STATIONARY SOLUTIONS OF A FREE BOUNDARY PROBLEM MODELING GROWTH OF ANGIOGENESIS TUMOR WITH INHIBITOR

. We consider a free boundary problem modeling the growth of angiogenesis tumor with inhibitor, in which the tumor aggressiveness is modeled by a parameter µ . The existences of radially symmetric stationary solution and symmetry-breaking stationary solution are established. In addition, it is proved that there exist a positive integer m ∗∗ and a sequence of µ m , such that for each µ m ( m > m ∗∗ ), the symmetry-breaking stationary solution is a bifurcation branch of the radially symmetric stationary solution.


1.
Introduction. This paper is concerned with a free boundary problem modeling the tumor growth with angiogenesis and inhibitor. Angiogenesis is an essential process in wound healing and new birth. Tumor-induced angiogenesis is a process that tumor cells secrete cytokines that stimulate the vascular system to grow toward the tumor. As a result of angiogenesis, the tumor possesses its own vasculature, then the nutrient may be supplied to tumor via the capillary network. In mathematical model of tumor-induced angiogenesis, the distribution of nutrient in the tumor satisfies the following reaction-diffusion equation [2,4,19,22]: where Ω(t) is the tumor domain at time t with a moving boundary ∂Ω(t), σ denotes the concentration of a nutrient which diffuses throughout the tumor, with diffusion coefficient D 1 , the termΓ 1 (σ B −σ) accounts for the transfer of nutrient by means of the vasculature stemming from tumor-induced angiogenesis,Γ 1 is the transfer rate of nutrient-in-blood-tissue, and σ B is the concentration of nutrient in the vasculature. The last term on the right side describes the nutrient consumption by tumor cells at the rate of λ 1 . Tumor inhibitors come from some blood-borne anti-cancer drugs or immune system of the body. Let β denote the concentration of inhibitor. Assume that the similar effect governs the evolution of tumor inhibitor. Then it follows that [2,5,8]: where D 2 ,Γ 2 , β B and λ 2 are similarly defined as those for the nutrient concentration σ. AndΓ 2 = 0 if the inhibitor is secreted by neighbouring healthy cells, in response to the "foreign" body, delivered by diffusion across the tumor boundary.
Using non-dimensional scales [2,4,5], we rewrite the equations (1) and (2): where δ 1 , δ 2 are small parameters. The pressure p stems from the transport of cells which proliferate or die. Formulated by the conservation of mass divu = µ(σ −σ − τ β) with Darcy's law u = −∇p, where u is the velocity of tumor cells, the term µ(σ −σ) on the right side is the proliferation rate, µ is a parameter expressing the "intensity" of the expansion by mitosis,σ is a threshold concentration and the term µτ β is the death rate of tumor cells caused by the inhibitor.
Since the nutrient enters tumor by the vascular system and diffusion across the boundary, using homogenization [18,32], it is assumed that σ satisfies the boundary condition: where n is the outward normal,σ is the nutrient concentration outside the tumor, α(t) is the rate of nutrient supply to the tumor, which may vary in time, and angiogenesis results in an increase in it; conversely, if the tumor is treated with anti-angiogenic drugs, it will decrease and the starved tumor will shrink. In this paper, we assume that α(t) ≡ α is a positive constant. As for the inhibitor, it is assumed to be secreted by neighbouring healthy cells, in response to the "foreign" body, delivered by diffusion across the tumor boundary, then β =β on ∂Ω(t), whereβ is the inhibitor concentration outside the tumor. Due to cell-to-cell adhesiveness and the continuity of the velocity field through the boundary [2], we derive the boundary conditions of p: where κ is the mean curvature, and V n is the velocity of the free boundary in the direction n.
During the last few decades, many mathematical models in the form of free boundary problems of partial differential equations have been proposed to model the growth of tumors; see survey papers [1,11,12,13,25] and the references therein, also the recent papers [9,18,29,30,31,34], [23]- [27]. Among those, some theoretical and numerical results are established for the problem (3)-(9) with different boundary conditions. If the boundary condition (6) is replaced by σ =σ (which is formally the case α(t) = ∞) and the inhibitor is absent (β = 0), it is proved in [10,14,15,19,20] that under the assumptionσ <σ, there exists a unique radially symmetric stationary solution on B Rs , and a branch of symmetry-breaking stationary solutions bifurcates from the above radially symmetric stationary solution for each µ n (R s )(n ≥ 2) with free boundary where Y n,0 is the spherical harmonic of order (n, 0). Moreover, Friedman and Hu [14] showed that the bifurcation branches stemming from µ n (R s )(n ≥ 3) are all unstable. The above results have been extended to the tumor model where the nutrient consumption rate and the proliferation rate are general nonlinear functions, tumor growth with a necrotic core, tumor growth in fluid-like tissue, tumor cord, multilayer model and so on; see the papers [6,7,9,16,23,33]. Moreover, Zhou and Wu [27,30,34], Xu et al. [31] established the asymptotic stability and bifurcation analysis for the case that the boundary condition (6) is nonlinear with Gibbs-Thomson relation. In the presence of inhibitor, the asymptotic stability and bifurcation of radially symmetric stationary solution were studied in [5,8,26,28,29]. It was shown that there exists a threshold value γ * for the surface tension coefficient which separates instability from stability for the radially symmetric equilibrium with respect to small enough non-radial perturbations, and a branch of non-radial solutions bifurcates from the radially symmetric solution for each γ n with n > n * * .
In this paper, we are interested in the tumor model with the boundary condition (6) stemming from angiogenesis. For the problem (3)-(9) in the absence of inhibitor, Lam and Friedman [18] established the existence of the radially symmetric stationary solution and the boundedness of free boundary for the bounded α(t). Recently, Huang, Zhang and Hu [24] showed that when α(t) ≡ α, there exists a branch of symmetry-breaking stationary solutions bifurcating from the radially symmetric stationary solution for each µ n (n ≥ 2). Motivated by the above papers, we study the stationary solution of the problem (3)-(9) and seek the effect of inhibitor on the growth of dormant tumor with angiogenesis. Namely, we consider the following problem: We firstly obtain the existence of the radially symmetric solution of the problem (10)- (16) for all µ, then prove that there exist a positive integer m * * and a sequence of µ m , such that for each µ m (m > m * * ), there exists a branch of symmetry-breaking solutions bifurcating from the above radially symmetric solution. We also prove that µ m is increasing with respect to the supply of inhibitorβ.
Noticing that the aggressiveness of tumor growth in this paper is measured by the parameter µ, the larger the µ is the more aggressive the tumor is. It is seen from our results that when µ is at µ m for large enough m, the tumor will develop finger and invasive; in addition, for the same branch protrusion, the more supply of inhibitors would require bigger tumor aggressiveness parameter to form the symmetry-breaking stationary state. Namely, increasing the amount of inhibitor can not only reduce the size of the stable dormant tumor ( [23,24]), but also play a positive role in stabilizing radially symmetric tumors.
The rest of this paper is organized as follows. In Section 2, we establish the existence of radially symmetric solution to the problem (10)-(16); in Section 3, we solve the linearized problem of the problem (10)-(16) at the radially symmetric solution; then we obtain the existence of symmetry-breaking solution by the bifurcation theorem in Section 4.
Proof. Obviously, we only need to show the existence of solutions to the problem (17)- (21). It is seen that (σ s , β s , p s ) given by (22)-(24) satisfies the problem (17)- (20). It remains to verify that the boundary condition (21) is satisfied, namely, (25) has at least one positive solution. Denote g(r) = coth r − 1 r .

A FREE BOUNDARY PROBLEM FOR GROWTH OF ANGIOGENESIS TUMOR 2597
For s > 0, it follows that g (s) > 0, (g(s)/s) < 0 and g(s) satisfies [25]: Then, σ s (r) can be rewritten as We compute Substituting (27) and (28) into (25), we get which is equivalent to the following Then from (26), we see that Sinceσ −σ − τβ > 0, α > 0,σ > 0, we conclude that T (s) = 0 has at least one positive root on (0, ∞) by using the continuity of T (s) and the intermediate value theorem. The proof is complete.
3. Linearized problem. In this section, we consider the linearization of the problem (10)-(16) at radially symmetric solution (σ s , β s , p s ) with radius R s and solve it by employing spherical harmonics and modified Bessel functions. Let (σ, β, p) be the solution of the problem (10)-(16) on the domains with boundaries ∂Ω ε : r = R s +R; hereR = εS(θ, ϕ). For simplicity, we denote R = R s . Assume that (σ, β, p) has the expansion as follows: where σ 1 , β 1 , p 1 are the functions to be determined.
Substituting (29)-(31) into (10)- (15), collecting all ε-order terms, we obtain the linearized problem which is satisfied by σ 1 , β 1 , p 1 : ∂ϕ 2 is the Laplace operator on the unit sphere. Here we used the fact that (σ s , β s , p s ) is a solution of the problem (17)-(21) and on the boundary [21]: Before solving the problem (32)-(37), we recall some properties of the spherical harmonic functions and the modified Bessel functions.
Let Y m,l (θ, ϕ) denote the spherical harmonic functions. Then the family {Y m,l } forms a normalized complete orthonormal basis for L 2 (Σ), where Σ is the unit sphere, and ∆ ω Y m,l = −m(m + 1)Y m,l .
4. Symmetry-breaking solutions. In this section, we reduce the problem (10)-(16) to a bifurcation problem by taking the aggressive parameter µ as a bifurcation parameter, then obtain the existence of symmetry-breaking solutions by using the following Crandall-Rabinowitz bifurcation theorem [3].  Then (0, µ 0 ) is a bifurcation point of the equation F (x, µ) = 0 in the following sense: In a neighborhood of (0, µ 0 ) the set of solutions of F (x, µ) = 0 consists of two C p−2 smooth curves Γ 1 and Γ 2 which intersect only at the point (0, µ 0 ); Γ 1 is the curve (0, µ) and Γ 2 can be parameterized as follows: For eachR = εS(θ, ϕ), µ, define a function F by Then (σ, β, p) given by (29)-(31) is a solution of the problem (10)- (16) if and only if F (R, µ) = 0 for someR and µ. In order to apply the Crandall-Rabinowitz theorem, we need to compute the Fréchet derivative of F (R, µ). From the definition of F (R, µ), we have the Fréchet derivative: Setting S(θ, ϕ) = Y m,l (θ, ϕ) as that in the last section, we compute the two terms on the right side of the above equality: Therefore, we have The equation We now proceed to establish our main lemma.
In the following, we show that (0, µ m ) is the bifurcation point of the equation F (R, µ) = 0 for sufficiently large m.
Furthermore, for the above aggressiveness parameter µ m , we can show that it is a monotonous function of the inhibitor supplyβ. Proof. In order to prove this lemma, we only need to verify that the following function is decreasing inβ: Combining the definition q m (R), we get .