Analysis of a Delayed Free Boundary Problem with Application to a Model for Tumor Growth of Angiogenesis

In this paper, we consider a time-delayed free boundary problem with time dependent Robin boundary conditions. e special case where 푛 = 3 is a mathematical model for the growth of a solid nonnecrotic tumor with angiogenesis. In the problem, both the angiogenesis and the time delay are taken into consideration. Tumor cell division takes a certain length of time, thus we assume that the proliferation process leg behind as compared to the process of apoptosis. e angiogenesis is reflected as the time dependent Robin boundary condition in the model. Global existence and uniqueness of the nonnegative solution of the problem is proved. When 푐 > 0 is sufficiently small, the stability of the steady state solution is studied, where is the ratio of the time scale of diffusion to the tumor doubling time scale. Under some conditions, the results show that the magnitude of the delay does not affect the final dynamic behavior of the solutions. An application of our results to a mathematical model for tumor growth of angiogenesis is given and some numerical simulations are also given.


Introduction
In the past a few decades, there are a lot of focus on mathematical models with regard to tumor growth for biological and mathematical interests. Many researchers developed various mathematical models from different aspects to detail the process of tumor growth (see, e.g., [1][2][3][4][5][6][7][8]). e tumor growth process can be classified into two different stages: the stage without a necrotic core (see, e.g., [2,[9][10][11][12][13]) and the stage with a necrotic core (see, e.g., [3,[14][15][16]). Almost all mathematical models are established by using reaction-diffusion dynamics and mass conservation law for the processes of proliferation and apoptosis.
is paper focus on a time-delayed free boundary problem with the time dependent Robin boundary condition. e model is as follows: where 휎(푟, 푡) and 푅(푡) are two unknown functions. is a positive constant. and are given functions, and e special case where 푛 = 3 is a mathematical model describing the growth of a nonnecrotic tumor with angiogenesis. In particular, when 푛 = 3, the biological meaning is as follows: is the nutrient concentration at time and radius . 푅(푡) represents the outer radius of tumor at time . represents the ratio between time scale of the diffusion and time scale of the tumor doubling, and is a constant represents the time delay in the process of proliferation, i.e., is the average time required from the beginning of cell division to the completion of division. In order to obtain nutrients, tumors attract blood vessels at a rate proportional to , so that (휕휎/휕푟) + 훽(휎 −휎) = 0 holds on the boundary, where 휎 is the nutrients concentration outside the tumor. It should be pointed out that the boundary condition (3) is a time dependent Robin boundary condition since the boundary changes with time. Equation (4) describes the changes of the volume of the tumor. Equations (3), (2), (5), and (6) are boundary and initial conditions. 푓, g, and ℎ are given functions. 푓(휎) represents the nutrient consumption rate. It is assumed that the rate of nutrient consumption by tumor cells is an increasing function of nutrient concentration. g(휎) represents the proliferation rate of tumor cells and ℎ(휎) represents the apoptosis rate of tumor cells. It is reasonable to assume that the rate of tumor cell proliferation is an increasing function of nutrient concentration and the rate of tumor cell apoptosis is a nonincreasing function of nutrient concentration.
e motivation for studying this model is as follows: Experiments have shown that changes in the proliferation rate modify apoptotic cell loss which does not occur immediatelythere exists a time delay for this modification (see [1]), i.e., the proliferation process lags behind as compared to the process of apoptosis. As a result of this research, many researchers have grown interest in the study of mathematical models for tumor growth with time delays (see, e.g., [6,11,[17][18][19] and their references). e idea of considering the time delay in the process of proliferation is motivated by the work of Byrne [1], Cui and Xu [11], Foryś and Bodnar [18] and Xu et al. [20] where either linear or constant functions 푓, g, and ℎ are considered in the above mentioned papers. e motivation of considering the nonlinear functions 푓, g, and ℎ is from the work of Cui [10] (where 푛 = 3 and 휏 = 0, i.e., the time delay is not considered). In this paper, we study a more general case, which not only considers both time-delay and nonlinear functions 푓, ℎ, and ℎ, but also takes as any positive integer greater than or equal to 3. e main aim of this paper is to study the time-delayed problem (1)-(6) for Robin boundary conditions and general nonlinear functions 푓, g, and ℎ.
It also should be pointed out that only Dirichlet boundary conditions are considered in [1,[10][11][12]20]. In the recent work of Friedman and Lam [21], the authors studied the special case of the problem (1)- (6) where 휏 = 0, the functions and g are linear and ℎ is a constant (but where is a given function of ). e special cases of the model have been extensively studied by many researchers, such as for linear functions and ℎ(휎) = 휇휎, where 휆, 휇, and 휎 are positive constants, Xu et al. [20] have studied the model with Gibbs-omson relation, which appears as the Dirichlet boundary condition. In [20], by rigorous mathematical derivation and using theories of functional differential equations, the authors studied the asymptotic behavior of steady state solutions. roughout this paper, we suppose that the functions 푓, g, and ℎ satisfy the following conditions: Moreover, we suppose the initial value functions and satisfy the following conditions: e paper is arranged as follows: Section 2 provides proof for the existence and uniqueness of a global solution to problem (1)- (6). Section 3 is devoted to studing asymptotic behavior of the solutions to problem (1)- (6). In the final section, an application of our results to a mathematical model for tumor growth of angiogenesis is given and some numerical simulations are also given.

Complexity
where 푀 1 = −ℎ(0). Moreover, from the the inequality on the le -hand side of (12), we can get It infer that when 푡 ≥ 휏, Noticing the inequality on the righthand side of (12) and (11) follows from (10). is completes the proof. ☐ Let 푇 > 0 which will be given later. Consider the problem (1)-(6), by (19), it is equivalent to the following problem: We define the following metric space 푀 , 푑 : e set consists of vector functions (휎(푟, 푡), 푅(푡)) satisfying 푢(푠, 푡) = 휓(푠푅(푡), 푡), 0 < 푠 < 1, −휏 ≤ 푡 ≤ 0.  (1)  where 푈(푟, 푅(푡)) is the unique solution to the following problem: where we use the monotonicity of the functions 푓, g and ℎ. It follows that us, ̃ satisfies the condition (I). Taking similar arguments as that in [24], it is not hard to prove is a contractive mapping for 푇 > 0 is sufficiently small. By the Banach fixed point theorem, we have the local existence and uniqueness of a solution to the problem (1)- (6). To prove global existence and uniqueness, we only need to prove that it is impossible for the local solution to blow up or tend to zero in a finite time. is follows from the priori estimates (see Lemma 1). e proof of eorem 1 is complete. ☐
Next, if we take the parameter values as follows: where one can get all conditions of eorem 3 satisfied. As can be seen from Figure 3, whether the initial value is taken 푥 0 = 12 or 50, all the solutions eventually tend to zero, which verifies the results of eorem 3. It can be seen from eorems 2 and 3 that time delay does not affect the final tendency of tumor growth to the steady state or to disappear. In the following, by using the Figures 4-6, we show that the time delay has an effect on the speed of tumor growth towards to the steady state solution or toward extinction. In Figures 4-6, except for the size of time delay, the other parameters take the same value (please refer to captions of Figures 4-6). In Figures 4 and 6, the top curve of three curves corresponds to the larger where 휏 = 9, the bottom curve of the three curves corresponds to a smaller where 휏 = 3, the remaining curve corresponds 휏 = 6. In Figure 5, the top curve of three curves corresponds to the smaller where 휏 = 3, the bottom curve of the three curves corresponds to a larger where 휏 = 9, the remaining curve corresponds 휏 = 6. From Figures 4-6, we see that when other conditions remain unchanged, the larger the time delay, the slower the tumor tends to the steady state solution or tends to disappear.

Data Availability
No empirical data were used for this study.  Next, using Matlab R2016a, we will do some numerical simulation of the tumor growth model discussed above. First, we take the following parameter values: e steady state solution is determined by (124). Let where and are as before. In Figure 1, we plot the curve of (the blue curve). As can be seen from Figure 1, noting the red curve is the curve of 휂/3, where 휂 =휎/휎, there is only one