A Mathematical Model of Prostate Tumor Growth Under Hormone ...

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Jan 5, 2010 - Abstract This paper extends Jackson's model describing the growth of a prostate tumor with hormone therapy to a new one with hypothetical ...
J Nonlinear Sci (2010) 20: 219–240 DOI 10.1007/s00332-009-9056-z

A Mathematical Model of Prostate Tumor Growth Under Hormone Therapy with Mutation Inhibitor Youshan Tao · Qian Guo · Kazuyuki Aihara

Received: 16 July 2008 / Accepted: 29 July 2009 / Published online: 5 January 2010 © Springer Science+Business Media, LLC 2009

Abstract This paper extends Jackson’s model describing the growth of a prostate tumor with hormone therapy to a new one with hypothetical mutation inhibitors. The new model not only considers the mutation by which androgen-dependent (AD) tumor cells mutate into androgen-independent (AI) ones but also introduces inhibition which is assumed to change the mutation rate. The tumor consists of two types of cells (AD and AI) whose proliferation and apoptosis rates are functions of androgen concentration. The mathematical model represents a free-boundary problem for a nonlinear system of parabolic equations, which describe the evolution of the populations of the above two types of tumor cells. The tumor surface is a free boundary, whose velocity is equal to the cell’s velocity there. Global existence and uniqueness of solutions of this model is proved. Furthermore, explicit formulae of tumor volume at any time t are found in androgen-deprived environment under the assumption of radial symmetry, and therefore the dynamics of tumor growth under androgen-deprived

Communicated by P.K. Maini. Y. Tao () Department of Applied Mathematics, Dong Hua University, Shanghai 200051, P.R. China e-mail: [email protected] Q. Guo Department of Mathematics, Division of Computation Science, Shanghai Normal University, Shanghai 200234, P.R. China e-mail: [email protected] K. Aihara Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan e-mail: [email protected]

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therapy could be predicted by these formulae. Qualitative analysis and numerical simulation show that controlling the mutation may improve the effect of hormone therapy or delay a tumor relapse. Keywords Prostate cancer · Partial differential equations · Free-boundary problem · Hormone therapy · Inhibitors Mathematics Subject Classification (2000) 35Q80 · 35R35 · 92C60 1 Introduction More than 670,000 men are diagnosed with prostate cancer every year in the world, accounting for one in nine of all new cancers in males. It is the secondly most common cancer in men after lung cancer (Ferlay et al. 2002). The highest incidence rates are in the United States and Sweden, and the lowest rates are in China and India (Ferlay et al. 2002). Substantial increase in incidence has been reported in recent years for many countries around the world (Hsing et al. 2000). Located under the bladder, the chestnut-shaped prostate gland produces seminal fluid. A prostate tumor is caused by an abnormal and uncontrolled growth of cells. It can either be malignant or benign. Although the cause of prostate cancer is unknown, it is thought that many factors such as age, family, history, race, and diet may influence the development of prostate cancer1 (Simon et al. 2002). The molecular mechanisms of prostate cancer are under intensive investigation. On one hand, loss of expression of tumor suppressor genes such as p27 gene is strongly associated with the development of prostate cancer (Macri and Loda 1998). On the other hand, there have been various specific growth factors implicated in the growth of prostate cancer (Chung 1995; Holland and Frei 2001; Ware 1993). The primary treatments of prostate cancer include surgery, radiation therapy, hormone therapy, chemotherapy, and cryoablation, which are used alone or in combination (see footnote 1). Tumor cells of prostate cancer are crucially hormone-sensitive, which means that they depend on the male hormone (androgen) for tumor growth. In this paper, we focus on hormone therapy of prostate cancer, or androgen deprivation therapy (ADT), which can be achieved easily by medical castration (see Jackson 2004a, 2004b and references therein). Total androgen blockage (TAB) which further combines anti-androgens with castration is also widely used. After transiently positive response to ADT or TAB, a relapse often happens due to emergence of an androgen-independent (AI) cancer cells (Jackson 2004a, 2004b). The studies in Ellis et al. (1996), Jackson (2004a, 2004b), Liu et al. (1996) imply that polyclonality and decreased apoptosis of AI cells are possible mechanisms for a tumor relapse following the hormone therapy. Jackson (2004a, 2004b) developed an interesting mathematical model of prostate cancer which is closely related to experimental studies, and investigated possible mechanisms of an AI tumor relapse. The model is a free-boundary problem for a system of partial differential equations which describe the evolution of the densities 1 http://www.besttreatments.co.uk. Prostate cancer, Best treatments.

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of androgen-dependent (AD) cells and androgen-independent (AI) cells within a tumor. Jackson’s model well agrees with the experimental observation and suggests that an androgen-independent relapse is associated with a decrease in apoptosis without an increase in proliferation (Jackson 2004a, 2004b). Jackson (2004a, 2004b) proposed the first PDE model for the AI response. Ideta et al. (2008) proposed an ODE model for the AI response and IAS (intermittent androgen suppression) by considering mutation, whereas Shimada and Aihara (2008) proposed an ODE model for the AI response and IAS by considering the nonlinear competition between the two types of cells (AD and AI). We are proposing a PDE model with mutation and mutation inhibitors for the AI response. In this study we extend Jackson’s model to a new one that includes the mutation by which AD cells mutate into AI cells (Ideta et al. 2008), since a tumor relapse may be associated with gene mutation (Chung 1995; Holland and Frei 2001; Ware 1993). The newly proposed model also introduces possible inhibition of the mutation, since signal transduction inhibitors, for instance, may inhibit the mutation and reduce the fraction of AI cells (Kamradt and Pienta 1999). Cancers are complex biological systems that appear with multiscale features: genes, cells, and biological tissues, corresponding to the molecular, cellular, and tissue scales, as documented in Bellomo and Delitala (2008), Bellomo et al. (2008). Hence, cancer models, according to the classification of scales, are classified into the following four kinds of models: microscopic models (at the molecular and the cellular scales; see Bellomo and Delitala 2008; Bellomo et al. 2008; De Angelis and Jabin 2003 for instance), macroscopic models (at the tissue scale; see Ambrosi and Preziosi 2002; Bellomo and Delitala 2008; Bellomo et al. 2008; Bresch et al. 2008; Friedman 2007; Guo et al. 2008; Greenspan 1976; Jackson 2002; Jackson 2004a, 2004b; Jackson and Byrne 2000; Tao 2009; Tao and Chen 2006; Tao and Guo 2007; Tao et al. 2004, 2009 for instance), super-macroscopic models (at the tissue scale with a spatial homogeneity assumption; see Ideta et al. 2008; Shimada and Aihara 2008 for instance), multiscale models (the overall system is viewed as a system of subsystems with specific scales; see Bellomo and Delitala 2008; Bellomo et al. 2008; Byrne et al. 2006; Ribba et al. 2006 for instance). Indeed, multiscale models represent good approximation of biological reality; however, the study of these models is nowadays mainly focused on numerical aspects, and they are often analytically intractable. Since this paper is focused on studying the dependence of the tumor dynamics on the model parameters, in particular, on the mutation parameter, we shall use the macroscopic scale to represent tumor growth. Here we should note that using a macroscopic scale also for the mutation by which AD tumor cells mutate into AI ones is a crude approximation of biological reality as it neglects heterogeneity of biological functions (Bellomo and Delitala 2008). Macroscopic models consider the evolution of cellular density (the number of cells per unit volume) or of the boundary of a tumor (when strong geometrical assumptions are supposed) (Bellomo and Delitala 2008; Bellomo et al. 2008; Bresch et al. 2008; Friedman 2007). They cannot always account correctly for the microscopic aspects of tumor growth (Bellomo and Delitala 2008; Bellomo et al. 2008; Bresch et al. 2008), as they deal with averages over a large number of cells. Mathematical modeling of tumor growth at the macroscopic scale is usually based on partial differential equations, such as reaction–diffusion equations (Byrne and Chaplain 1996;

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Friedman 2007; Greenspan 1976), advection equations (Ambrosi and Preziosi 2002) or reaction–diffusion–advection equations (Jackson 2002, 2004a, 2004b; Jackson and Byrne 2000; Tao 2009; Tao and Chen 2006; Tao and Guo 2007; Tao et al. 2004, 2009; Ward and King 2003). In this paper, our newly proposed macroscopic model is based on reaction–diffusion–advection equations. In this model, we treat the tumor mass as an incompressible fluid, as done in Friedman (2007), Greenspan (1976), Jackson (2004a), Ward and King (2003). However, we should note that the general trend in this area is to model tumor tissues as viscoelastic materials (see Bresch et al. 2008 for instance), since this viscoelastic model accounts, the same time, for microscopic (cell-cycle) and macroscopic (cellular-adhesion, mechanical effects) aspects of the tumor growth (Bresch et al. 2008). On the other hand, we also should note that it is very challenging to develop a rigorous mathematical theory for such viscoelastic tumor models. Our newly proposed model is a simple PDE model for prostate tumor growth under continuous androgen suppression (CAS) therapy. As afore-mentioned, the relapse of tumor is a crucial problem in hormonal therapy of prostate cancer. There are two possible mechanisms of the tumor recurrence (Isaacs and Coffey 1981). One is mutation or adaptation: AD cells mutate into AI cells under the androgen-deprived condition. The other is selection or competition: AI cells are minor but exist from the beginning of the therapy and will dominate under the androgen suppression condition. The above two mechanisms are not exclusive, and they have different effects on the tumor relapse. Hence, different mathematical models are needed (Ideta et al. 2008; Shimada and Aihara 2008). Recent experimental and clinical studies (see Bruchovsky et al. 2006, 2007 for instance) suggested that IAS therapy may prolong or prevent the relapse when compared with CAS therapy. As summarized above, Jackson (2004a, 2004b), as far as we know, developed the first PDE model of CAS therapy for prostate cancer. Ideta et al. (2008) initially proposed an ODE model of IAS therapy for prostate cancer with mutation. Shimada and Aihara (2008) established an ODE model of IAS therapy for prostate cancer with competition. Guo et al. (2008) extended the ODE model (Ideta et al. 2008) to a PDE model, whereas Tao et al. (2009) extended the ODE model (Shimada and Aihara 2008) to a PDE model. The essential difference between the PDE model (Guo et al. 2008) and the ODE model (Ideta et al. 2008) is that the equation for change of AD (or AI) cell density is nonlinear due to spatial motion of cells in the PDE model, while the corresponding one in the ODE model (Ideta et al. 2008) is linear (see Guo et al. 2008 for a more detailed explanation). Furthermore, Guo et al. (2008) numerically studied the following important issue for IAS therapy: how to optimally plan the IAS therapy. For the above reasons, there are also essential differences between two papers (Shimada and Aihara 2008; Tao et al. 2009). In a word, there are the following three essential differences between existing mathematical models for prostate cancer: the difference between CAS models and IAS models, the difference between mutation models and competition models, and the difference between PDE models and ODE models. One goal of this paper is to perform a rigorous nonlinear analysis of the new model. First, we will give the results on the existence and uniqueness of global solutions of the model; then, we will find explicit formulae of tumor volume at any time t in androgen-deprived environment, which, as far as we know, is quite novel

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for partial differential equation models of tumor growth. The other goal is to investigate the dependence of the tumor growth on the model parameters, in particular, on the parameter of hypothetical mutation inhibitors. The present paper considers a PDE model of CAS therapy for prostate cancer with mutation and hypothetical mutation inhibitors. The mathematical model, tumor dynamics, and numerical simulations in this paper differ from those Tao et al. (2009) who dealt with a PDE model with competition. On the other hand, the proof of the global existence of solutions to the model in this paper is technically similar to that given in Tao et al. (2009), so we will omit it. We also should mention that Guo et al. (2008) only studied mathematical modeling and numerical simulation for a PDE model of IAS therapy for prostate cancer with mutation. As afore-mentioned, the present paper finds novel explicit formulae of tumor growth in an androgen-deprived environment, which is a great difference between this paper and the others (Guo et al. 2008; Tao et al. 2009). This paper is organized into six sections. Section 2 presents the model. Section 3 introduces changes of variables to transform the problem in a moving domain into a new one in a fixed domain and describes the results on the global existence and uniqueness of a solution to the model. Section 4 finds explicit formulae of tumor volume at any time t in androgen-deprived environment. Section 5 numerically studies the model. Finally, Sect. 6 provides a discussion related to the model and to our qualitative analysis. 2 Model Following Jackson (2004a, 2004b), the prostate tumor is viewed as a densely packed and radially symmetric sphere of radius R(t) which contains both AD and AI cell types whose volume fractions are p(r, t) and q(r, t), respectively. Since the level of androgen within the tissue under consideration can be controlled by medical means (see Bruchovsky et al. 2006, 2007; Ideta et al. 2008, Jackson 2004a, 2004b), we use a super-macroscopic scale for androgen (that is, we neglect spatial heterogeneity of androgen within the tissue). The tumor is modeled as an incompressible fluid with velocity u (u := u · r/|r|) generated by cell proliferation and death, which are dependent upon the androgen level a(t). For solid tumor growth, it is widely assumed that cell movement has two components: (1) motion due to the velocity u(r, t) (Jackson 2002; Jackson and Byrne 2000; Tao and Guo 2007; Ward and King 2003) and (2) random motion (Byrne and Chaplain 1996; Friedman 2007; Greenspan 1976). We exploit the spherical symmetry of the problem by assuming henceforth that the variables p, q, and u depend only on (r, t), where r is the radial distance from the center of the tumor, and t is time. As an extension of the former model Jackson (2004a, 2004b), the new model consists of the following equations for the tumor cell populations:  ∂p 1 ∂  2 (r, t) + 2 r u(r, t)p(r, t) ∂t r ∂r       Dp ∂ ∂p r 2 (r, t) + αp a(t) p(r, t) − δp a(t) p(r, t) = 2 ∂r r ∂r   − (1 − I )β a(t) p(r, t),

(2.1)

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 1 ∂  2 ∂q (r, t) + 2 r u(r, t)q(r, t) ∂t r ∂r     Dq ∂ ∂q = 2 r 2 (r, t) + (1 − I )β a(t) p(r, t) ∂r r ∂r     + αq a(t) q(r, t) − δq a(t) q(r, t).

(2.2)

In (2.1) and (2.2), Dp and Dq are the random motility coefficients of the AD and AI cells; αp (a(t)), αq (a(t)), δp (a(t)), and δq (a(t)) are their respective proliferation and apoptosis rates; β(a(t)) is the mutation rate by which AD cells mutate into AI ones; the intensity of the inhibitors which reduce the mutation rate is represented by parameter I , which ranges from zero to one. I = 0 corresponds to no inhibition of the mutation, and I = 1 corresponds to the prefect inhibition of the mutation. So, the rate at which cells mutate from the AD to AI types should be represented by 1 − I (not I ). The cell proliferation and apoptotic rates and the mutation rate are assumed to be dependent on the local androgen concentration a(t). We assume that the total number of cells is constant per unit volume within the tumor (see Bresch et al. 2008; Jackson 2004a, 2004b; Ward and King 2003 and references therein), i.e., p + q = k ≡ constant.

(2.3)

Equations (2.1) and (2.2) with assumption (2.3) yield      1 ∂ k ∂  2 2 ∂p r u(r, t) = (Dp − Dq ) 2 r (r, t) + αp a(t) p(r, t) 2 ∂r r ∂r r ∂r      + αq a(t) k − p(r, t) − δp a(t) p(r, t)    − δq a(t) k − p(r, t) .

(2.4)

By the radial symmetry assumption of the problem, ∂q ∂p (0, t) = (0, t) = u(0, t) = 0. ∂r ∂r

(2.5)

To close the system of equations, we need to impose boundary and initial conditions. Boundary Conditions We assume that there is no-flux of cancer cells across the outer boundary of the tumor, i.e.,   ⎧

dR(t) ∂p ⎪ ⎪ = 0, ⎪ ⎨ p(r, t) dt − p(r, t)u(r, t) − Dp ∂r (r, t) r=R(t)  

⎪ ∂q dR(t) ⎪ ⎪ − q(r, t)u(r, t) − D (r, t) q(r, t) = 0, ⎩ q dt ∂r r=R(t)

(2.6)

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which, together with (2.3), yields the following free boundary conditions: ⎧ dR(t)   ⎪ − u R(t), t = 0, ⎨ dt ⎪ ⎩ ∂p R(t), t  = 0. ∂r

(2.7)

Initial Conditions We prescribe the initial data as follows: R(0) = R0 ,

p(r, 0) = p0 (r)

for 0 ≤ r ≤ R0 .

(2.8)

Remark 2.1 The no-flux boundary condition (2.6) is written down by considering the relative velocity of cells on the outer boundary of the growing (or shrinking) tumor. This kind of no-flux boundary conditions for diffusion–advection equations in a moving domain {r ≤ R(t)}, as far as we know, was initially clarified by Tao (2009). We note here that since p + q = k throughout the tumor, it is not necessary to impose an additional initial condition for q in (2.8). We also note that (2.2) is a consequence of (2.1), (2.3), and (2.4), so that in what follows we may drop this equation. After introducing the nondimensional variables (Jackson 2004a) and using the parameter values given in Jackson (2004a), the model (2.1)–(2.8) can be rewritten as follows:  1 ∂  2 ∂p r u(r, t)p(r, t) (r, t) + 2 ∂t r ∂r       1 ∂ 2 ∂p = 2 r (r, t) + αp a(t) p(r, t) − δp a(t) p(r, t) ∂r r ∂r   − (1 − I )β a(t) p(r, t),  1 ∂  2 r u(r, t)p(r, t) 2 r ∂r     = αp a(t) p(r, t) + 1 − p(r, t) − δp a(t) p(r, t)    − δq a(t) 1 − p(r, t) ,   dR = u R(t), t , dt R(0) = 1, p(r, 0) = p0 (r),

(2.9)

(2.10) (2.11) (2.12)

∂p (0, t) = 0, u(0, t) = 0, ∂r  ∂p  R(t), t = 0, ∂r

(2.13) (2.14)

where 0 < ε1  1 is some constant, the functions a(t), αp (a(t)), δp (a(t)), δq (a(t)), and β(a(t)) take the following specific forms (Jackson 2004a, 2004b): a(t) = e−bt + as ,

t ≥ 0,

(2.15)

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  αp a(t) = θ1 + (1 − θ1 )

a(t) , a(t) + K

  δp a(t) = δ1 ω1 + (1 − ω1 )

  δq a(t) = δ2 ω2 + (1 − ω2 )

(2.16)

a(t) , a(t) + K

(2.17) (2.18)

    a(t) , β a(t) = β1 1 − 1 + as

(2.19)

where b, K, δ1 , δ2 , ω1 , and ω2 are some positive constants with the following assumed conditions: 0 ≤ as < 1,

0 ≤ θ1 < 1,

δ 1 < δ2 ,

ω1 > 1,

and ω2 < 1.

(2.20)

In (2.15) we assume that the hormonal treatment is initiated at the time t = 0. Equation (2.19) assumes that the mutation rate β(a(t)) is decreasing with increasing the local androgen concentration a(t) (Ideta et al. 2008). The parameter as > 0 corresponds to ADT, and as = 0 corresponds to TAB. The parameter θ1 represents the proliferation rate of AD cells in an androgen-deprived state, and 0 ≤ θ1 < 1 represents that the deprivation of the androgen decreases the proliferation rate of the AD cells. The assumption of δ1 < δ2 is due to the fact that the AD cells are dominant in an androgen-rich condition (Ellis et al. 1996; Jackson 2004a, 2004b). The assumption of ω1 > 1 and ω2 < 1 represents that the deprivation of the androgen increases the apoptosis rate of the AD cells but reduces that of AI cells (Jackson 2004a). In this paper we will prove the existence of global solutions of this model (2.9)– (2.14). Furthermore, we will find explicit formulae of tumor volume at any time t in androgen-deprived environment by a rigorous nonlinear analysis.

3 Global Solution To transform the moving domain {r < R(t)} into a fixed domain, as shown in Tao and Chen (2006), we introduce a transformation of variables (r, t, p, u, R) → (ρ, t, m, v, R) as follows: ρ = r/R(t), t = t, R(t) = R(t),     m(ρ, t) = p ρR(t), t , v(ρ, t) = u ρR(t), t /R(t).

(3.1)

In terms of the new variables, the system (2.9)–(2.14) takes the following form in {0 < ρ < 1, t > 0}:  

∂m ∂m ε1 1 ∂ 2 ∂m (ρ, t) + v(ρ, t) − ρv(1, t) (ρ, t) − 2 ρ (ρ, t) ∂t ∂ρ ∂ρ R (t) ρ 2 ∂ρ         = αp a(t) + δq a(t) − 1 − δp a(t) m(ρ, t) 1 − m(ρ, t)   − (1 − I )β a(t) m(ρ, t), (3.2)

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m(ρ, 0) = m0 (ρ),

(3.3)

∂m ∂m (0, t) = (1, t) = 0, ∂ρ ∂ρ  ρ     1 αp a(t) m(s, t) + 1 − m(s, t) − δp a(t) m(s, t) v(ρ, t) = 2 ρ 0    − δq a(t) 1 − m(s, t) s 2 ds, dR(t) = R(t)v(1, t), dt R(0) = 1,

(3.4)

(3.5) (3.6) (3.7)

where we have used the fact that v(0, t) = 0 in deriving (3.5). If we regard (3.2) as a one-dimensional parabolic equation with the spatial variable ρ, then the coefficient 2/ρ of ∂m/∂ρ in   ∂ 2 m 2 ∂m 1 ∂ 2 ∂m ρ m ≡ 2 ρ ≡ + ∂ρ ρ ∂ρ ρ ∂ρ ∂ρ 2 has a singularity at tumor center ρ = 0. Therefore, we cannot directly apply an Lp -estimate to (3.2). However, using the boundedness of the function v(ρ, t)/ρ, the above singularity can be eliminated in the three-dimensional Cartesian coordinate form. We shall use the following notation:   B1 (0) = y ∈ R3 : |y| < 1 , QT = B1 (0) × (0, T ),   Wk2,1 (QT ) = m(y, t) ∈ Lk (QT ) : myi , myi yj , mt ∈ Lk (QT ) , where 1 ≤ k ≤ ∞, i, j = 1, 2, 3. We shall assume that 0 ≤ m0 (ρ) ≤ 1,

  m0 (ρ) ∈ Wk2 B1 (0) ,

∂m0 ∂m0 (0) = (1) = 0. ∂ρ ∂ρ

(3.8)

0 Here ∂m ∂ρ (0) = 0 is a consequence of the radial symmetry assumption, and Wk2 (B1 (0)) := {ϕ(y)|ϕ, Dy ϕ, Dy2 ϕ ∈ Lk (B1 (0))}, where k > 5, and the derivatives are in the weak sense. The main result of this section is as follows:

Theorem 3.1 Under assumptions (3.8), there exists a unique solution (m(ρ, t), v(ρ, t), R(t)) of (3.2)–(3.7) for all t > 0; furthermore, R(t) ∈ C 1 [0, ∞), v(ρ, t) ∈ C 1 ([0, 1] × [0, ∞)), m(ρ, t) ∈ Wk2,1 (QT ) (k > 5) for any T > 0, and

for some β > 0.

0 ≤ m(ρ, t) ≤ 1,   v(ρ, t) ≤ β,

(3.10)

e−βt ≤ R(t) ≤ eβt ,

(3.11)

(3.9)

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The proof of Theorem 3.1 is based on a fixed-point argument and parabolic Lp -theory (see Ladyzenskaja et al. 1968 for instance) and it is technically similar to that of Theorem 4.1 in Tao et al. (2009), so we omit it here.

4 Formula of Tumor Dynamics The dynamics of tumor growth for a general case with a = a(t) = 0 given in (2.15) will be numerically studied in the next section. In this section we shall focus on the dynamical behavior of prostate tumor growth in androgen-deprived environment, i.e., in the following case: a = 0,

(4.1)

and therefore αp (a) = θ1 ,

δp (a) = δ1 ω1 ,

δq (a) = δ2 ω2 ,

and β(a) = β1 .

(4.2)

Under assumption (4.1), (3.2)–(3.7) can be rewritten as follows:  

∂m ∂m ε1 1 ∂ 2 ∂m (ρ, t) + v(ρ, t) − ρv(1, t) (ρ, t) − 2 ρ (ρ, t) ∂t ∂ρ ∂ρ R (t) ρ 2 ∂ρ   = (θ1 + δ2 ω2 − 1 − δ1 ω1 )m(ρ, t) 1 − m(ρ, t) − (1 − I )β1 m(ρ, t), (4.3) m(ρ, 0) = m0 (ρ),

(4.4)

∂m ∂m (0, t) = (1, t) = 0, ∂ρ ∂ρ  ρ 1 v(ρ, t) = 2 θ1 m(s, t) + 1 − m(s, t) − δ1 ω1 m(s, t) ρ 0   − δ2 ω2 1 − m(s, t) s 2 ds,

(4.5)

(4.6)

dR(t) = R(t)v(1, t), dt R(0) = 1.

(4.7) (4.8)

Set n(ρ, t) ≡ 1 − m(ρ, t) and let 

1

Vm (t) =: 4πR (t) 3

 2

m(ρ, t)ρ dρ, 0

1

Vn (t) =: 4πR (t) 3

n(ρ, t)ρ 2 dρ,

0

where Vm (t) and Vn (t) are the volumes occupied by AD cells and AI cells at time t, respectively. Then 4 (4.9) V (t) =: Vm (t) + Vn (t) ≡ πR 3 (t) 3 is the tumor volume at time t. In this section we shall derive formulae of the tumor volume (or the tumor radius) at time t. In fact, we prove the following:

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Theorem 4.1 (I) If Condition 1: θ1 − δ1 ω1 − (1 − I )β1 = 0

and 1 − δ2 ω2 = 0

holds, then we have V (t) =

4π + Vm (0)(1 − I )β1 t. 3

(4.10)

(II) If Condition 2: θ1 − δ1 ω1 − (1 − I )β1 = 0

and 1 − δ2 ω2 = 0

holds, then we have V (t) = Vm (0) + Vn (0)e(1−δ2 ω2 )t +

(1 − I )β1 Vm (0) (1−δ2 ω2 )t e − 1 . (4.11) 1 − δ2 ω2

(III) If Condition 3: θ1 − δ1 ω1 − (1 − I )β1 = 0

and 1 − δ2 ω2 = 0

holds, then we have V (t) = Vn (0) + Vm (0)e[θ1 −δ1 ω1 −(1−I )β1 ]t  (1 − I )β1 Vm (0)  [θ1 −δ1 ω1 −(1−I )β1 ]t + −1 . e θ1 − δ1 ω1 − (1 − I )β1

(4.12)

(IV) If Condition 4: θ1 − δ1 ω1 − (1 − I )β1 = 0,

1 − δ2 ω2 = 0,

and θ1 − δ1 ω1 − (1 − I )β1 − 1 + δ2 ω2 = 0 holds, then we have

V (t) = Vn (0) + (1 − I )β1 Vm (0)t e(1−δ2 ω2 )t + Vm (0)e[θ1 −δ1 ω1 −(1−I )β1 ]t .

(4.13)

(V) If Condition 5: θ1 − δ1 ω1 − (1 − I )β1 = 0,

1 − δ2 ω2 = 0,

and θ1 − δ1 ω1 − (1 − I )β1 − 1 + δ2 ω2 = 0 holds, then we have V (t) = Vm (0)e[θ1 −δ1 ω1 −(1−I )β1 ]t + Vn (0)e(1−δ2 ω2 )t (1 − I )β1 Vm (0)e(1−δ2 ω2 )t θ1 − δ1 ω1 − (1 − I )β1 − 1 + δ2 ω2  [θ −δ ω −(1−I )β −1+δ ω ]t  1 2 2 −1 . × e 1 1 1 +

(4.14)

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Proof By (4.3) and (4.5)–(4.7), integration by parts, and direct calculations,  1  1 ∂m 2 1 ˙ ˙ ρ dρ mρ 2 dρ + R 3 (t) Vm (t) = 3R 2 (t)R(t) 4π 0 0 ∂t  1  1 ∂m 2 ρ dρ = 3R 3 (t)v(1, t) mρ 2 dρ + R 3 (t) 0 0 ∂t  1 = 3R 3 (t)v(1, t) mρ 2 dρ 0



1

(θ1 + δ2 ω2 − 1 − δ1 ω1 )m(1 − m) − (1 − I )β1 m ρ 2 dρ

+ R 3 (t) 0



1

+ ε1 R(t) 0



  ∂ 2 ∂m ρ dρ ∂ρ ∂ρ

1

∂m dρ ρ 2 v(ρ, t) − ρ 3 v(1, t) ∂ρ 0  1 = 3R 3 (t)v(1, t) mρ 2 dρ − R 3 (t)

0



1

(θ1 + δ2 ω2 − 1 − δ1 ω1 )m(1 − m) − (1 − I )β1 m ρ 2 dρ

+ R 3 (t) 0

  ∂m 1 + ε1 R(t) ρ 2 ∂ρ ρ=0 

1 − R 3 (t) ρ 2 v(ρ, t) − ρ 3 v(1, t) m(ρ, t) ρ=0  1  1

∂ 2 ρ v(ρ, t) dρ − 3R 3 (t)v(1, t) + R 3 (t) m mρ 2 dρ ∂ρ 0 0  1

= R 3 (t) (θ1 + δ2 ω2 − 1 − δ1 ω1 )m(1 − m) − (1 − I )β1 m ρ 2 dρ 0



1

+ R 3 (t)



m θ1 m + 1 − m − δ1 ω1 m − δ2 ω2 (1 − m) ρ 2 dρ

0



1

θ1 − δ1 ω1 − (1 − I )β1 mρ 2 dρ

= R 3 (t) 0

=

1 θ1 − δ1 ω1 − (1 − I )β1 Vm (t), 4π

that is,

V˙m (t) = θ1 − δ1 ω1 − (1 − I )β1 Vm (t).

(4.15)

Vm (t) = Vm (0)e[θ1 −δ1 ω1 −(1−I )β1 ]t .

(4.16)

This further yields

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231

We also derive from (4.9), (4.6), (4.7), and (4.15) that ˙ − V˙m (t) V˙n (t) = 4πR 2 (t)R(t)

= 4πR 3 (t)V (1, t) − θ1 − δ1 ω1 − (1 − I )β1 Vm (t)  1

= 4πR 3 (t) θ1 m + 1 − m − δ1 ω1 m − δ2 ω2 (1 − m) ρ 2 dρ 0



1

θ1 − δ1 ω1 − (1 − I )β1 mρ 2 dρ

− 4πR 3 (t) 0



1

= 4πR 3 (t) 0

(1 − δ2 ω2 )nρ 2 dρ



1

+ 4πR (t) 3

(1 − I )β1 mρ 2 dρ

0

= (1 − δ2 ω2 )Vn (t) + (1 − I )β1 Vm (t). So, we get the following ODE equation for Vn (t): V˙n (t) = (1 − δ2 ω2 )Vn (t) + (1 − I )β1 Vm (t).

(4.17) 

This, together with (4.8)–(4.9) and the expression (4.16), yields (4.10)–(4.14).

Remark 4.1 We note that the necessary condition for successful treatment was derived in Jackson (2004a) (see Jackson 2004a, p. 196). However, Theorem 4.1 finds the explicit formulae of the tumor volume at time t in androgen-deprived environment, and therefore the dynamics of the tumor growth in androgen-deprived environment can be predicted by these formulae. Although the formulae were derived under the assumption of radial symmetry, it may be useful for predicting the long-term behavior of tumor growth. Remark 4.2 Conditions 1–4 are all special cases, and only Condition 5 is a general case. For the typical parameter values given in Jackson (2004a) δ1 =

0.3812 , 0.4621

δ2 =

0.4765 , 0.4621

θ1 = 0.8,

ω1 = 1.35,

ω2 = 0.25,

the condition θ1 − δ1 ω1 < 0

and 1 − δ2 ω2 > 0

(4.18)

holds, which means that the “net” growth rate of AD cells is negative, whereas the “net” growth rate of AI cells is positive in androgen-deprived environment. It follows from (4.18) that θ1 − δ1 ω1 − 1 + δ2 ω2 < 0. We now rewrite the formula (4.14) as follows:

(4.19)

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 (1 − I )β1 Vm (0) e[θ1 −δ1 ω1 −(1−I )β1 ]t θ1 − δ1 ω1 − (1 − I )β1 − 1 + δ2 ω2   (1 − I )β1 Vm (0) e(1−δ2 ω2 )t + Vn (0) − θ1 − δ1 ω1 − (1 − I )β1 − 1 + δ2 ω2

 V (t) =

Vm (0) +

=: A(t) + B(t).

(4.20)

If we further assume that Vn (0) > 0, that is, that the initial volume occupied by AI cells is nonzero, then by (4.18) and (4.19) we have A(t) → 0 and B(t) → +∞,

as t → +∞.

Therefore, by (4.20), V (t) → +∞ as t → +∞.

(4.21)

This implies a tumor relapse under androgen-deprived therapy. We here should note that (4.21) holds in a mathematical sense, and it may not be biologically plausible behavior. In fact, the host will die when the tumor is large enough. However, (4.21) predicts eventual failure of the androgen-deprived therapy. To better understand the time scale on which the asymptotic behavior in (4.21) is achieved, we will further give a numerical example. We first note that the asymptotic behavior of tumor growth is very sensitive to the programmed death rate δ2 of AI cells, so we regard it as a varying parameters. Taking the following typical parameter values given in Jackson (2004a), θ1 = 0.3,

ω1 = 1.35,

ω2 = 0.9,

δ1 = 0.3812/0.4621,

β1 = 0.9,

and taking I = 0.8 and δ2 = 1.07, 1.06, 1.05, 1.04, 1.03, respectively, we easily check that (4.18), (4.19), and Condition 5 hold. For the above parameter values, Vm (0) = 3.8 and Vn (0) = 0.388 (Vm (0) + Vn (0) ≡ V (0) = 43 π · 13 ≈ 4.188), the required time t ∗ is given as t ∗ = 314, 252, 211, 182, 160 (days), respectively, for the tumor growing very large with a radius R(t) = 30 mm. The above numerical example also suggests that increasing the apoptosis rate δ2 of AI cells a little bit will greatly delay the relapse of a tumor. Remark 4.3 Since A(t) → 0 and B(t) → +∞ as t → +∞,

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233

the term B(t) will dominate the growth of a tumor. We also note that −

(1 − I )β1 (δ1 ω1 − θ1 ) + (1 − δ2 ω2 ) ≡1− , θ1 − δ1 ω1 − (1 − I )β1 − 1 + δ2 ω2 (1 − I )β1 + (δ1 ω1 − θ1 ) + (1 − δ2 ω2 )

which is decreasing with increasing I ∈ [0, 1] by (4.18). This fact suggests that controlling the mutation may delay the tumor relapse under the androgen-deprived therapy. Remark 4.4 Denote “the net” growth rate of AD cells = the proliferation rate of AD cells − the death rate of AD cells − the mutation rate = θ1 − δ1 ω1 − (1 − I )β1 , “the natural net” growth rate of AI cells = the proliferation rate of AI cells − the death rate of AI cells = 1 − δ2 ω2 , “the net” growth rate of AI cells = the mutation rate + “the natural net” growth rate of AI cells = (1 − I )β1 + (1 − δ2 ω2 ), where the mutation rate = (1 − I )β1 by which AD cells mutate into AI cells. Condition 1 means that the “net” growth rate of AD cells and the “natural net” growth rate of AI cells are both equal to zero, under which the tumor relapse cannot be avoided due to the mutation of AD cells (0 ≤ I < 1) that results in an increase of AI cells, as shown in (4.10); however, (4.10) also clearly suggests that controlling the mutation (i.e., increasing the value of I ) may delay (for 0 < I < 1) or prevent (for I = 1) the relapse. In fact, by (4.10), as t → +∞, V (t) → +∞ if 0 < I < 1

4 but V (t) ≡ π 3

if I = 1.

Condition 2 means that the “net” growth rate of AD cells is equal to zero and the “natural net” growth rate of AI cells is positive (we assume that 1 − δ2 ω2 > 0 for the typical parameter values given in Jackson 2004a), under which the AI cells will dominate the tumor growth as shown in (4.11), and the tumor relapse cannot be avoided. In fact, by (4.11) and 1 − δ2 ω2 > 0, V (t) → +∞

as t → +∞.

Condition 3 means that the “net” growth rate of AD cells is nonzero (we assume that it is negative for typical parameter values given in Jackson 2004a) and the “natural net” growth rate of AI cells is zero, under which the tumor growth could be

234

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controlled as shown in (4.12). In fact, by θ1 − δ1 ω1 − (1 − I )β1 < 0 and (4.12), V (t) → Vn (0) +

(1 − I )β1 Vm (0) (1 − I )β1 − (θ1 − δ1 ω1 )

as t → +∞.

Condition 4 means that the “net” growth rate of AD cells is equal to the “natural net” growth rate of AI cells, and they are positive, under which both AD cells and AI cells contribute to the tumor growth as shown in (4.13), and the tumor relapse could not be avoided. In fact, by (4.13) and θ1 − δ1 ω1 − (1 − I )β1 = 1 − δ2 ω2 > 0, V (t) → +∞

as t → +∞.

Condition 5 means that the “net” growth rate of AD cells and the “natural net” growth rate of AI cells are both nonzero, and they are not equal. For typical parameter values given in Jackson (2004a), we assume that the former is negative and the latter is positive (as in (4.18)), under which the AI cells will dominate the tumor growth as shown in (4.14), and the tumor relapse could not be avoided as shown in (4.21).

5 Numerical Simulation: Effects of Mutation Inhibitor In Sect. 4 we found the explicit formulae of the tumor volume (or tumor radius), and therefore tumor dynamics can be predicted by these formulae in androgen-deprived environment (i.e., a ≡ 0). In this section we will numerically study the model (3.2)– (3.7) for different values of I with 0 ≤ I ≤ 1 and a(t) in (2.15) (i.e., a > 0). There are two main goals in this section: One is to compare the effect of TAB treatment with that of ADT treatment; the other is to investigate the dependence of the tumor growth on the model parameters θ1 , ω1 , ω2 , and I . The typical parameter values for our numerical analysis are as follows Jackson (2004a): ε1 = 0,

δ1 = 0.3812/0.4621 ≈ 0.8249,

K = 1,

b = 1.

δ2 = 0.4762/0.4621 ≈ 1.0305,

In the following simulations, we will regard θ1 , ω1 , ω2 , β1 , and I as varying parameters. Figure 1 compares the effects of TAB treatment and ADT treatment for two sets of parameters θ1 and ω2 and for different initial cell densities. Figure 1 shows that, overall, ADT treatment may be more effective than TAB treatment with respect to an AI tumor relapse in an androgen-deprived state, although TAB treatment can be initially better than ADT treatment in some cases (see Fig. 1C). Figure 1 may support a possible strategy of intermittent androgen suppression (IAS), which is a form of androgen ablative therapy delivered intermittently with off-treatment periods (Bruchovsky et al. 2006, 2007; Ideta et al. 2008). Figure 2 clearly shows that the tumor growth is very sensitive to parameter ω2 . For the typical parameter value ω2 ∈ [0.25, 1.0] (Jackson 2004a), Fig. 2A also implies an AI tumor relapse. To improve the efficacy of TAB treatment, we should try to raise the value of ω2 (Jackson 2004a, 2004b); this may be realized by combining with another

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235

Fig. 1 The comparison between the effect of TAB treatment and that of ADT treatment where as = 0.2 for ADT, as = 0 for TAB, ω1 = 1.35 (Jackson 2004a), and I = 1

Fig. 2 The sensitivity of the tumor growth to parameter ω2 where ω1 = 1.35 (Jackson 2004a), I = 1, as = 0

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Fig. 3 The sensitivity of the tumor growth to parameter θ1 , where ω1 = 1.35 (Jackson 2004a), ω2 = 0.9, I = 1, as = 0, m0 (ρ) = 0.1 + 0.2ρ 2

therapy such as differentiation therapy (see Carducci et al. 1996; Holland and Frei 2001 and the next discussion section). Figure 2B indicates that the tumor could be well controlled for larger ω2 (ω2 > 0.97). Figures 3 and 4 show that the tumor growth is insensitive to parameters θ1 and ω1 , since AI cells will be dominant and an AI relapse occurs for typical parameter values given in Jackson (2004a) (see Remark 4.2). However, Figs. 3 and 4 also show that decreasing the proliferation rate of AD cells and increasing the apoptosis rate of AD cells are always a better choice. Figure 5 shows that the tumor growth is to some extent sensitive to the inhibitor parameter I and implies that controlling mutation may improve the effect of TAB treatment or delay a tumor relapse.

6 Discussion In this paper we have analyzed an newly extended mathematical model of prostate tumor growth under hormone therapy with mutation inhibitors. The tumor contains two types of cancer cells, namely androgen-dependent (AD) cells and androgenindependent (AI) cells which are undetectable prior to treatment (Ellis et al. 1996; Jackson 2004a, 2004b) but grow even in androgen-poor conditions during continuous hormone therapy. The model is formulated as a free-boundary problem for a nonlinear system of parabolic equations which describe the evolution of the cell populations within a tumor.

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237

Fig. 4 The sensitivity of the tumor growth to parameter ω1 , where θ1 = 0.3, ω2 = 0.9, I = 1, as = 0, m0 (ρ) = 0.1 + 0.2ρ 2

We first claimed that the model admits a unique global solution; since the proof is technically similar to that given in Tao et al. (2009), we omitted it. Then, we found explicit formulae of the tumor volume at any time t in androgen-deprived environment, and therefore the long-term behavior of tumor growth could be predicted by these formulae. The formulae suggest that novel therapeutic strategies should aim at decreasing the net growth rates of AD cells and AI cells. In fact, new approaches to prostate cancer therapy like differentiation therapy (see Holland and Frei 2001, Chap. 108) may afford possibility of better therapeutics against prostate cancer in this direction by raising the value of parameter ω2 and lowering the value of parameter θ1 . Another therapeutic strategy may involve signal transduction inhibitors because many growth factors are relate to the growth and progression of prostate cancer (Holland and Frei 2001). Signal transduction inhibitors may reduce the fraction of AI cells (Kamradt and Pienta 1999). Our new model here considers gene mutation (see Ideta et al. 2008 for details) and possibility of inhibiting the mutation. Qualitative analysis (see Remark 4.2) suggests that a tumor relapse could not be avoided under androgen-deprived therapy. This suggestion may support a possible strategy of intermittent androgen suppression (IAS), which is a form of androgen ablative therapy delivered intermittently with off-treatment periods (Bruchovsky et al. 2006, 2007; Ideta et al. 2008). Qualitative analysis (see Remark 4.3) and numerical simulation show that controlling the mutation may delay the tumor relapse under hormone therapy. Jackson (2004a, 2004b) proposed the first PDE model for the AI response. Ideta et al. (2008) proposed an ODE model for the AI response and IAS by considering mutation. In this paper we are proposing a PDE model with mutation and mutation inhibitors for the AI response.

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Fig. 5 The sensitivity of the tumor growth to the inhibitor parameter I , where as = 0, θ1 = 0.3, ω1 = 1.35, ω2 = 0.9, β1 = 0.9, m0 (ρ) = 0.5 + 0.4ρ 2

There are some model extensions that are of interest. Modeling combined therapies against prostate cancer using hormone therapy and differentiation therapy will be an interesting future problem. Modeling the onset of prostate neoplasia or establishing a multiscale model of prostate tumor growth, as suggested in two recent review articles (Bellomo and Delitala 2008; Bellomo et al. 2008), may be another very interesting and challenging future problem.

Acknowledgements The authors would like to thank two anonymous referees and Prof. P. Maini of Oxford University for their valuable comments. The first author (Tao) and the second author (Guo) are partially supported by the National Natural Science Foundation of China (NSFC 10571023), the first author (Tao) is also supported by the Natural Science Foundation of Shanghai (09ZR1401200), and the third author (Aihara) is partially supported by JST (Japan Science and Technology Agency).

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