Bifurcation analysis on a hybrid systems model of intermittent hormonal therapy for prostate cancer
ABSTRACT Hybrid systems are widely used to model dynamical phenomena that are characterized by interplay between continuous dynamics and discrete events. An example of biomedical application is modeling of disease progression of prostate cancer under intermittent hormonal therapy, where continuous tumor dynamics is switched by interruption and reinstitution of medication. In the present paper, we study a hybrid systems model representing intermittent androgen suppression (IAS) therapy for advanced prostate cancer. Intermittent medication with switching between ontreatment and offtreatment periods is intended to possibly prevent a prostatic tumor from developing into a hormonerefractory state and is anticipated as a possible strategy for delaying or hopefully averting a cancer relapse which most patients undergo as a result of longterm hormonal suppression. Clinical efficacy of IAS therapy for prostate cancer is still under investigation but at least worth considering in terms of reduction of side effects and economic costs during offtreatment periods. In the model of IAS therapy, it depends on some clinically controllable parameters whether a relapse of prostate cancer occurs or not. Therefore, we examine nonlinear dynamics and bifurcation structure of the model by exploiting a numerical method to clarify bifurcation sets in the hybrid system. Our results suggest that adjustment of the normal androgen level in combination with appropriate medication scheduling could enhance the possibility of relapse prevention. Moreover, a twodimensional piecewiselinear system reduced from the original model highlights the origin of nonlinear phenomena specific to the hybrid system.

Article: Mathematically modelling and controlling prostate cancer under intermittent hormone therapy.
[Show abstract] [Hide abstract]
ABSTRACT: In this review, we summarize our recently developed mathematical models that predict the effects of intermittent androgen suppression therapy on prostate cancer (PCa). Although hormone therapy for PCa shows remarkable results at the beginning of treatment, cancer cells frequently acquire the ability to grow without androgens during longterm therapy, resulting in an eventual relapse. To circumvent hormone resistance, intermittent androgen suppression was investigated as an alternative treatment option. However, at the present time, it is not possible to select an optimal schedule of on and offtreatment cycles for any given patient. In addition, clinical trials have revealed that intermittent androgen suppression is effective for some patients but not for others. To resolve these two problems, we have developed mathematical models for PCa under intermittent androgen suppression. The mathematical models not only explain the mechanisms of intermittent androgen suppression but also provide an optimal treatment schedule for the on and offtreatment periods.Asian Journal of Andrology 03/2012; 14(2):2707. · 2.14 Impact Factor  SourceAvailable from: europepmc.org[Show abstract] [Hide abstract]
ABSTRACT: If a mathematical model is to be used in the diagnosis, treatment, or prognosis of a disease, it must describe the inherent quantitative dynamics of the state. An ideal candidate disease is prostate cancer owing to the fact that it is characterized by an excellent biomarker, prostatespecific antigen (PSA), and also by a predictable response to treatment in the form of androgen suppression therapy. Despite a high initial response rate, the cancer will often relapse to a state of androgen independence which no longer responds to manipulations of the hormonal environment. In this paper, we present relevant background information and a quantitative mathematical model that potentially can be used in the optimal management of patients to cope with biochemical relapse as indicated by a rising PSA.Journal of Molecular Cell Biology 05/2012; 4(3):12732. · 7.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The behaviors of system which alternate between Duffing oscillator and van der Pol oscillator are investigated to explore the influence of the switches on dynamical evolutions of system. Switches related to the state and time are introduced, upon which a typical switched model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point and associated Floquet multipliers are calculated, based on which twoparameter bifurcation sets of the switched system are obtained, dividing the parameter space into several regions corresponding to different types of attractors. It is found that cascading of perioddoubling bifurcations may lead the system to chaos, while fold bifurcations determine the transition between period3 solution and chaotic movement.Communications in Nonlinear Science and Numerical Simulation 03/2014; 19(3):750757. · 2.77 Impact Factor
Page 1
Physica D 237 (2008) 2616–2627
www.elsevier.com/locate/physd
Bifurcation analysis on a hybrid systems model of intermittent hormonal
therapy for prostate cancer
Gouhei Tanakaa,∗, Kunichika Tsumotoa,b, Shigeki Tsujia,b, Kazuyuki Aiharaa,b
aInstitute of Industrial Science, University of Tokyo, Tokyo 1538505, Japan
bERATO Aihara Complexity Modelling Project, JST, Tokyo 1510065, Japan
Received 16 August 2007; received in revised form 30 January 2008; accepted 22 March 2008
Available online 4 April 2008
Communicated by A. Mikhailov
Abstract
Hybrid systems are widely used to model dynamical phenomena that are characterized by interplay between continuous dynamics and discrete
events. An example of biomedical application is modeling of disease progression of prostate cancer under intermittent hormonal therapy, where
continuous tumor dynamics is switched by interruption and reinstitution of medication. In the present paper, we study a hybrid systems model
representing intermittent androgen suppression (IAS) therapy for advanced prostate cancer. Intermittent medication with switching between on
treatment and offtreatment periods is intended to possibly prevent a prostatic tumor from developing into a hormonerefractory state and is
anticipated as a possible strategy for delaying or hopefully averting a cancer relapse which most patients undergo as a result of longterm hormonal
suppression. Clinical efficacy of IAS therapy for prostate cancer is still under investigation but at least worth considering in terms of reduction of
side effects and economic costs during offtreatment periods. In the model of IAS therapy, it depends on some clinically controllable parameters
whether a relapse of prostate cancer occurs or not. Therefore, we examine nonlinear dynamics and bifurcation structure of the model by exploiting
anumericalmethodtoclarifybifurcationsetsinthehybridsystem.Ourresultssuggestthatadjustmentofthenormalandrogenlevelincombination
with appropriate medication scheduling could enhance the possibility of relapse prevention. Moreover, a twodimensional piecewiselinear system
reduced from the original model highlights the origin of nonlinear phenomena specific to the hybrid system.
c ? 2008 Elsevier B.V. All rights reserved.
PACS: 05.45.a; 87.19.Xx; 87.53.Tf
Keywords: Hybrid systems; Piecewise linear systems; Grazing bifurcations; Limit cycle; Chaos; Prostate cancer; Intermittent androgen suppression
1. Introduction
Following great progresses of life science and nonlinear
science in recent years, a mathematical approach is becoming
a more promising methodology for advanced studies in
biomedical science. In particular, much attention has been
paid to mathematical and computational modeling of cancer
dynamics involving nonlinear biological interactions [1,2]. In
the context of prostate cancer, for example, dynamical systems
have been extensively used to describe tumor growth [3,4] and
temporal variations of biomarkers [5–7] in efforts to predict
∗Corresponding author. Tel.: +81 3 5452 6693; fax: +81 3 5452 6694.
Email address: gouhei@sat.t.utokyo.ac.jp (G. Tanaka).
medical conditions and help appropriate diagnoses. The main
concern of these studies is to understand the mechanism of
treatmentresistant tumor growth, or a cancer relapse, after
a remission period induced by hormone deprivation therapy.
A possible strategy to delay or prevent the progression to
hormoneresistance caused by prolonged hormone suppression
is to incorporate treatment interruption, which is known as
intermittent hormonal therapy repeating cycles of ontreatment
and offtreatment periods [8–15]. In this paper, we deal with
nonlinear dynamics and bifurcations in a mathematical model
of intermittent hormonal therapy for prostate cancer.
Prostate cancer is a disease characterized by uncontrolled
growth of cancer cells within the prostate gland in males.
The prostate gland is dependent on hormonal secretion by
01672789/$  see front matter c ? 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physd.2008.03.044
Page 2
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
2617
the testes for growth and development. When production of
the male hormone (androgen) decreases, the prostate begins
to degenerate. Since this androgendependent (AD) nature of
the prostatic cells is shared by unusual malignant tumor cells,
namely AD cells of prostate cancer, androgen deprivation has
been the mainstay for treating advanced or metastatic prostate
cancer [16]. Most androgens synthesized and secreted from the
testes can be suppressed by surgical or chemical castration. The
influence of adrenal androgens remaining after castration can
be eliminated by the addition of nonsteroidal androgen receptor
antagonists, or antiandrogens. Combination of castration and
antiandrogens, referred to as combined or maximum androgen
blockade, can facilitate tumor regression [17]. The serum
prostatespecific antigen (PSA), which is a widely used
biomarkerforprostatecancer,enablesmonitoringofthedisease
progression through blood tests. Androgen deprivation initiated
in a state with a high PSA level promotes apoptotic death
of malignant cells and leads to a significant decrease of the
PSA value implying tumor regression. However, continuation
of androgen suppression not only causes side effects from
the treatment but also mostly results in a relapse by recurrent
tumor growth after a remission period [18]. A relapse of
prostate cancer is considered to be primarily due to the
progression to an androgenindependent (AI) state where the
tumor is no longer responsive to androgen deprivation [19].
Many efforts have been made to understand the properties of
androgen receptors and signaltransduction pathways leading
to a hormonerefractory prostate cancer [20–23]. Intermittent
androgen suppression (IAS) is an alternative approach to
maintain tumor sensitivity to androgen deprivation with the
possibility to prolong a relapse [8–15]. Although clinical
trials are still in progress for establishing the clinical efficacy
of IAS therapy [11,14,15], it has already been confirmed
that side effects and economic costs are at least reduced in
comparison with continued androgen suppression. Following
these advances in experimental and clinical studies of IAS, a
mathematical model of IAS therapy for prostate cancer was
presented [24–26].
IAS therapy aims at delaying prostate cancer relapse by
keeping androgen dependency of the tumor through repeating
ontreatment and offtreatment cycles. Correspondingly, the
IAS therapy model alternates between two different kinds of
dynamics. The androgen concentration tends to recover and
maintain the normal level a0 (nmol/l) during offtreatment
periods whereas it decays to almost zero during ontreatment
periods. These different androgen levels make much difference
in tumor dynamics regulated by the androgen concentration
and in variations of the serum PSA concentration. Interruption
and reinstitution of therapeutic administration for androgen
suppression are determined based on changes of the serum PSA
level in the model as well as in real treatments. It is assumed in
our model that administration is suspended when a decreasing
PSA level drops to a lower threshold value r0(ng/ml) whereas
it is resumed when an increasing PSA level reaches an upper
threshold value r1 (ng/ml) [24–26]. If this dosing strategy
is successful in relapse prevention, then the serum PSA
level remains around a region between r0and r1. Otherwise,
recurrent tumor growth eventually happens and thereby the
serum PSA level continues to increase explosively. In the model
study, occurrence and prevention of a relapse are characterized
by divergent and nondivergent solutions, respectively. Thus,
parameter conditions for relapse prevention can be revealed by
investigating the transition between two qualitatively different
regimes. In this paper, we examine bifurcation phenomena with
respect to changes of clinically controllable parameters such as
a0, r0, and r1. For this purpose, we utilize shooting methods
for locating bifurcation sets of a limit cycle in hybrid systems.
Moreover, the origin of nonlinear and chaotic dynamics of the
IAS therapy model is clarified through return plot analysis of
a piecewise linear system derived from the original model by
simplification.
Hybrid dynamical systems involving both continuous and
discrete variables have been pervasive especially in computer
science and control engineering [27–29]. Analysis and control
of nonlinear phenomena in hybrid dynamical systems are
practicallysignificantforindustrialapplicationssuchasprocess
control [30,31], mechanical systems [32], robotic control [33],
power systems [34], and power electronics [35,36]. A number
of methodologies have been developed for analysis and control
designofhybrid(piecewisesmooth)dynamicalsystems[37].In
nonlinear science, stability and bifurcations in hybrid systems
have been the main focus since early studies [38,39], because
interactions between a trajectory and borders for discrete events
can bring about complex phenomena. Recently much attention
has been paid to grazing phenomena [40] and bordercollision
bifurcations [41]. Return map analysis is often effective for
understanding chaotic behavior in lowdimensional switched
dynamical systems [42]. Numerical methods for locating
bifurcation points of a periodic solution have been proposed by
appropriately formulating conditional equations [43]. Similar
methods have been developed to formulate grazing phenomena
of periodic and nonperiodic trajectories [44].
Tumor growth under intermittent therapy can also be viewed
as a hybrid system composed of continuous tumor dynamics
and discrete events with interruption and reinstitution of
medication. In the model study, the clinical issue of how to
schedule on–off cycles of administration for preventing an
explosion of malignant cell populations can be reduced to
a mathematical problem of how to adequately set adjustable
parameter values for confining a solution in a finite region in
continuous state space. Therefore, it is significant to understand
nonlinear phenomena peculiar to hybrid systems, such as a
grazing bifurcation which occurs when a trajectory tangentially
hits a boundary set related to a discrete event [40]. Present
research on the IAS therapy model not only supports a
possible clinical advantage of IAS therapy but also highlights
the complexity generated as a fundamental feature of hybrid
systems.
In Section 2, we will describe shooting methods to specify
bifurcation sets of a limit cycle in a hybrid system by defining
an appropriate Poincar´ e map. Results of bifurcation analysis
of the IAS therapy model will be shown in Section 3. The
essential mechanism of nonlinear and complex behavior in the
IAS therapy model will also be elucidated by analysis of a
Page 3
2618
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
simplified model. Finally, the last section will be devoted to
conclusions.
2. Method
2.1. Hybrid systems
In this section, we introduce a method to analyze
bifurcations of a hybrid system consisting of two ordinary
differential equations described as follows:
Su:dx
where t ∈ R, x ∈ Rn, and u ∈ {0,1} denote time, continuous
state vector, and the binary discrete variable, respectively. For
simplicity, we assume that the vector field fu (u = 0,1)
defined on Rnis a C∞class map for any state variables and
parameters. We consider a hybrid automaton [37] exhibiting
transitionsbetweensubsystems S0and S1,whichdescribestime
evolution of the continuous state x and the discrete state u.
We assume that switching from S1−uto Su(u = 0,1) occurs
when a trajectory in continuous state space hits the following
codimensionone surfaces:
dt
= fu(x),
u = 0,1,
(1)
Πu= {x ∈ Rn qu(x) = 0},
where qu is a scalarvalued function of the continuous state
vector x. The solution of subsystem Su is represented as
follows:
u = 0,1,
(2)
x(t) ≡ ϕu(t,x0),
where x0= x(0) is an initial condition of the continuous state.
2.2. Limit cycle and Poincar´ e map
u = 0,1,
(3)
Let us begin with constructing a Poincar´ e map in order to
formulate the conditions of a limit cycle and its bifurcations
in a hybrid system. We consider a situation where a trajectory
of the hybrid system (1) exhibits a nearly periodic motion
with repeating transitions between S0and S1alternately. Fig. 1
shows a schematic illustration of a part of a trajectory crossing
the sections Π0and Π1alternately in continuous state space. A
trajectory leaving from the initial point x0on Π0intersects with
Π1atx1withtravelingtimeτ0underthedynamicsinsubsystem
S0. At the moment of intersection, the subsystem changes from
S0to S1, and then the trajectory returns to Π0with traveling
time τ1. We define maps from one section to the other one as
follows:
T0: Π0→ Π1;x0?→ x1≡ ϕ0(τ0,x0),
T1: Π1→ Π0;x1?→ ϕ1(τ1,x1).
By treating Π0as a Poincar´ e section, a Poincar´ e map T :
Π0→ Π0is defined by a composite of two submaps (4) and (5)
as follows [43]:
(4)
(5)
T = T1◦ T0,
where the total return time is given by τ = τ0+ τ1. It should
(6)
Fig. 1. Schematic illustration for construction of the Poincar´ e map P on the
local coordinate for a trajectory of the hybrid system (1), where the trajectory
under consideration exhibits a nearly periodic motion crossing two sections of
discrete events alternately.
be noted here that the global coordinate x0 ∈ Π0 ⊂ Rn
is redundant to identify any point on the codimensionone
Poincar´ e section. Thus, we define a local coordinate w ∈ Σ ⊂
Rn−1and introduce the projection map h and its inverse h−1:
h : Π0→ Σ;x0?→ w,
h−1: Σ → Π0;w ?→ x0,
such that the Poincar´ e map on the local coordinate is
represented as follows:
(7)
(8)
P : Σ → Σ;
Similarly, a Poincar´ e map for an mfolded limit cycle crossing
the Poincar´ e section for m times is given by Pm= h◦Tm◦h−1.
w ?→ h ◦ T ◦ h−1(w).
(9)
2.3. Stability and local bifurcation
A limit cycle in continuous state space corresponds to a
fixed point of the Poincar´ e map P, which satisfies the following
equation:
Flc(w) ≡ P(w) − w = 0.
In general this equation is not analytically solvable as usual for
nonlinear systems; it can be numerically solved with respect to
(n −1) unknown variables w by using numerical methods such
as Newton’s method. The Jacobian matrix of Flcrequired for
the numerical computation is given by:
(10)
DFlc(w) =∂P
where Indenotes the n×n identity matrix. See Appendix A for
details on the calculation procedure of firstorder derivatives of
the Poincar´ e map P.
Stability of a fixed point of P, which is equivalent to that of
a corresponding limit cycle, can be evaluated by characteristic
(Floquet) multipliers of the Jacobian matrix of P at the fixed
point, which are the solutions of the following characteristic
equation:
?∂P
If all the exponents are located inside the unit circle on the
complex plane, then the fixed point is stable. Loss of stability
of the fixed point occurs when one of the exponents crosses
∂w− In−1,
(11)
χ(µ) ≡ det
∂w− µIn−1
?
= 0.
(12)
Page 4
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
2619
Fig. 2. A grazing bifurcation point λ can be characterized by a trajectory
tangent to the eventtriggering section Π0at x = x0. A trajectory hits the
section at λhit> λ whereas it misses to touch the section at λmiss< λ.
Fig. 3. Construction of maps¯Tkand cumulative traveling time ¯ τkfor an m
folded limit cycle.
the unit circle outward from the inside by changing values of
system parameters, which is accompanied with a codimension
one local bifurcation.
For specifying a local bifurcation of a fixed point, the fixed
point condition (10) is combined with the eigenvalue condition
as follows:
?P(w) − w
These equations are simultaneously solved with respect to n
unknown variables (w,λ) where λ is a bifurcation parameter.
We fix the value of µ for a specific type of local bifurcation:
µ = 1 for saddlenode bifurcation, µ = −1 for period
doubling bifurcation, and µ
bifurcation. The Jacobian matrix of Flbneeded to solve Eq. (13)
by Newton’s method is written as follows:
The first and secondorder derivatives of P required to
calculate components of the matrix (14) are described in
Appendix A.
Flb(w,λ) ≡
χ(µ)
?
= 0.
(13)
=
eiθfor Neimark–Sacker
DFlb=
∂P
∂w− In−1
∂χ(µ)
∂w
∂P
∂λ
∂χ(µ)
∂λ
.
(14)
2.4. Grazing bifurcation
A grazing bifurcation exhibited by hybrid systems is
concerned with a sudden change in the interaction between a
trajectory and an eventtriggering boundary for switching of
subsystems [44]. When a trajectory is transversal but almost
tangent to the Poincar´ e section Π0as illustrated in Fig. 2, a
slight change of a system parameter value can lead to loss
of the intersection and induce a change of dynamics. This
critical change in system behavior, called grazing bifurcation,
can be characterized by a trajectory tangent to the event
triggering section. If such a qualitative change occurs for a
periodic solution, then it is called periodic grazing [44]. At
the right grazing bifurcation with respect to the section Π0,
the directional vector of the trajectory is orthogonal to the
normal vector of the section at the point of tangency. Since the
transversality condition (A.14) at Π0(see Appendix A) does
not hold for a tangency point of a grazing trajectory, we need
to sort out the conditions of periodic grazing by regarding the
return time of a limit cycle as an unknown variable.
We consider a grazing bifurcation of an mfolded limit
cycle corresponding to an mperiodic point of the Poincar´ e
map for a natural number m in the following. Let us suppose
that a grazing trajectory starting from x0on Π0transversally
intersects with Π0and Π1for (2m − 1)times alternately and
then finally becomes tangent to Π0. For convenience, we define
the following maps:
¯T2k: Π0→ Π0;x0?→ x2k≡ (T1◦ T0)k(x0),
k = 0,1,...,m − 1,
¯T2k+1: Π0→ Π1;x0?→ x2k+1≡ T0◦ (T1◦ T0)k+1(x0),
k = 0,1,...,m − 1,
as illustrated in Fig. 3. Correspondingly, the traveling time from
xk to xk+1 is denoted by τk(xk) for k = 0,1,...,2m − 1.
Then, the cumulative traveling time from x0to xk, denoted by
¯ τk, satisfies the following formula:
¯ τk+1(x0) = ¯ τk(x0) + τk(xk),
where ¯ τ0(x0) ≡ 0. The above setting provides a map dependent
on the total return time τ and the initial condition x0as follows:
(15)
(16)
k = 0,...,2m − 1,
(17)
¯T(τ,x0) = ϕ1(τ − ¯ τ2m−1(x0),x2m−1),
where τ > ¯ τ2m−1(x0).
By using the map¯T, the conditions of a grazing bifurcation
of an mfolded periodic solution can be formulated as follows:
∂x
The first equation means that a trajectory leaving from x0on Π0
is periodic with return time τ. The second equation indicates
that the trajectory is located on Π0at time τ. The third equation
represents the condition that the trajectory is tangent to section
Π0at time τ. The set of Eq. (19) are simultaneously solved
with respect to (n +2) unknown variables (x0,τ,λ) where λ is
a bifurcation parameter. The Jacobian matrix of Fgbrequired to
numerically solve Eq. (19) is given by:
(18)
Fgb(x0,τ,λ) ≡
¯T(τ,x0) − x0
q0(¯T(τ,x0))
∂q0
· f1(¯T(τ,x0))
= 0.
(19)
DFgb
=
∂¯T
∂x0
∂q0
∂x
?∂2ϕ1
− In
∂¯T
∂x0
∂¯T
∂τ
∂¯T
∂λ
∂q0
∂x
?∂2ϕ1
∂¯T
∂τ
∂q
∂x
?∂2ϕ1
∂¯T
∂λ+∂q0
∂¯T
∂λ
∂q0
∂x
·
∂t∂x
∂¯T
∂x0
?
∂q0
∂x
·
∂t∂x
∂¯T
∂τ
?
∂q0
∂x
·
∂t∂x
∂λ+∂2ϕ1
∂t∂λ
?
.
(20)
Page 5
2620
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
Refer to Appendix B for details of the calculation procedure of
the derivatives of¯T.
3. Results
3.1. Mathematical model of IAS therapy
The numerical method for bifurcation analysis introduced
in the previous section is used to investigate the mathematical
model of IAS therapy for prostate cancer [24,25]. The
purpose of this section is to elucidate how dynamical behavior
is influenced by some control parameters with respect to
clinical importance in prostate cancer therapy. As introduced
in Section 1, at least two different kinds of prostatic cells
are involved in androgen suppression therapy. AD cells are
sensitivetoandrogensuppressionandarelikelytoinducetumor
regression by their apoptosis, while AI cells are insensitive to
androgen deprivation and considered to be responsible for a
relapse. The IAS therapy model represents growth of a tumor
consisting of a mixed dynamical assembly of AD and AI cancer
cells under intermittent administration. Population growths of
both AD and AI cells are dependent on androgen concentration
a (nmol/l) whose dynamics relies on the binary variable u
indicating whether medication of maximum androgen blockade
is administered (u = 1) or not (u = 0). Administration is
reinstated when an increasing PSA concentration y (ng/ml)
rises up to the upper threshold valuer1, whereas it is interrupted
when a decreasing PSA level falls to the lower threshold value
r0.
Due to these discrete events concerning administration, the
IAS therapy model is formulated as a hybrid system [24,25]
consisting of subsystems described as follows:
dxD
dt
dxI
dt
da
dt
where continuous state variables xD
populations of AD and AI cancer cells, respectively. The
discrete variable u indicates onadministration periods for u =
1 or offadministration ones for u = 0. As in clinical practice,
suspension and reinstitution of administration are conducted in
the following way:
?1 → 0
Serum PSA concentration y serves as a biomarker for growth
of a prostatic tumor as follows: y = f (xD,xI) = cDxD+cIxI
where cD= cI= 1 for the sake of simplicity. Net growth rates
of AD and AI cells are denoted by gDand gI, respectively, and
the rate of mutation from AD cells to AI ones is indicated by
m. These factors deeply affecting tumor dynamics are given by
functions of androgen concentration as follows [24,25]:
= (gD(a) − m(a))xD,
= m(a)xD+ gI(a)xI,
= −γ(a − a0(1 − u)),
(21)
and xI
represent
u :
when y = r0and dy/dt < 0
when y = r1and dy/dt > 0.
0 → 1
(22)
gD(a) = αD
?
k1+(1 − k1)a
a + k2
?
− βD
?
k3+(1 − k3)a
a + k4
?
,
(23)
(24)
gI(a) = αI(1 − ea) − βI,
m(a) = m1
?
1 −a
a0
?
.
(25)
Here gD(a) is the difference between proliferation and
apoptosis rates of AD cells, which is approximately equal
to αD− βD in an androgenrich environment. Values of αD
and βD can be estimated from the experimental data [45]
on cell proliferation and apoptosis rates of AD cells for
hormonally untreated patients, respectively. In an androgen
depleted environment, AD cells are not able to proliferate
and their apoptosis rate can be estimated from real data [12–
15] of the serum PSA during androgen suppression. Plausible
functions with these conditions can be realized by adjusting
parameters k1, k2, k3, and k4. Growth rate of AI cells is also
dependent on the androgen level [46] because their growth
is still dependent on the androgen receptor, although details
of their androgen dependence remain unclear. Therefore we
assume that gI(a) is a linear function of the androgen level [24,
25] for the sake of simplicity, which is equal to αI− βIin an
androgendepleted environment. Values of αI and βI can be
estimated from experimental data [45] for hormonally failing
patients. The androgen dependence of the proliferation rate of
AI cells is controlled by the parameter e. We also assume that
the mutation rate m(a) is linearly increasing with a decrease of
androgen level, because continuation of androgen suppression
is considered to enhance the mutation from AD cells to AI ones.
The IAS therapy model is regarded as a hybrid automaton
with continuous state vector x = (xD,xI,a) and discrete state
variable u. Eventtriggering sections related to discrete events
(22) are defined as follows:
Π0= {x  q0(x) = f (xD,xI) − r0= 0},
Π1= {x  q1(x) = f (xD,xI) − r1= 0}.
A Poincar´ e map T with respect to the Poincar´ e section Π0
can be derived as demonstrated in Section 2. By choosing the
following transformation between local and global coordinates
on Π0:
(26)
(27)
h : Π0→ Σ;(xD,xI,a) ?→ (w1,w2) = (xD,a),
h−1: Σ → Π0;(w1,w2) ?→ (xD,xI,a)
= (w1,r0− w1,w2),
we obtain the Poincar´ e map P = h ◦ T ◦ h−1on the local
coordinate. Consequently, bifurcation analysis of a limit cycle
in the IAS therapy model is reduced to that of a fixed or a
periodic point of P.
(28)
(29)
3.2. Bifurcation phenomena
In the previous study [24,25], we have investigated the IAS
therapy model, mainly focusing on the effect of net growth rate
of AI cells. On the other hand, in the present paper, we clarify
the detailed influence of clinically controllable parameters such
Page 6
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
2621
as threshold values of the serum PSA, r0 and r1, and the
normal androgen level a0. The other parameters are fixed as
follows [24,25]:
γ = 0.08,
αD= 0.0204,
βI= 0.0168,
k1= 0,
The values of γ, m1, and e are set so that a relapse occurs in
numerical simulations around several years after the initiation
of continuous androgen suppression, which are clinically
plausible. We adopt growth rates of AD and AI cells in the
bone metastasis case [45]. The values of αDand βDand those
of αI and βI are estimated from experimental data [45] for
hormonally untreated patients and hormonally failing patients,
respectively. We set k1= 0 because AD cells do not multiply
without androgens. The value of k2is set to make the growth
rate curve of the AD cells plausible. The value of βDk3, which
is the apoptosis rate of AD cells in an androgendepleted
environment, is estimated by fitting an exponential function
to the decreasing serum PSA concentration during androgen
suppression in actual data for simplicity [12–15]. The value of
k4is set so that the evolution of AD cells changes from decrease
toincreaseatarounda = 5.See[24,25]fordetaileddiscussions
on parameter setting.
Fig. 4 shows a bifurcation diagram for variations of a0
and r0, indicating the parameter region of relapse prevention
together with bifurcation sets of limit cycles. The colored
region corresponds to relapse prevention, where a solution is
not divergent but confined within a finite region. The diagram
shows that the lower threshold value r0to restart administration
should be relatively small for preventing a relapse by the
following reasons. The first reason is that the range of a0for
relapse prevention is wider for a smaller value of r0as shown
in Fig. 4. In this regard, however, a too small value of r0
resulting in a relapse should be avoided. The second reason
is that if the lower threshold value r0 is near the upper one
r1, the frequency of the switching of ontreatment and off
treatment periods is too high to adequately evaluate progression
of the disease and efficacy of IAS. The normal androgen level
a0 also largely affects the tumor growth and the possibility
of relapse prevention. In fact, when a0 is fixed to be less
than a certain value, a relapse is inevitable for any choice
of r0value. This property suggests that the normal androgen
level during offtreatment periods is related to the potency of
IAS therapy. Although the mainstay for treatment of advanced
prostate cancer is androgen suppression, some experimental
studies suggested that AI tumors that have escaped from
androgen deprivation therapy might be inhibited by high levels
of androgen supplementation [47,48]. From a mathematical
viewpoint, the reason why a diverging solution is unavoidable
for a low value of a0is that the net growth rate of AI cells
is consistently positive independently of the androgen level as
given by Eq. (24) and thereby the flow in continuous state space
is always expanding. This result supports the possibility that
androgen supplementation during offtreatment periods might
m1= 0.00005,
βD= 0.0076,
e = 0.015,
αI= 0.0242,
k2= 2,
k3= 8,
k4= 0.5.
(30)
Fig. 4. Bifurcation diagram of the IAS therapy model (21)–(25) with r1= 30.
The parameter region of relapse prevention corresponds to that of nondivergent
solutions. Perioddoubling and grazing bifurcations of an mfolded limit cycle
are denoted by PDmand Gm, respectively. Bifurcation phenomena along the
horizontal arrowed line at r0= 25 are shown in Fig. 5.
Fig. 5. 1parameter bifurcation diagram of the IAS therapy model (21)–(25)
with r0= 25 and r1= 30. For a fixed value of a0, the intersecting points
with the Poincar´ e section Π0are plotted. Stable and unstable limit cycles are
indicated by solid and dashed lines, respectively. Perioddoubling and grazing
bifurcation points are denoted by PDmand Gm, respectively.
be beneficial for delaying or averting a relapse depending on
the net growth rate of AI cells.
Let us further examine detailed bifurcations of limit cycles
in Fig. 4. Solid and dashed curves indicate perioddoubling
and grazing bifurcation sets, respectively. Fig. 5 shows a one
parameter bifurcation diagram along the horizontal arrowed
line at r0 = 25 in Fig. 4. With decrease of a0, a stable
limit cycle undergoes successive perioddoubling bifurcations
and develops into chaotic solutions. Fig. 6(a)–(c) show a limit
cycle, a perioddoubled limit cycle, and a chaotic attractor,
respectively. The limit cycle which loses its stability at the first
perioddoubling bifurcation PD1finally disappears through a
grazing bifurcation G1as shown in Fig. 6(e). In a similar way,
the perioddoubled limit cycle generated at PD2also exhibits
a loss of stability and vanishes at another grazing bifurcation
G2as shown in Fig. 6(d). The chaotic attractor resulting from
the perioddoubling cascade is considered to disappear due
to transient grazing [44] by phase space analysis. In fact,
for a certain fixed parameter value, it depends on the initial
condition on Π0whether a trajectory returns to Π0again or
Page 7
2622
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
Fig. 6. Oribital motions of the IAS therapy model (21)–(25) with r0= 25 and
r1= 30: (a) A stable limit cycle with a0= 36.4; (b) A stable 2folded limit
cycle with a0= 35.2; (c) A chaotic solution with a0= 34.9; (d) A grazing
bifurcation of an unstable twofolded limit cycle with a0= 33.5; (e) A grazing
bifurcation of an unstable limit cycle with a0= 30.5.
not. Qualitatively different motions are separated at a boundary
set on the section, where a trajectory is grazing with respect
to the section. Therefore, a contact between the edge of a
chaotic attractor and the boundary set of initial conditions on
the section can cause a disappearance of the chaotic attractor.
We will characterize transient grazing with a simplified model
in Section 3.3. The pair of a grazing bifurcation point Gmand
a perioddoubling one PDmseems to be nested with increase
of m for variation of the parameter a0. The mechanism of this
structure is analogous to that of a similar bifurcation pattern
with interplay between border collisions and a perioddoubling
cascade [49].
Next we fix the value of the normal androgen level a0and
focus on how to set threshold values of the serum PSA level
to stop and restart administration. Bifurcation diagrams for
different values of a0 are shown in Fig. 7. The size of the
parameter region for relapse prevention becomes larger as a0
increases. We have more choices of threshold values for dosing
by setting the normal androgen level as high as possible within
a clinically feasible range. Boundaries of the parameter area for
relapse prevention are approximately given by perioddoubling
bifurcation sets PD1. Fig. 7 shows that the ratio between r0
and r1is critical rather than the respective absolute values of
r0 and r1 under the dosing strategy (22). This result stems
from the scaling property of the IAS therapy model. Namely,
Eqs. (21) and (22) are invariant under the following variable
transformation:
(xD,xI,r0,r1) → (pxD, pxI, pr0, pr1),
where p is any nonzero real value. It should be noted, however,
that this scaling property seems not realistic for a large p.
The transformation holds if the serum PSA concentration is
given as a linear sum of the populations of the AD and AI
cells, i.e. y = cDxD+ cIxI. Thus it may be generally more
appropriate to consider the serum PSA level y = f (xD,xI) as
a nonlinear function of the populations of cancer cells in more
realistic modeling.
(31)
3.3. Origin of nonlinearity
AsimplifiedversionoftheIAStherapymodelisinvestigated
in order to clarify the origin of nonlinear phenomena and
transient grazing in the hybrid system. Let us suppose, for the
sake of simplicity, that androgen concentration instantaneously
changes to steady values in response to switching between on
administration and offadministration periods, i.e. γ → ∞ in
Eq. (21). Androgen concentration a is then considered to take
alternative values 0 and a0. Androgen concentration switches
Fig. 7. Bifurcation diagrams of the IAS therapy model (21)–(25). Parameter regions for relapse prevention correspond to those of nondivergent solutions. Period
doubling and grazing bifurcations of a limit cycle are denoted by PD1and G1, respectively. (a) a0= 28. (b) a0= 30. (c) a0= 32.
Page 8
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
2623
from a0to 0 when an increasing PSA level y reaches the upper
threshold value r1, whereas it switches from 0 to a0when a
decreasing PSA level touches the lower threshold value r0.
Under this assumption, the IAS therapy model given by Eqs.
(21) and (22) is reduced to a piecewise linear system described
as follows:
dxD
dt
dxI
dt
= (gD(a) − m(a))xD,
(32)
= m(a)xD+ gI(a)xI,
?0 → a0
The simplified model can be viewed as a hybrid system
consisting of the following two linear systems:
?xD
d
dt
xI
m1
w1
xI
(33)
a :
when y = r0and dy/dt < 0
when y = r1and dy/dt > 0.
a0→ 0
(34)
d
dt
xI
?
?
=
?v0
?v1
0
0
w0
??xD
??xD
xI
?
?
,
with a = a0,
(35)
?xD
=
0
,
with a = 0,
(36)
where v0 = gD(a0), w0 = gI(a0), v1 = gD(0) − m1, and
w1 = gI(0). Eventtriggering sections are given by Eqs. (26)
and (27) with the continuous state x = (xD,xI).
Analytical solutions of individual linear equations (35) and
(36) can be easily obtained. We assume that a trajectory starting
from (x0,r0− x0) on Π0reaches Π1with traveling time τ0and
returns to Π0with traveling time τ1. For such a trajectory, a
onedimensional Poincar´ e map (return map) can be constructed
with respect to an initial condition x0[42] as follows:
R(x0) = ev1τ1+v0τ0x0,
where τ0and τ1are the implicit functions of x0, satisfying
(37)
f (ev0τ0x0,ew0τ0(r0− x0)) − r1= 0,
f (ev1τ1z(x0),α(ev1τ1− ew1τ1)z(x0) + ew1τ1(r1− z(x0)))
−r0= 0,
whereα = m1/(v1−w1)and z(x0) = ev0τ0x0.Periodicmotions
can be well understood by observing a sequence of intersecting
points between a trajectory and the section Π0. Fig. 8 shows re
turn plots calculated from Eqs. (37) to (39) for different param
eter values of a0. The fixed point of the return map corresponds
to a limit cycle of the simplified model. In Fig. 8(a) and (b), the
fixed point is globally stable in the domain of (0,r0) for the ini
tial condition x0. In Fig. 8(c), on the other hand, the return plot
ends at a certain value of x0, at which transient grazing occurs.
Transient grazing is characterized by a trajectory of the system
touching the section Π0tangentially [44]. Namely, the return
ing trajectory loses an intersection with the section Π0at the
point of transient grazing.
The location of a trajectory starting from the initial point
(x0,r0−x0)attimeτ > τ0isdenotedby(ϕD(x0,τ),ϕI(x0,τ))
where
(38)
(39)
ϕD(x0,τ) = ev1(τ−τ0)+v0τ0x0,
(40)
Fig. 8. Return plots of the simplified model (32)–(34) withr0= 5 andr1= 30,
which is represented by Eq. (37). (a) a0= 30. (b) a0= 25. (c) a0= 20.
Fig. 9. A trajectory exhibiting transient grazing. The initial condition is given
by the value of x0at which the return plot ends in Fig. 8(c).
ϕI(x0,τ) = α(ev1(τ−τ0)− ew1(τ−τ0))ev0τ0x0
+ew1(τ−τ0)+w0τ0(r0− x0).
Using these notations, we can formulate conditions for a
trajectory that exhibits transient grazing with respect to Π0as
follows:
?
which are solved with respect to unknown variables (x0,τ).
Fig. 9 shows a grazing trajectory with initial condition x0at
the endpoint of the return plot in Fig. 8(c).
Next we consider bifurcations of a limit cycle of the
simplified model. Conditional equations for a perioddoubling
bifurcation of the fixed point of the return map with a
bifurcation parameter λ are described as follows:
?R(x0) − x0
which are solved with respect to unknown variables (x0,λ). By
combining the conditional equation (42) for transient grazing
with a fixed point condition, on the other hand, the conditions
for a grazing bifurcation of a limit cycle can be formulated as
follows:
(v1+ m1)ϕD(x0,τ) + w1ϕI(x0,τ)
= 0,
which are solved with respect to unknown variables (x0,τ,λ).
Fig. 10 shows a bifurcation diagram of the simplified model
(32)–(34) with r1 = 30, indicating the parameter region of
relapse prevention. The return plots at the parameter values
indicated by (a)–(c) of Fig. 10 are shown in Fig. 8(a)–(c),
respectively. The perioddoubling bifurcation denoted by PD1
runs along the boundary of the parameter region for relapse
prevention. We can also confirm that the parameter set of the
grazing bifurcation denoted by G1exists very close to that
(41)
Ftg(x0,τ) ≡
ϕD(x0,τ) + ϕI(x0,τ) − r0
(v1+ m1)ϕD(x0,τ) + w1ϕI(x0,τ)
?
= 0, (42)
Fpd(x0,λ) ≡
R?(x0) + 1
?
= 0,
(43)
Fpg(x0,τ,λ) ≡
ϕD(x0,τ) − x0,
ϕD(x0,τ) + ϕI(x0,τ) − r0
(44)
Page 9
2624
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
Fig. 10. Bifurcation diagram of the simplified model (32)(34) with r1= 30.
The parameter region of relapse prevention corresponds to that of nondivergent
solutions. Perioddoubling and grazing bifurcations of a limit cycle are denoted
by PD1and G1, respectively. The inset for enlargement shows that the grazing
bifurcation set exists very close to the perioddoubling one. The return plots
in Fig. 8(a)–(c) are observed at the parameter values indicated by (a)–(c),
respectively.
of the perioddoubling bifurcation as shown in the inset of
Fig. 10. From comparison between Figs. 4 and 10, we can
see that the parameter dependency of the region of relapse
prevention is quite similar between the IAS therapy model and
its simplified version. Therefore, the dynamical property of the
IAS therapy model is considered to be nearly equivalent to that
of the simplified system consisting of two linear systems. It
suggests that nonlinear phenomena of the IAS therapy model
originate from the nonlinearity generated by switching between
subsystems rather than from that of individual subsystems.
4. Conclusions
We have studied nonlinear dynamics and bifurcations in
the hybrid systems model representing intermittent androgen
suppression therapy for advanced prostate cancer. First, we
have introduced a numerical method to specify a limit cycle
and its bifurcation sets in a general hybrid automaton. Stability
and bifurcations of a limit cycle have been reduced to those
of a fixed or a periodic point of the Poincar´ e map. Then,
the method has been applied to analysis of the IAS therapy
model [24,25]. We have considered the effect of clinically
controllable parameters on relapse prevention, where a solution
is not divergent but confined in a finite region of continuous
state space. It has been shown that the possibility of relapse
prevention can be enhanced by an increase of the normal
androgen level during offmedication periods. This result has
suggested that androgen supplementation can be effective for
relapse prevention depending on the net growth rate of a tumor.
We have shown that the ratio between the upper threshold
value of the serum PSA level to restart medication and the
lower one to interrupt medication plays a significant role in
the dosing strategy that we have adopted in this paper. From
bifurcation analysis, we have found that the model generates
complex chaotic solutions concerned with the nested structure
of a pair of perioddoubling and grazing bifurcations. Such
a chaotic attractor should also be effective for IAS therapy
because it remains within a bounded region of state space like
stable limit cycles, although it may be difficult to clinically
observe nonlinear behavior just by several on–off cycles in real
treatment.
Furthermore, we have investigated a piecewise linear system
reduced from the IAS therapy model. Return plot analysis
has been helpful for understanding the nonlinearity generated
by the hybrid system and transient grazing. It has also been
clarified that the bifurcation structure of the IAS therapy model
is approximately inherited by the simplified model.
Hybrid or switched dynamical systems would be a useful
framework in modeling intermittent therapy which is worth
considering not only for prostate cancer but also for other
diseases. Treatment interruption has been practiced based on
empirical or heuristic factors so far. Currently, systematic
understanding of disease progression under intermittent therapy
is becoming more and more important. In this regard,
medication control based on a mathematical model of
intermittent therapy could open a new frontier for pursuit of
better treatment in biomedical science. For developing this
attempt further, modeling of biomedical systems should be
carefully advanced in collaboration with medical doctors and
biomedical researchers as an interdisciplinary research.
Acknowledgments
The authors are grateful to Prof. Nicholas Bruchovsky
of The Prostate Center at Vancouver General Hospi
tal, Dr. Takashi Shimada of University of Tokyo, and
Dr. Yoshito Hirata of ERATO Aihara Complexity Modelling
Project of JST for valuable comments and fruitful discussions.
Appendix A. Derivatives of the Poincar´ e map P
We need to numerically compute first and the second
order derivatives of the Poincar´ e map P with respect to the
initial condition w and a system parameter λ in order to solve
conditional equations for a limit cycle or its bifurcation by
Newton’s method. From Eq. (9), the derivatives of P are
represented by those of T as follows:
∂P
∂w=∂h
∂P
∂λ=∂h
∂x
∂T
∂x0
?∂T
∂2T
∂x2
∂h−1
∂w,
(A.1)
∂x
∂λ+∂T
?∂h−1
?
∂x0∂λ
∂x0
∂h−1
∂λ
?2
∂h−1
∂w
?
,
(A.2)
∂2P
∂w2=∂h
∂x
0
∂w
,
(A.3)
∂2P
∂w∂λ=∂h
∂x
∂2T
+∂2T
∂x2
0
∂h−1
∂w
∂h−1
∂λ
?
.
(A.4)
Page 10
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
2625
From Eq. (6), the derivatives of T are represented by those of
the submaps T0and T1as follows:
∂T
∂x0
∂T
∂λ=∂T1
=∂T1
∂x1
?∂T0
?∂T0
?
∂x0
?
?
,
(A.5)
∂x1
∂λ
+∂T1
?
∂λ,
(A.6)
∂2T
∂x2
0
=∂T1
∂x1
∂2T0
∂x2
?∂2T0
∂2T1
∂x1∂λ
0
+∂2T1
∂x2
?
?∂T0
1
?∂T0
∂x0
?2
,
(A.7)
∂2T
∂x0∂λ
=
∂T1
∂x1
∂x0∂λ
+∂2T1
∂x2
?
1
?∂T0
∂x0
??∂T0
∂λ
?
+
∂x0
.
(A.8)
From Eqs. (4) and (5), the derivatives of each submap Tu
(u = 0,1) are described as follows:
∂Tu
∂xu
∂xu
∂Tu
∂λ ∂λ
∂2Tu
∂x2u
∂x2u
∂xu∂t
∂xu
∂2Tu
∂xu∂λ
∂xu∂t
∂2τu
∂xu∂λ.
=∂ϕu
+ fu∂τu
∂xu,
(A.9)
=∂ϕu
=∂2ϕu
+ fu∂τu
+∂2ϕu
∂2ϕu
∂xu∂λ+∂2ϕu
∂λ,
(A.10)
∂τu
+∂2ϕu
∂t∂xu
∂τu
∂xu
+ fu∂2τu
∂x2u
,
(A.11)
=
∂τu
∂λ+∂ϕu
∂t∂λ
∂τu
∂xu
+fu
(A.12)
A trajectory leaving from the initial condition xuon Πu(u =
0,1) is assumed to reach the other section Πu? with traveling
time τuunder the system Suas follows:
qu?(ϕu(τu(xu),xu)) = 0,
where u?= 1 − u. If a trajectory transversally intersects with
the section,
(A.13)
∂qu?
∂xu
· fu?= 0.
(A.14)
Under this assumption, by differentiating both sides of Eq.
(A.13) with respect to the initial condition xu, the derivatives
of τuare explicitly described as follows:
∂τu
∂xu
= −
1
∂qu?
∂xfu
1
∂qu?
∂xfu
?∂qu?
?∂qu?
∂qu?
∂x
∂x
∂ϕu
∂xu
?
+∂qu?
,
(A.15)
∂τu
∂λ
= −
∂x
∂ϕu
∂λ
?∂2ϕu
∂λ
?
,
(A.16)
∂2τu
∂x2u
= −
1
∂qu?
∂xfu
∂x2u
+∂2ϕu
∂xu∂t
∂τu
∂xu
+∂2ϕu
∂t∂xu
∂τu
∂xu
?
,
(A.17)
∂2τu
∂xu∂λ
= −
1
∂qu?
∂xfu
?∂2ϕu
∂qu?
∂x
×
∂xu∂λ+∂2ϕu
∂xu∂t
∂τu
∂λ+∂2ϕu
∂t∂λ
∂τu
∂xu
?
.
(A.18)
The derivatives of ϕucan be calculated from first and second
order variational equations given as follows:
?∂ϕu
d
dt
∂λ ∂xu
∂λ
?∂2ϕu
d
dt
∂xu∂λ∂xu
∂xu∂λ
∂2fu
∂xu∂λ∂xu
d
dt
∂xu
?
?
=∂fu
∂xu
=∂fu
?
?
?∂ϕu
?∂ϕu
?∂2ϕu
∂fu
∂xu
?
?
,
(A.19)
?∂ϕu
+∂fu
?
∂λ,
(A.20)
d
dt
∂x2u
=∂fu
∂xu
∂x2u
?∂2ϕu
+∂2fu
∂x2u
?
?∂ϕu
?∂ϕu
∂xu
?2
,
(A.21)
?∂2ϕu
=+∂2fu
∂x2u
?
?∂ϕu
∂xu
??∂ϕu
∂λ
?
(A.22)
+
.
The fourthorder Runge–Kutta method is employed for
numerical integration of the above equations with the following
initial conditions:
????t=0
∂x2u
t=0
where In and On indicate n × n identity and null matrices,
respectively.
∂ϕu
∂xu
∂2ϕu
= In,
∂ϕu
∂λ
????t=0
= On,
????t=0
????
= On,
∂2ϕu
∂xu∂λ
= On,
(A.23)
Appendix B. Derivatives of the map¯T
From Eq. (18), the derivatives of¯T with respect to the initial
condition x0, the return time τ, and the bifurcation parameter λ,
are calculated as follows:
∂¯T
∂x0
∂¯T
∂τ
∂¯T
∂λ=∂ϕ1
=∂ϕ1
∂x1
∂¯T2m−1
∂x0
− f1∂ ¯ τ2m−1
∂x0
,
(B.1)
= f1,
(B.2)
∂x1
∂¯T2m−1
∂λ
− f1∂ ¯ τ2m−1
∂λ
+∂ϕ1
∂λ.
(B.3)
In a similar way, derivatives of ¯ τ2m−1 can be sequentially
calculated by using the following recurrence formula:
∂ ¯ τk
∂x0
∂ ¯ τk
∂λ
=∂ ¯ τk−1
∂x0
=∂ ¯ τk−1
+∂τk
∂xk
+∂τk
∂xk
∂¯Tk−1
∂x0
∂¯Tk−1
∂λ
,
k = 1,...,2m − 1,
(B.4)
∂λ
+∂τk
∂λ,
k = 1,...,2m − 1,
(B.5)
which are derived from Eq. (17). From Eqs. (15) and (16),
derivatives of ¯T2m−1 can be calculated by using recurrence
Page 11
2626
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
equations for derivatives of¯T2kand¯T2k+1given as follows:
∂¯T2k
∂x0
∂x2k−1
∂¯T2k
∂λ∂x2k−1
=
∂T1
?∂¯T2k−1
?∂¯T2k−1
∂x0
?
?
,
k = 0,1,...,m − 1,
+∂T0
(B.6)
=
∂T1
∂λ∂λ,
k = 0,1,...,m − 1,
(B.7)
∂¯T2k+1
∂x0
∂¯T2k+1
∂λ
=
∂T0
∂x2k
∂T0
∂x2k
?∂¯T2k
?∂¯T2k
∂x0
?
?
,
k = 0,1,...,m − 1,
+∂T1
(B.8)
=
∂λ∂λ,
k = 0,1,...,m − 1.
(B.9)
The derivatives of T0and T1can be calculated by Eqs. (A.9)
and (A.10).
References
[1] H.M. Byrne, T. Alarcon, M.R. Owen, S.D. Webb, P.K. Maini, Modelling
aspects of cancer dynamics: A review, Phil. Trans. R. Soc. Lond. A 364
(2006) 1563–1578.
[2] A.L. Garner, Y.Y. Lau, D.W. Jordan, M.D. Uhler, R.M. Gilgenbach,
Implications of a simple mathematical model to cancer cell population
dynamics, Cell Prolif. 39 (1) (2006) 15–28.
[3] T.L. Jackson, A mathematical model of prostate tumor growth and
androgenindependent relapse, Disc. Cont. Dyn. Syst.Series B 4 (1)
(2004) 187–201.
[4] T.L. Jackson, A mathematical investigation of the multiple pathways to
recurrent prostate cancer: Comparision with experimental data, Neoplasia
6 (6) (2004) 697–704.
[5] K.R. Swanson, L.D. True, D.W. Lin, K.R. Buhler, R. Vessella, J.D.
Murray, A quantitative model for the dynamics of serum prostatespecific
antigen as a marker for cancerous growth, Am. J. Pathol. 158 (6) (2001)
2195–2199.
[6] P.W.A. Dayananda, J.T. Kemper, M.M. Shvartsman, A stochastic model
for prostatespecific antigen levels, Math. Biosci. 190 (2004) 113–126.
[7] D. Veestraeten, An alternative approach to modelling relapse in cancer
with an application to adenocarcinoma of the prostate, Math. Biosci. 199
(2006) 38–54.
[8] K. Akakura, N. Bruchovsky, S.L. Goldenberg, P.S. Rennie, A.R. Buckley,
L.D. Sullivan, Effects of intermittent androgen suppression on androgen
dependent tumors: Apoptosis and serum prostatespecific antigen, Cancer
71 (9) (1993) 2782–2790.
[9] A.M.Evens,T.M.Lestingi,J.D.Bitran,Intermittentandrogensuppression
as a treatment for prostate cancer: A review, The Oncologist 3 (1998)
419–423.
[10] M.S. Bhandari, J. Crook, M. Hussain, Should intermittent androgen
deprivation be used in routine clinical practice? J. Clin. Oncol. 23 (32)
(2005) 8212–8218.
[11] N. Mottet, C. Lucas, E. Sene, C. Avances, L. Maubach, J.M. Wolff,
Intermittent androgen castration: A biological reality during intermittent
treatment in metastatic prostate cancer? Urol. Int. 75 (2005) 204–208.
[12] N. Bruchovsky, L.H. Klotz, M. Sadar, J.M. Crook, D. Hoffart, L. Godwin,
M. Warkentin, M.E. Gleave, S.L. Goldenberg, Intermittent androgen
suppression for prostate cancer: Canadian prospective trial and related
observations, Mol. Urol. 4 (3) (2000) 191–199.
[13] N. Bruchovsky, S.L. Goldenberg, N.R. Mawji, M.D. Sadar, Evolving
aspects of intermittent androgen blockade for prostate cancer: Diagnosis
and treatment of early tumor progression and maintenance of remission.
In: Andrology in the 21st Century, Proceedings of the VIIth International
Congress of Andrology, 2001, pp. 609–623.
[14] N. Bruchovsky, L. Klotz, J. Crook, S. Malone, C. Ludgate, W.J. Morris,
M.E. Gleave, S.L. Goldenberg, Final results of the canadian prospective
phase II trial of intermittent androgen suppression for men in biochemical
recurrence after radiotherapy for locally advanced prostate cancer, Cancer
107 (2) (2006) 389–395.
[15] N. Bruchovsky, L. Klotz, J. Crook, S. Larry, S.L. Goldenberg, Locally
advanced prostate cancer – biochemical results from a prospective phase
II study of intermittent androgen suppression for men with evidence of
PSA relapse after radiotherapy, Cancer 109 (5) (2007) 858–867.
[16] D.G. McLeod, Hormonal therapy: Historical perspective to future
directions, Urology 61 (2003) 3–7.
[17] L. Klotz, Combined androgen blockade in prostate cancer: Metaanalyses
and associated issues, BJU Int. 87 (2001) 806–813.
[18] E.D. Crawford, M. Rosenblum, A.M. Ziada, P.H. Lange, Overview:
Hormone refractory prostate cancer, Urology 54 (Suppl. 1) (1999) 1–7.
[19] M. Diaz, S.G. Patterson, Management of androgenindependent prostate
cancera, Cancer Control 11 (6) (2004) 364–373.
[20] B.J. Feldman, D. Feldman, The development of androgenindependent
prostate cancer, Nat. Rev. Cancer 1 (2001) 34–45.
[21] S.M. Dehm, D.J. Tindall, Regulation of androgen receptor signaling in
prostate cancer, Expert Rev. Anticancer Ther. 5 (1) (2005) 63–74.
[22] J. Edwards, J. Bartlett, The androgen receptor and signaltransduction
pathways in hormonerefractory prostate cancer. Part 1: Modifications to
the androgen receptor, BJU Int. 95 (2005) 1320–1326.
[23] M.E. Taplin, Drug insight: Role of the androgen receptor in the
development and progression of prostate cancer, Nat. Clin. Pract. Oncol.
4 (4) (2007) 236–244.
[24] A.M. Ideta, G. Tanaka, T. Takeuchi, K. Aihara, A mathematical model of
intermittent androgen suppression remedy for prostate cancer, Technical
Report, Department of Mathematical Informatics, Graduate School of
Information Science and Technology, University of Tokyo, 200632, May
2006.
[25] A.M. Ideta, G. Tanaka, T. Takeuchi, K. Aihara, A mathmatical model of
intermittent androgen suppression for prostate cancer, J. Nonlinear Sci.
(in press).
[26] K. Aihara, G. Tanaka, T. Suzuki, Y. Hirata, A hybrid systems approach
to hormonal therapy of prostate cancer and its nonlinear dynamics, in:
Noise and Fluctuations: 19th International Conference on Noise and
Fluctuations, Sep 2007, pp. 479–482.
[27] A.J. van der Schaft, J.M. Schumacher, An Introduction to Hybrid
Dynamical Systems, in: Lecture Notes in Control and Information
Sciences, vol. 251, Springer, 2000.
[28] A.V. Savkin, R.J. Evans, Hybrid Dynamical Systems: Controller and
Sensor Switching Problems, SpringerVerlag, 2001.
[29] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston,
2003.
[30] H.E. Garcia, A. Ray, R.M. Edwards, A reconfigurable hybrid systems and
its application to power plant control, IEEE Trans. Control Syst. Tech. 3
(2) (1995) 157–170.
[31] B. Lennartson, M. Tittus, B. Egardt, S. Pettersson, Hybrid systems in
process control, Contr. Syst. Mag. 16 (1996) 45–56.
[32] B. Brogliato, Nonsmooth Mechanics: Models, Dynamics, and Control,
2nd edition, SpringerVerlag, New York, 1999.
[33] M. Spong, M. Vidyasagar, Robot Dynamics and Control, Wiley, New
York, 1989.
[34] I. Hiskens, Power system modeling for inverse problems, IEEE Trans.
Circuits Syst.I 51 (2004) 539–551.
[35] S. Banerjee, G.C. Verghese (Eds.), Nonlinear Phenomena in Power
Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, IEEE
Press, New York, 2001.
[36] M. di Bernardo, C.K. Tse, Chaos in power electronics: An overview,
in: Chaos in Circuits and Systems, World Scientific, Singapore, 2002,
pp. 317–340.
[37] P.J. Antsaklis, X.D. Koutsoukos, Hybrid systems: Review and recent
Page 12
G. Tanaka et al. / Physica D 237 (2008) 2616–2627
2627
progress, in: T. Samad, G. Balas (Eds.), SoftwareEnabled Control:
Information Technology for Dynamical Systems, IEEE Press, 2003,
pp. 272–298.
[38] S.D. Johnson, Simple hybrid systems, Int. J. Bifurcat. Chaos 4 (6) (1994)
1655–1665.
[39] J. Guckenheimer, S. Johnson, Planar hybrid systems, in: Hybrid Systems
II, in: Lecture Notes in Computer Science, vol. 999, 1995, pp.
202–225.
[40] M. di Bernardo, C.J. Budd, A.R. Champneys, Grazing and border
collision in piecewisesmooth systems, Phys. Rev. Lett. 86 (12) (2001)
2553–2556.
[41] H.E. Nusse, E. Ott, J.A. Yorke, Bordercollision bifurcations: An
explanation for observed bifurcation phenomena, Phys. Rev. E 49 (1994)
1073–1077.
[42] T. Saito, S. Nakagawa, Chaos from a hysteresis and switched circuit,
Philos. Trans. R. Soc. Lond. A 353 (1995) 47–57.
[43] T. Kousaka, T. Ueta, H. Kawakami, Bifurcation of switched nonlinear
dynamical systems, IEEE Trans. Circ. Syst.II 46 (7) (1999) 878–885.
[44] V. Donde, I.A. Hiskens, Shooting methods for locating grazing
phenomena in hybrid systems, Int. J. Bifurcat. Chaos 16 (3) (2006)
671–692.
[45] R.R. Berges, J. Vukanovic, J.I. Epstein, M. CarMichel, L. Cisek, D.E.
Johnson, R.W. Veltri, P.C. Walsh, J.T. Isaacs, Implication of cell kinetic
changes during the progression of human prostatic cancer, Clin. Cancer
Res. 1 (1995) 473–480.
[46] J. Kokontis, K. Takakura, N. Hay, S. Liao, Increased androgen receptor
activity and altered cmyc expression in prostate cancer cells after
longterm androgen deprivation, Cancer Res. 54 (1994) 1566–1573.
[47] Y.Umekita,R.A.Hiipakka,J.M.Kokontis,S.Liao,Humanprostatetumor
growth in athymic mice: Inhibition by androgens and stimulation by
finasteride, Proc. Natl. Acad. Sci. 93 (1996) 11802–11807.
[48] R.T. Prehn, On the prevention and therapy of prostate cancer by androgen
administration, Cancer Res. 59 (1999) 4161–4164.
[49] Y. Ma, H. Kawakami, Bifurcation analysis of switched dynamical systems
with periodically moving borders, IEEE Trans. Circ. Syst.I 51 (6) (2004)
1184–1193.