM FOR TUMOR GROWTH 3537
The paper is organized as follows: In Section 2 the existence and uniqueness of a local and global solution to problem (1)-(5) is proved. Section 3 is devoted to the quasi-stationary case c = 0. In section 4, we discuss the asymptotic behavior of the solutions to problem (1)- (5). In the last section, we give a conclusion.
Lemma 2.1. The function p(x) = x coth x − 1 x 2 has the following properties: (2) x 3 p(x) is strictly monotone increasing for x > 0.
Proof. The proof of (1) can be found in [14] and the proof of (2) can be found in [10].
(iv) For any t > 0, Proof. (i) By the maximum principle, we immediately have 0 ≤ σ(r, t) ≤σ, for which implies that R(t) ≥ R(0) exp(− µσt 3 ) and where η = R 3 . By Lemma 3.1 in [10] and Theorem 3.1 in Chapter 1 [17], we have Then (iii) By the left-hand side of the inequality (7), we have (R exp( From the right-hand side of the inequality (7), we know
(iv) It can be easily got from (iii). Proof. For arbitrary T > 0, we introduce a metric space (M T , d) as follows: The set M T consists of vector functions (σ(r, t), R(t)) satisfying The metric d is defined by It is clear that (M T , d) is a complete metric space. Define a mapping F : (σ, R) → (σ,R) in the following way.
∂σ ∂r (0, t) = 0,σ(R(t), t) = G(t), 0 < r < R(t), t > 0, Define σ(r, t) = G(t) for r ≥ R(t). Using similar arguments [9], we can prove F is a contraction for T > 0 is small. Therefore, Banach fixed point theorem implies the local existence and uniqueness of a solution to the problem (1)- (5). By a continuation theorem, to prove global existence and uniqueness, we only need to prove that the local solution can not blow up or tend to zero in a finite time interval. This can be got from Lemma 2.2(ii).
3. The quasi-stationary case c = 0. In this section, we study the the quasistationary case c = 0. To reduce the number of coefficients, by a resealing arguments, we always set λ = µ =σ = 1 as [19].
The solution of (1)-(2) is Substituting (15) into (3), we have where Thus, for the quasi-stationary case, we only need to study Eq.(16) with initial condition (5). Denote and The function (18) has properties as follows, please see [18] or [19]: Recall that the stationary solution which is denoted by (σ s (r), R s ) for R s > 0 satisfies the following two equations.
(ii)If 0 <σ < θ * , problem (1)-(5) has two positive stationary solutions (σ s1 (r), Lemma 3.2. Consider the initial value problem of a delay differential equatioṅ Assume that the function f is defined and continuously differentiable in R + × R + and strictly monotone increasing in the second variable, we have following results: be the solution of the problem of (21), (22) Proof. For the proof of (1) and (2), please see Lemma 3.4 in [10]. Next, following the idea of [10], we prove (3).
On the other hand, which is in contradiction with (23). Thus the claim is true.
In the following, we prove: If 0 ≤ x 0 (t) < x s for −τ ≤ t ≤ 0, then lim t→∞ x(t) = 0. Consider the following initial problem: By the assumption we see that f (x, x) < 0 for x = x s . Thus x (0+) = f (C, C) < 0. It follows that there exists a small δ > 0 such that x (t) < 0 for 0 < t < δ. We claim that for any t > 0, if , then which contradicts the condition x (t) ≥ 0. Hence the assertion holds. It follows immediately that x(t) < C < x s for all t > 0. The above assertion ensures that if [t 1 , t 2 ] is an interval in which x(t) is monotone nondecreasing then t 2 − t 1 < τ . Thus only two cases are possible: either (i) there exists a t 0 such that x(t) is monotone nonincreasing for t ≥ t 0 , or (ii) for any t 0 > 0, x(t) is oscillating in (t 0 , ∞). In the first case one can easily deduce that lim t→∞ x(t) = 0 holds. In what follows we consider the second case. We only need to prove that lim sup t→∞ Denote x * = lim sup t→∞ x(t). Clearly, we can find a monotone increasing sequence of numbers {t n } such that t n → ∞, x(t n ) → x * , and every t n is a local maximum point of the function x = x(t). Since x (t n ) = 0, we have x(t n ) < x(t n − τ ) (n = 1, 2, · · · ), which implies that The proof is similar to that of (3), we omit it here. This completes the proof.
In quasi-stationary case c = 0, the nonnegative solution of (16) exists for t ≥ −τ with nonnegative initial value ϕ and the dynamics of solutions to this equation is as follows: (I) Ifσ > θ * , for any nonnegative initial value ϕ, lim t→∞ R(t) = 0.
Proof. Let η = R 3 , then (16) takes the form as follows and the initial condition η(t) = ϕ 3 (t) for −τ ≤ t ≤ 0, where η τ = η(t − τ ). It is obvious that every solution to equation (28) exists for all t > 0, because we may rewrite this equation in the following functional form: (29) and solve it using the step method (see, e.g., [17]) on intervals [nτ, (n + 1)τ ], n ∈ N . Therefore, the solution of (16) exists for t ≥ −τ . Now, for any nonnegative initial value ϕ, we prove that every solution to equation (16) is nonnegative. Actually, since η = R 3 , we only need to prove every solution to equation (28) is nonnegative for any nonnegative initial value ϕ. Let By Lemma 2.1, noticing the properties of function H, it is easy to get G(s) > 0 for s > 0. By Lemma 1.1 in [1], one can get that the solution to equation (28) is nonnegative for any nonnegative initial value ϕ.
In the following, we will study the dynamics of solutions to equation (16). By direct computation, we have it immediately follows that (S1) Ifσ > θ * , then f (x, x) < 0 for all x > 0.
By Lemma 3.1 in [10], we can get η(t) ≤ ω(t). Thus, lim t→∞ η(t) = 0. Then lim t→∞ R(t) = 0 follows. This completes the proof. For some T > 0 and ε > 0 Assume further that for 0 ≤ r ≤ R 0 , Then, there exists a positive constant C independent of c, T, L, M and R 0 but depend of ε and γ such that for arbitrary 0 ≤ r ≤ R(t), 0 ≤ t ≤ T .

SHIHE XU, MENG BAI AND FANGWEI ZHANG
It follows that where we have used the fact that x 3 p(x) is monotone increasing (see Lemma 2.1 (2)). Since then there exists c 0 > 0 sufficiently small such that R (T ) < 0 when c ∈ (0, c 0 ]. It is a contraction to R (T ) ≥ 0. Therefore, the assertion R(t) ≤ K holds. Let |ϕ|e M τ 3 + δ ≤ K < R s1 . By Lemma 2.2 (iv), noticing that µ = 1, we have (36) is not valid for some t, then there exists T > τ such that for 0 ≤ r ≤ R(t), 0 ≤ t < T and 0 < c ≤ c 0 . Therefore, for t > τ , It follows that where we have used the fact that 3p( then there exists c 0 > 0 sufficiently small such that R (T ) < 0 when c ∈ (0, c 0 ]. It is a contraction to R (T ) ≥ 0. Therefore, the assertion R(t) ≤ K holds.
Then also the inequalities where C 3 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 α and c. Consider the initial value problem By (P3), it is easy to get that there exists c 0 > 0 such that for c ∈ (0, c 0 ], the Eqs. G(x, x)) ± Cαc = 0 has two positive solutions R ± s1 and R ± s2 respectively, and satisfy By similar proof as that of Theorem 3.3, it is not hard to get that the nonnegative solution of (45) exists for t ≥ −τ with nonnegative initial value (46) and By the fact that p(x) is monotone decreasing for all x > 0, we can get Since ∂f ∂y > 0 and G(x, y) = f (x, y)/x, we can get ∂G ∂y > 0, thus G(x, y) ± Cαc is monotone increasing in y. By comparison principle (cf. [10] Lemma 3.1), we have for t ≥ 0. By linearizing Eqs.(45) at the stationary point R + s2 and R − s2 respectively, one can get where From [19] we know that Q (R ± s2 ) < 0, Q(R ± s2 ) > 0. Since for c ∈ (0, c 0 ] and α ∈ (0, α 0 ]. This implies that all complex roots of the characteristic equations of Eqs.(50) and (51) are located in the left half plane. Therefore, noticing (47), there exist positive constants M ,θ 0 and T 0 such that for t > T 0 , Then we can get that for any t ≥ T 0 , By the mean value theorem and the fact that for 0 ≤ r ≤ R(t). It follows that In particular, |σ(r, 0) − v(r, 0)| ≤ Cα for 0 ≤ r ≤ R(0). Since |R (t)| ≤ α for all t ≥ 0, by Lemma 4.2 there exists a positive constant c 0 independent of c and α such that Denote for t ≥ 2τ . By the mean value theorem and the fact R s1 + δ ≤ R(t) < 1/δ for t ≥ 0, we can get for t ≥ T 0 + τ . Then |R (t)| ≤ Cα(c + e −θ1t ) follows from R (t) = R(t)h(t). Let Then |σ(r, t) − σ s2 (r)| ≤ Cα(c + e −θt ) follows from (53), (55) and |σ(r, t) − σ s2 (r)| ≤ |σ(r, t) − v(r, t)| + |v(r, t) − σ s2 (r)|. The proof of Lemma 4.4 is completed.

5.
Conclusion. In this paper we study a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. It is assumed that the process of proliferation is delayed compared with apoptosis. The delay represents the time taken for cells to undergo mitosis. We mainly study the asymptotic behavior of the solution, and prove that in the case c (the ratio of the diffusion time scale to the tumor doubling time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a dormant state as t → ∞. The results show that dynamical behavior of solutions of the model are similar to that of solutions for corresponding nonretarded problems. More precisely, the stability of solutions to quasi-stationary case, please compare Theorem 3.1 in [19]. For the stability of solutions to (1)-(5), please see Theorem 4.5 in [19]. We hope that the analysis methods and the results would be useful to analysis of other similar retarded differential equations.