INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 19 (2006) 419–440
NONLINEARITY doi:10.1088/0951-7715/19/2/010
An elliptic–hyperbolic free boundary problem modelling cancer therapy Youshan Tao and Miaojun Chen Department of Applied Mathematics, Dong Hua University, Shanghai 200051, People’s Republic of China E-mail:
[email protected]
Received 23 May 2005, in final form 27 October 2005 Published 14 December 2005 Online at stacks.iop.org/Non/19/419 Recommended by M P Brenner Abstract In this paper we study a free boundary problem modelling the growth of an avascular tumour with drug application. The tumour consists of two cell populations: live cells and dead cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations. The tumour surface is a moving boundary, which satisfies an integro-differential equation. The nutrient concentration and the drug concentration satisfy nonlinear diffusion equations. The nutrient drives the growth of the tumour, whereas the drug is capable of killing cells with Michaelis–Menten kinetics. We prove that this free boundary problem has a unique global solution. Furthermore, we investigate the combined effects of a drug and a nutrient on an avascular tumour growth. We prove that the tumour shrinks to a necrotic core with radius Rs > 0 and that the global solution converges to a trivial steady-state solution under some natural assumptions on the model parameters. We also prove that an untreated tumour shrinks to a dead core or continually grows to an infinite size, which depends on the different parameter conditions. Mathematics Subject Classification: 35Q80, 35R35, 92A15, 92C50
1. Introduction Over the past three decades, mathematical models for tumour growth have been developed and studied; for example, see [1–7, 9, 10, 12–15, 18–23] and the references given there. These models are based on mass conservation laws and on the reaction-diffusion process for cell densities and nutrient concentration within the tumour. These models make the following general assumptions. 0951-7715/06/020419+22$30.00 © 2006 IOP Publishing Ltd and London Mathematical Society Printed in the UK
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The cells within the tumour are in one of three phases: proliferating, quiescent or necrotic. Living cells die as a result of a cell-loss mechanism (apoptosis) and quiescent cells may also die as a result of starvation (necrosis). All living cells undergo mitosis, but proliferating cells undergo mitosis at abnormally large rates, which depend on the nutrient concentration. Finally, the tumour is uniformly packed with cells, and there are no voids within the tumour. There are basically two kinds of partial differential equation (PDE) models: reaction– diffusion models and convection-dominated models. The reaction–diffusion models generally consist of an ordinary differential equation (ODE), coupled to one or more reaction–diffusion equations (RDEs) [2–5, 9, 12, 13]. The ODE derives from mass conservation applied to the tumour and describes the evolution of the tumour boundary, whereas the RDEs describe the distribution, within the tumour, of nutrients and growth inhibitory factors. An important feature of the reaction–diffusion models is that there is no explicit mention of the constituent tumour cells in these models. Instead, it is assumed that the nutrient level at each point within the tumour provides an index of the cell type. Thus, high nutrient levels indicate cell mitosis, moderate levels indicate quiescence and low levels necrosis. In reaction–diffusion models the different populations of cells are assumed to be separated by interfaces. The convection-dominated models generally consist of an ordinary differential equation (ODE), coupled to one or more reaction–diffusion equations (RDEs) and two or more convection equations (CEs) [6, 7, 10, 14, 15, 18–22]. The ODE is the equation of continuity: the velocity of the tumour boundary is the same as the cell velocity at the boundary. The RDEs describe the distribution, within the tumour, of nutrients and drugs. The CEs derive from mass conservation law for the different phases of the cells. The tumour volume is modelled as an incompressible fluid, through which the cells travel via a convective field whose velocity is V. Local changes in cell population lead to variations in the internal pressure, which in turn induce motion of the tumour cells. Since the diffusion coefficients of the tumour cells are much smaller than the diffusion coefficient of the much smaller and more motile drug (or nutrient) molecules [14, 15, 20, 21], cellular diffusion can be neglected compared with the convective motion. That is, the active movement of the cells is the convective motion. An important feature of the convection-dominated models is the explicit mention of the different phases of tumour cells. In these models the different populations of cells are assumed to be continuously present everywhere in the tumour, at all times. In vitro observations [17] suggest that in early stages solid tumours remain approximately spherically symmetric as they grow. Therefore, most of the models [2, 4, 5, 7,9,10, 12–15, 18–22] focus on the existence and stability of radially-symmetric solutions of the model equations. Since tumours grown in vitro have a nearly spherical shape, it is important to determine whether radially-symmetric tumours are asymptotically stable. If the radiallysymmetric configuration is stable with respect to all asymmetric perturbations then the tumour maintains a radially-symmetric structure—this corresponds to the growth of a benign tumour. The instability of the radially-symmetric configuration with respect to some asymmetric perturbations usually indicates tumour invasion [3]. So, the stability of the radially-symmetric configuration with respect to asymmetric perturbations is very important in cancer biology. Since the study of the stability needs more delicate analysis and techniques of PDEs, the stability of the radially-symmetric solution with respect to asymmetric perturbations is also very interesting in the PDE field. Byrne and Chaplain [3] studied the linear stability of the radially-symmetric solution with respect to asymmetric perturbations. Recently, Friedman and co-authors (e.g. see [1]) have studied the nonlinear stability of the radially-symmetric solution with respect to asymmetric perturbations.
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The nonlinear analysis of the fully multi-dimensional tumour models is very challenging in mathematics. To our knowledge the global existence of the solutions for the fully multidimensional tumour models is still open. However, there appear some numerical works on the two-dimensional tumour models (e.g. see [23]). Zheng et al [23] recently employed the level set method to numerically study a two-dimensional tumour model. There are many mathematical models of drug therapies involving tumours (see the references given in [21]). However, most of these models consist of systems of ODEs describing the temporal response to a drug of a population, or multiple populations, of tumour cells, using the linear of logistic kinetics; very few models consider spatial variations, such as those resulting from drug diffusion into tumour masses. Recently, PDE mathematical models of tumour response to drug treatment have begun to appear in the literature; for example, see [14, 15, 18, 21, 22]. The models in [14, 15] consider vascular tumours with drug treatment. The drugs can be transferred from intratumoural blood vessels into tumour tissue. These models in [14, 15] neglected the effect of a nutrient on tumour growth. However, for an avascular tumour, a drug and a nutrient penetrate tumour tissue mainly by the mechanism of diffusion. In this situation, the effect of a nutrient on tumour growth cannot be neglected. The model [21] considered the combined effects of a drug and a nutrient on an avascular tumour growth. Furthermore, in [21] the kinetics of drug action on cells is assumed to be the nonlinear Michaelis–Menten kinetics. In this paper we consider the radially-symmetric case and we aim at establishing the rigorous mathematical analysis of the model; in particular, we study the asymptotic behaviour of the tumour radius. These kinds of problems are interesting (e.g. see [2, 7, 9]) and are generally not trivial. We use the Leray–Schauder fixed theorem (cf. [8, chapter 7]) and the contraction mapping principle to prove the global existence; then we study the asymptotic behaviour of the tumour boundary by auxiliary function techniques and the maximum principle of elliptic equations. The structure of this paper is as follows. In section 2 we describe the model. In section 3 we transform the problem into a system of equations in a fixed domain. In section 4 we prove global existence and uniqueness of the solution. To this aim, a semi-linear elliptic problem, a first order hyperbolic problem and an elliptic–hyperbolic system are studied in this section. The large time behaviour of the free boundary and the global solution are studied in section 5. In section 6 we also investigate the evolution of an untreated tumour. Finally, we close this paper with a conclusion section, in which we give some physical interpretation of the results that are derived in this paper. 2. The model In this paper we consider a mathematical model which describes the response of a tumour spheroid to anti-cancer drugs. The cells within the tumour are in one of two states: live or dead. We shall denote the corresponding cell densities (cells/unit volume) by n and m, respectively. The proliferation or death of the live cell is dependent on the nutrient concentration c and the drug concentration w. Live cells proliferate at a rate kp (c), undergo natural death at a rate kd (c) and become dead at a rate Kf (w) due to the drug; the constant K is the maximum possible rate of drug induced cell death. The forms of the functions kp , kd and f are taken to be Michaelis-Menten kinetics, namely wc w Ac σc , kd (c) = B 1 − , f (w) = , kp (c) = cp + c cd + c wc + w where A, B, cp , cd , σ and wc are positive constants. Fick’s law is assumed to describe the diffusion of the nutrient, which, for reasons stated in [20], is consumed at a rate proportional
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to the rate of cell mitosis, namely βkp (c)n (where β is a positive constant). Fick’s law is also assumed to describe the diffusion of the drug, which is degraded only when it attacks a living cell, giving a maximum degradation rate of K/ω; any other process of drug breakdown is neglected. Here the dimensionless constant ω can be viewed as a measure of the drug’s effectiveness, with increasing ω implying that less drug is consumed during the cell killing process. Due to proliferation and death of live cells, there are volume changes leading to movement described by a velocity field v (cf [20, 21]). Assuming spherical symmetry, we can then write the conservation of mass laws for densities n and m of the live cells and the dead cells, for the nutrient concentration c and for the drug concentration w within the tumour region in the following form: ∂n 1 ∂(r 2 vn) + = [kp (c) − kd (c) − Kf (w)]n, (2.1) ∂t r 2 ∂r ∂m 1 ∂(r 2 vm) (2.2) + 2 = [kd (c) + Kf (w)]n, ∂t r ∂r ∂c D ∂ ∂c 1 ∂(r 2 vc) (2.3) + = 2 r2 − βkp (c)n, ∂t r 2 ∂r r ∂r ∂r ∂w ∂w 1 ∂(r 2 vw) Dw ∂ K + 2 = 2 r2 − f (w)n. (2.4) ∂t r ∂r r ∂r ∂r ω We assume that the spheroid is made up of live cells and dead cells. Anti-cancer drugs are often relatively small molecules, having diffusive properties similar to glucose (see [21] and references therein). Such drugs will therefore be treated in much the same way as the nutrient, contributing negligible volume to the spheroid. Defining VL and VD to be the inverse densities (volume/cell) of a live and dead cell, respectively, we have VL n + VD m = 1. (2.5) If we multiply equation (2.1) by VL , multiply equation (2.2) by VD , add the two resulting equations and make use of (2.5), we get 1 ∂(r 2 v) (2.6) = {VL kp (c) − (VL − VD )[kd (c) + Kf (w)]}n. r 2 ∂r This equation can be used to replace the conservation law (2.2) for m. We take the boundary conditions, at the free (moving) boundary r = R(t), to be c = c0 , (2.7) (2.8) w = w0 (t), Figure? dR = v, (2.9) dt where c0 is the external nutrient concentration, assumed fixed, w0 (t) is a non-negative function describing the manner of drug administration and R(t) is the (unknown) spheroid radius. Finally, we impose the following initial conditions: R(0) is prescribed, n(r, 0), c(r, 0) and w(r, 0) are given for 0 r R(0), (2.10) where 0 < VL n(r, 0) < 1, 0 c(r, 0) c0 . (2.11) The model (2.1)–(2.11) was introduced by Ward and King [21]. In this paper we prove global existence and uniqueness of the solution; we also prove that the tumour shrinks to a dead core with radius Rs > 0 and that the global solution converges to a trivial steady-state solution under an explicit parameter condition.
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Remark 2.1. In general, for three-dimensional spheroid growth, equation (2.9) cannot hold. To determine five variables n, m, c, w and V, additional equations are needed to close the system. One possible closure involves using Darcy’s law (V = −µ∇p, where µ denotes the motility of the tumour cells) to relate velocity V to pressure p (e.g. see [3, 6]). The threedimensional model takes the following form: ∂n + ∇ · (nV) = [kp (c) − kd (c) − Kf (w)]n in (t), ∂t
(2.12)
∂m + ∇ · (mV) = [kd (c) + Kf (w)]n in (t), ∂t
(2.13)
∂c + ∇ · (cV) = Dc − βkp (c)n in (t), ∂t
(2.14)
K ∂w + ∇ · (cV) = Dw w − f (w)n in (t), ∂t ω
(2.15)
VL n + VD m = 1 in (t),
(2.16)
V = −µ∇p in (t),
(2.17)
µp = −{VL kp (c) − (VL − VD )[kd (c) + Kf (w)]}n in (t),
(2.18)
where (t) denotes the tumour domain at time t. We impose the following boundary conditions: c = c0 on ∂(t),
(2.19)
w = w0 (t) on ∂(t),
(2.20)
p = p0 on ∂(t),
(2.21)
where we assumed that on the tumour boundary p matches continuously with the external pressure field where p = p0 . Furthermore, ∂p on ∂(t), (2.22) ∂ν where ν is the outward normal and Vν is the velocity of the free boundary ∂(t) in the direction ν. We finally prescribe initial conditions: Vν = −µ
n|t=0 = n0 , c|t=0 = c0 , w|t=0 = w0 in (0), where (0) is given.
(2.23)
For the three-dimensional model, one may prove the local existence of the solution (cf [6]). However, the global existence is open. Under conditions of spherical symmetry, we can eliminate the internal pressure field p and derive the model (2.1)–(2.10) from the general model (2.12)–(2.23). 3. Transformation Denote non-dimensional variables with bars c w c¯ = , w¯ = , n¯ = VL n, c0 wc r R t¯ = At, r¯ = , R¯ = , r0 r0
v¯ =
v , r0 A
why?
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where r0 = (3VL /4π)1/3 is the radius of a single cell. We chose r0 as a representative length scale for consistency with [20, 21]; the radius R(0) at the start of treatment is typically much larger than r0 . After the above re-scalings, the system (2.1)–(2.11) takes the following form: ∂ n¯ ∂ n¯ ¯ c, + v¯ = (a( ¯ c, ¯ w) ¯ − b( ¯ w) ¯ n) ¯ n, ¯ (3.1) ∂ t¯ ∂ r¯ ¯ ∂ c¯ 1 ∂ ∂ c¯ 1 ∂(¯r 2 v¯ c) ¯ c) ε1 + 2 = 2 r¯ 2 − k( ¯ n, ¯ (3.2) ∂ t¯ r¯ ∂ r¯ r¯ ∂ r¯ ∂ r¯ ¯ ¯ ∂ w¯ 1 ∂(¯r 2 v¯ w) 1 ∂ K¯ 2 ∂w ε2 + 2 = 2 r¯ − f¯(w) ¯ n, ¯ (3.3) ∂ t¯ r¯ ∂ r¯ r¯ ∂ r¯ ∂ r¯ α 1 ∂(¯r 2 v) ¯ ¯ c, = b( ¯ w) ¯ n, ¯ r¯ 2 ∂ r¯ in {r < R(t), t > 0}, ¯ t¯), t¯) = 1, c( ¯ R(
(3.4)
(3.5)
¯ t¯), t¯) = w¯ 0 (t¯) 0, w( ¯ R( ¯ t¯) dR( ¯ t¯), t¯), = v( ¯ R( dt¯ n(¯ ¯ r , 0) = n¯ 0 (¯r ),
(3.6) ¯ R(0) = R¯ 0 ,
(3.7)
0 < n¯ 0 (¯r ) < 1 for 0 r¯ R¯ 0 ,
(3.8)
c(¯ ¯ r , 0) and w(¯ ¯ r , 0) are prescribed for 0 r¯ R¯ 0 ,
(3.9)
where α = ωDw VL /Ar02 , K¯ = Kwc /A, ε1 = Ar02 /D, ε2 = Ar02 /Dw , and ¯ (w), ¯ − k¯d (c) ¯ − Kf ¯ a( ¯ c, ¯ w) ¯ = k¯p (c) ¯ c, ¯ (w)], b( ¯ w) ¯ = k¯p (c) ¯ − (1 − δ)[k¯d (c) ¯ + Kf ¯ ¯ c) k( ¯ = β¯ k¯p (c), ¯
where ¯ = k¯p (c)
c¯ , c¯p + c¯
c¯ B 1−σ ¯ = k¯d (c) , A c¯d + c¯
f¯(w) ¯ =
w¯ , 1 + w¯
r 2 βA cp cd VD , β¯ = 0 , c¯p = , c¯d = . VL DVL c0 c0 c0 For the typical parameters given in [20, 21], ε1 and ε2 are typically small compared with other parameters, and therefore we simplify (3.2) and (3.3) to their quasi-steady form as done in [21]. (Very recently, Bueno et al [2] also used a quasi-steady approach to study a mathematical model describing the growth of tumours.) We note that if (3.2) and (3.3) are quasi-steady-state approximated then we should drop the initial condition (3.9). ¯ t¯)} into the fixed region It will be convenient to transform the region {0 r¯ R( {0 ρ 1} by ¯ t¯) ρ = ρ(¯r , t¯) = r¯ /R( δ=
and set t = t¯ and n(ρ, ˜ t) = n(¯ ¯ r , t¯), w(ρ, ˜ t) = w(¯ ¯ r , t¯),
c(ρ, ˜ t) = c(¯ ¯ r , t¯), ¯ t¯) v(ρ, ˜ t) = v(¯ ¯ r , t¯)/R(
for 0 ρ 1. Using the quasi-steady-state approximation to (3.2) and (3.3), in terms of the new variables and after dropping the tildes of n(ρ, ˜ t), c(ρ, ˜ t), w(ρ, ˜ t), v(ρ, ˜ t) and the bars
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¯ k, ¯ k¯p , k¯d , K, ¯ β, ¯ c¯p , c¯d for notations’ convenience, the system (3.1)–(3.8) takes the of a, ¯ b, following form in {0 < ρ < 1, t > 0}: K 1 ∂ 2 ∂w (3.10) ρ = R 2 (t)f (w)n, ρ 2 ∂ρ ∂ρ α w(1, t) = w0 (t), 1 ∂ 2 ∂c ρ = R 2 (t)k(c)n, ρ 2 ∂ρ ∂ρ
(3.11)
c(1, t) = 1,
(3.13)
∂n ∂n + [v(ρ, t) − ρv(1, t)] = a(c, w)n − b(c, w)n2 , ∂t ∂ρ
(3.14)
n(ρ, 0) = n0 (ρ),
(3.15)
0 < n0 (ρ) < 1,
˙ = R(t)v(1, t), R(t) R(0) = R0 > 0, ρ 1 v(ρ, t) = 2 b(c(s, t), w(s, t))n(s, t)s 2 ds, ρ 0
(3.12)
(3.16) (3.17)
where we note that (3.17) implies v(0, t) = 0, as the denominator is O(ρ 2 ) while the numerator is O(ρ 3 ). Throughout this paper, we assume that w0 (t) ∈ C[0, +∞),
n0 (ρ) ∈ C 1 [0, 1].
In section 4 we establish global existence and uniqueness for the system (3.10)–(3.17). The rest of this paper is devoted to the study of the asymptotic behaviour of the free boundary r = R(t). 4. Global existence and uniqueness We first establish the local existence and uniqueness of the solution. The hyperbolic equation (3.14) has the feature that the lines ρ = 0 and ρ = 1 are characteristic curves which are vertical on the line t = 0, so no difficulties will arise on the boundaries ρ = 0 and ρ = 1. The coefficients in (3.14) defining the characteristics are more than Lipschitz, due to the special structure given in (3.17). Thus the local existence and uniqueness can be proved by standard methods in studying 1-space dimensional free boundary problems. Recently, local existence and uniqueness for a higher dimensional (non-radially symmetric) elliptic– hyperbolic free boundary cancer model was studied by Chen and Friedman [6]. However, in our proof of the local existence and uniqueness, we will derive some estimates which will be needed in studying the global existence and especially the asymptotic behaviour. So, in the following we still give the main ideas and steps of the proof of the local existence and uniqueness. Denote B M ≡1+ +K A and introduce the following space of functions v(ρ, t): ET = v(ρ, t) ∈ C 1,0 ([0, 1] × [0, T ]); v(0, t) = 0, v(ρ, 0) = v0 (ρ), |v(ρ, t)| M, ∂v(ρ, t) 3M , ∂ρ
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where
ρ 1 b(c(s, 0), w(s, 0))n0 (s)s 2 ds. v0 (ρ) = 2 ρ 0 Here (c(s, 0), w(s, 0)) is the solution of the elliptic system (3.10)–(3.13) at time t = 0. C 1,0 ([0, 1] × [0, T ]) is the space of all functions v(ρ, t) which are continuous together with their first-order ρ derivative ∂v(ρ, t)/∂ρ, and the norm in this space is defined by ∂v vC 1,0 ≡ vC 0 [0ρ1,0tT ] + . ∂ρ 0 C [0ρ1,0tT ]
Since R(0) > 0, the space ET is non-empty if T is sufficiently small. We shall first solve the system (3.10)–(3.17) for a small time interval 0 t T , using the contraction mapping principle in the space ET . Given any v ∈ ET we get R(t) by solving the ODE problem (3.16) and proceed to solve the elliptic–hyperbolic system (3.10)–(3.15) by a fixed-point method. We then define a mapping F : v(ρ, t) → u(ρ, t)
by
ρ 1 b(c(s, t), w(s, t))n(s, t)s 2 ds. (4.1) ρ2 0 We shall prove that F is a contraction if T is sufficiently small, and this will complete the proof of local existence and uniqueness for the system (3.10)–(3.17); global existence will be proved by continuing the local solution. For any given v ∈ ET , we introduce the forward characteristic curves ξ = ξ(t; ρ0 ) of the first-order hyperbolic equation (3.14) by dξ (4.2) = v(ξ, t) − ξ v(1, t), ξ(0; ρ0 ) = ρ0 (0 ρ0 1), dt where ∂ (v(ξ, t) − ξ v(1, t)) 4M (4.3) ∂ξ u(ρ, t) =
by the assumption v ∈ ET . (4.3) implies that there exists a unique C 1 solution to (4.2). In view of (4.2) and v(0, t) = 0, the lines ξ = 0 and ξ = 1 are characteristic curves, and the characteristic curves ξ = ξ(t; ρ0 ) do not exit the interval 0 < ξ < 1 if 0 < ρ < 1. The above considerations imply that the solution of the first-order hyperbolic problem (3.14) and (3.15) in {0 ρ 1, t > 0} can be determined by the initial data n0 (ρ0 ) for 0 ρ0 1. For any given v ∈ ET , we get R(t) by solving the ODE problem (3.16), and we shall establish the local existence and uniqueness of the solution of the elliptic–hyperbolic system (3.10)–(3.15) using a fixed-point method. To this end, we introduce a convex set ∂n ST = n(ξ(t), t) : n(ξ(t), t), (ξ(t), t) ∈ C[0, T ], n(ξ(0), 0) = n0 (ρ0 ), ∂ρ ∂n ht ∂n 0 < n(ξ(t), t) < 1, (ξ(t), t) Le , (ξ(t), t) 2M(L + 1)eht , ∂ρ ∂t here ξ(t)is defined by(4.2) , where L = 2n0 (ρ)C[0,1] and h > 2M is some positive constant; and the norm ∂n (ξ(t), t) , n ∈ ST . nST = n(ξ(t), t)C[0,T ] + ∂ρ C[0,T ]
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Given any n(ξ(t), ˜ t) ∈ ST , let (w, c) be the solution of (3.10)–(3.13) with n replaced by n, ˜ and let n be the solution of (3.14) and (3.15) with this (w, c). Define the mapping H : n˜ → n(n˜ ∈ ST ).
We shall prove that H is a contraction if T is sufficiently small, and this will complete the proof of local existence and uniqueness for the elliptic–hyperbolic system (3.10)–(3.15) for any given v ∈ ET . We begin by studying the time-dependent semi-linear elliptic problem (3.10)–(3.13) in Bρ (0, 1) =: {0 ρ 1} with n replaced by n. ˜ This problem can be rewritten as − w(ρ, t) +
K 2 w(ρ, t) ˜ t) R (t)n(ξ(t), = 0, α 1 + w(ρ, t)
w(1, t) = w0 (t), − c(ρ, t) + βR 2 (t)n(ξ(t), ˜ t)
(4.4) (4.5)
c(ρ, t) = 0, cp + c(ρ, t)
(4.6)
c(1, t) = 1,
(4.7)
where = (∂ 2 /∂x 2 ) + (∂ 2 /∂y 2 ) + (∂ 2 /∂z2 ), ρ = x 2 + y 2 + z2 and the time variable t can be regarded as a parameter. Theorem 4.1. Assume that w0 (t) > 0 is a continuous and bounded functions in [0, ∞). Then, for any fixed t ∈ [0, T ], there exists a unique solution w(ρ, t) ∈ C 2,γ (Bρ (0, 1)) to problem (4.4) and (4.5) and a unique solution c(ρ, t) ∈ C 2,γ (Bρ (0, 1)) to problem (4.6) and (4.7), and the following estimates hold: √ w0 (t) sinh( K/αR0 eMt ρ) (4.8) w(ρ, t) w0 (t), √ ρ sinh( K/αR0 eMt ) ∂ w(ρ, t) M0 e2Mt , (4.9) ∂ρ ∞ L (Bρ (0,1)) √ sinh( βR0 eMt ρ) c(ρ, t) 1, (4.10) √ ρ sinh( βR0 eMt ) ∂ c(ρ, t) M0 e2Mt , (4.11) ∞ ∂ρ L (Bρ (0,1)) where 0 < γ < 1 and M0 > 0 are some constants independent of T and M is defined as before. Furthermore, w(ρ, t) and c(ρ, t) are increasing in variable ρ, and w(ρ, t), c(ρ, t), ∂w(ρ, t)/∂ρ and ∂c(ρ, t)/∂ρ are continuous in variable t ∈ [0, T ]. Proof. We prove the existence of a solution to problem (4.4) and (4.5) by the Leray–Schauder fixed-point theorem (cf [8, chapter 7] or [11, chapter 11]). Take a Banach space X = w|w ∈ C 1,γ (Bρ (0, 1)), 0 w sup w0 (t) t∈[0,T ]
and let N be a mapping of X × [0, 1] into X: w = N (w, ˜ λ),
(4.12)
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where w˜ belongs to X, λ is a real parameter which varies in [0, 1] and w is the solution to the following problem: − w(ρ, t) +
KR 2 (t)n(ξ(t), ˜ t) w(ρ, t) = 0 inBρ (0, 1), α(1 + λw(ρ, ˜ t))
w(1, t) = λw0 (t).
(4.13) (4.14)
By the standard elliptic Schauder theory, for any fixed t ∈ [0, T ], there exists a unique solution w(ρ, t) ∈ C 2,γ (Bρ (0, 1)) → C 1,γ (Bρ (0, 1)) to problem (4.13) and (4.14). Furthermore, by the maximum principle we get 0 w(ρ, t) w0 (t). Therefore (a) the mapping (4.12) is well defined and N is a compact mapping for fixed λ ∈ [0, 1]. Furthermore, by the Schauder estimate and the maximum principle, one can easily check that ˜ λ) is continuous in X; (b) for any fixed λ, N (w, ˜ λ) is continuous in λ; (c) for w˜ in bounded sets of X, N (w, (d) w − N (w, 0) = 0 has a unique solution w(ρ, t) ≡ 0; (e) there exists a (finite) constant M1 such that every possible solution w of w − N (w, λ) = 0 (w ∈ X, λ ∈ [0, 1]) satisfies w C 1,γ (Bρ (0,1)) M1 . By (a)–(e) and the Leray–Schauder fixed theorem, the mapping N has a fixed point in X. That is, there exists a solution w(ρ, t) ∈ C 1,γ (Bρ (0, 1)) to the problem (4.4) and (4.5). Furthermore, by the Schauder estimate, we can raise the regularity of w(ρ, t) to C 2,γ (Bρ (0, 1)). The uniqueness of the solution of the problem (4.4) and (4.5) follows from R(t) > 0, n(ξ(t), ˜ t) > 0 and the maximum principle. Clearly, by the maximum principle, the solution w(ρ, t) of the problem (4.4) and (4.5) satisfies 0 w(ρ, t) w0 (t).
(4.15)
In the following we will establish a refined lower bound on w(ρ, t). We first note that, by (3.16) and v ∈ ET , 0 < R(t) = R0 e
t 0
v(1,τ ) dτ
R0 eMt .
(4.16)
By 0 < n˜ < 1, (4.15), (4.16), (4.4) and (4.5) we find − w(ρ, t) +
K 2 2Mt R e w(ρ, t) 0 inBρ (0, 1), α 0
w(1, t) = w0 (t).
(4.17) (4.18)
Let w∗ (ρ, t) be the solution to the following problem: − w∗ (ρ, t) +
K 2 2Mt R e w∗ (ρ, t) = 0 in Bρ (0, 1), α 0
w∗ (1, t) = w0 (t).
(4.19) (4.20)
By the comparison principle, we get
√ w0 (t) sinh( K/αR0 eMt ρ) w(ρ, t) w∗ (ρ, t) ≡ > 0. √ ρ sinh( K/αR0 eMt )
This completes the proof of the estimate (4.8).
(4.21)
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Next, we prove the estimate (4.9). We rewrite the problem (4.4) and (4.5) as K − w(ρ, t) + R 2 (t)n(ξ(t), ˜ t)w(ρ, t) = f (ρ, t) in Bρ (0, 1), α w(1, t) = w0 (t),
429
(4.22) (4.23)
where K 2 w 2 (ρ, t) R (t)n(ξ(t), . ˜ t) α 1 + w(ρ, t) By 0 < n(ξ(t), ˜ t) < 1, (4.15), (4.16) and the standard Lp -estimate of elliptic type equations (e.g. cf [16]) we get f (ρ, t) =:
wW 2,p (Bρ (0,1)) M2 (wLp (Bρ (0,1)) + f Lp (Bρ (0,1)) + w0 (t)W 2,p (Bρ (0,1)) ) M2 (2w0 (t) K + w02 (t)R 2 (t)) M3 (1 + R 2 (t)) M4 e2Mt (4.24) α where M4 is some positive constant depending on supt∈[0,T ] w0 (t) and R0 , but M4 is independent of T . By (4.24) and the Sobolev imbedding theorem (cf [11, corollary 7.11]) W 2,p () → C 1 () (p > N =: the dimension of the domain ), we easily get the estimate (4.9). To prove that w(ρ, t) is increasing in ρ, differentiating equation (4.4) with respect to ρ and setting x(ρ, t) ≡ ∂w(ρ, t)/∂ρ, we get − x(ρ, t) +
KR 2 (t)n(ξ(t), ˜ t) x(ρ, t) = 0 in Bρ (0, 1). α(1 + w(ρ, t))2
(4.25)
Note that ∂w(1, t) >0 (4.26) ∂ρ by (4.4) and (4.5) and the strong maximum principle (cf [11, chapter 3]). Proceeding as in the proof of (4.21), we can construct a positive sub-solution x∗ (ρ, t) of problem (4.25) and (4.26). Therefore ∂w(ρ, t) ≡ x(ρ, t) x∗ (ρ, t) > 0. ∂ρ That is, w(ρ, t) is increasing in ρ. Finally, we establish the continuity of w(ρ, t) and ∂w(ρ, t)/∂ρ in the variable t. For any t1 , t2 ∈ [0, T ], from (4.4) and (4.5), n˜ ∈ ST , (4.15) and using the Lp -estimate, we can get x(1, t) =
w(ρ, t1 ) − w(ρ, t2 )W 2,p (Bρ (0,1)) C(T )(|w0 (t1 ) − w0 (t2 )| + |R(t1 ) − R(t2 )| +|ξ(t1 ) − ξ(t2 )| + |t1 − t2 |).
(4.27)
So that, by (4.27) and the Sobolev imbedding theorem, W 2,p () → C 1 () (p > N), and the continuity of function w0 (t), R(t) and ξ(t), we can establish the continuity of w(ρ, t), ∂w(ρ, t)/∂ρ in the variable t. In the same way, we can prove the existence and uniqueness of the solution c(ρ, t) ∈ C 2,γ (Bρ (0, 1)) to problem (4.6) and (4.7), the continuity of c(ρ, t), ∂c(ρ, t)/∂ρ in the variable t, the monotonicity of c(ρ, t) in the variable ρ and the estimates (4.10) and (4.11). For any given v ∈ ET , we next solve the first-order hyperbolic problem (3.14) and (3.15) with (w, c) being the solution to problem (4.4)–(4.7). Here we rewrite this problem as follows: ∂n ∂n (4.28) + [v(ρ, t) − ρv(1, t)] = a(c, w)n − b(c, w)n2 , ∂t ∂ρ 0 < n0 (ρ) < 1. (4.29) n(ρ, 0) = n0 (ρ), Proceeding as in the proof of theorem 7.1 in [19], we get the following theorem.
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Theorem 4.2. For any v(ρ, t) in ET there exists a unique solution n(t) ˆ of (4.28) and (4.29) such that ∂ nˆ ∂n n(t) ˆ ≡ n(ξ(t), t), (t) ≡ (ξ(t), t) ∂ρ ∂ρ are in C([0, T ]), and 0 < n(t) ˆ < 1, ∂ nˆ (t) Leht , ∂ρ
∂ nˆ (t) 2M(L + 1)eht , ∂t
(4.30) (4.31)
where L = 2n0 (ρ)C[0,1] , and h > 2M is some positive constant which is independent of T . For any given v ∈ ET and R(t) defined by (3.16), we can now prove the local existence and uniqueness of the solution of the elliptic–hyperbolic system (3.10)–(3.15) by theorems 4.1 and 4.2 and a fixed-point method. Theorem 4.3. Assume that w0 (t) > 0 is continuous and a bounded function in [0, ∞). If T is small enough then for any v(ρ, t) in ET and R(t) defined by (3.16), there exists a unique solution (w(ρ, t), c(ρ, t), n(ρ, t)) of (3.10)–(3.15) such that w(ρ, t), c(ρ, t) ∈ C 2,γ (Bρ (0, 1)) (0 < γ < 1, t ∈ [0, T ] as a parameter), w(ρ, t), c(ρ, t), ∂w(ρ, t)/∂ρ, ∂c(ρ, t)/∂ρ and ∂ nˆ ∂n (t) ≡ (ξ(t), t) ∂ρ ∂ρ are in C([0, T ]), and w(ρ, t) and c(ρ, t) are increasing in variable ρ. Furthermore n(t) ˆ ≡ n(ξ(t), t),
0 < w(ρ, t) w0 (t),
(4.32)
0 < c(ρ, t) 1,
(4.33)
0 < n(ρ, t) < 1, ∂ w(ρ, t) M0 e2Mt , ∞ ∂ρ L (Bρ (0,1)) ∂ c(ρ, t) M0 e2Mt , ∞ ∂ρ L (Bρ (0,1)) ∂ nˆ ∂ nˆ (t) 2M(L + 1)eht , (t) Leht , ∂t ∂ρ
(4.34) (4.35) (4.36) (4.37)
where ξ = ξ(t; ρ0 ) (0 ρ0 1) is the forward characteristic curve of equation (3.14) initiating at the point (ρ0 , 0), and the constants M0 , M, L, h are as before. Proof. Given a n(ξ(t), ˜ t) ∈ ST , let (w, c) be the solution of (3.10)–(3.13) with n replaced by n, ˜ and let n be the solution of (3.14) and (3.15) with this (w, c). Defining the mapping, H : n˜ → n
(n˜ ∈ ST ),
by theorem 4.2 we see that H maps ST into itself. To prove that H is a contraction, we take n˜ 1 and n˜ 2 in ST , and we set n1 = Hn˜ 1 ,
n2 = Hn˜ 2 .
Proceeding as in the proof of theorem 8.1 in [19], we can prove that n1 − n2 ST C(T ) n˜ 1 − n˜ 2 ST ,
(4.38)
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where C(T ) → 0 as T → 0. We conclude from (4.38) that H is a contraction provided T is sufficiently small. Hence, there exists a unique local solution of (3.10)–(3.15) for any given v ∈ ET and R(t) defined by (3.16). Finally, the estimates (4.32)–(4.37) follow from theorems 4.1 and 4.2. We finally establish the existence and uniqueness of the free boundary problem (3.10)–(3.17) Theorem 4.4 (Global existence). Assume that w0 (t) > 0 is continuous and bounded functions in [0, ∞). Then, the free boundary problem (3.10)–(3.17) admits a unique solution (w(ρ, t), c(ρ, t), n(ρ, t), v(ρ, t), R(t)) for 0 ρ 1, 0 t < ∞ such that w(ρ, t), c(ρ, t) ∈ C 2,γ (Bρ (0, 1)) (0 < γ < 1, t ∈ [0, ∞) as a parameter), w(ρ, t), c(ρ, t), n(ρ, t), ∂w(ρ, t)/∂ρ, ∂c(ρ, t)/∂ρ, ∂n(ρ, t)/∂ρ are continuous in variable t ∈ C[0, ∞), and w(ρ, t) and c(ρ, t) are increasing in variable ρ. Furthermore, for 0 t < ∞, the estimates (4.32)–(4.37) hold and R(0)e−Mt R(t) R(0)eMt .
(4.39)
Proof. We first establish the local existence and uniqueness of the solution, then we continue the solution to all t > 0. Consider the mapping F :v→u
(v ∈ ET ),
where u is defined by (4.1). We shall prove that F maps ET into itself provided T is sufficiently small. By (4.1) and theorem 4.3 we easily check that u(ρ, t) ∈ C 1,0 ([0, 1] × [0, T ]), |u(ρ, t)
u(0, t) = 0,
1 b(c, w)L∞ M, 3
∂u(ρ, t) 4 ∂ρ 3 b(c, w)L∞ 3M,
u(ρ, 0) = v0 (ρ),
(4.40) (4.41) (4.42)
provided T is sufficiently small. We conclude from (4.40)–(4.42) that u ∈ ET , so that F maps ET into itself provided T is sufficiently small. Using theorem 4.3 and proceeding as in the proof of theorem 9.1 in [19] (or theorem 8.1 in [10]), we can also prove that F is a contraction provided T is sufficiently small. We therefore conclude that there exists a unique solution to (3.10)–(3.17) for 0 t T . Furthermore, the estimate (4.39) follows from (3.16) and v ∈ ET , and the estimates (4.32)–(4.37) follow from theorem 4.3. The solution established above for small times can be extended step-by-step to all t > 0 provided we can prove that if the solution exists for 0 t < T , T > 0 arbitrary, then a priori estimates (4.32)–(4.37) and ∂v(ρ, t) |v(ρ, t) M, (4.43) ∂ρ 3M hold. Clearly, by the proof of theorem 4.1 we find that the estimates (4.32)–(4.34) hold. Then, by (4.32)–(4.34) and (3.16), we easily get (4.43). Using (4.32)–(4.34), (4.43) and proceeding as in the proof of theorems 4.1 and 4.2, we can further prove the estimates (4.35)–(4.37).
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5. Large time behaviour In this section we will study the asymptotic behaviour of the free boundary and the global solution. This problem is interesting, and it is generally not trivial. From (3.16) and (3.17) we get 1 1 dR(t) = R(t) b(c, w)ns 2 ds = R(t) [kp (c) − (1 − δ)kd (c) − (1 − δ)Kf (w)] dt 0 0 ×ns 2 ds, (5.1) where c = c(s, t), w = w(s, t), n = n(s, t) and c B c , kd (c) = 1−σ , kp (c) = cp + c A cd + c
f (w) =
w . 1+w
(5.2)
In what follows we will take the following typical parameter values B/A = 1,
0 < σ 1,
0 0,
w(ρ, t) > 0,
c(ρ, t) > 0.
(5.4)
To study the large time behaviour of the free boundary R(t), we first establish the following lemma. Lemma 5.1. Assume that w0 (t) 1, σ 1 1 + sup w0 (t) , + K> (1 − δ)cp cd t∈[0, ∞) α
K(1 + cp ) . β
(5.5) (5.6)
(5.7)
Then dR(t) < 0 for all t > 0. dt
(5.8)
Proof. From (3.10), (3.12) and (5.4) we easily get Kf (w) w(ρ, t) = . c(ρ, t) αk(c)
(5.9)
Using (4.32), (4.33) and (5.2) we get K(cp + c) w K(cp + 1) w w Kf (w) = · · =: λ , αk(c) αβ(1 + w) c αβ c c
(5.10)
where λ 1 by assumption (5.7). Combining (5.9) and (5.10) and noting λ 1, 0 < n < 1 and c/c = βR 2 (t)n/(cp + c), we get − (w − c) +
βR 2 (t)n (w − c) 0. cp + c
(5.11)
By (5.5) we have w(1, t) − c(1, t) = w0 (t) − 1 0.
(5.12)
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From (5.11), (5.12) and the maximum principle, we have w(ρ, t) − c(ρ, t) 0,
i.e. 0
0. dt This completes the proof of lemma 5.1.
In the following we will establish a positive lower bound on R(t). Lemma 5.2. There exists a r0 > 0 such that R(t) r0 .
(5.15)
Proof. Set m ≡ 1 − n and let
1
Vn ≡ 4πR (t) 3
Vm ≡ 4π R (t)
2
3
n(ρ, t)ρ dρ, 0
1
m(ρ, t)ρ 2 dρ, 0
where Vn and Vm are the volumes occupied by the live tumour cells and the dead tumour cells, respectively. By (3.14), (3.16) and (3.17) and direct calculations, 1 1 1 1 ˙ ∂m 2 ˙ Vm (t) = 3R 2 (t)R(t) mρ 2 dρ + R 3 (t) mρ 2 dρ ρ dρ = 3R 3 (t)v(1, t) 4π 0 0 ∂t 0 1 1 +R 3 (t) [b(c, w)n2 − a(c, w)n]ρ 2 dρ + R 3 (t) [ρ 2 v(ρ, t) − ρ 3 v(1, t)] 0
0
1 1 ∂n × dρ = 3R 3 (t)v(1, t) mρ 2 dρ + R 3 (t) [b(c, w)n2 − a(c, w)n]ρ 2 dρ ∂ρ 0 0 1 1 3 2 2 3 −R (t) [b(c, w)nρ − 3ρ v(1, t)]n dρ = 3R (t)v(1, t) mρ 2 dρ 0
1
+3R 3 (t)v(1, t) = R (t) 3
0 1
nρ 2 dρ − R 3 (t)
0 1
a(c, w)nρ 2 dρ 0
[b(c, w) − a(c, w)]nρ 2 dρ,
(5.16)
0
where c = c(ρ, t), w = w(ρ, t) and n = n(ρ, t). By (4.32) and (4.33) and (5.3) we easily find that σ b(c, w) − a(c, w) = δ[kd (c) + Kf (w)] δkd (c) δ 1 − =: δ0 > 0. (5.17) 1 + cd
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From (3.14), |a(c, w)| M, |b(c, w)| M (where M is defined in the beginning of section 4) and 0 < n < 1, we get n(ξ(t), t) n0 (ρ0 )e−2Mt n˜ 0 e−2Mt ,
(5.18)
where n˜ 0 =: inf 0ρ1 n0 (ρ) > 0. Combining (5.16), (5.17) and (5.18) we get 4π 3 V˙m (t) δ0 n˜ 0 e−2Mt · R (t) δ0 n˜ 0 e−2Mt Vm (t), 3 and therefore Vm (t) Vm (0)e(δ0 n˜ 0 /2M)(1−e
−2Mt
)
Vm (0)eδ0 n˜ 0 /4M =:
4π r03 > 0, 3
if t is large enough. So 4π r03 4πR 3 (t) ≡ Vn (t) + Vm (t) Vm (t) . 3 3 That is, R(t) r0 > 0. This completes the proof of lemma 5.2.
By lemmas 5.1 and 5.2 we immediately get the following theorem. Theorem 5.3. Under the assumptions (5.5)–(5.7), there exists a Rs : 0 < Rs < R(0) such that lim R(t) = Rs .
t→+∞
(5.19)
Further, we can study the large time behaviour of the global solution of the system (3.10)–(3.17) established in theorem 4.4 We will prove the following theorem. Theorem 5.4. In addition to the assumptions (5.5)–(5.7), we also assume that lim w0 (t) = w0 > 0.
t→+∞
(5.20)
Then, the unique global solution (w(ρ, t), c(ρ, t), n(ρ, t), v(ρ, t), R(t)) of the system (3.10)–(3.17) established in theorem 4.4 will tend to the trivial steady-state solution (ws (ρ), cs (ρ), ns (ρ), vs (ρ), Rs ) = (w0 , 1, 0, 0, Rs ). Proof. By b(c, w) < 0 (see (5.14)), a(c, w) b(c, w), 0 < n < 1 and (3.14) we find that along the characteristic curve Dn b(c, w)n(1 − n) 0. Dt Combining (5.21) and 0 < n < 1 we can conclude that lim n(ξ(t), t) = ns 0 exists.
t→∞
(5.21)
(5.22)
We claim that ns = 0. Indeed, from (5.14) we have b(c, w) −b0 w,
(5.23)
where b0 > 0 is some positive constant. By lemma 5.1 and theorem 5.3 we find that 0 < R(t) < R(0).
(5.24)
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Let w(ρ, t) be the solution to the following problem: − w(ρ, t) +
K 2 R w(ρ, t) = 0 in Bρ (0, 1), α 0
w(1, t) = w0 (t),
(5.25) (5.26)
where R0 = R(0). We easily find that √ w0 (t) sinh K/αR0 ρ . w(ρ, t) = √ ρ sinh K/αR0
(5.27)
Note that sinh ρ eρ − e−ρ ≡ ρ 2ρ is increasing in ρ > 0 and limρ→0+ g(ρ) = 1. So that, by (5.27) we get √ K/αR0 inf w0 (t) =: w˜ 0 > 0 . w(ρ, t) √ sinh K/αR0 t∈[0,∞) g(ρ) =:
By the comparison principle, we find that w(ρ, t) is a sub-solution of the problem (3.10) and (3.11). That is, w(ρ, t) w(ρ, t) w˜ 0 > 0.
(5.28)
We also note that by (5.21) n(ξ(t), t) n(ξ(0), 0) = n0 (ρ0 ) sup n0 (ρ) < 1, 0ρ1
and therefore ns sup n0 (ρ) < 1.
(5.29)
0ρ1
Combining (5.21)–(5.23), (5.28) and (5.29) we easily find that ns = 0.
(5.30)
Further, by limt→∞ n(ξ(t), t) = 0, (5.19), (3.10)–(3.13) and (3.17) we find that lim (w(ρ, t), c(ρ, t), v(ρ, t)) = (w0 , 1, 0).
t→∞
This completes the proof of theorem 5.4.
Remark 5.1. Assumption (5.5) means that the concentration of drug surrounding the tumour should be larger than the nutrient concentration for controlling a tumour; assumption (5.6) means that the rate of cell death induced by drug cannot be too small in order to control a tumour; assumption (5.7) means that the less the drug is consumed during the cell kill process, the better the efficacy of treatment that may be expected. In the following we investigate the existence and uniqueness of the steady-state solution. Definition. We denote (ws (ρ), cs (ρ), ns (ρ), vs (ρ), Rs ) by a steady-state solution of the free boundary problem (3.10)–(3.17), if there exists a finite number Rs > 0 such that (ws (ρ), cs (ρ), ns (ρ), vs (ρ), Rs ) satisfies the following system in {0 < ρ < 1}: K ws = Rs2 f (ws )ns , (5.31) α ws (1) = w0 ,
(5.32)
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cs = Rs2 k(cs )ns ,
(5.33)
cs (1) = 1,
(5.34)
vs
dns = a(cs , ws )ns − b(cs , ws )n2s , dρ
(5.35)
1 d 2 (ρ vs ) = b(cs , ws )ns , ρ 2 dρ
(5.36)
vs (0) = 0,
(5.37)
vs (1) = 0.
Clearly, (ws (ρ), cs (ρ), ns (ρ), vs (ρ), Rs ) = (w0 , 1, 0, 0, Rs ) is a trivial steady-state solution. Furthermore, we have the following theorem. Theorem 5.5. The trivial steady-state solution (ws (ρ), cs (ρ), ns (ρ), vs (ρ), Rs ) = (w0 , 1, 0, 0, Rs ) is the unique steady-state solution of the system (3.10)–(3.17). Furthermore, under the assumptions of theorem 5.4, this steady-state solution is globally stable. Proof. We only need to prove the uniqueness. Suppose (ws (ρ), cs (ρ), ns (ρ), vs (ρ), Rs ) is a steady-state solution of the system (3.10)–(3.17). We infer from (5.35) and (5.36) that 1 d 2 1 d 2 (ρ vs ) = δkp (cs )ns + (1 − δ) 2 (ρ vs ns ). 2 ρ dρ ρ dρ Integrating (5.38) and using (5.37) we get 1 δ ρ 2 kp (cs )ns dρ = 0.
(5.38)
(5.39)
0
This, together with δ > 0 and kp (cs )ns 0, yields kp (cs )ns = 0.
(5.40)
Inserting (5.40) into (5.33) and noting k(cs ) = βkp (cs ) we have cs = 0, which, together with (5.34) and the maximum principle, yields cs (ρ) ≡ 1.
(5.41)
Combining (5.40) and (5.41), we conclude that ns ≡ 0.
(5.42)
From this, (5.31) and (5.32) and (5.36) and (5.37), we immediately get ws (ρ) ≡ w0 ,
vs (ρ) ≡ 0.
(5.43)
Finally, the global stability of the unique steady-state solution follows from theorems 5.3 and 5.4. This completes the proof of theorem 5.5. Remark 5.2. In theorem 5.5, the uniqueness of the steady-state solution of the system (3.10)–(3.17) means that (ws (ρ), cs (ρ), ns (ρ), vs (ρ)) = (w0 , 1, 0, 0) is unique. However, the radius Rs (0 < Rs < R0 ) of a dead core, which is dependent on the initial conditions, may be arbitrary.
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6. The growth of an untreated tumour In this section we study the evolution of an untreated tumour. In the limit of vanishing drug concentration (w(1, t) = w0 → 0) or vanishing drug effects (K → 0), the tumour growth tends to that of an untreated tumour. In the following we will take K = 0, and therefore c c − 1−σ ≡ a(c), a(c, w) = cp + c cd + c c c b(c, w) = − (1 − δ) 1 − σ ≡ b(c). cp + c cd + c Theorem 6.1. Assume that K = 0 and σ 1 < 0. − (1 − δ) 1 − 1 + cp 1 + cd
(6.1)
Then, there exists a Rs : 0 < Rs < R(0) such that lim R(t) = Rs .
t→+∞
(6.2)
Proof. Note that b(c) is increasing in c. By (5.1), (6.1) and 0 < c 1 (see theorem 4.4) we get 1 1 dR(t) b(c)ns 2 ds R(t) b(1)ns 2 ds < 0. (6.3) = R(t) dt 0 0 From lemma 5.2 and (6.3) we immediately get (6.2).
Remark 6.1. Clearly, if σ and δ are small then condition (6.1) holds. This condition means that the maximum rate of the cell death is larger than the maximum rate of the cell proliferation. Therefore, we may expect that the live cells will eventually die out. Indeed, this can be proved proceeding as in the proof of theorem 5.4. Next, we will find parameter conditions under which the untreated tumour continually grows to an infinite size. Theorem 6.2. Assume that K = 0, δ = 1 and 1 σ + − 1 > 0. 1 + cp 1 + cd
(6.4)
Then R(t) is increasing and lim R(t) = ∞.
t→+∞
Proof. Note that if δ = 1 then we have b(c) = c/(cp + 1) > 0. By (5.1) we have 1 dR(t) = R(t) b(c)ns 2 ds 0, dt 0
(6.5)
(6.6)
so that R(t) is increasing in t. As R(t) increases, the nutrient concentration c(ρ, R(t)) decreases and, therefore, so does b(c). Thus, it is not clear whether R(t) tends to ∞ or whether it remains bounded. We shall prove that the first alternative occurs under the assumptions K = 0, δ = 1 and (6.4). By (6.6), R(t) is increasing. If (6.5) does not hold then lim R(t) = R∞ < ∞.
t→+∞
(6.7)
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We shall prove that (6.7) leads to a contradiction. Let c(ρ) be the solution to the following problem: 2 c(ρ) = 0 inBρ (0, 1), − c(ρ) + βR∞
(6.8)
c(1) = 1.
(6.9)
By the comparison principle, we find that c(ρ) is a sub-solution of the problem (3.12) and (3.13). So, √ sinh βR∞ ρ c(ρ, t) c(ρ) = . (6.10) √ ρ sinh βR∞ Note that c(1) = 1 and c(ρ) is independent of t. Therefore, by the continuity of c(ρ) we find that for any small ε > 0, there exists a 0 < ρ(ε) ˜ < 1 such that ˜ ρ 1. 1 − ε < c(ρ) 1 for ρ(ε)
(6.11)
Setting m = 1 − n, by (3.14) we easily get along the characteristic curve b−a Dm = (b − a) + (a − 2b)m + bm2 = b(m − 1) m − . Dt b
(6.12)
By (6.10), (6.11) and the expressions of a(c) and b(c) we find that for ρ(ε) ˜ ρ1 b(c) − a(c) σ m(ε), (6.13) (1 + cp ) 1 − 1 + cd b(c) where m(ε) ≡ (1 − ε + cp )[1/(1 − ε) − σ/(1 − ε + cd )] < 1 (if ε is small enough ) by assumption (6.4). We also note that 0 < m < 1 (see theorem 4.4) and that the equilibrium point (b − a)/b to the equation (6.12) is asymptotically stable. Therefore, if t is large enough then 1 + m(ε) < 1 for ρ(ε) ˜ ρ 1. 0 < m(ρ, t) 2 That is, 1 − m(ε) > 0 for ρ(ε) ˜ ρ 1. (6.14) n(ρ, t) ≡ 1 − m(ρ, t) 2 Noting that b(c) > 0 is increasing in c and combining (6.6), (6.10) and (6.14) we have for t T (where T > 0 large enough), 1 1 1 2 2 ˙ b(c)ns ds R(t) b(c)ns ds b(c(0))R(t) ns 2 ds b(c(0))R(t) R(t) = R(t) 0
0
0
3 ˜ ] b(c(0))(1 − m(ε))[1 − (ρ(ε)) × R(t) ≡ κ(ε)R(t), ns 2 ds 6 ρ(ε) ˜ 1
(6.15)
where κ(ε) > 0. (6.15) leads to R(t) R(T )eκ(ε)(t−T ) → +∞, which contradicts (6.7).
Remark 6.2. For the typical parameter values σ = 0.9, cp = 0.1, cd = 0.05 given in [21], condition (6.4) holds. From theorem 6.2 and the continuity of the solution on the parameters K and δ, we conclude that if K and 1 − δ are small and (6.4) holds then the tumour continually grows to an infinite size.
Free boundary problem modelling cancer therapy
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7. Conclusion In this paper we analyse an existing free boundary problem that describes the growth of an avascular tumour undergoing drug treatment. The free boundary problem consists of an ODE and an elliptic–hyperbolic system. Using the Leray-Schauder fixed theorem, the Schauder estimate, the Lp -theory, the Sobolev imbedding theorem, the maximum principle, the characteristic curve method and the contraction mapping principle, we prove that the free boundary problem has a unique global solution. Our analytical results show that the nutrient concentration c and the drug concentration w increase monotonically from r = 0 to r = R(t) (see theorem 4.1). This agrees qualitatively with experimental results (see [21], figure 4) and the references cited in [3]). Furthermore, if we assume that (i) the administrated drug concentration w0 (t) is larger than the external nutrient concentration c0 = 1 (see (5.5)); (ii) the rate of cell death induced by drug is relatively ‘large’ (see (5.6)); (iii) the drug is ‘effective’ (see (5.7), in which α can be viewed as a measure of the drug’s effectiveness) then we prove that all live tumour cells will eventually be killed and the tumour shrinks to a necrotic core. That is, the global solution converges to a trivial steady solution (see theorems 5.4 and 5.5). We also discuss the growth of the untreated tumours. If the rate of the natural death of the live cells is larger than the proliferation rate of the live cells (see (6.1)) then the untreated tumour also shrinks to a necrotic core (see theorem 6.1). If the rate of cell mitosis is larger than the rate of cell death then the untreated tumour continually grows to an infinite size (see theorem 6.2). Finally, we remark that the global existence of the solutions for the three-dimensional model is still open. Acknowledgments The authors are supported by NSFC 10571023 and by Shanghai Educational Development Foundation. The authors are grateful to the referees for their helpful comments. References [1] Bazaliy B and Friedman A 2003 Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumour growth Indiana Univ. Math. J. 52 1265–304 [2] Bueno H, Ercole G and Zumpano A 2005 Asymptotic behaviour of quasi-stationary solutions of a nonlinear problem modelling the growth of tumours Nonlinearity 18 1629–42 [3] Byrne H M and Chaplain M A J 1996 Modelling the role of cell-cell adhension in the growth and development of carcinomas Math. Comput. Modelling 12 1–17 [4] Byrne H M and Chaplain M A J 1995 Growth of nonnecrotic tumours in the presence and absence of inhibitors Math. Biosci. 130 151–81 [5] Byrne H M and Chaplain M A J 1996 Growth of necrotic tumors in the presence and absence of inhibitors Math. Biosci. 135 187–216 [6] Chen X and Friedman A 2003 A free boundary problem for elliptic-hyperbolic system: an application to tumor growth SIAM J. Math. Anal. 35 974–86 [7] Cui S and Friedman A 2003 A hyperboic free boundary problem modeling tumor growth Interfaces Free Boundaries 5 159–81 [8] Friedman A 1964 Partial Differential Equations of Parabolic Type (Englewood Cliffs, NJ: Prentice-Hall) [9] Friedman A and Reitich F 1999 Analysis of a mathematical model for the growth of tumors J. Math. Biol. 38 262–84
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