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Research article

Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation

  • Received: 02 May 2019 Accepted: 30 July 2019 Published: 14 August 2019
  • In this paper we consider a time-delayed mathematical model describing tumor growth with angiogenesis and Gibbs-Thomson relation. In the model there are two unknown functions: One is σ(r,t) which is the nutrient concentration at time t and radius r, and the other one is R(t) which is the outer tumor radius at time t. Since R(t) is unknown and varies with time, this problem has a free boundary. Assume α(t) is the rate at which the tumor attracts blood vessels and the Gibbs-Thomson relation is considered for the concentration of nutrient at outer boundary of the tumor, so that on the outer boundary, the condition σr+α(t)(σN(t))=0,  r=R(t) holds, where N(t)=ˉσ(1γR(t))H(R(t)) is derived from Gibbs-Thomson relation. H() is smooth on (0,) satisfying H(x)=0 if xγ, H(x)=1 if x2γ and 0H(x)2/γ for all x0. In the case where α is a constant, the existence of steady-state solutions is discussed and the stability of the steady-state solutions is proved. In another case where α depends on time, we show that R(t) will be also bounded if α(t) is bounded and some sufficient conditions for the disappearance of tumors are given.

    Citation: Shihe Xu, Junde Wu. Qualitative analysis of a time-delayed free boundary problem for tumor growth with angiogenesis and Gibbs-Thomson relation[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7433-7446. doi: 10.3934/mbe.2019372

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  • In this paper we consider a time-delayed mathematical model describing tumor growth with angiogenesis and Gibbs-Thomson relation. In the model there are two unknown functions: One is σ(r,t) which is the nutrient concentration at time t and radius r, and the other one is R(t) which is the outer tumor radius at time t. Since R(t) is unknown and varies with time, this problem has a free boundary. Assume α(t) is the rate at which the tumor attracts blood vessels and the Gibbs-Thomson relation is considered for the concentration of nutrient at outer boundary of the tumor, so that on the outer boundary, the condition σr+α(t)(σN(t))=0,  r=R(t) holds, where N(t)=ˉσ(1γR(t))H(R(t)) is derived from Gibbs-Thomson relation. H() is smooth on (0,) satisfying H(x)=0 if xγ, H(x)=1 if x2γ and 0H(x)2/γ for all x0. In the case where α is a constant, the existence of steady-state solutions is discussed and the stability of the steady-state solutions is proved. In another case where α depends on time, we show that R(t) will be also bounded if α(t) is bounded and some sufficient conditions for the disappearance of tumors are given.


    Tumor growth is a complex process. Many mathematical models have been established from different aspects to describe this process in recent years (see, e.g., [1,2,3,4,5,6]). There are several distinct stages of tumor growth, starting from the early stage without angiogenesis (see, e.g., [2,7,8,9,10,11]) to the process with angiogenesis (see, e.g., [12]). Experiments have shown that changes in the rate of tumor cell proliferation can lead to changes in the rate of apoptosis, which does not happen immediately: There is a time lag between the two changes (see[1]). Rencently, the study of mathematical models for tumor growth with time delays has attracted many interests of other researchers (see, e.g., [9,13,14,15]). In this paper we consider a mathematical model for tumor growth with angiogenesis and Gibbs-Thomson relation. The model has a time delay and a free boundary that depends on time.

    First, let us introduce the mathematical model. Suppose the tumor is spherical and occupies the following space:

    {r|rR(t),r=x21+x22+x23},

    where R(t) is an unknown function which represents the outer tumor radius at time t. In the model there is another unknown function σ(r,t) which is the concentration of nutrient at time t and radius r. It is assumed that the consumption rate of nutrient is proportional to the nutrient concentration at the corresponding local place, and the proportionality constant is Γ. Then σ satisfies the equation:

    cσt=1r2r(r2σr)Γσ,0<r<R(t),t>0,

    where

    c=Tdiffusion/Tgrowth>0 (1.1)

    is a constant which is the ratio of time scale of the nutrient diffusion to time scale of the tumor growth (see [2,13,10]). By the mass conservation law, R satisfies

    ddt(4πR33)=S(t)Q(t),

    where S(t) denotes the net rates of proliferation and Q(t) is the net rates of natural apoptosis. From [1,9,13], we know that there is a time delay in the process of cell proliferation. The size of the time delay is the time required from the beginning of cell division to the completion of division. Assume that the proliferation rate is proportional to the corresponding local nutrient concentration and the coefficient of proportionality is μ, then

    S(t)=2π0π0R(tτ)0μσ(r,tτ)r2sinθdrdθdφ

    where τ is the time delay in the process of cell proliferation. Suppose the rate of the apoptotic cell loss is μ˜σ, then

    Q(t)=2π0π0R(t)0μ˜σr2sinθdrdθdφ.

    Assume the boundary value condition on r=R(t) is as follows:

    σr+α(t)(σN(t))=0,

    where α(t) depends on the density of the blood vessels. Thus, it is a positive valued function. N(t) is a given function which is the supply of nutrients outside the tumor. The special cases of the above boundary value has been studied extensively. For the case where α(t)= and N(t) is a constant, Friedman and Reitich [10] gave a strict mathematical analysis of the spherically symmetric case of the corresponding model. They proved the existence, uniqueness and asymptotic stability of spherically symmetric steady state solutions. Cui[16] studied the generalized model of [10] where the cell proliferation rate and nutrients consumption rate are assumed to be nonlinear functions. From [17,18,19], we know that the energy would be consumed in maintaining the tumors compactness by cell-to-cell adhesion on the external boundary of the tumor and the nutrient acts as the source of energy, and the nutrient concentration on the external boundary satisfies a Gibbs–Thomson relation given by

    N(t)=ˉσ(1γR(t))H(R(t))

    which denotes the concentration of nutrients at external boundaries, where γ>0 is a constant describing the cell-to-cell adhesiveness. H() is smooth on R+=(0,) such that H(x)=0 for xγ, H(x)=1 for x2γ and 0H(x)2/γ for x0. N(t) is induced by Gibbs–Thomson relation ([18,19,11]).

    We will consider the problem together with the following initial conditions

    σ0(r,t)=ψ(r,t),r[0,R(t)],t[τ,0],
    R0(t)=φ(t),t[τ,0].

    We assume that ψ and φ satisfy compatibility conditions. From [2,7] we know that Tdiffusion1min and Tgrowth1day, noticing (1.1), so that c1. In this paper we only study the limiting case where c=0. The model studied in this paper is as follows

    1r2r(r2σr)=Γσ,0<r<R(t),t>τ, (1.2)
    σr+α(σˉσ(1γR(t))H(R(t)))=0,r=R(t),t>τ, (1.3)
    ddt(4πR3(t)3)=4π(R(tτ)0μσ(r,tτ)r2drR(t)0μ˜σr2dr),t>0, (1.4)
    σ0(r,t)=ψ(r,t),0rR(t),τt0, (1.5)
    R0(t)=φ(t),τt0. (1.6)

    Consider the compatibility of the problem, we assume Eqs (1.2) and (1.3) are hold for t>τ. The model (1.2)–(1.6) with the boundary condition (1.3) replaced by σ=ˉσ(1γR(t))H(R(t)) on r=R(t) (which is the case α=) was studied in [19] and in the model studied in [19], the time delay is not considered. In [19], the author established the global existence and uniqueness of the solution and discussed the stability of the stationary solutions. The model where N(t) is a constant has been studied by Friedman and Lam [12]. In [12], the authors mainly discussed the dynamics behavior of the solutions.

    The structure of this paper is as below. In section 2, some preliminaries are given. In section 3, the existence and uniqueness of the solution to Eqs (1.2)–(1.6) is proved. Section 4 is devoted to studying the case where α(t)= constant. In section 5, we discuss the case when α(t) is bounded. In the last section, we give some computer simulations and dscussions.

    First we introduce some functions which will be used in our analysis:

    p(x)=xcothx1x2, q(x)=xp(x), g(x)=p(x)α+q(x), f(x)=α(1γx)g(x)H(x)

    and

    J(x)=α[(x2γx)p(x)+γp(x)]+(2γxx2)p2(x),

    where α is a positive constant.

    Lemma 2.1. (1) p(x)<0 for all x>0, and limx0+p(x)=13,limxp(x)=0.

    (2) The function q(x)=xp(x) satisfies:

    (i) q(0)=0, (ii) limxq(x)=1, (iii) q(0)=1/3, (iv) q(x)>0 for x0.

    (3) k(x)=x3p(x) is strictly monotone increasing for x>0.

    (4) (xp(x)p(x))<0 and 2<xp(x)p(x)<1 for x>0.

    (5) For any α,γ>0, J(x)<0 for x>γ.

    (6) For any α>0, (x3g(x))>0 for x>0.

    Proof. (1) and (2) can be found in Lemma 2.1 and Lemma 2.2 [12]. For the proof of (3), please see [9]. In the following we prove (4), (5) and (6).

    (4) From Lemma 3.3 [7], we know (xp(x)p(x))<0 for x>0 and

    xp(x)p(x)=2(sinh3xx3coshx)(x2+xcoshxsinhx2sinh2x)sinhx2.

    By a simple calculation, we have limxxp(x)p(x)=2. Thanks to the fact limx0xp(x)p(x)=1 (see the proof of Lemma 2.1 in [20]), then 2<xp(x)p(x)<1 for x>0 follows.

    (5) Let L1(x)=(x2γx)p(x)+γp(x) and L2(x)=(2γxx2)p2(x). Then J(x)=αL1(x)+L2(x). In the following we prove that Li(x)<0 (i=1,2) for all x>γ. First, we prove L1(x)<0 for all x>γ. Direct computation yields

    L1(x)=2xp(x)+(x2γx)p(x) (2.1)
    =p(x)(2x+(xγ)xp(x)p(x)). (2.2)

    Since (xp(x)p(x))<0 and 2<xp(x)p(x)<1, one can get that

    (xγ)xp(x)p(x)>2(xγ)

    if x>γ. It follows that 2x+(xγ)xp(x)p(x)>2x2(xγ)=2γ>0. By the fact that p(x)<0, we have L1(x)<0. And then we prove L2(x)<0 for x>γ. Since

    L2(x)=2(γx)p2(x)+2(2γxx2)p(x)p(x)=2p(x)[γ(p(x)+2xp(x))x(p(x)+xp(x))]2p(x)[γ(p(x)+xp(x))x(p(x)+xp(x))]=2p(x)(γx)(xp(x)),

    where we used the fact p(x)<0, by the facts that p(x)>0 and (xp(x))>0, we can get that for x>γ, L2(x)<0. Thus,

    J(x)=αL1(x)+L2(x)<0

    for x>γ.

    (6) Since

    (x3g(x))=(k(x)α+q(x))=k(x)(α+q(x))q(x)k(x)(α+q(x))2=αk(x)(α+q(x))2+(k(x)q(x))q2(x)(α+q(x))2=αk(x)(α+q(x))2+2xq2(x)(α+q(x))2,

    it verifies that (x3g(x))>0 noticing k(x)>0. This completes the proof of Lemma 2.1.

    The following lemma will be used to prove the global stability of steady-states.

    Lemma 2.2. ([21]) Consider the following initial value problem:

    ˙x(t)=G(x,xτ),t>0 (2.3)
    x(t)=x0(t),τt0. (2.4)

    where xτ=x(tτ)). Assume G is defined on R+×R+ and continuously differentiable. Assume further that G is strictly monotone increasing in the variable xτ, then one can get the following results:

    (1) Assume G(x,x)=0 has a unique positive solution xs in (c,d) such that G(x,x)>0 for a<x<xs and G(x,x)<0 for xs<x<b, where c,d are two positive constants. If x(t) be the solution to the problem of (2.3), (2.4), assume that x0(t)C[τ,0], and c<x0(t)<d for τt0. Then limtx(t)=xs.

    (2) Assume G(x,x)<0 for all x>0, then limtx(t)=0.

    (3) Assume G(x,x)=0 has a unique positive solution xs in (0,xs] such that G(x,x)<0 for 0<x<xs and G(0,0)=0. Let x(t) be the solution to the problem of (2.3), (2.4), assume that x0(t)C[τ,0] and 0x0(t)<xs for τt0, then limtx(t)=0.

    (4) Assume G(x,x)=0 has a unique positive solution xs in [xs,) such that G(x,x)<0 for x>xs. Let x(t) be the solution to the problem of (2.3), (2.4), assume that x0(t)C[τ,0] and x0(t)>xs for τt0, then limtx(t)=xs.

    Theorem 3.1. Assume φ(t) is continuous and nonnegative on [τ,0]. Suppose α(t) is continuous and positive on [τ,], then there exists a unique nonnegative solution to problem (1.2)–(1.6) on interval [τ,).

    Proof. Using scale transformation of the spatial variable, one can assume Γ=1. Then, the solution to (1.2) and (1.3) is

    σ(r,t)=αˉσα+q(R)ζ(r)ζ(R(t))(1γR(t))H(R(t)), (3.1)

    where ζ(x)=sinhx/x. Substituting (3.1) into (1.4), we obtain

    dRdt=μˉσR[α(tτ)α(tτ)+q(Rτ)R3τp(Rτ))R3(1γRτ)H(Rτ)˜σ3ˉσ]=:G(R(t),Rτ). (3.2)

    where Rτ=R(tτ) and

    G(x,y)=μˉσx[α(tτ)α(tτ)+q(y)y3p(y))x3(1γy)H(y)˜σ3ˉσ].

    Denote x=R3, then Eq (3.2) becomes

    dxdt=3μˉσF(3x(tτ))μ˜σx(t), (3.3)

    where

    F(3x(tτ)=α(tτ)x(tτ)p(3x(tτ))α(tτ)+q(3x(tτ))(1γ3x(tτ))H(3x(tτ)).

    Then, the initial condition of x(t) have the following form

    x0(t)=[φ(t)]3,τt0. (3.4)

    By the step method it is obvious that the problem (3.3), (3.4) has a unique solution x(t) which exists on [0,), since we can rewrite the problem (3.3), (3.4) in the following functional form:

    x(t)=x0(0)e˜σt+3μˉσt0e˜σ(ts)F(3x(sτ))ds,

    and solve it by using the step method (see, e.g., [22]) on intervals [nτ,(n+1)τ],nN. Thus, the solution of (3.3) exists on [τ,).

    Next, we prove that the solution is nonnegative. Consider the following auxiliary problem:

    dxdt=μ˜σx(t),t>0;x(0)=x00.

    The solution is x(t)=x0eμ˜σt0. By Theorem 1.1 [23], to prove the solution of problem (3.3), (3.4) is nonnegative on the interval on which it exists, we only need to prove that F(x)0 for all x0 which follows from Lemma 2.1 and the fact H0. This completes the proof.

    In this section, we study the case where α(t)= constant denoted by α. We will discuss the existence and stability of stationary solutions. The stationary solution denoted by (σs(r),Rs) of the problem (1.2)–(1.6) must satisfy the following equations:

    1r2r(r2σs(r)r)=Γσs(r),0<r<Rs, (4.1)
    σs(r)r+α(σs(r)ˉσ(1γRs)H(Rs))=0,r=Rs, (4.2)
    Rs0μσs(r)r2drRs0μ˜σr2dr=0. (4.3)

    Using spatial scale transformation, one may assume Γ=1. Then, the solution to (4.1) and (4.2) is

    σs(r)=αˉσα+q(Rs)ζ(r)ζ(Rs)(1γRs)H(Rs), (4.4)

    where ζ(x)=sinhx/x and by (4.3), Rs satisfies

    f(Rs)=˜σ3ˉσ. (4.5)

    Theorem 4.1. Let x be the unique solution to J(x)=0, then there exists a unique positive constant ϑ which is determined by

    ϑ=3f(x)

    such that the following conclusions are valid:

    (i) If 0<˜σ<ϑˉσ, there exist two different stationary solutions (σs1(r),Rs1) and (σs2(r),Rs2) to problem (1.2)–(1.6), where Rs1<Rs2.

    (ii) If ˜σ=ϑˉσ, there exists a unique stationary solution (σs(r),Rs) to problem (1.2)–(1.6).

    (iii) If ˜σ>ϑˉσ, there is no stationary solutions to problem (1.2)–(1.6).

    Proof. Through a simple calculation, one can get that if γ<x2γ,

    f(x)=αx2[γg(x)+(x2γx)g(x)]+α(1γx)g(x)H(x)=αx2[α+xp(x)]2J(x)+α(1γx)g(x)H(x)

    and if x2γ,

    f(x)=αx2[γg(x)+(x2γx)g(x)]=αx2[α+xp(x)]2J(x)

    where

    J(x)=α[(x2γx)p(x)+γp(x)]+(2γxx2)p2(x).

    Since 0H(x)2/γ for x0, noting J(x) is strictly monotone decreasing and

    limx2γJ(x)=αγ[2γp(2γ)+p(2γ)]=αγ[xp(x)]|x=2γ=αγq(x)|x=2γ>0,

    where we have used q(x)>0 for x0 (see Lemma 2.1 (2)), then f(x)>0 for γ<x2γ. By a simple computation, limxγ+J(x)=αγp(γ)+γ2p2(γ)>0. Since limxq(x)=limxxp(x)=1, there exists a positive constant M0, such that xp(x)>12. Choose M1=max{M0+1,3γ,2γ(α+1)}, it follows that

    [αγ+(2γM1)M1p(M1)]p(M1)<(αγ+12(2γM1))p(M1)(αγ+12(2γ2γ(α+1)))p(M1)=0.

    Then

    J(M1)=α[(M21γM1)]p(M1)+[αγ+(2γM1)M1p(M1)]p(M1)<0.

    Since J(x)<0 for x>γ, by the mean value theorem, we have there exists a unique constant x(γ,M1) such that J(x)=0; J(x)<0 for x>x and J(x)>0 for γ<x<M1. It follows that

    f(x)=αx2[α+xp(x)]2J(x)+α(1γx)g(x)H(x){>0,γ<x<x,=0,x=x,<0,x>x.

    Then

    ϑ=:3f(x)=3maxx[γ,M1]f(x).

    Since f(x)=α(1γx)g(x) and αg(x)<p(x)<1/3, one can get

    ϑ=3f(x)(0,1).

    Noticing (4.5), we can get (ⅰ), (ⅱ) and (ⅲ). This completes the proof.

    For simplicity of notation, in the following of the paper, let's denote |φ|=maxτt0φ(t) and write minτt0φ(t) simply as minφ.

    Theorem 4.2. For any continuous nonnegative initial value function φ, the nonnegative solution of (3.2) and (1.6) exists for tτ and the dynamics of solutions to problem (3.2) and (1.6) are as follows:

    (I) If ˜σ>ϑˉσ, then limtR(t)=0.

    (II) If ˜σ=ϑˉσ, when |φ|<Rs, then limtR(t)=0 and when minφ>Rs, we have limtR(t)=Rs.

    (III) If 0<˜σ<ϑˉσ, when |φ|<Rs1, we have limtR(t)=0 and when minφ>Rs1, then limtR(t)=Rs2.

    Proof. By Theorem 3.1, for any continuous initial value function φ, we know that the nonnegative solution of (3.2) and (1.6) exists for tτ. Next, we study the dynamics of solutions to Eq (3.2) where α is a constant. By a direct computation, one can get

    Gy=μαˉσx2[γyg(y)H(y)+H(y)y3g(y)(1γy)+(y3g(y))(1γy)H(y)]. (4.6)

    Noting properties of the function H and Lemma 2.1, n, one can easily get Gy>0. By properties of f(x) and noticing the fact

    G(x,x)=μˉσx[f(x)˜σ3ˉσ], (4.7)

    it immediately follows that

    (P1) If ˜σ>ϑˉσ, we have f(x,x)<0 for all x>0.

    (P2) If ˜σ=ϑˉσ, we have f(x,x)<0 for all xRs.

    (P3) If 0<˜σ<ϑˉσ, we have f(x,x)<0 for all x<Rs1, f(x,x)>0 for all Rs1<x<Rs2 and f(x,x)<0 for all x>Rs2.

    By (P1)–(P3) and Lemma 2.2, we can readily get (I)–(III). This completes the proof.

    Remark. Since f,ϑ and Rs are functions depending on α and γ, we can rewrite them as f(x,α,γ), ϑ(α,γ) and Rs(α,γ). Simple computation yields

    fα=p(x)q(x)(α+q(x))2(1γx)H(x)>0,fγ=αxg(x)H(x)<0,x>γ, (4.8)

    which implies that ϑα0 and ϑγ0. Moreover, for 0<˜σ<ϑˉσ, by (4.8) we also can get that Rs1α0, Rs2α0, Rs1γ0 and Rs2γ0. From Theorem 4.2, we see that the stable stationary solution Rs2 is decreasing in γ and increasing in α. Two examples are given (see Figure 1 for γ=2 and α=5,8 and Figure 2 for α=8 and γ=2.5,3). From Figures 1 and 2, it is obvious that the stable stationary solution Rs2 (if it exists) is decreasing in γ and increasing in α.

    Figure 1.  An example of the function f for γ=2 and α=5,8 respectively.
    Figure 2.  An example of the function f for α=8 and γ=2.5,3 respectively.

    If α(t) is bounded, there exists two constants m,M (0m<M) such that mα(t)M. By Eq (3.2), we can get

    dRdtμˉσR[MM+q(Rτ)R3τp(Rτ)R3(1γRτ)H(Rτ)˜σ3ˉσ]. (5.1)

    Consider the following initial value problem

    dRdt=μˉσR[MM+q(Rτ)R3τp(Rτ)R3(1γRτ)H(Rτ)˜σ3ˉσ],t>0, (5.2)
    R0(t)=φ(t),τt0. (5.3)

    Define

    G1(x,y)=μˉσx[MM+q(y)y3p(y))x3(1γy)H(y)η/3].

    where η=˜σ/ˉσ. By similar arguments as that in section 4, one can get there exists a unique positive constant X satisfies the following equation:

    J1(x)=M[(x2γx)p(x)+γp(x)]+(2γxx2)p2(x)=0.

    Let f1(x)=M(1γx)g(x)H(x), one can get the following two lemmas by similar arguments as that in section 4.

    Lemma 5.1. Let X be the unique solution to J1(x)=0, then there exists a unique positive constant θ which is determined by

    θ=3f1(X)

    such that the following conclusions are valid:

    (i) If 0<˜σ<θˉσ, there exist two stationary solutions (σMs1(r),RMs1) and (σMs2(r),RMs2) to problem (5.2)–(5.3), where RMs1<RMs2.

    (ii) If ˜σ=θˉσ, there exists a unique stationary solution (σMs(r),RMs) to problem (5.2)–(5.3).

    (iii) If ˜σ>θˉσ, there is no stationary solution to problem (5.2)–(5.3).

    Lemma 5.2. For any nonnegative initial value function φ, the nonnegative solution of (5.2) and (5.3) exists for tτ and the following conclusions can be drawn:

    (I) If ˜σ>θˉσ, then limtRM(t)=0.

    (II) If ˜σ=θˉσ, when |φ|<RMs, then limtRM(t)=0 and when minφ>RMs, we have limtRM(t)=RMs.

    (III) If 0<˜σ<θˉσ, when |φ|<RMs1, we have limtRM(t)=0 and when minφ>RMs1, then limtRM(t)=RMs2.

    Theorem 5.3. For any nonnegative continuous φ(t), the nonnegative solution of (5.2) and (5.3) exists for tτ. Moreover, if α(t) is uniformly bounded, the dynamics of solutions to this problem are as follows:

    (1) If ˜σ>θˉσ, then limtR(t)=0.

    (2) If ˜σ=θˉσ, when |φ|<RMs, then limtR(t)=0 and when minφ>RMs, we have lim suptR(t)RMs.

    (3) If 0<˜σ<θˉσ, when |φ|<RMs1, we have limtR(t)=0 and when minφ>RMs1, then lim suptR(t)RMs2.

    Proof. By (5.1) and the comparison principle (see, e.g. Lemma 3.1 in [9]), we just need to prove Gy>0. Actually,

    G1y=μMˉσx2[γyg1(y)H(y)+H(y)y3g1(y)(1γy)+(y3g1(y))(1γy)H(y)] (5.4)

    where g1(y)=1M+q(y). Noting properties of the function H and Lemma 2.1, one can easily get G1y>0. The comparison principle (see, e.g. Lemma 3.1 in [9]) implies that

    R(t)RM(t). (5.5)

    By Lemma 5.2, noticing R(t)0 and taking upper limits for both R(t) and RM(t) as t, one can get (1), (2) and (3). This completes the proof.

    In this section, by using Matlab R2016a, we give some numerical simulations of solutions to Eq (3.3) for special parameter values (see Figures 3 and 4). We use matlab to find some special cases where the steady-state solutions are larger than 2γ. In this case H(x)=1, therefore

    Figure 3.  The curve of the function f for ˉσ=5,μ=1,˜σ=1,α=8,γ=2.
    Figure 4.  An example of the dynamics of solution to Eq (3.3) for ˉσ=5,μ=1,˜σ=1,α=8,γ=2,τ=3 and x0=100,1600, respectively.
    f(x)=α(1γx)g(x). (6.1)

    If the parameters in Eq (3.3) are taken values as

    ˉσ=5,μ=1,˜σ=1,α=8,γ=2,τ=3,x0=100,1600, (6.2)

    by using Matlab R2016a, we can solve the equation

    f(3x)=˜σ3ˉσ.

    It has two positive solutions: The smaller one is xs1=25.03 and the larger one is xs2=870.26 (The curve of f is shown in Figure 3). Noticing (4.5), we have Rs1=325.03 and Rs2=3870.26>2γ=4.

    Since ϑ>30.085 (see Figure 3), it is obvious that

    0<˜σ=1<30.0855=1.275<ϑˉσ.

    Therefore, the conditions of Theorem 4.2 (Ⅲ) are satisfied. Since the initial conditions x0=100,1600 are larger than Rs1 which third power is smaller than 100. Thus all the solutions of Eq (3.3) tend to Rs2=3870.26. The dynamics of solution to Eq (3.3) for the parameters in Eq (3.3) are taken values as (6.2) are given by Figure 4.

    In this paper considering the time-delay in the cell proliferation process and the Gibbs-Thomson relation in the boundary condition, we study a time-delayed free boundary problem describing tumor growth with angiogenesis. We prove the existence and uniqueness of time-varying solutions. When α is a positive constant, we discuss the existence of steady-state solutions and the number of steady-state solutions (Theorem 4.1), and prove the asymptotic behavior of solutions (Theorem 4.2). From the biological point of view, the result of Theorem 4.2 (Ⅰ) means when ˜σ which represents the rate of apoptosis of tumor cells is less than a critical value ϑˉσ, the tumor will disappear. The result of Theorem 4.2 (Ⅱ) means when ˜σ is equal to the critical value ϑˉσ, for small initial functions satisfying maxτt0φ(t)<Rs, the tumor will disappear; for large initial functions satisfying minτt0φ(t)>Rs, the tumor will not disappear and will tend to the unique steady-state. The result of Theorem 4.2 (Ⅲ) means that when ˜σ is relatively small (less than a critical value ϑˉσ), whether the tumor will disappear or not depends on the value of the initial value function. When the minimum value of the initial value function is greater than Rs1, the tumor tends to the larger steady-state solution; when the maximum value of the initial value function is less than Rs1, the tumor will disappear. We prove that with the increase of α, the larger steady-state solution will increase when nutrients is sufficient (see Remark in section 4). According to the conclusion of the Theorem 4.2 (Ⅲ) and the Remark in section 4, increasing α can increase the final volume of the tumor. Noting that α represents the ability of tumor to attract blood vessels, this indicate that the larger α is, the stronger it is to attract blood vessels, and the larger the tumor will be, which is consistent with biological phenomena. We also prove that when α(t) is bounded, tumors will not increase indefinitely and will remain bounded.

    This work is partially supported by NNSF of China (11301474), NSF of Guangdong Province (2018A030313536). The authors would like to thank the editor and the referees for their very helpful suggestions on modification of the original manuscript.

    The authors declare that there is no conflict of interest regarding the publication of this paper.



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