A Systems Biology Approach in Therapeutic Response Study for Different Dosing Regimens-a Modeling Study of Drug Effects on Tumor Growth using Hybrid Systems.

Abstract

Motivated by the frustration of translation of research advances in the molecular and cellular biology of cancer into treatment, this study calls for cross-disciplinary efforts and proposes a methodology of incorporating drug pharmacology information into drug therapeutic response modeling using a computational systems biology approach. The objectives are two fold. The first one is to involve effective mathematical modeling in the drug development stage to incorporate preclinical and clinical data in order to decrease costs of drug development and increase pipeline productivity, since it is extremely expensive and difficult to get the optimal compromise of dosage and schedule through empirical testing. The second objective is to provide valuable suggestions to adjust individual drug dosing regimens to improve therapeutic effects considering most anticancer agents have wide inter-individual pharmacokinetic variability and a narrow therapeutic index. A dynamic hybrid systems model is proposed to study drug antitumor effect from the perspective of tumor growth dynamics, specifically the dosing and schedule of the periodic drug intake, and a drug's pharmacokinetics and pharmacodynamics information are linked together in the proposed model using a state-space approach. It is proved analytically that there exists an optimal drug dosage and interval administration point, and demonstrated through simulation study.

Full text

Page 1

Open Access
Full open access to this and
thousands of other papers at
http://www.la-press.com.
Cancer Informatics 2012:11 41–60
doi: 10.4137/CIN.S8185
This article is available from http://www.la-press.com.
© the author(s), publisher and licensee Libertas Academica Ltd.
This is an open access article. Unrestricted non-commercial use is permitted provided the original work is properly cited.
Cancer Informatics
O r I g I N A L r e S e A r C h
Cancer Informatics 2012:11
41
A systems Biology Approach in Therapeutic Response study
for Different Dosing Regimens—a Modeling study of Drug
effects on Tumor Growth using Hybrid systems
Xiangfang Li1, Lijun Qian2, Michale L. Bittner3 and edward r. Dougherty1,3,4
1Department of electrical and Computer engineering, Texas A&M University, College Station, TX 77843, USA.
2Department of electrical and Computer engineering, Prairie View A&M University, Prairie View, TX 77446, USA.
3Computational Biology Division, Translational genomics research Institution, Phoenix, AZ 85004, USA. 4Department of
Bioinformatics and Computational Biology, University of Texas M.D. Anderson Cancer Center, houston, TX 77030, USA.
Corresponding author email: edward@ece.tamu.edu
Abstract: Motivated by the frustration of translation of research advances in the molecular and cellular biology of cancer into treatment,
this study calls for cross-disciplinary efforts and proposes a methodology of incorporating drug pharmacology information into drug
therapeutic response modeling using a computational systems biology approach. The objectives are two fold. The first one is to involve
effective mathematical modeling in the drug development stage to incorporate preclinical and clinical data in order to decrease costs
of drug development and increase pipeline productivity, since it is extremely expensive and difficult to get the optimal compromise of
dosage and schedule through empirical testing. The second objective is to provide valuable suggestions to adjust individual drug dosing
regimens to improve therapeutic effects considering most anticancer agents have wide inter-individual pharmacokinetic variability and
a narrow therapeutic index. A dynamic hybrid systems model is proposed to study drug antitumor effect from the perspective of tumor
growth dynamics, specifically the dosing and schedule of the periodic drug intake, and a drug’s pharmacokinetics and pharmacodynam-
ics information are linked together in the proposed model using a state-space approach. It is proved analytically that there exists an
optimal drug dosage and interval administration point, and demonstrated through simulation study.
Keywords: drug effect, drug efficacy region, dosing regimens, hybrid systems, systems biology, tumor growth

Page 2

Li et al
42
Cancer Informatics 2012:11
Introduction
The past three decades have seen spectacular advances
in our understanding of the molecular and cellular
biology of cancer. However, data suggest that the
overall success rate for oncology products in clinical
development is ∼10%, and the cost of bringing a new
drug to market is over US $1 billion.1 Oncology drug
development is such an expensive and prolonged
process, typically, a new drug requiring on average
10 years.2,3 New tools are needed to accelerate the
drug discovery process and increase productivity.4,5
While producing information both at the basic and
clinical level is no longer the issue,6 the effective inte-
gration of data and knowledge from many disparate
sources will be crucial to future cancer research.7,8
Systems biology approaches promise to have a
profound impact on medical practice by bringing
together efforts from cross disciplinary scientists and
permitting a comprehensive evaluation of underlying
predisposition to disease, disease diagnosis, disease
progression and disease treatment.9–11
While providing the right drug for the right patient
is very important, finding the right dose for each
patient is also critical but tricky.12 Finding a dose and
dose range of a drug candidate that are both effica-
cious and safe is a fundamental objective through
the drug discovery process.13 Dose finding happens
throughout the long process of drug discovery, from
non clinical development to multi-phase clinical trials.
Even after the drug is approved and available on the
market, new drug doses are still studied carefully
and the level of investigation depends on responses
observed from the general patient population. When
necessary, dose adjustment based on post-marketing
information is still a common practice. However, it is
extremely expensive and difficult to get the optimal
compromise of dosage and schedule through empiri-
cal testing. Modeling and simulation analysis, which
can evolve and be continuously updated throughout
different stages to incorporate relevant new data,
will help to make crucial decisions earlier, with more
certainty, and at lower cost, and hence can add value
in all stages of drug development.5,14
The complexity of cancer itself and the heteroge-
neity of therapeutic responses may make dosing study
more complicated. For example, most anticancer
agents have wide inter-individual pharmacokinetic
(PK) variability and a narrow therapeutic index.15
Recent works have shown that many patients who are
currently being treated with 5-fluorouracil (5-FU) are
not being given the appropriate doses to achieve opti-
mal plasma concentration. Of note, only 20%–30%
of patients are treated in the appropriate dose range,
approximately 40%–60% of patients are being under-
dosed, and 10%–20% of patients are overdosed.16
Traditionally, the standard approach for calculating
5-FU drug dosage, as with many anticancer agents,
has been done by normalizing dose to body surface
area (BSA), which is calculated from the height and
weight of the patient;16 however, studies have shown
that this is inadequate.17 For example, dosing based
on BSA is associated with considerable variability
in plasma 5-FU levels by as much as 100-fold,15,17
and such variability is a major contributor to toxic-
ity and treatment failure.16 Since there are many fac-
tors collaboratively affecting drug effect variability,18
a general approach is needed to facilitate quantita-
tive thinking to drug administration regimens. Drug
dosing regimens could be tailored to each individual
patient based on feedback information from the
treatment. One challenge of such modeling is how to
link relevant biomarkers19 or surrogate endpoints to
treatment outcome as feedback information in order
to give valuable dosing suggestions.
Traditional design of the dosing regimen based
on achieving some desired target goal such as rela-
tively constant serum concentration may be far from
optimal owing to the underlying dynamic biologi-
cal networks. For example, Shah and co-workers20
demonstrate that the BCR-ABL inhibitor dasatinib,
which has greater potency and a short half-life, can
achieve deep clinical remission in CML patients
by achieving transient potent BCR-ABL inhibition,
while traditional approved tyrosine kinase inhibitors
usually have prolonged half lives that result in con-
tinuous target inhibition. A similar study of whether
short pulses of higher dose or persistent dosing with
lower doses have the most favorable outcomes has
been carried out by Amin and co-workers21 in the
setup of inactivation of HER2-HER3 signaling. For
best results, models should be selected based on the
underlying mechanism of drug action. For example,
detailed dynamic signal transduction models are
needed to accommodate target-receptor interaction
and feedback loops for analyzing dosing effect in the
above examples.

Page 3

A therapeutic response study for different dosing regimens
Cancer Informatics 2012:11
43
Computational systems biology is emerging
as a valuable tool in therapeutics to address these
challenges.10,22–25 This approach provides functional
understanding of disease-drug interaction and marks
a shift from the traditional “black-box” approach.
In this study, a general methodology incorporating
dynamic drug pharmacology information into drug
therapeutic response modeling using computational
systems biology is proposed. The process begins with
building a quantitative model of a biological system.
Then, by incorporating related pharmacology infor-
mation relevant to the target system, a new computa-
tional model under drug perturbation can be built. We
believe that with the help from the theoretical model-
ing proposed in this study and through an iterative
process with experimentalists to refine the model, the
proposed methodology has the potential to supply
better recommendations for dosage and frequency.
Modeling
A good model should be based on a sound under-
standing of the biological problem, hold a realistic
mathematical representation of the biological phe-
nomena, and possess a tractable solution.26 A bio-
logical interpretation of the deductions resulting from
such a model can yield non-intuitive insights, as well
as provide a predictive framework,10 a vital issue in
cancer treatment. In recent years it has become clear
that carcinogenesis is a complex process, both at the
molecular and cellular levels.25,27 Modeling biologi-
cal systems to develop computer models of disease
that can be used to understand disease mechanisms
and to test in silico approaches for treating disease is
a key issue in moving forward. Recent mathematical
advances have made it more feasible to model cancer
from a mathematical viewpoint. There are numerous
works modeling cancer at multiple levels and scales,
ranging from molecules to cells to tissues.7,28 For
example, multi-scale models have been developed
that can capture interactions across different spatial
and temporal scales.8 A number of researchers have
recommended hybrid or hierarchical systems to com-
bine the strengths of both discrete and continuous
approaches.8,29,30
Biological systems are naturally nonlinear; however,
purely nonlinear continuous models of biological sys-
tems can be too large and complex for simulation and
analysis. On the other hand, a linear continuous model
or a fully discrete approximation of the model can
sometimes lose crucial and pertinent information.31
Hybrid systems32 provide a rigorous foundation for
modeling biological systems at desired levels of
abstraction, approximation, and simplifications.33 For
example, systems that exhibit multi-scale dynamics
can be simplified by replacing certain slowly changing
variables by their piecewise constant approximations.
Additionally, sigmoidal nonlinearities are commonly
observed in biology and the corresponding models
often use sigmoidal functions. These can be approxi-
mated by discrete transitions between piecewise-linear
regions. In some instances, nondeterministic upper
and lower bounds are more useful than determinis-
tic approximations because they capture all critical
behavior of the system.33–35 The hybrid systems model
encapsulates a broad space of models and systems. For
example, the Lac operon system has been well studied
both experimentally and using continuous models.36,37
A hybrid model and use of a reachability algorithm
were validated by comparison with experimental data
and continuous models.38 Other biological hybrid sys-
tems analyzed in similar ways include the Delta-Notch
decision process,39,40 genetic regulatory networks of
carbon starvation,41 nutritional stress response42 in
E. coli, and our previous work on drug effect model-
ing under genetic regulatory networks.43 In this study,
we adopt hybrid systems models to accommodate the
hybrid nature of disease progression and therapeutic
responses. Specifically, a tumor growth model under
drug perturbation is studied to demonstrate how to
integrate diverse data and ultimately predict outcomes
for clinical purposes.
Tumor growth model using hybrid
systems
Cancer research has been a fertile ground for mathe-
matical modeling.44 A number of mathematical tumor
growth models have been reported in the literature,
reflecting different paradigms. Empirical models use
mathematical equations to describe the tumor growth
curve without in-depth mechanistic description of the
underlying physiological processes. Initially, models
were used to conceptualize the simple exponential
growth of solid tumors.45–47 Subsequently, sigmoi-
dal functions such as logistic, Verhulst, Gompertz,
and von Bertalanffy were used for the description
of reduced growth in the later stages as the tumor

Page 4

Li et al
44
Cancer Informatics 2012:11
cells outgrew their blood supply, producing central
necrosis.48–51 A drawback of this model class is
that it is not straightforward to predict modifica-
tion of the growth curve under drug perturbation.
Functional models, conversely, are based on mecha-
nistic descriptions of biological processes underlying
tumor growth. Such models require a set of assump-
tions involving cell cycle kinetics (proliferating vs.
quiescent cells) and biochemical processes, such as
those related to antiangiogenic and/or immunologi-
cal responses.52,53 Owing to the biological complexity
they try to capture, these models have a much larger
number of parameters compared with the empirical
models. Hence, in addition to the standard tumor
growth measurements, further data are needed, such
as flow cytometry analysis and measurements of bio-
chemical and immunological markers, to avoid iden-
tification problems due to the over parametrization.53
The situation becomes even more complex when the
effect of treatment with an anticancer drug is consid-
ered on account of the incomplete knowledge of the
mode of action in vivo.
It is important to realize that all models have limi-
tations, including those in oncology: simple models
may produce insights and describe existing data, but
they risk oversimplification and oversight of critical
variables; on the other hand, it is generally difficult to
fit functional models versus experimental data since
over parametrization can be avoided only if further
“microscopic” observations are available. Hence, it
is a challenge to achieve a correct balance between
empirical and functional models. In this paper, we
adopt a model that is a compromise between empiri-
cal and mechanism-based approaches.54,55 The model
is based on a system of ordinary deferential equations
that link the dosing regimen of a compound to the
tumor growth in xenograft mice, with tumor growth
in untreated animals being described by exponential
growth followed by a linear growth phase. In treated
animals, the tumor growth rate decreases proportion-
ally to both drug concentration and the number of
proliferating tumor cells. It relies on a few identifiable
parameters, the estimation of which requires only the
data typically available in the preclinical setting.
There are two parameters related to drug effect: c(t),
the drug concentration, and k2, a constant measuring
drug potency.55 In their later study,56 good correlation
was achieved with a novel approach proposed to
predict the expected active dose in humans from the
studies mentioned above.54,55
Although modeling in tumor growth has attracted
a lot of attention, most of the aforementioned efforts
have not explored the impact of drug effects. There
is some work assuming the system is at steady state,
which means that the concentrations of active drugs
at the active site are constant. In some drug effect
models,55,57,58 drug effect is assumed to be related
to drug concentration and number of tumor cells;
however, if we would like to compare drug effects
for different dosing regimens to consider issues as
whether we give patient frequent small or infrequent
large drug dosage given fixed total drug intake, more
realistic and dynamic drug effect models are needed.
This study proposes a model dynamically linking dis-
ease progression, in which hybrid systems are adopted
to accommodate disease progression and therapeutic
responses. Specifically, we adapt the tumor growth
model proposed in54 to hybrid systems model to
accommodate the tumor growth dynamics in different
stages and augment it with a drug effect model related
to PK and pharmacodynamics (PD). In this proposed
framework, PK and PD are linked by a state-space
approach, where drug concentration will fluctuate
between dosages and drug efficacy will change with
drug concentration. Our main aim is to model drug
effect on tumor growth for different treatment dosing
regimens given related pharmacology information.
Unperturbed growth model
(without drug treatment)
Following the same biology setup as Magni et al,54
unperturbed and perturbed growth models are formu-
lated to model tumor growth dynamics without treatment
and with treatment, respectively. Tumor growth is mod-
eled by an exponential growth phase followed by a linear
growth phase for the unperturbed growth model. A hybrid
systems model is proposed to accommodate tumor growth
dynamics in different stages. It takes the form

 ww s w
u
s w
(
uuu
w
=+
−
βθ
,
βθ
,
10
()),
w
+

(1)
where wu denotes unperturbed tumor weight, β1
and β0 are parameters characterizing the rates of
exponential and linear growth. s+(.) is the unit step
function defined by

Page 5

A therapeutic response study for different dosing regimens
Cancer Informatics 2012:11
45

sx
x
x
+
=
<
≥



( , )
θ
,
θ
θ
0
1

(2)
s−(.) = 1 − s+(.), and θw is the corresponding threshold
value at which tumor growth switches from expo-
nential to linear growth. To assure the continuity of
derivatives in equation (1) at θw, θw = β0/β1 can be
derived. Given current progress in tumor growth
modeling, the tumor growth characteristics might be
quite different in different situations. The proposed
model based on hybrid systems can be extended to
accommodate more complicated cases, such as more
growth stages with different growth rates.
Perturbed growth model
(with drug treatment)
All the tumor cells are assumed to be proliferating
in the unperturbed model. With drug treatment, it is
assumed that cells affected by drug action stop pro-
liferating and pass through different stages character-
ized by progressive degrees of damage and eventually
they die.55 A transit compartment model is used for
the cells’ progression to death under drug treatment:
 x x s w
1 1
β
x
w
swx
pw
p
pw
u
10
1
1
=+−
−
+
β
(,)(,)
θθγ1
(3)

 x211 2
k x
= γ1
ux
−

(4)

 x3=−
k x
1
(x
23
)

(5)



(6)

 xk x
1
(
x
nnn
=−
−
1
) (7)

wx
pi
i
n
=
=∑
1

(8)
x1(0) = w0 (9)
x2(0) = x3(0) = … xn(0) = 0 (10)
where x1 indicates the portion of proliferating cells
within the total tumor weight wp with drug treatment.
x1(t) will go through exponential and then linear
growth similar to the unperturbed tumor model,
where β1 and β0 denote the respective growth param-
eters. In these equations, wp is the total tumor weight,
represented by the sum of cells in the various stages,
and w0 is the tumor weight at the inoculation time
(t = 0). Since not all cells are proliferating, the linear
growth rate is slowed down by the ratio of the pro-
liferating cells over the total tumor cells x wp
model assumes that the drug elicits its effect related
to the number, x1, of proliferating cells and γ1
the drug effect coefficient and will be defined in the
next section (drug treatment model), which is closely
related to drug efficacy (PD) and fluctuates based on
changes of drug concentration (PK). The damaged
tumor cells proceed through progressive degrees of
damage through n different stages with rate constant
k1. The term k1xn represents the weight of cells that die
in each unit of time.
1/
. The
u. γ1
u is
Drug treatment model
The basis of clinical pharmacology is the fact that the
intensities of many pharmacological effects are func-
tions of the amount of drug in the body and, more
specifically, the concentration of drug at the effect
site.59 For a long time, PK and PD had been consid-
ered as separate disciplines; however, the information
provided by these disciplines is limited if regarded
in isolation. On one hand, PK is characterized as
what the body does to the drug, and it denotes the
concentration-time course of drugs in different body
fluids. On the other hand, PD is assessed as what the
drug does to the body, and it characterizes the intensity
of effects resulting from certain drug concentrations at
the assumed effect site. In order to describe the time
course of drug effect in response to different dosing
regimens, the integrated PK/PD model is indispens-
able, which builds the bridge between these two clas-
sical disciplines of pharmacology.60 Following each
dosing regimen, instead of a two dimensional dose-
concentration (PK) and concentration-effect (PD) rela-
tionship, our proposed approach enables a description
of a three dimensional dose-concentration-effect rela-
tionship. Specifically, PK and PD are linked through
a state-space approach to facilitate the description and
prediction of the time course of drug effects resulting
from different drug administration regimens.
PK/PD modeling is an active research area in
pharmacology.5,59 Application of such concepts has

Page 6

Li et al
46
Cancer Informatics 2012:11
been identified as potentially beneficial in all phases
of preclinical and clinical drug development.61,62
This work is our first attempt in the direction of
quantitative drug effect modeling. Although we
make some assumptions about concentration-effect
and dose-concentration curves in this paper, the
methodology proposed is flexible enough such that
many specific PK/PD data can be accommodated in
the proposed framework. If the mathematical for-
mulation becomes too complicated and theoretical
analysis is not possible, an extensive simulation
study can be carried out for available PK/PD data.
Periodic drug intake: pharmacokinetics (PK)
model
We consider a periodic drug intake scenario. One
could use a detailed theoretical or empirical pharma-
cokinetic description of time dependent drug concen-
tration at the site of action in a simulation study. We
prefer to keep the model mathematically tractable so
that we can perform a strict theoretical analysis and
thereby gain insights. Thus, we assume the concentra-
tion has exponential decay. Since we are using hybrid
systems, the PK model can be extended to include
more complicated cases, such as the case where the
drug concentration will first exponentially increase,
then slowly change (equilibrium), and then expo-
nentially decrease.63 The model used for drug intake
and concentration levels is illustrated in Figure 1. We
denote the period of drug intake for the two cases as
τ1 and τ2, respectively. Without loss of generality, it
is assumed that τ1 = Mτ2, where M . 1 is an integer.
It is also assumed that u1(kτ1) = ζ1 = Mu2(lτ2) = Mζ2,
where k and l are non-negative integers, and ζ1 and
ζ2 are dosages in cases 1 and 2, respectively. This
means that, in the long run, the patient takes the same
total drug amount in both cases. It is assumed that
the concentration level of the drug at the effect site
follows exponential decay during each period, ie,
ui( )te
i
=ζ
is the degradation factor. Note that Figure 1 does not
show the case where there is “leftover” from the pre-
vious dosage when the patient is taking the current
dosage.
()
t k
−
di
−λτ, where kτi # t # (k + 1)τi and λd
Drug efficacy and potency: pharmacodynamics
(PD) model
The PK model provides the concentration time
course resulting from the administered dose and the
continuous description of concentration will serve
as input function for the PD model, which relates
the concentration to the observed effect. Generally,
the magnitude of pharmacological effect increases
monotonically with increased dose, eventually
reaching a plateau level where further increases in
dose have little additional effect.13 The classic and
most commonly used concentration-effect model
is the Hill equation,64 also called the sigmoidal
Emax model65 or logistic model.66 The relationship
between the concentration of the drug candidate and
its effect is most often nonlinear. In some cases, the
curve even looks like a “roller coaster”, which is
referred to as the “double Hill Model”.67 One com-
mon method is to replace certain slowly changing
variables by their piecewise linear approximation.
In this study, we use hybrid systems to approximate
the sigmoidal Emax PD model (see Fig. 3). The Emax
model has the general form:

E
EC
+
ECC
m
m
50
m
=
max
,
(11)
where Emax is the maximum effect, C is the concentra-
tion, EC50 is the concentration necessary to produce
50% of Emax, and m represents a sigmoidity factor or
steepness of the curve.
We assume a threshold of concentration below
which the drug candidate is ineffective (such dose is
often called the minimum effective dose (MinED)). We
assume another threshold called the maximum effec-
tive dose (MaxED), above which there is no clinically
significant increase in pharmacological effect. We use
a linear curve to approximate the concentration-effect
τ1
τ1
τ1
τ2
τ2
τ2
τ2
τ2
τ2
time
Drug concentration level
u1
u2
Figure 1. The concentration level of drug under periodic drug intake. Two
cases are shown: (1) large dose with longer period; (2) small dose with
shorter period.

Page 7

A therapeutic response study for different dosing regimens
Cancer Informatics 2012:11
47
curve between MinED and MaxED. We assume that
the drug effect coefficient γ1
centration u through a sigmoid function and can be
approximated by the curve shown in Figure 2. The
corresponding relationship can be expressed as
u is related to the con-
γ1
uu
1
u
1
u
≤
q u
q
u
>
u
=
<
−
−
≤
θ



0
θ
θ
θ
θ
θ
θ
(
()
)
,
(12)
where qu
cient and the drug concentration (in the linear range).
This reflects the fact that the drug only starts to take
effect when its concentration level is above a lower
threshold (θ, corresponding to MinED) and its effect
saturates when its concentration level exceeds an upper
threshold (θ , corresponding to MaxED). Note that the
sigmoidal Emax model can be well approximated by the
proposed PD model. By taking the derivative of E with
respect to C and evaluating it at EC50, we obtain the
slope as q
mE
EC . The upper and lower
bound should satisfy qu
ple of the sigmoidal Emax model when m = 4 and our
proposed PD model are plotted together in Figure 3,
where it is observed that our proposed model closely
resembles the sigmoidal Emax model. Furthermore, by
tuning the parameters in the proposed model, we may
approximate many different types of PD models in
the literature.
1 is the ratio between the drug effect coeffi-
u
1=
max/4
50
(
θ
1=−=
)
θ
Emax. An exam-
Drug effect analysis
Based on the proposed perturbed growth model
(Eqs. (3) to (7)), the drug effect is related to the
number of proliferating tumor cells x1 and drug
effect coefficient γ1
ber of the proliferating cells dominate the changes
u. Since the changes of the num-
of all the cells under drug treatment (please refer to
Appendix A for proof), we will study the drug effect
on the number of proliferating cells in our analytical
study. We first decouple the growth phase into two
stages based on tumor weight.55 In the first stage of
the tumor growth, when tumor weight x1 , θw, the
model with drug treatment is given by

 xxx
11 11
=−βγ1
u

(13)
where γ1
ceding assumptions, the hybrid systems model can be
updated by incorporating Figures 1 and 2 into Eq. 13.
We consider a realistic setting where a patient takes the
drug periodically. For each period kτi # t # (k + 1)τi,
i = 1, 2, ..., representing different dosage and schedule
arrangements,
u is defined by Eq. (12). Based on the pre-

 xxq u
1
s us uxqs ux
u
iii
u
1i11 111
=−−−−
+−+
βθθθθ
(
θθ
()( , )( , )) ( , )
(14)

u t
i
e
i
t k
−
di
( )
()
=
−
ζ
λτ
(15)
where ui(t) is the drug concentration level at the
assumed effect site.
State-space analysis
In our proposed model, there are both continuous
quantitative changes (eg, the drug concentration
level) and discrete transitions (eg, PD model). As is
common in hybrid systems, there are both continu-
ous and discrete states. The entire state space may
be divided into different domains according to the
value of the discrete state. When the quantitative
change of the continuous state meets certain criteria,
it will cause a discrete transition from one domain
02468 1012 141618 20
0
0.2
0.4
0.6
0.8
1
1.2
EC50
θ
θ
Emax
Emax/2
Figure 3. Sigmoidal Emax model (m = 4), and approximation by our PD
model.
u
0
0
q1 (θ–θ)
u –
–
–
–
θ
γ1
u
θ
Figure 2. The Concentration-effect curve.

Page 8

Li et al
48
Cancer Informatics 2012:11
to another. Specifically, after each drug intake, the
drug concentration at the effect site is dynamically
changing following the PK model, with the changing
concentration falling into different ranges (domains).
The tumor growth dynamics will change according to
different PD model at each domain. A state space and
trajectory plot of the state of the proliferating tumor
growth and drug concentration level under periodic
drug intake are illustrated in Figure 4. There are
five domains in the state space, with D1, D3, D5 not
being transient.
The figure shows the case when the drug is effec-
tive and the initial drug concentration level is larger
than the upper threshold θ (that means the state tra-
jectory starts from Domain D5) and the sample trajec-
tory of the state corresponds to two periods of drug
intake. We observe that, when the state transits in
each period under periodic drug intake, it may pass
through different domains (depending on the drug
concentration decay along time). When the drug con-
centration is higher than θ (MinED), the drug has an
anti-tumor effect. The tumor weight may decrease
(the tumor growth level is pushed to the left) depend-
ing on drug efficacy and concentration during the
transit time through domains D5 and D3; however,
the tumor will grow during the transit time through
domain D1, where the drug is not effective because its
concentration is below MinED. In order for the drug
to be effective, the push to left side should be stronger
than the push to the right side. This means that we
should have x1((k + 1)τ) # x 1 (kτ), so that after each
treatment period the number, x1, proliferating tumor
cells will decrease.
Depending on the initial drug intake, the state
trajectory may start from different domains. For example,
if the initial conditions are x1 = x1(kτi) and ui = ζi . θ (as
in Fig. 4), then the state trajectory starts from domain D5
(Case 1). If the initial condition is θ , ui = ζi , θ , then
the state trajectory starts from domain D3 (Case 2). The
state trajectory starting from D1 corresponds to the case
where the drug concentration is too low to be effective,
and therefore has no therapeutic effect.
Case 1: state trajectory starts from domain D5
We define t1 as the traveling time from the initial condi-
tion to the boundary between D5 and D3, and t2 as the
traveling time from the initial condition to the bound-
ary between D3 and D1. The traveling time within D3
is therefore t2 − t1. Since we are considering the case
that state trajectory starts from domain D5, the initial
conditions are x1 = x1(kτi) and ui = ζi . θ .
For kτi # t # (k + 1)τi, i = 1,2., the corresponding equa-
tions and solutions in each domain are given by
− D5 (from time kτi to t1, ie, kτi # t # t1):

 x xqx
x t
1
x k
x
1 1
(
e
e
u
1
i
q
d
u
1
ki
∫
t
11 11
−
1
1
=−−⇒
=
=
−

β
β
()
( )()
)
()
θθ
τ
τ
θ θ
τ
σ
11
(( ))(

)
k
i
qt k
−
u
i
θ θ−τβ −

(16)

u t
i
e
i
t k
−
di
( )
()
=
−
ζ
λτ
(17)
In order to reduce x1, we need
(
q

β
τ
11
0
−−
)
−<
tk
u
i
() (
)
θθ

(18)
this implies

β11
<−
qu()
θθ
(19)
−D3 (from time t1 to t2, ie, t1 # t # t2):

 x x q u
1
x
u
11 11
=−−⇒β
()
θ

x t
1
x t e
1
( )
x t e
1
( )
qed
q
u
1
d
t
t
t
1
1
1
1
11
1
( )
()
()
=
∫
=
−−




+
−−
β
β
(
θθσ
λσ
u u
u
1
d
dt t
λ
(
t t
−
q
λ
e
θ
θ
) ()()
)
+−
−−
1
1
1

(20)
u
χ1(kτ)χ1
χ1((k+2)τ)
u(kτ) = u(k+1)τ
= u((k+2)τ–δ )
u((k+1)τ–δ )
χ1((k+1)τ)
θ
θ
D5
D4
D3
D2
D1
Figure 4. The trajectory of the state (tumor weight growth and drug con-
centration level) under periodic drug intake ζi . θ. δ is a very small
positive number.

Page 9

A therapeutic response study for different dosing regimens
Cancer Informatics 2012:11
49

u t
i
e
dt t
(
λ
( )
)
=
−−
θ
1
(21)
In order to reduce x1, we need

(β111
1
1
10
−−+−<
−−
qtt
q
λ
d
e
u
u
dt t
(
θ
θ
λ
) ()()
)
(22)
−D1 (from time t2 to (k + 1)τi, ie, t2 # t # (k + 1)τi):

 xxx t
1
x t e
1
( )
t t
−
11 1 2
12
=⇒=β
β
,( )
() (23)

u t
i
e
dt t
(
λ
( )
)
=
−−
θ
2
(24)
Since the drug dosage is below the effective level (θ ),
the drug is not effective on the tumor, as expected.
For the drug to be effective, both the inequalities
(22) and (19) must be satisfied; however, they are just
loose bounds. We could deduce the necessary and
sufficient condition for the effectiveness of the drug
by expressing the inequality x1((k + 1)τ) # x1(kτ) in
terms of the dose period τ and unit dose ζ, so that
after each treatment period the number of proliferat-
ing tumor cells x1 will decrease. When the initial con-
ditions are x1 = x1(kτi) and ui
governing the state trajectory from time kτi to time
(k + 1)τi are given by
i
= ζθ
> , the equations
x t
1
x k
1
(e
i
qtk
u
1
i
1
11
( ))
(() ()
=
−−−
τ
θ θτβ

(25)
x t
1
x t e
1
( )
qtt
q
λ
e
u
1
u
1
d
dt
λ
t
21
1
121
21
( )
(
) ()()
()
=
+−+−
−−
βθ
θ

(26)
xkx t e
1
( )
i
k
t
i
12
1
1
1
2
(( ) )
τ
(( )
)
+=
+
−
τ
β

(27)

θζ
λτ
=
−−
i
tk
e
di
()
1

(28)

θθ
λ
=
−−
e
dt
t()
21

(29)
and can be simplified to

xkx k
1
(e
ii1
1 (() )
τ
)
+=τ
Ψ (30)
Ψ = +−−+−

β1
1
1
τ
λ
θ θ
ln
θ θ
ln
ζθθ
i
u
d
i
q
ln
( )()

(31)
For the drug to be effective, we need Ψ , 0, so
that the number of proliferating tumor cells x1 will
decrease following each period of drug intake.
Case 2: state trajectory starts from domain D3
When the drug dosage is below θ but is above θ
the initial conditions are given by x1 = x1(kτi) and
ui = θ , ζi , θ . In this case, for kτi # t # (k + 1)τi,
i = 1, 2., the corresponding equations and solutions of
the domains are given by
−D3 (from time kτi to t2):
 xxq u
1
x
u
11 11
=−−⇒β
()
θ
x t
1
x k
1
(e
x k
1
(e
i
qed
i
u
1
i
d
ki
τ
ki
∫
t
1
( ))
)
()
()
(
=
=
−−




−−
τ
τ
ζθσ
λσ
τ
β
β β11
1
λ
1
+−+−
−−
qt k
q
e
u
i
u
i
d
d
ki
τ
θτ
ζ
λ
) ()()
()t

(32)

u t
i ie
dt ki
(
−λ
( )
)
=
−
ζ
τ

(33)
−D1 (from time t2 to (k + 1)τi):

 x xxx t e
1
( )
t t
−
11 112
12
=⇒=β
,
()
β

(34)

u t
i
e
t t
−
d
( )
()
=
−
θ−
λ
2

(35)
The equations governing the state trajectory from
time kτi to time (k + 1)τi are given by

x t
1
x k
1
(e
i
qt k
−
q
e
u
1
i
u
1
λ
i
d
dt ki
(
−λ
2
1
1
( ))
()()(
))
=
++
−−
τ
θτ
ζ
τ
β
(36)

xkx t e
1
( )
i
kt
i
12
1
1
1
2
(() )
τ
(())
+=
+−τβ

(37)

θζ
λτ
=
−−
i
tk
e
di
()
2
(38)
This can be simplified to

x
k
x k
1
(
e
i
i1
1()
)
+
() =
τ
τ
Ψ (39)
Ψ =+−−+−
[]
β1
1
1
(
τ
λ
ζζθ θ
ln
ζζθ
i
u
d
iiii
q
lnln
)( ) (40)

Page 10

Li et al
50
Cancer Informatics 2012:11
For the drug to be effective, we need Ψ , 0, so that
the number of proliferating tumor cells will decrease
following each period of drug intake.
Tumor growth minimization
We have proved mathematically that the reduction
of the number of proliferating tumor cells, defined as
x1((k + 1)τi) − x1(kτi), is a strictly convex function68
of time interval τi. The detailed proof is given in the
Appendix A. This implies that the function of tumor
size reduction, which should be negative for a success-
ful treatment, has a “U” shape and has a unique global
minimum point,68 where x1((k + 1)τi)−x1(kτi) , 0 is
the smallest that corresponds to the maximum reduc-
tion in tumor size (where |x1((k + 1)τi)−x1(kτi)| is the
largest).
Results
In order to validate the analytical results on drug effect
for different dosing regimens, we firstly perform
numerical simulations using predefined parameters to
validate the analytical results. Then we proceed with
parameter estimation on synthetic data sets generated
based on the experimental study.54,55 This second step
is critical to facilitate the use of hybrid mathematical
model to biologist. We also demonstrate that simi-
lar conclusion on drug efficacy region can be drawn
based on the synthetic data sets generated using the
parameters from real experiments.
Simulations using predefined parameters
In order to validate the analytical results on drug
effect, we firstly perform numerical simulations
using MATLAB/SIMULINK, based on the detailed
transit compartment model presented from Eqs. (3)
to (7). The cells affected by drug action stop prolif-
erating and pass through four different stages, x1,
x2, x3, and x4, characterized by progressive degrees
of damage, where x1 indicates the portion of pro-
liferating cells and wp is the total tumor weight.
Specifically, the parameters in the simulation are
β1 = 1.0, β0 = 0.2, k1 = 1.0, θw = 40. For the PD
model, we follow Eq. (12) and set the parameters as
qu
1
0 211 021
===
.,. ,
θθ
and
consider periodic drug intake and the drug concen-
tration level follows an exponential decay during
each period, as illustrated in Figure 1. The decay
rate is λd = 0.5.
Observation 1: To compare the effect of different
dosages and frequencies given a certain total drug
intake, we define the density of drug intake as α = ζi/τi,
where ζ is the dosage and τ is the dosing period. In prac-
tice, α is related to drug toxicity level. The time course
of responses of the tumor weight change (including
the 4 different stages based on damages) under three
different dosing regimens (all with total drug intake
α = 3.0) are compared: small frequent dosage (dos-
age = 15 and period (τ) = 5), medium and less frequent
dosage (dosage = 24, and period (τ) = 8), and large
. For the PK model, we
01020304050607080
0
20
40
60
80
100
Weight
Total weight of tumor cells
x1: proliferating tumor cells
x2: damaged cells
x3: damaged cells
x4: dead cells
Figure 5. τ = 5, ζ = 15, Ψ = 0.7.
01020304050607080
0
10
20
30
40
50
Time
Concentration
Drug concentration level
01020304050607080
0
5
10
15
20
25
30
Weight
Total weight of tumor cells
x1: proliferating tumor cells
x2: damaged cells
x3: damaged cells
x4: dead cells
Figure 6. τ = 8, ζ = 24, Ψ = −0.24.
01020304050607080
0
10
20
30
40
50
Time
Concentration
Drug concentration level
01020304050607080
0
20
40
60
80
100
Weight
Total weight of tumor cells
x1: proliferating tumor cells
x2: damaged cells
x3: damaged cells
x4: dead cells
Figure 7. τ = 15, ζ = 45, Ψ = 1.5.
010203040
Time
50607080
0
10
20
30
40
50
Concentration
Drug concentration level

Page 11

A therapeutic response study for different dosing regimens
Cancer Informatics 2012:11
51
Observation 3: There are many factors that
affect drug response, inter-individual PK vari-
ability being one of them. Thus, it is important to
check how different PK parameters change drug
effect. In this study, we plot the percentage of tumor
weight change versus τ for different PK decay rates
(λd = 0.42, 0.44, 0.46, 0.48, 0.5) in Figure 9. It is
observed that the effect of PK (specifically, the
decay rate λd) on tumor reduction is significant.
When the drug decay is slow, say λd = 0.42, the
tumor weight will decrease much faster than when
drug decay is fast, say λd = 0.5. This confirms our
hypothesis that the drug effect is closely related to
the PK parameters, which is one reason for the het-
erogeneity of therapeutic responses. Hence, if we
could estimate inter-individual PK variability based
on accurate measurement and interpretation of drug
concentration in biological fluids and perform cor-
responding therapy assessment to model disease
progression, then it is possible that we could adjust
dosing regimens during treatment based on such
feedback information for each individual following
the methodology presented in this study, thereby
improving the drug’s therapeutic effect.
Simulations using synthetic data
generated from experimental study
In this part of the study, synthetic data sets are pro-
duced from experimental study conducted by Magni,
Simeoni and et al54,55 firstly. Then we perform param-
eter estimation based on the synthetic data sets using
infrequent dosage (dosage = 45 and period (τ) = 15).
These are shown in Figures 5–7, respectively.
Figures 5–7 show the responses of tumor under
three dosing regimens (all with same total drug intake
α = 3.0). The left figure (Fig. 5) corresponds to the small
frequent dosing, the right figure (Fig. 7) corresponds
to the large infrequent dosing, and the case of interme-
diate dosing in between (Fig. 6). Other parameter set-
ting for the above figures: β1 = 1.0, β0 = 0.2, k 1 = 1.0,
θw = 40, qu
1
0 211 0
==
.,. ,
θ
We observe that the changes of total tumor weight
wp follow similar patterns with the changes of the
number of proliferating cells x1, which confirms our
theoretical analysis. Moreover, the results clearly
show that dosing regimens play a critical role in
disease treatment, even when the total drug in take
remains the same. Both the small frequent dosing
(Fig. 5) and large in frequent dosing case (Fig. 7) do
not reduce the tumor size effectively. Only in the case
with moderate dosage and interval (Fig. 6), are both
the number of proliferating cells and the total tumor
weight reduced effectively. At the same time, the
results demonstrate what is predicted in the analytical
results: we need Ψ , 0 so that the tumor will degrade
following each period of drug intake.
Observation 2: To further verify that the change
of tumor weight between treatments, ∆x1 = x1((k + 1)
τ) − x1(kτ), is a strictly convex function of time
interval, we plot the percentage of tumor weight
change versus dosing period τ in Figure 8 for
different total drug intakes with α = 2.8, 2.9, 3.0,
3.1, 3.2. For each curve with fixed total drug intake,
it can be seen that ∆x1 is indeed convex, and an
optimal choice of drug administration can be made
based on the point of maximum tumor reduction.
There exist some “sweet spots” (defined as “drug
efficacy regions”) of drug administration that will
satisfy the condition Ψ , 0. The three special
dosing cases (Figs. 5–7, with α = 3.0) can be tested
in the curve and it is easily confirmed that only
the moderate dosage and interval case with τ = 8
falls into the drug efficacy region. Furthermore,
Figure 8 illustrates the tradeoff between efficacy
and toxicity. When the total drug intake α increases,
the drug efficacy region gets larger accordingly. At
another extreme, the drug efficacy region may not
exist when α gets too small.
21 0 5.
==
,
θλ
and
d
.
2468101214
−100
−50
0
50
100
150
200
250
300
350
400
Period of drug intake (τ )
Percentage of change in tumor
weight per period
α increases
α = 2.8
α = 2.9
α = 3.0
α = 3.1
α = 3.2
Figure 8. The percentage of tumor weight change change vs. τ and total
drug intake α, where decay rate λd is same).

Page 12

Li et al
52
Cancer Informatics 2012:11
nonlinear least square method.69 Finally we show that
similar observations on drug efficacy can be obtained
using these synthetic data sets.
generation of synthetic data
Although the experimental data sets54,55 are not pub-
licly available, the authors54,55 provided the parame-
ter values such that the tumor growth model in their
papers matches their experimental data very well.
Hence, we perform numerical simulation using the
model given in the study54,55 to produce synthetic
data sets. Specifically, the PK data (drug plasma
concentration) is generated by using the model of
c(t) given by Equations (17)–(19) on page 138 of
Magni et al 54 and the corresponding parameter val-
ues given in Table 2 on page 140 of Magni et al.54
The tumor growth data during the entire treatment
process is generated by firstly using the unperturbed
model given by Magni et al54 for the first 15 days,
then using the perturbed model given by Magni et
al54 with the input from the PK data for 32 days (day
16 to day 47). Then the treatment is stopped from
day 48 and on. In order to model the drug effect
due to different drug plasma concentration, we also
include the sigmoidal Emax model as given by Eq.
(11) in our numerical simulation. The SIMULINK
block diagrams for generating PK data and the
entire treatment process are given in Figures 10 and
11, respectively. The generated synthetic data of a
typical run is plotted in Figure 12 for the case of
taking drug every day from day 16 to day 47.
It is observed that the tumor grows exponentially
for the first 15 days, then the weight of proliferating
cells x1 dropsfrom 2 to 1.25 when drug is taken from
day 16 to day 47. At the same time, the entire tumor
starts growing slower and eventually reduced and
stabilized. During each day, due to the PK/PD profile,
x1 reduces sharply when the initial drug concentration
is high, then x1 starts to increase because the drug
concentration decreases exponentially.
Parameter estimation
Because of the nonlinear nature of the model, we
applied nonlinear least square method69 for parameter
estimation. Specifically, we use the “nlinfit” func-
tion in MATLAB statistical toolbox. nlinfit returns
the least square parameter estimates, ie, it finds the
parameters that minimize the sum of the squared dif-
ferences between the observed responses and their
fitted values. It uses the Gauss-Newton algorithm
with Levenberg-Marquardt modifications for global
2468101214
−100
−50
0
50
100
150
200
Period of drug intake (τ )
Percentage of change in tumor
weight per period
γu increases
γu = 0.5
γu = 0.44
γu = 0.42
γu = 0.46
γu = 0.48
Figure 9. The percentage of tumor weight change vs. τ and decay rate
λd, where total drug intake is the same (α = 3).
Figure 11. SIMULINK block diagram for generating tumor growth data
during the entire treatment process.
Figure 10. SIMULINK block diagrams for generating PK data.

Page 13

A therapeutic response study for different dosing regimens
Cancer Informatics 2012:11
53
05101520 253035404550
0
0.5
1
1.5
2
2.5
3
3.5
Weight
Total weight of tumor cells
x1: proliferating tumor cells
x2: damaged cells
x3: damaged cells
x4: dead cells
05 1015202530 35 40 4550
0
1
2
3x 104
Day
Concentration
level
Drug concentration level
Figure 12. The tumor growth data from a typical run for the case of taking drug every day from day 16 to day 47.
convergence. The detailed steps for parameter esti-
mation is illustrated below.
1. Use nlinfit function to estimate the exponential
growth parameter β1 based on the measurements
of the tumor size from the first 15 days and the
unperturbed tumor growth model.
2. Now by plugging in the estimated values of β1,
γ1
k
and
based on the measurements of the tumor size from
day 16 today 47 (when drug is taken) and the per-
turbed tumor growth model.
u
1 can be estimated using nlinfit function
Note that since xi(t) cannot be experimentally mea-
sured, it is not feasible to estimate the time-varying
parameter γ1
ments using nonlinear least square method. Instead
we consider the average effect of the drug and esti-
mate the average value of γ1
square method is applicable. In case that the states xi(t)
and the time-varying parameter γ1
mated, Kalman filter70 can be applied. Kalman filter-
ing provides minimum-mean-square-error estimation
of the state of a stochastic system disturbed by Gauss-
ian white noise, since Gaussian white noise is added
to the parameters for each treated subject when creat-
ing the synthetic data. This will be part of our future
work.
The plot of nonlinear least square curve fitting
for parameters β1 and k1 are given in Figures 13
and 14, respectively. It can be seen that β1 can be
accurately estimated without much error. The true
value is β1 = 0.349 and the mean of the estimates is
^
β = 0.344
1
ut ( ) directly from tumor size measure-
ut ( ) so that nonlinear least
ut ( ) need to be esti-
. This is because the unperturbed tumor
growth model for the first 15 days is a simple expo-
nential curve that can be easily fitted. However, the
error for estimating k1 is large due to the complicated
dynamics when drug is applied, and nonlinear curve
fitting may give inaccurate estimates because only
approximate expression can be obtained for the tumor
growth, as also observed by Magni, Simeoni and
et al.54,55 The true value of k1 is 0.405, while the mean
of the estimates is ˆ
.
k1
0 616
=
.
Drug efficacy under different administrations
In order to obtain insights on the drug efficacy under
various dosage and frequency schedules, we study
the drug effect for 5 different dosing regimens with
the same total drug intake, specifically, (1) once per
day with the dosage given by Magni, Simeoni and
et al54,55; (2) double dosage given every two days; (3)
4 times dosage given every 4 days; (4) 8 times dos-
age given every 8 days; (5) 16 times dosage given
every 16 days; respectively, during the 32-day (day
16 today 47) treatment process. The detailed plots
051015
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Day 1 to day 15
Measurements
Curve from model
Figure 13. Curve fitting for parameter β1.

Page 14

Li et al
54
Cancer Informatics 2012:11
for case (1) are given in Figure 12, the detailed plots
for the rest cases are given in Figure 16 to 19 from
Appendix B. The results are compared in Figure
15. It is observed that taking drug every 4 to 8 days
seems to reduce the tumor the most, and without
much oscillations. Of course, which dosing regi-
men to choose is depending on many other practical
considerations, including toxicity. It is demonstrated
that although the total drug intake during the 32-day
treatment process remains the same, different dosage
and frequency schedules do have significant impact
on the tumor growth, which is consistent with what
we obtained analytically and observed before using
predefined parameters.
Observation 4: Through this study based on syn-
thetic data generated from experimental study,54,55 it
is clear that the parameters can be estimated by the
measurements of tumor weights along the treatment
process. This would enable the proposed hybrid sys-
tem model to be applied to study drug effects in real-
world experiments. We believe that it is feasible to
refine the model with the experimentalist through an
iterative process, then such model can be used to pre-
dict the drug effect and provide better recommenda-
tion for different dosing regimens.
conclusion
A proof-of-concept study of quantitative drug effect
modeling has been carried out using hybrid systems.
Specifically, the PK/PD data are linked together with
tumor growth dynamics in our analysis of therapeu-
tic effects. This is a small step towards quantitative
modeling of drug effect and we have kept the exam-
ples simple so that they are mathematically tractable
and valuable insights can be obtained from the ana-
lytical results. For example, we have demonstrated
that drug effect is closely associated with different
dosing regimens and individual PK/PD characteris-
tics, and the simulation results match the theoreti-
cal analysis. Although the examples in this paper
are simple, the proposed framework for quantita-
tive modeling of drug effect is flexible enough to be
able to incorporate many practical PK/PD data as
well as different models for tumor growth if desired.
Of course, when more complicated PK/PD data
and tumor growth models are used in the proposed
framework, analytical results may not be attainable
and one may have to rely on a simulation tool built
on the proposed framework to obtain the drug effect
for different dosing regimens and individual PK/PD
characteristics.
Acknowledgements
Xiangfang Li has been supported by the National
Cancer Institute (2 R25CA090301-06). The authors
would like to thanks anonymous reviewers’ valuable
suggestions to make this a better paper.
Disclosures
Author(s) have provided signed confirmations to the
publisher of their compliance with all applicable legal
and ethical obligations in respect to declaration of
conflicts of interest, funding, authorship and contrib-
utor ship, and compliance with ethical requirements
in respect to treatment of human and animal test
subjects. If this article contains identifiable human
subject(s) author(s) were required to supply signed
patient consent prior to publication. Author(s) have
confirmed that the published article is unique and not
under consideration nor published by any other pub-
lication and that they have consent to reproduce any
1520253035404550
2
2.2
2.4
2.6
2.8
3
3.2
3.4
Day 16 to day 47
Measurements
Curve from model
Figure 14. Curve fitting for parameter k1.
0510152025
Day
3035404550
0
0.5
1
1.5
2
2.5
3
3.5
Tumor size
Per day
Every 2 days
Every 4 days
Every 8 days
Every 16 days
Figure 15. Comparison of tumor growth under different dosage and
frequency schedule from day16 today 47.

Page 15

A therapeutic response study for different dosing regimens
Cancer Informatics 2012:11
55
copyrighted material. The peer reviewers declared no
conflicts of interest.
References
1. Hait WN. Anticancer drug development: the grand challenges. Nature
Reviews Drug Discovery. 2010;9:253–4.
2. Kola I, Landis J. Can the pharmaceutical industry reduce attrition rates? Nat
Rev Drug Discov. 2004;3(8):711–5.
3. Giersiefen H, Hilgenfeld R, Gukkuscg A. Modern methods of drug dis-
covery: A introduction. In Modern Methods of Drug Discovery. Edited by
Hillisch A, Hilgenfeld R, Germany: Birkhuser Verlag; 2002.
4. Kamb A. What’s wrong with our cancer models? Nat Rev Drug Discov.
2005;4(2):161–5.
5. Rajman I. PK/PD modeling and simulations: utility in drug development.
Drug Discov Today. 2008;13(7–8):341–6.
6. Quaranta V, Weaver A, Cummings P, Anderson A. Mathematical modeling
of cancer: The future of prognosis and treatment. Clinica Chimica Acta.
2005;357:173–9.
7. Searls DB. Data Integration: Challenges for drug discovery. Nature Reviews
Drug Discovery. 2005;4:45–58.
8. Alexander R, Anderson A, Quaranta V. Integrative mathematical oncology.
Nature Reviews Cancer. 2008;8:227–34.
9. Hood L, Perlmutter RM. The impact of systems approaches on bio-
logical problems in drug discovery. Nature Biotechnology. 2004;22(10):
1215–7.
10. Kumar N, Hendriks B, Janes K, Graaf D, Lauffenburger D. Applying com-
putational modeling to drug discovery and development. Drug Discovery
Today. 2006;11:806–11.
11. Butcher E, Berg E, Kunkel E. System biology in drug discovery. Nature
Biotechnology. 2004;22:1253–9.
12. Jones D. Steps on the road to personalized medicine. Nature Reviews Drug
Discovery. 2007;6:770–1.
13. Ting N. Introducton and new drug development process. In Dose Finding in
Drug Development. Edited by Ting N, New York, NY: Springer 2006.
14. Gieschke R, Steimer JL. Pharmacometrics: modeling and simulation tools to
improve decision making in clinical drug development. European Journal
of Drug Metabolism and Pharmaco Kinetics. 2000;25:49–58.
15. Undevia S, abd M Ratain GGA. Pharmacokinetic variability of anticancer
agents. Nature Reviews Cancer. 2005;5:447–58.
16. Saif MW, Choma A, Salamone S, Chu E. Pharmaco kinetically guided dose
adjustment of 5-fluorouracil: a rational approach to improving therapeutic
outcomes. J Natl Cancer Inst. 2009;101(22):1543–52.
17. Baker S, Verweij J, Rowinsky E, et al. Role of body surface area in dosing of
investigational anticancer agents in adults, 1991–2001. J Natl Cancer Inst.
2002;94(24):1822–3.
18. Sangkuhl K, Berlin DS, Altman RB, Klein TE. Pharm GKB: Understand-
ing the effects of individual genetic variants. Drug Metabolism Reviews.
2008;40(4):539–51.
19. Group BDW. Biomarkers and surrogate endpoints: proposed definitions and
conceptual framework. Clin Pharmacol Ther. 2001;69:89–95.
20. Shah N, Kasap C, Weier C, et al. Transient potent BCR-ABL inhibition is
sufficient to commit chronic myeloid leukemia cells irreversibly to apoptosis.
Cancer Cell. 2008; 14(6):485–93.
21. Amin D, Sergina N, Ahuja D, et al. Resiliency and vulnerability in
the HER2-HER3 Tumorigenic Driver. Science Transitional Medicine.
2010;2(16):1–9.
22. Rajasethupathy P, Vayttaden S, Bhalla U. Systems modeling: a pathway to
drug discovery. Current Opinion in Chemical Biology. 2005;9:400–6.
23. Butcher E. Can cell systems biology rescue drug discovery? Nature Reviews
Drug Discovery. 2005;4:461–67.
24. Deisboeck TS, Zhang L, Martin S. Advancing cancer systems biology:
Introducing the Center for the Development of a Virtual Tumor, CViT.
Cancer Informatics. 2007;5:1–8.
25. Materi W, Wishart DS. Computational systems biology in cancer:
Modeling methods and applications. Gene Regulation and Systems Biology.
2007;1:91–110.
26. Dougherty ER. Translational Science. Epistemology and the investigative
process. Current Genomics. 2009;10(2):102–9.
27. Hanahan D, Weinberg RA. The hallmarks of cancer. Cell. 2000;
100:57–70.
28. Ideker T, Lauffenburger D. Building with a scaffold: emerging strate-
gies for high-to low-level cellular modeling. Trends in Biotechnology.
2003;21(6):255–62.
29. Coveney PV, Fowle PW. Modelling biological complexity: a physical
scientists perspective. Journal of The Royal Society Interface. 2005;2:
267–80.
30. Sorger PK. A reductionist’s systems biology: Opinion. Current Opinion in
Cell Biology. 2005;17:9–11.
31. Szallasi Z, Stelling J, Periwal V. System Modeling in Cell Biology: From
Concepts to Nuts and Bolts. Cambridge, MA: MIT Press 2006.
32. Schaft A, Schumacher J. An Introduction to Hybrid Dynamical Systems.
Lecture Notes in Control and Information Sciences, 251, London: Springer
2000.
33. Lincoln P, Tiwari A. Symbolic Systems Biology: Hybrid Modeling and
Analysis of Biological Networks. In Hybrid Systems: Computation and
Control HSCC, Volume 2993. Edited by Alur R, Pappas G, Springer
2004:660–72.
34. Glass L, Kauffman S. The logical analysis of continuous non-linear bio-
chemical control networks. Journal of Theoretical Biology. 1973;39:
103–29.
35. de Jong H. Modeling and simulation of genetic regulatory systems: A litera-
ture review. Journal of Computational Biology. 2002;9:67–103.
36. Santillan M, Mackey M. Quantitative approaches to the study of bistabil-
ity in the lac operon of escherichia coli. J. Roy. Soc., Interface the Royal
Society. 2008;5.
37. Muller-Hill B. The lac operon: A short history of agenetic paradigm.
Walterde Gruyter. 1996.
38. Halasz A, Kumar V, Imielinski M, et al. Analysis of lactose metabolism in
E. coli using reachability analysis of hybrid systems. IET Systems Biology.
2007;1:120–48.
39. Ghosh R, Tiwari A, Tomlin C. Automated symtolic reachability analy-
sis; with application to delta-notch signaling automata. Hybrid Systems:
Compuation and Control. 2003;233–48.
40. Ghosh R, Tomlin C. Symbolic reachable set computation of piecewise
affine hybrid automata and its application to biological modelling: Delta-
Notch protein signalling. IET Systems Biology. 2004;1:170–83.
41. Drulhe S, Ferrari-Trecate G, de Jone H, Viari A. Reconstruction of switch-
ing thresholds in piecewise-affine models of genetic regulatory networks.
Hybrid Systems: Compuation and Control. 2006;184–99.
42. Batt G, Ropers D, de Jone H, Geiselmann J, Page M, Schneider D.
Qualitative analysis and verification of hybrid models of genetic regulatory
networks: Nutritional stress response in Escherichia coli. Hybrid Systems:
Compuation and Control. 2006;134–50.
43. Li X, Qian L, Bittner M, Dougherty E. Characterization of drug efficacy
regions based on dosage and frequency schedules. IEEE Transaction on
Biomedical Engineering. 2010;58(3):488–98.
44. Swanson KR, Bridge C, Murray JD, Alvord EC. Virtual and real brain
tumors: using mathematical modeling to quantify glioma growth and
invasion. Journal of the Neurological Sciences. 2003;216:1–10.
45. Collins VP, Loeffler RK, Tivey H. Observations on growth rates of human
tumors. Am J Roentgenol Radium Ther Nucl Med. 1956;76(5):988–1000.
46. Kusama S, Spratt JS, Donegan WL, Watson FR, Cunningham C.
The gross rates of growth of human mammary carcinoma. Cancer.
1972;30(2):594–9.
47. Goldie JH, Coldman AJ. Drug Resistance in Cancer: Mechanisms and
Models. Cambridge University Press. 1998.
48. Bajzer Z, Marusic M, Vuk-Pavlovi S. Conceptual frameworks for math-
ematical modeling of tumor growth dynamics. Mathematical and Computer
Modelling. 1996;23(6):3146.
49. Norton L. A Gompertzian model of human breast cancer growth. Cancer
Res. 1988;48:7067–71.
50. Spraft JA, von Fournier D, Spraff JS, Weber EE. Decelerating Growth and
human breast cancer. Cancer. 1993;71:2013–19.