Content uploaded by Rafic Younes
Author content
All content in this area was uploaded by Rafic Younes on Feb 13, 2015
Content may be subject to copyright.
Volume 8(1) 074-095 (2015) - 74
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
Research Article Open Access
Sbeity and Younes, J Comput Sci Syst Biol 2015, 8:2
http://dx.doi.org/10.4172/jcsb.1000173
Review Article Open Access
Computer Science
Systems Biology
Keywords: Tumor; Mathematical model; Drug therapy; Protocols
treatment; Optimal control model
Introduction
Cancer is one of the major diseases that limited the human life;
it is treated with surgery, radiation, chemotherapy, hormones, and
immunotherapy. According to estimates from the International Agency
for Research on Cancer (IARC), there were 12.7 million new cancer
cases in 2008 worldwide. e corresponding estimates for total cancer
deaths in 2008 were 7.6 million (about 21,000 cancer deaths a day). By
2030, the global burden is expected to grow to 21.4 million new cancer
cases and 13.2 million ancer deaths simply due to the growth and aging
of the population, as well as reductions in childhood mortality and
deaths from infectious diseases in developing countries [1].
e development work in cancer prevention, detection, treatment,
and management is recently very advanced. Mathematical modeling of
cancer is one of the important methodology that have contributed in
this domain. With their basis in clinical studies, multiple discussions,
revisions and evaluations, mathematical models help researchers
understand the eects that various factors, such as tumor growth and
drug infusion rates, have on optimal treatment plans. e models are
used to spotlight the importance of looking at cancer treatment as a
formal optimization problem. e number of cancerous cells, toxicity,
and drug resistance are the key factors in planning a chemotherapy
treatment. Most of the studies examining chemotherapy treatment
optimizations use various ways to model the interactions among these
key factors. Certain authors consider the minimal number of cancer
cells at the end of therapy to be a sucient indicator of treatment
quality (e.g., Costa and Boldrini [2]), but the negative cumulative eect
of the administered drug(s) should also be explicitly included in the
performance index.
e mathematical modeling of the various phases of solid-tumor
growth has itself been developing and expanding over the years.
However, over the last 20 years or so, models of cancer invasion
have begun to appear in the research literature [3-7]. In [8], a
hybrid, multiscale cellular-automaton model was presented with
the goal of studying the eects of various combinations of cell-cycle
specic chemotherapy drugs in the presence of internal and external
heterogeneity.
Actually, surgery and radiation therapy are the most common
direct therapies for curing solid tumors [9]. However, when cancer
reaches the metastasis stage, when cancerous cells from the primary
tumor are transmitted to other parts of the body, a systemic treatment
such as chemotherapy must be applied to the diused cancerous cells.
Surgery is performed when a tumor is found in only one area, and it
is likely that all of the tumor can be removed; occasionally radiation
therapy is also used during or aer an operation. When the tumor is
entirely excised, we do not need to study the treatment optimization
problem. However, when chemotherapy is used as treatment, there is
an evolution of the number of cancer cells with time, as shown in Figure
1. Because chemotherapy works like a two-sided sword, destroying the
normal cells while annihilating the cancerous cells, we cannot inject
the drug frequently. erefore, the administration of chemotherapy
involves a tradeo between cancerous cell reduction and tissue toxicity,
which is a function of the size of the destroyed normal cell population
and the drug dosage limits. For this reason, we are interested in
optimizing cancer chemotherapy.
Randomized clinical trials are the standard method for the
evaluation of chemotherapy treatment plans. For example, in reference
[10], Andre et al. conduct a trial to compare the eectiveness of two-
drug chemotherapy with various dosages, frequencies and treatment
durations. Apart from the dosages and time duration, the choice of
drug combination is another essential factor considered in clinical
trials [11].
e choice of treatment depends on the type of cancer, its stage
and its grade. e oncologist will also consider the overall health of
*Corresponding author: Hoda Sbeity, L.I.S.V., Université de Versailles, Vélizy
78140, France, Tel: +33621805766; E- ma il : sbeity_hoda@hotmail.com
Received December 08, 2014; Accepted December 17, 2014; Published
February 01, 2015
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for
Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-095.
doi:10.4172/jcsb.1000173
Copyright: © 2015 Sbeity H, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Abstract
Tumors in humans are believed to be caused by a sequence of genetic abnormality. Understanding these
sequences is important for improving cancer treatments. Biologists have uncovered some of the most basic
mechanisms by which normal stem cells develop into cancerous tumors. These biological theories can then be
transformed into mathematical models. In this paper, we review the mathematical models applied to the optimal
design of cancer chemotherapy. However, chemotherapy is a complex treatment mode that requires balancing the
benets of treating tumors with the adverse toxic side effects caused by the anti-cancer drugs. Some methods of
computational optimization have proven useful in helping to strike the right balance. The purpose of this paper is
to discuss the limitations of the existing theoretical research and provide several directions to improve research in
optimizing chemotherapy treatment planning using real protocol treatments dened by the oncologist.
Review of Optimization Methods for Cancer Chemotherapy Treatment
Planning
Hoda Sbeity * and Rac Younes
L.I.S.V, Université de Versailles, Vélizy 78140, France
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 75
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
the patient and their medical history, personal preferences and other
relevant factors. Typically, chemotherapy is administered into the veins
through an injection—this mode of injecting the drug into the vein
is known as intravenous chemotherapy. Chemotherapy can also be
administered orally with the help of tablets and capsules; this method
is known as oral chemotherapy. Chemotherapy drugs can be injected
directly into the muscles, known as intramuscular injection, or beneath
the skin’s surface, called subcutaneous injection. In intravenous
chemotherapy, the drug reaches the blood stream directly [12].
Although clinical trials have been used to determine the most
reliable and ecient chemotherapy treatment plans, they are limited by
high costs, long trial times, and the diculty of having to test multiple
options. In addition to the treatment cost and the eectiveness of the
chemotherapy plan, its feasibility should also be evaluated. All of these
steps multiply the cost; for this reason, we are studying chemotherapy
treatment planning as an optimization problem using a mathematical
model. As several researchers note, further renement of chemotherapy
will require attention to rigorously derived models because clinical
empiricism can be an inecient method of understanding and
developing a treatment strategy [13,14].
To this end, Mathematical modeling provides a low-cost method to
evaluate dierent treatment strategies more eciently, quantifying the
relationships among several important factors, such as the population
of cancerous cells, toxicity, and drug resistance. Mathematical models
also aid researchers in understanding the eects of other variables, such
as the tumor growth and drug infusion rates, on the performance of the
optimal treatment plan. erefore, there is a growing interest among
researchers on the problem of chemotherapy treatment optimization.
A close collaboration with an oncologist will improve the model,
making the research more practical and meaningful.
In general, many researchers develop mathematical models to
simulate the pharmacokinetic and pharmacodynamic processes.
Pharmacokinetic processes describe the distribution and metabolism
of the drug, and pharmacodynamic processes characterize the eects of
a drug on cancerous cells and normal cells [15]. e researchers then
apply computational methods to solve these models and compute the
optimal chemotherapy plan, which generally species the combination,
frequency, and dose of drug administration.
In real cases, the oncologist uses for a given cancer a standard
chemotherapy treatment dened from the results of many clinical trials
[16,17]. For example, for a patient with breast cancer, we have many
standard protocol treatments. e oncologist uses the predetermined
treatment dosage.
Most cancer modeling studies consider treatment in the form of
a generic ecacy term that represents the eectiveness of the drug.
Various treatment types have been used, such as chemotherapy as a
means to reduce cancer cells [18-23] in an attempt to simulate clinical
practice as closely as possible. In [18], the authors demonstrate that,
in the chemotherapy treatment, the eect of the drugs was shown to
reduce not only cancer cells but also normal and immune cells, as is
the case in reality.
e concept of applying optimal control to various disease states
began by the mid-1970s, and since then, it has become the subject
of various publications [24-27]. In [28], engineering optimal control
theory is applied to investigate the drug regimen for reducing an
experimental tumor cell population. In [29], several optimal-control
problems resulting from the simplest models of cancer chemotherapy
leading to singular control solutions was proven.
Swan [30] presents a study that has used engineering optimal
control theory for a chemotherapy problem. It involves a human tumor
and minimizes the total amount of used drug for a specied value
of tumor cell population. e study by Swan is critical for the basic
understanding and comprehension of the early mathematical modeling
approaches on chemotherapy treatment planning problem. e rst
published review of optimal control problems in the general area of
cancer research appeared in [31]. In later papers, for example [32],
Swan provides evidence for the use of continuous delivery of drugs. An
excellent general reference for this whole topic is [26]. But in [33], Zietz
and Nicolini attempted a compromise between toxicity and cell kill by
using an objective function that is a combination of tumor cell nal and
normal population. As a dierent methodology, like in [34] and [35],
the application of drugs is matched up with the progression of the cells
through the cell cycle. In [34] the optimal period for drug application
corresponds approximately to the normal cell cycle time.
In the literature, one can nd several modeling approaches related
to cancer chemotherapy problems and to optimal drug regimens.
Several of these are mentioned in [36], and we can nd many dierent
approaches and models for the job of reducing the tumor burden while
keeping the side eects of the drugs suciently in check.
References in recent years, such as Jinghua Shi et al. [37] and
Nanda et al. [38], represent various optimal control models of cancer
chemotherapy. Most of these references concern the treatment of solid
tumors or cancer treatment in general.
In [39], the authors review many topics about i) the modelization
of the cell cycle as an object control under the action of anticancer
chemotherapy, which includes modeling of drug resistance; ii) the
modelization of tumor angiogenesis and antiangiogenic therapy; and
iii) the modelization of therapies targeting specic cellular regulatory
networks, involved in carcinogenesis and cancer growth and
progression. All of these topics exist in the literature and are discussed
to help improve the situation and realize the potential for cancer
treatment. is paper is rich with references on mathematical models
and does not take into account the optimization part.
In another work [23], the authors conrm that the two major
obstacles against successful chemotherapy of cancer are the cell-cycle-
phase dependence of treatment and the emergence of resistance of
050 100 150 200 250 300
0
0.5
1
1.5
2
2.5
3
3.5 x 10
11
Time (Days)
Number of cancer cells
Surgery treatment
Chemiotherapie treatment protocol
Surgery treatment
Figure 1: Comparison of the evolution of cancer cells using chemotherapy
or surgery treatment.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 76
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
they consider models based on amplication of the resistance gene
up to a very large number of copies. is approach is to study basic
mathematical properties of the models to help in understanding the
control problem.
e simplest mathematical models that describe the optimal control
of cancer chemotherapy treat the entire cell cycle as one compartment
[51]. In reality, each drug aects the cell evolution in a particular
phase. erefore, it makes sense to combine these drugs to produce
the greatest cumulative eect on the cancer population. However,
as 20 years have already passed, updating the level of knowledge in
chemotherapy-treatment planning research is necessary. Clare et al.
[13] introduce several models in their review of the application of
mathematical models on breast cancer and discuss the mathematical
modeling of adjuvant chemotherapy as one of the subsections. Parker
and Doyle [52] provide a comprehensive review of the articles using
mathematical modeling for drug delivery with only a subsection on
the optimal cancer chemotherapy. is review describes the primary
factors considered in the chemotherapy treatment models without
surveying the literature in mathematical modeling systematically.
In the previous discourse, we discussed and appraised some
mathematical models of cancer treatment. We noted the benets
of using the theory of optimal control to determine chemotherapy
scheduling, and we underscored the need for cancer treatment to be
viewed as a formal optimization problem.
e optimization model
Optimal control theory is widely used to model chemotherapy
treatment planning problems. In this paper, we rst explain the
problem formulation. en, we discuss the model describing the
tumor growth and give an explication of several conditions that aect
the optimal control results, such as the objective function and the
treatment’s function, duration and dosage.
To properly pose the optimal-control problem, we must dene the
goal we wish to minimize (i.e., the objective functional, the treatment
dosage and duration). Hence, an optimization problem consists of a
set of constraints that must be fullled and an objective function to
be minimized (or maximized) with respect to a set of independent
variables.
Let us describe the continuous optimal problem as follows:
[ ( ), ( )] ( , , )
Tf
jf
ff f
Ti
MinJ Fut w t dt f u w T= +
∫
(1)
Subject to:
min max
min max
( , ,) 0
(,)
= >
=
<<
<<
j
j
j
dw
fct w u t t
dt
du fct u t
dt
u uu
w ww
(2)
Where
j
w
represents the dierent type of cells (i.e., cancer, normal
or immune cells) described in the model studied,
u
is the dosage of the
drug and
f
f
is the nal value function.
is paper will be organized according to the denition of the
cancer cells to cytotoxic agents. ese two problems are understood
by applying optimal control theory to mathematical models of cell
dynamics. e authors have dened a mathematical model that can
be used to pose and solve an optimal chemotherapy problem under
evolving resistance and estimated the parameters of the constructed
models.
In [40], Clairambault presents a review concerning several
problems encountered by biologists and physicians who address
natural cell proliferation and disruptions of its physiological control
in cancer disease. He concludes that the modeling of cell proliferation
and drug disposition by continuous time evolution equations can help
by i) studying the theoretical eects of the control functions; ii) proving
the feasibility or unfeasibility of proposed therapeutic strategies, under
accurate biological theoretical assumptions on the biological framework
used; and iii) proposing optimized multidrug multi-targeted therapies
according to criteria and constraints dened by oncologists. In this
paper, the author describes in detail the biological phenomena related
to cancer and treatment without going into the mathematical details
of in which case they are mentioned and whether they are mentioned
briey or in a general and simple way.
Finally, in [41], Billy et al. review i) some models that describe
the evolution of cancer cells under the inuence of anticancer drugs,
which have been used or may be used to tackle the general problem of
therapeutic optimization in oncology; and ii) theoretical therapeutic
optimization methods that can be used in the context of various models
of cell-population growth. en, they present several techniques
used for the identication of the parameters of population-dynamics
models used in chemotherapy, and they focus on a novel method of
optimization under unwanted toxicity constraints. is method is
based on the optimization of eigenvalues in an age-structured model of
cell-population dynamics.
e following is another set of references where drug therapies
for cancers that involve tumors, are studied: Chakrabarty and Hanson
[42], Duda [43], Fister and Panetta [19], Murray [43,36] and Swan [27]
on cancer chemotherapy. For alternate types of objective functions, we
can cite the work of Swierniak, Polanski, and Kimmel [44], Murray [43]
and Ledzewicz and Schättler [20]. e books by Sethi and ompson
[45], Cohen [46], Clark [47] and Eisen [48] provide the background for
the optimal control theory we use and include some simple biological
applications. e book of Martin and Teo [49] applies optimal control
to several detailed models of cancer tumors, and provides a survey of
many research results of optimal control applied to cancer.
Mathematical work that has been performed in the optimal control
setting includes two non-cell-cycle-specic (drugs that are eective
in all of the phases of the cell cycle) models by Murray [43,36]. Swan
[27] provides a good review of the role of optimal control in non-
cell-cycle-specic cancer chemotherapy: he not only elaborates on
miscellaneous-growth kinetic models and cell-cycle models, but he
also briey describes important issues in chemotherapy treatment
planning, such as drug resistance that requires the use of quantitative
research. Swierniak, Polanski, and Kimmel [44] use optimal control
theory on a cell-cycle-specic chemotherapeutic model. Swierniak,
Polanski, and Kimmel [44], along with Swan [27], investigate the eects
of the drugs on the normal tissue and use this to limit the drug strength,
but only for non-cell-cycle-specic treatments. erefore, we develop
an optimal-control problem that will directly determine the eects of
the cell-cycle-specic treatments on the normal tissue, and we attempt
to relate the mathematical results to known clinical information. In
another work [50], the authors stress the aspect of drug resistance;
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 77
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
mathematical optimization problem. For this reason, we will review i)
a set of deterministic models that describe the evolution of cancer cells;
ii) a set of mathematical models that describe the pharmacokinetics
and pharmacodynamics of injected chemotherapy drugs; iii) a set of
objective functions, considered in the literature, and aect the results
of the optimization method; and iv) various optimization methods
found in the literature. All these points are reviewed to optimize
the theoretical drug delivery and the number of cancer cells in
chemotherapy treatment.
e cancer models presented include deterministic, stochastic,
compartmental, spatial, and hybrid models. ey provide the reader
with a state of the art overview of the rapidly evolving eld of cancer
modeling and treatment optimization.
e remainder of this paper is organized into sections. In the section
of review of Cancer Modeling, we review the dierent deterministic
mathematical models used to describe tumor growth under the
inuence of chemotherapy. In the section of review of treatments,
we review the treatment type, the administration method, the drug
combination and the treatment protocols for the chemotherapy. In
the section of Review of the Objective Function, we review the various
objective functions that aect the results of the optimization method.
In the section of Solution Methods, we summarize the role of optimal
control and the optimal control models applied to the chemotherapy
treatment planning and group them according to the solution methods
used. Finally, in the section of Discussion and Conclusions, we describe
the challenges of using mathematical modeling and discuss the gap
between theoretical research and their clinical applications. We also
provide several future research directions.
Review of Cancer Modeling
Because there are three distinct stages (avascular, vascular, and
metastatic) to cancer development, researchers oen concentrate their
eorts on answering specic questions on each of these stages [53]. is
review aims to describe the current state of the mathematical modeling
of avascular tumor growth, i.e., tumors without blood vessels. is is
because, when attempting to model any complex system, it is wise to
attempt to understand each of the components as well as possible before
they are all put together. Avascular tumor growth is much simpler to
model mathematically, and yet it contains many of the phenomena that
we will need to address in a general model of tumor growth.
Cancer modeling has a wide variety of forms. Indeed, it can involve
almost any type of applied mathematics, as shown in Figure 2. We use
probability models to understand how genetic mutations lead to cancer
progression, metastasis and resistance to therapy. Ordinary dierential
equations can be used to study the growth of tumor cell populations,
oen leading to a conclusion of Gompertzian growth [54].
Partial dierential equation models (PDE) using cell densities
and nutrient concentrations as state variables can be used to analyze
various spatio-temporal phenomena [23].
e agent-based model (ABM) that treats cells as discrete objects
with predened rules of interaction can oer an improvement over
PDE methods in some situations, such as the study of angiogenesis
[55, 56], the development of new blood vessels to bring nutrients to
a growing tumor [57]. In [58], an extensive theoretical investigation
of the process of tumor-induced angiogenesis is presented, and the
results from computational simulations of the mathematical model
have highlighted a number of important new targets for therapeutic
intervention.
In general, PDE models are used with hyperthermia cancer
treatment [59], and the ABM model is used with immunotherapy
treatment [60]. e ABM model oen contains components of multi-
scale models, which have the ability to simulate tumor properties
across multiple scales in space and time [61,62].
Another phenomenon is incorporated into the process of cancer
development: metastasis [63,64]. In case of the formation of another
cancer type via metastasis phenomena, we resolve this problem using
two dierent protocol treatments for each cancer to minimize the
number of cancer cells with the minimum treatment dosage.
In this paper, we are interested in reviewing the models used to
describe tumor growth with chemotherapy treatment, which can be
viewed as a formal optimization problem.
e mathematical models that describe tumor growth are
divided into ve categories:
• Deterministic model with non-linear dierential
equations: In this model, we used a system of ordinary dierential
equations (ODE) to describe the evolution of cells reected by a well-
dened curve.
( , ,)
dw f wut
dt
(3)
Where
()ft
may take dierent non-linear forms.
Compartmental model: e cell cycle is modeled as compartments,
each of which either describe dierent cell phases or combine phases
of the cell cycle into clusters. e function
()ft
is linear, and the
optimization solutions are analytical.
Stochastic model: Observations of cancer cell cycles are oen
presented in an erratic fashion. For this reason, we are trying to
introduce probabilities, such as those used in stochastic models.
e various stochastic models that are able to describe biological
processes, include the following:
• Moran model [65]
• Wright-Fisher model [66]
• Galton–Watson branching process [67]
• Markov chain processes [68]
• Model of Moolgavkar, Venzon, and Knudson [69]
ere are attempts to their development, to use them in the
Figure 2: General schema of the different cancer modeling.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 78
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
description of cancer process.
• Spatio-temporal model [70-72]: In this model, the evolution
of cancer cells is studied in space as well as time. We can add Laplacian
terms into the dierent equations of the deterministic models to study
the spatial evolution of cancer cells.
2
( , ,)
∂= +∇
∂
l
wf wut d w
t
(4)
Where
l
d
is the diusive coecient for the described cell. e
Laplacian terms denoted by
2
w∇
capture the diusive properties of
the specic cell in arbitrary spatial dimensions.
An analysis of these models was carried out in a one-dimensional
geometrical setting, in which the operator ∇2 was set equal to the second
order partial derivative
22
x∂∂
of the normal and abnormal cell
population with x representing the real number line. A validation of the
analysis was extended to higher spatial dimensions, and simulations of
the model were considered in two-and three-dimensional space and in
radial geometries. In reference [73], a continuous three-dimensional
model of avascular glioma spatiotemporal evolution is developed.
Based on the experimental validation, as well as the evaluation by
clinical experts, the proposed model may provide an essential tool for
the patient-specic simulation of various tumor evolution scenarios
and the reliable prognosis of glioma spatio-temporal progression.
• Hybrid model: this model is similar to the non-linear
dierential equation model, but we added a term for white noise or
some other parameter (such as birth rate) that varies according to the
probability law. is reects the external randomness that aects the
tumor growth behavior.
( , , ,)
θ
=
dw f wu t
dt
(5)
Where
θ
represents the random perturbations in our system.
Referring to many studies [23,39-41] on the application of the
optimization method for the various mathematics models that describe
the evolution of cancer cells, we conclude that the optimization method
i) is not largely applied to the stochastic models, ii) is not applied to
PDE models and iii) is largely applied to the ODE models. e studies
concerning the application of the optimization method on the ODE
models was not applicable in the real case of used treatment protocols.
For this reason, in this paper, we include a bibliography with a
simple denition for the dierent deterministic mathematical models
(nonlinear dierential equation, hybrid and compartmental models); ii)
the treatment type, the administration method, the drug combination
and the treatment protocols, for the chemotherapy; iii) the various
objective functions that aect the results of the optimization method;
and iv) the solution methods of the optimizations models, to determine
chemotherapy scheduling, viewed as a formal optimization problem.
is study concerning the optimization of cancer-treatment protocols
can become in the future a standard element to help oncologists predict
the minimum dosage for a patient based on theoretical considerations
such as those reviewed in this paper.
Deterministic model
Mathematical models describing the cancer-evolution phenomena
can be used as an important tool in therapy planning. Several models
of cancer at various stages have been formulated over many years of
research. In this paper, we focus on models in the form of ordinary
dierential equations (ODEs), controlling cancer growth on a cell-
population level. Many references, such as [25,28,30], studied cancer
growth using Gompertzian growth [26]. Other ODE models have also
been used to describe the exchange between two cells populations,
proliferating and quiescence ones [25]. Other ODE models have
considered some form of tumor immune interaction [31,36]. Normal
cells have also been modeled in many forms to model toxicity
disadvantage [25,74].
We must divide the deterministic model into three types:
• Non-linear dierential equations
• Compartmental model
• Hybrid model
Non-linear dierential equations: is model must be further
divided based on the number of dierential equations, which are
used to describe the evolution of several cell types (including cancer,
normal and immune cells). Historically, these models are cited as
below in the various articles but scientically they are not related.
We start with the simplest model:
Model with one dierential equation: e most frequently used
model in oncology for growth retardation in malignant tumors is
the Gompertz equation [75], developed by the 19th century English
actuary, Benjamin Gompertz [54,76]. e equation is characterized by
two parameters, the initial specic rate of exponential growth and the
rate of exponential fall in the initial growth rate. Unlike the logistic
function, which is symmetrical about the inexion, the Gompertz
function is asymmetrical about the inexion, which is always 0.37 of
the asymptotic value [76].
Typically a tumor will grow very rapidly with the growth rate
proportional to the tumor size. However, as tumors increase, the
growth rate decreases as tumor size increases. In order to reect this,
dierential equation models have been proposed. It is customary to
assume that tumor volume and number of tumor cells are proportional
and so the model which follows describes
()dC t dt
, where
()Ct
represents the number of tumor cells at time t. ree models have been
widely used in the study of tumor growth: exponential, logistic and
Gompertz growth. e eect of chemotherapy (pharmacodynamics)
is incorporated into tumor growth model by adding a kill term to the
dierential equation by assuming that there is a concentration
()ut
of a cytotoxic drug at time
t
and the presence of the drug will cause a
decline in tumor cell population jointly proportional to concentration
()ut
and the population size
()Ct
at any given instant. We will also
assume that there is a threshold drug concentration level,
ε
below
which no tumor cells are killed. e Gompertz growth equation is the
most commonly used model of tumor growth and takes with treatment
the form:
log( ) ( ) ( )
θ
λ εε
= −− −
dC C ku Hu
dt C
(6)
Where
λ
is a positive constant related to the growth function
and
θ
is the largest possible size of the tumor.
k
is a constant of
proportionality, and
H
is the Heaviside unit function [77].
Among the proposed models, those based on Gompertz growth
are frequently proposed [78]. It models the growth of a population
consisting of a group of individuals of one or more similar species
in the absence of migration and interaction with other species, by an
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 79
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
S-shaped function
( )
C Ct=
that is the solution of:
( )
( ) log( )
αβ
=−−
dC C ut C C
dt
(7)
Within the context of tumor growth,
C
denotes the number
of tumor cells at time
t
,
0
C
is the size of the tumor at the start of
treatment,
()ut
denotes the drug concentration at the site of action.
()Ct
is assumed to be continuous and dierentiable [79]. Parameters
α
and
β
(measured in
1
t
−
), denote growth and decay rates,
respectively, they characterize the evolution of dierent tumor types.
During chemotherapy it is assumed that the basic growth kinetics of
the tumor will be perturbed by the action of the cytotoxic drug
()ut
.
Swan and Vincent [12] used assumptions about Gompertzian
growth of immunoglobulin G (IgG) multiple myeloma cells and
developed a single dierential equation of drug action on such cells.
e equation describing the drug action is as follows:
1
2
log( )
θ
α
= − +
k uC
dC C
dt C k u
(8)
Where, the parameter
θ
represents the greatest size of the tumor;
and
1
k
and
2
k
are the positive constants in the saturation type loss
term in the equation.
Model with two dierential equations: e cell growth of various
types of cells, whether they are immune, cancer or normal cells have
certain features in common. erefore, a mathematical model of a
particular cell growth may be used to model other types of cell growth,
provided some basic changes can be made.
A few models of acute myeloblastic leukemia (AML) were proposed
by Afenya [80]. Normal and leukemic cells were assumed side-by-side
with the two cell populations obeying Gompertzian dynamics but
with the leukemic cells exercising inhibition over the normal cells. e
kinetic equations and steady-state properties of one of the models are
analytically obtained as:
log( ) ( )
log( ) ( ) ( , )
θ
αβ
α
= −−
= −− − +
A
dC C C ku t C
dt C
N
dN bN cNC hu t N f C N
dt N
(9)
where
A
N
and
θ
are the asymptotic bounds (limits) on, or carrying
capacities of, the normal and leukemic cell populations, respectively,
and
( )
,f CN is a function that can take many form:
()
(, )=
mCNH C
fCN
rN
(10)
a
is the intrinsic growth rate or growth speed of the normal cells,
and
b
is their death rate.
c
is a measure of the degree of inhibition of
the normal cells with respect to the leukemic cells,
k
is the fraction
of leukemic cells that are killed due to the drug eect, and
h
is the
fraction of the normal cells that are destroyed by the lethal binding
eects of the drugs (it is assumed that
kh>
).
r
is the maximum
fraction of normal cells that should be injected per unit time to ensure
normal cell regrowth. It can also be interpreted to be the maximal rate
per unit time at which growth factors are infused. We assume that, as
the normal cell population decreases because of the drug eect, the
regenerative process through which normal cells are injected or growth
factors are infused could be directly related to this normal population.
is is modeled by the term
rN
. e negative eects of the drugs on the
normal cells are modeled by the term
( )
hu t N
[78].
m
is the maximum
growth coecient of the normal cells and
( ) 1 (1 )H C pC= +
, which
could take a number of forms, is the growth fraction of the normal
cells, which is dependent on the leukemic population, and
p
can be
considered as an inhibitive parameter.
Model with three dierential equations: In this model, we let
()
n
Tt
denote the native
T
cell population, and
()
e
Tt
the eector T
cell population at time
t
[38,81]. We assume that the eector
T
cells
are specic to one type of cancer (e.g., chronic myelogenous leukemia,
CML), activated by the presence of the CML antigen. e state system
for our model is given by:
( )
2
2
12
()
()
1 ( ) log( ) ( )
αα γ
ηη
η
θ
α βγ
= − −−
++
=−−
+
=− −−
e
nnn ee ee e e
n
n nn nn
ce
dT CC
k T T u t d T CT
dt C C
dT C
S u t dT KT
dt C
dC u t C u t C CT
dt C
(11)
is model consists of a system of three nonlinear ordinary
dierential equations. Each equation represents the rate of change with
respect to time of one of the cell populations. All parameter descriptions
are in the cited reference [38].
Another model with three dierential equations is described as
follows by Villasana and Ochoa [82]:
( )
( )
( )
( )
4 1 21
()
1 343 1
24
()
13
() 2 () () () ( )
() () () () () (1 ) ()
() () ()
() () () () ()
() ()
() 1 ()
= − − −−
= −− − − − −
=+ −− −
++
−−
MI II
kut
I M MM M
IM
IM
IM
kut
dC t aC t cCTt dCt aCt
dt
dC t aCt dCt aCt cCtTt k e Ct
dt
Tt C t C t
dT t k cTtC t cC tTt
dt Ct C t
dTt k e Tt
(12)
is model divides the population of the tumor cells into interphase
cells (pre-mitotic, phase period comprising G1, S and G2) and mitosis
cells, which are represented by
()
I
Ct
and
()
M
Ct
, respectively. e term
()Tt
represents the population of immune cells that are the population
of cytotoxic T cells. Let
()ut
be the concentration of the drug present at
time
t
, and
τ
be the resident time of cells in the interphase stage. e
parameters
ρ
,
α
and
n
depend on the type of tumor being considered
and the health of the immune system [82,83].
Model with four dierential equations: Immunotherapies are
quickly becoming an important component in the multi-pronged
approaches being developed to treat certain forms of cancer [84]. e
goal of immunotherapy is to strengthen the body’s own natural ability
to combat cancer by enhancing the eectiveness of the immune system.
e importance of the immune system in ghting cancer has been
veried in the laboratory as well as with clinical experiments [85-88].
Additionally, it is known that those with weakened immune systems,
such as those suering from AIDS, are more likely to contract certain
rare forms of cancer. is phenomenon can be interpreted as providing
further evidence that the role played by the immune response in
battling cancer is critical [89,90]. rough the mathematical modeling
of tumor growth, the presence of an immune component has been
shown to be essential for producing clinically observed phenomena
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 80
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
such as tumor dormancy, oscillations in tumor size, and spontaneous
tumor regression.
e mathematical modeling of the entire immune system can
be an enormously intricate task, as demonstrated in [91], so models
that describe the immune system response to a tumor challenge must
necessarily focus on those elements of the immune system that are
known to be signicant in controlling tumor growth. In the work of de
Boer and Hogeweg [92], a mathematical model of the cellular immune
response was used to investigate such an immune reaction to tumors.
It was found that, initially, small doses of antigens do lead to tumor
dormancy.
is model describes the kinetics of four cell populations (tumor
cells and three types of immune cells), as well as the concentrations
of two drugs in the bloodstream, using a series of coupled ordinary
dierential equations that are based on the model developed by de
Pillis and Radunskaya [93]. e populations at time
t
are denoted by
the following terms [84]:
()Ct
, tumor cell population
()
NK
Tt
, total NK cell population
8
()
CD
Tt
, total CD8+ T cell population
()
L
Tt
, number of circulating lymphocytes (or white blood cells)
Both NK
T and
8CD
T
cells are capable of killing tumor cells, NK
T
cells are normally present in the body, even when no tumor cells are
present, and the active tumor-specic
8CD
T
cells are only present in
large numbers when tumor cells are present.
NK
T
and
8CD
T
cells become
inactive aer a certain number of encounters with tumor cells, and
the circulating lymphocyte levels can be used as a measure of patient
health.
e equations governing the population kinetics must take into
account a net growth term for each population, the fractional cell kill, the
per cell recruitment, the cell inactivation and the external intervention
with medication. We attempt to use the simplest expressions for each
term that still accurately reect the experimental data and population
interactions.
Bringing together the specic forms for each cell growth and
interaction term leads to the full system of equations:
( )
( )
( )
( )
( )
22
2
8
8 12 8
22
8
8
2
2
1 ()
(1 ) 1
1
1
ν
α
−
−
−
−
=−+ −+ + −
+
−− + +
+
= − − −− −
=−+ − − −
+
=−− −
CD
CD L NK L NK CD
uI CD
L CD L
I
u
L NK T
u
NK
L NK NK NK N NK
u
L
LC L
dT DC
mT j qT C rT r T C UT T
dt k D C
PT I
k eT t
gI
dC C bC T T c DC k e C
dt
dT C
eT fT g T pT C k e T
dt h C
dT a kT K e T
dt
(13)
All parameter descriptions are in Table 1 of the cited reference
[84]. e term I represents the immunotherapy drug concentration in
the bloodstream, and in our case we set I equal zero because we are
interested only in the chemotherapy drug concentration.
Finally, as stated above, Villasana and Ocho’s model does not
include quiescent tumor cells. In [94], we show how Yaa [95]
develops a delayed dierential equation model for the interactions of
proliferating and quiescent tumor cells. e concept of cancer cells in
the quiescent phase
()
Q
T
is introduced, and the model by Villasana,
cited above, is corrected:
( )
( )
( )
4 5 61 21
()
2
1 343 1
24
() 2 () ( ) () () () ( )
() () () () () (1 ) ()
() () () ()
() () () ()
() () ()
ττ
τ
ρ
α
−
= − −+ − − − −
= −− − − − −
++
=+ −−
+ ++
I
M I QI I I
kut
M
I MMM M
n
IMQ
IM
n
IMQ
dC t aC t aCt aCt cCTt dCt aCt
dt
dC t aCt dCt aCt cCtTt k e Ct
dt
TtCtCtCt
dT t k cT t C t cC tT
dt CtCtCt
( )
( )
()
4
6 13
6
5 6 45 5
()
() () () 1 ()
() ( ) () () () () 1 ()
τ
−
−
−
− −−
= −− − − − −
kut
Q
Qku
I QQ Q Q
t
cC tTt dTt k e Tt
dC t aCt aCt dCt cTtCt k e Ct
dt
(14)
Compartmental model: In the compartmental model, the cell
cycle is modeled in the form of compartments that describe dierent
cell phases or combine phases of the cell cycle into clusters. Each cell
passes through a sequence of phases from cell birth to cell division.
e starting point is the growth phase G1 aer which the cell enters the
phase S, in which the DNA synthesis occurs. en, the second growth
phase G2 occurs, in which the cell prepares for mitosis or phase M, in
which cell division occurs.
Each of the two ospring cells can either reenter phase G1 or may
simply lie dormant for some time in a separate phase G0 until reentering
G1, thus starting the entire process all over again.
e simplest mathematical models that describe the optimal
control of cancer chemotherapy treat the entire cell cycle as one
compartment, but solutions to these single-compartment models are
Drug combinations
Initials Combination Indications
ABVD
A: Adriamycin
B: bleomycin
V: Vinblastine
D: Dacarbazine
Hodgkin's disease [136]
AIM
A: Adriamycin
I: Ifosfamide
M: Mesna
Soft tissue cancer [138]
Alexanian therapy
M: Melphalan
P: Prednisone
Given orally each week
Myeloma [139]
BEP
B: Bleomycin
E: Etoposide
P: Cisplatin
Testicular cancer [140]
CAF
C: Cyclophosphamide
A: Adriamycin
F: Fluorouracil
Breast cancer [135]
MIC
M: Mitomycin-C
I: Ifosfamide + Mesna
C: Cisplatin
Non-small cell lung
cancer [141]
M-VAC
M: Methotrexate
V: Vinblastine
A: Adriamycin
C: Cisplatin
Bladder cancer [137]
CP C: Cyclophosphamide
P: Cisplatin Ovarian cancer [142]
MACOP-B
M: Methotrexate
A: Adriamycin
C: Cyclophosphamide
O: Vincristine
P: Prednisone
B: Bleomycin
High grade non-Hodgkin's
lymphoma [9]
Table 1: Different combinations of treatments used for certain cancers
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 81
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
not very informative due to the over-simplied nature of the model.
For the compartmental model, the problem of nding an optimal
cancer chemotherapy protocol is formulated as an optimal control
problem over the nite time interval of the xed therapy horizon. e
state variable is given by the average number of cancer cells, and the
control is the eect of the drug dosages on the respective subpopulation.
e goal is to maximize the number of cancer cells that the agent
kills and to appropriately minimize the number of cancer cells at the
end of the therapy session, while keeping the toxicity to the normal
tissues acceptable. e last aspect is modeled implicitly by including
an integral of the control over the therapy interval in the objective so
that minimizing controls will balance the quantity of drugs given with
the conicting objective to kill cancer cells. e analytical approaches
to these models are based on applications of the Pontryagin Maximum
Principle.
e general compartmental model can be described uniformly by
the following state equation [22]:
0
1
() (), (0)
=
=+=
∑
m
ii
i
Ct A uB Ct C C
(15)
where
012
[ .... ]
T
j
C CCC C=
is an innite dimensional state vector,
with
j
C
denoting the number of cancer cells in the
j th−
compartment,
1,...,jn=
. e control is a vector
( )
1,..., T
m
uu u=
with
i
u
denoting the
drug dosage administered. Let
A
and
i
B
,
1,...,im=
be constant
nn×
matrices, and the elements of matrices
A
and
i
B
are positive factors
related to mean transit time of cells through the
j th−
compartment.
It is important to note that model parameters satisfy the following
relations:
31
0aa>>
, and
2
0a<
. All of the matrices have negative
diagonal entries but nonnegative o-diagonal entries. e diagonal
entries correspond to the outows from the
i th−
compartments,
and the o-diagonal entries represent the inows from the
i th−
into
the
j th
−
compartment,
ij≠
. is is satised for each of the models
described below. More generally, if the condition that the rst orthant
of the control system is positively invariant were violated, this would be
a strong indication that the modeling is inconsistent.
We start from mathematical models describing the dynamics of
the cancer cell population under a single drug treatment, from the
simplest, two-compartmental, to innite-dimensional ones. Problems
of multidrug treatment and multidrug resistance are addressed
aerward. A model with an innite number of compartments was
considered in a series of papers by Kimmel, Swierniak, and coworkers
(e.g., references [22,23,42,51,96,97]), in the context of drug resistance
caused by the gene amplication dynamical process.
e innite-dimensional compartmental model of drug-resistance
evolution has been proposed to approximate the dynamic branching
random walk related to the process of gene amplication [98] and has
led to results not achievable in simplied nite-dimensional models.
ese results in turn have made it possible to formulate several
recommendations regarding treatment protocols. Optimization of
the model has become possible aer its transformation to the integro-
dierential form.
We briey recall some compartment models that t into this
general class. For a more detailed description of the models, we refer
the reader to Swierniak et al. [44].
In a two-compartment model, the phases G0, G1 and S are clustered
into the rst compartment, G2 and M are combined into the second
compartment, and only a killing agent
1
uu=
is considered. us,
2, 1nm= =
, and the matrices
A
and 1
BB=
are given by:
12 2
12
202
,
00
−−
= =
−
aa a
AB
aa
(16)
e
i
a
values are positive coecients related to the mean transit
times of cells through the
i th−
compartment.
e dierential equations of this model are:
1
11 2 2 1
2
11 22
2 (1 )=−+ −
= −
dC aC aC u
dt
dC aC aC
dt
(17)
In this three-compartment model, a blocking agent
2
u
ν
=
is
additionally considered which is active in the synthesis phase S; thus,
S is modeled as a separate compartment. Now,
3, 2
nm= =
, and the
matrices are given by:
13
12
23
3
1 22
2
02
0
0
00 2 0 0 0
00 0 , 0 0
00 0 0 0
−
= −
−
−
= =
−
aa
Aa a
aa
a
B Ba
a
(18)
us, the dierential equations of this model are:
1
11 33 1
2
11 2 2 2
3
22 2 33
2 (1 )
( 1)
(1 )
=−+ −
=+−
= −−
dC aC aC u
dt
dC aC a C u
dt
dC aC u aC
dt
(19)
In both models, the control
1
uu=
represents the dose of the
killing agent administered with the value
0=u
corresponding to no
treatment and
1u=
corresponding to a maximum dose. It is assumed
that the dose stands in direct relation to the fraction of cells that are
being killed in the G2/M phase. erefore, only the fraction
1u−
of the
outow of cells from the last compartment undergoes cell division and
reenters the rst compartment. However, all cells leave compartment
G2/M.
In the second model, the blocking agent
2
u
ν
=
is additionally
applied to slow the transit times of cancer cells during the synthesis
phase S. As a result, the ow of cancer cells from the second into the
third compartment is reduced by a factor of
1
ν
−
of its original ow
to
( ) ( )
2 2 max
(1 ) , 0 1t aC t
νν
− ≤<
. Here the control
() 0t
ν
=
corresponds
to no drug being applied, and a maximal reduction occurs with a full
dose
max
ν
.
In the models mentioned above, the compartments are divided
based on the cell cycle. Next, we must divide the compartments
depending on the number of cells sensible or resistant to the drug
injected [74]. e most basic model consists of only two compartments
representing the sensitive and drug resistant subpopulations. It has
been a subject analyzed by many researchers but arguably, the most
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 82
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
recent comprehensive studies can be found in Ledzewicz and Schättler
[20,21].
e model based on the assumption of the mutation being the
basis for drug resistance caused the cell to acquire one additional
copy of a certain gene. erefore, a sensitive mother cell produces two
daughter cells, one of which remains sensitive, while the other changes
into a resistant one with a probability of
γ
. Similarly, if a resistant
cell undergoes cell division, one of the ospring remains sensitive. e
other one may mutate back into a sensitive cell with a probability
d
(it
is assumed that no additional gene copies can be acquired).
Let us denote by
0
C
and
1
C
the average number of sensitive and
drug-resistant cells, respectively en, the system is described by the
following set of equations:
0 00 0 1
1 11 0 1
(1 2 ) (1 )
(1 )
λγ
λγ
=− −− +
= +− −
C u C u C dC
C C u C dC
(20)
is model represent the evolution of cancer cells using a single-
agent treatment; in Panetta [99], the simplest model possible was
analyzed using two treatment agents, and the mutation that makes cells
resistant to the rst drug is assumed to be irreversible. erefore, the
second sub population resistant to this rst drug
1
u
is assumed to be
sensitive to the second drug
2
u
, so the resulting mathematical model
is as follows:
0 0 10
1 10 1 2 1
()
()
λ
γλ
= −
= +−
C uC
C uC u C
(21)
where
γ
is a probability of a mutational event leading to drug
resistance. For this model, no formal optimization has been performed.
If we consider killing and blocking agent actions (denoted by
1
u
and
2
u
, respectively), the issue of resistance is much more complex.
However, if only the resistance to the killing agent is considered, then,
the compartments of cells in the phase G1, S, G2 and M, denoted by
0, 1, 2i=
, respectively, and by the
3i≥
compartments of resistant
cells, the following description is obtained:
0 00 0 22 3
1 1 11 0 0
2 2 2 1 11
3 33 2 2 4
11
2(1 )(1 )
(1 )
(1 )
( ) 2 (1 )
() 4
λ γλ
λλ
λλ
λγ
λ
+−
=− +− − +
=−− +
=− +−
=− −+ + − +
=− −+ + + ≥
i i ii i
C C u C dC
C uC C
C C uC
C C b d C u C bC
C C b d C dC bC i
(22)
e variables
i
C
denote the average number of cells in the
ith
compartment, e lifespan of all cells are independent, exponentially
distributed random variables with means
1
i
λ
for cells of type
i
. e
parameter
i
λ
is assumed to be a constant, not a function of
u
.
Another form of the compartmental model with two compartments
is denote as the model of Gyllenberg and Webb [79,100]; the creators
used this model to show that it yields Gompertz or logistic growth
under special parameter selection. e model consists of proliferating
and quiescent cell compartments; it allows for transition between the
compartments and cell death from either compartment. e transition
rate functions are considered to be functions of the total number of
cells. e following set of ordinary dierential equations describes the
model:
( )
0
0
00
() ()
() (() )
,1
βµ
µ
=−− +
= ++
=+ +=
P P Pi Q
Q P i qQ
PQ P
C r C C r CC
C r CC r C C
CC C C Q
(23)
P
C
and
Q
C
represent the normalized number of proliferating and
nonproliferating (quiescent) cells, and
0
P
µ
≥
and
0
q
µ
≥
represent the
death rate parameters for the
P
C
and
Q
C
compartments, respectively.
0
β
>
is the proliferation-rate parameter. e transition rate functions
are:
0
() 0rC ≥
, describing the transition from the proliferation
subpopulation into the quiescent subpopulation, and
() 0
i
rC ≥
, describing the transition into the proliferation subpopulation from
the quiescent subpopulation. By Combining the Gompertz and the
Gyllenberg–Webb model we have:
( )
( )
()
βµ
βµ µ µ
= − −Φ
=−+ −
P pP
p qP q
C CC
C CC
(24)
is idea was briey introduced in a recent paper [101]; we start by
dening the net transition rate function:
0
() () ()Φ= −
Pi Q
C r CC r CC
(25)
When
() 0CΦ>
, the net transition rate is from the proliferating
compartment into the quiescent compartment.
Analytical approaches to these models are based on applications of
the Pontryagin Maximum Principle [102].
A mathematical model of two ordinary dierential equations for
the interactions between the cancer cell growth and the activity of the
immune system during the development of cancer was proposed by
Stepanova [103]. is model is described as follows [104]:
2
()
()
αγ
µδ
= −−
= − − +−
x
x x cx xx
I x n nY
x
dC C F C CT kCu
dt
dT C C T d T S k Tu
dt
(26)
Where
T
represents the immunocompetent cell densities related
to various types of T cells activated during the immune reaction,
x
C
denotes the tumor volume,
1
δ
corresponds to a threshold
beyond which the immunological system becomes depressed by the
growing tumor, and the coecients
I
µ
and
δ
are used to calibrate
the interactions between the cancer and the immune cell, and in the
product with T collectively describe the state-dependent inuence
of the cancer cells on the stimulation of the immune system, e
coecients
x
k
and
Y
k
are chosen to normalize the control set, i.e.,
we assume that
01u<<
. In Stepanova’s original formulation,
F
is simply given by
()1
Ex
FC =
, in Gompertzian growth,
F
is given by
( ) log( )
Gx x
FC C C
∞
= −
, and in the logistic growth models,
F
is given
by
( ) 1( )
Lx x
FC C C
ν
∞
= −
, with
0
ν
>
.
Hybrid model: Discrete or continuous time equations are used
for the deterministic model. It is represented by ordinary dierential
equations (ODEs) with delay or partial dierential equations (PDE) [79]
or the agent-based model (ABM) [105]. Among the suggested models,
the simplest ones are those of one or several dierential equations.
Despite their simplicity, these models must predict the evolution of
many biological phenomena [106]. Models based on the deterministic
Gompertz law appear to be particularly compatible with the evidence of
tumor growth.
However, it is quite common that discrepancies are found to
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 83
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
exist between clinical data and theoretical predictions due to intense
environmental uctuations. For instance, Ferreira et al. [107] analyzed
the eect of various chemotherapeutic strategies on the vascular tumor
growth. ey conrmed that an environment such as chemotherapy
would aect tumor growth behavior and lead to morphological
transitions under certain conditions. erefore, a better model is
necessary to reect the external randomness that aects the tumor-
growth behavior.
Ferrante et al. [108] proposed a stochastic version of the Gompertz
law in which random uctuations of the model parameters are
considered. ey assume that the growth deceleration factor
β
(death
rate) does not change, whereas the variability of the environmental
conditions induces uctuations in the intrinsic growth rate
α
. e
intrinsic growth rate is assumed to vary in time according to the
following equation:
( ) ( )
θ α σε
= +tt
(27)
e parameter
α
is the constant mean value of
( )
t
θ
,
σ
is the
diusion coecient, and
( )
t
ε
is a Gaussian white noise process.
By adding the expression for
( )
t
θ
into Eq. (7) without treatment,
we have:
( )
( ) log
α σε β
=+−
dC tC C C
dt
(28)
Briey, all deterministic equations could then be written by adding
the term
( )
tC
σε
to the equation, which describes the evolution of
cancer cells, transforming them from deterministic to hybrid models.
Finally, the hybrid model is a combination between deterministic
and stochastic models. It appears in three forms:
α
and
β
are random and follow the normal distribution, with
means
α
and
β
and variances
α
σ
and
β
σ
α
and
β
are random and follow the uniform distribution
(
[ ]
min max
,
ααα
∈
and
[ ]
min max
,
βββ
∈
)
α
and
β
are constants, and a white noise term is added to the
equation
Beginning with the model described by Eq. (6),
λ
is a random
number. It follows either the normal distribution, with mean
λ
and
variance
λ
σ
, or the uniform distribution, with
[ ]
min max
,
λλλ
∈
, or
λ
and
k
are constants, and a white noise term is added. en, Eq. (6)
becomes:
log( ) ( ) ( ) log( )
θθ
λ ε ε σε
= − − −+
dC C ku Hu
dt C C
(29)
For the model described by Eq. (8),
α
is also a random number that
can be represented by the same three distribution options described
above, if we add a white noise to this model, the equation becomes:
1
2
log( ) ln( )
θθ
α σε
= −+
+
k uC
dC CC
dt C k u C
(30)
For a model with three dierential equations, described by Eq. (11),
we obtain the following equation, when
α
and
β
are set to constants
and white noise is added:
( )
1 2 12
1 ( ) log( ) ( ) (1 ( )) log( ) ( )
θθ
α β γ σε
=− − −+ − −
ce
dC ut C u t C CT C ut ut
dt C C
(31)
Review of Treatments
Standard chemotherapies are typically administered with a
constant dose scheduling [22,109-112]. In the administration of
cancer treatments, it is conventional that strategic dosing is used to
maximize anticancer-drug eects while minimizing host toxicity
[113]. Accordingly, many therapy schedules employ intensive therapy
initially, when the tumor is the largest, and then the dose is decreased
as the tumor is reduced.
Treatment type and administration method
Drug delivery is the method or process of administering a
pharmaceutical compound to achieve a therapeutic eect; drug-
delivery technologies modify the drug-release prole, absorption,
distribution and elimination for the benet of improving the product
ecacy and safety, as well as patient convenience and compliance.
e most common routes of administration are the oral, topical
(skin), transmucosal and inhalation routes. Many medications may
not be delivered using these routes because they might be susceptible
to enzymatic degradation or cannot be absorbed into the systemic
circulation eciently due to their molecular size. For this reason, the
drugs must be delivered by injection. Current eorts in the area of
drug delivery include the development of targeted delivery in which
the drug is only active in the target area of the body and of sustained-
release formulations in which the drug is released over a period of
time in a controlled manner from a formulation. To achieve ecient
targeted delivery, the designed system must avoid the host’s defense
mechanisms and circulate to its intended site of action [114]. One type
of sustained-release formulation includes liposomes.
We are interested i) in the dierent methods of chemotherapy-
drug administration that can be used in cancer treatments, and ii)
for each type of drug administration to describe the pharmacokinetic
(PK) models that predict the hematic-drug concentration aer their
administration.
Continuous-infusion drugs: An infusion is a method used to put
uids, including saline and drugs, directly into the bloodstream as a
body-wide way to ght cancer.
Because infusional chemotherapy is administered directly into the
blood, every cell in the body is exposed to the drugs. Cancer cells as
well as certain healthy cells may be aected. Blood counts may change
aer each treatment depending on the drugs given, so a test called a
complete blood count (CBC) will be administered to check the levels of
white and red cells as well as other elements in the blood.
Mathematical models used to describe the diusion throughout
the body of drugs that are administered via continuous infusion are
described below.
For comparison, we rst consider a therapy in which the dose
linearly increases with time [115]. is can be implemented by simply
replacing the constant dosage treatment
u
with the following:
max
( ) (1 )
β
= +ut u t
(32)
Next, we also examine the late logarithmic intensication therapy
proposed by Gonzalez et al. [116]. To model such a logarithmic therapy,
we replace the constant dosage
u
with the below term:
max
( ) log( )= +Γut u e t
(33)
where
e
is the ‘Neper constant’ equal to 2.718. We choose
Γ
and
β
to be two adjustable negative parameters that control the rate
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 84
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
because clinically, a therapy with a linearly increasing intensity could
be fatal to the patient.
Another form in the administration of drugs is used, the size-
dependent therapy [117]. For this, we replace the constant dosage
u
by the following equation:
22
11
()
2 max
max 0
2 max 2 max
11
22
()
σσ
ψ
−+
−−
=−+
++
Au t
AA
ut u e
Au Au
(34)
0
ψ
is the initial tumor size,
1
A
is the intrinsic growth rate of
the tumor (related to the initial mitosis rate) and
2
A
is the growth
deceleration factor (related to the antiangiogenic processes) and
σ
is the diusion coecient. Determining the schedule of treatment
depends mainly on the initial state of the cancerous cell population.
e function cited above describes the best way to administer a
treatment to the body. Next, we need to describe the pharmacokinetic
(PK) models that predict the hematic drug concentration aer their
administration. In the literature, the majority of therapeutic studies
did not include the pharmacokinetic action of the drug injected but
a general ecacy term is introduced, in the form
min max
u uu<<
, to
represent the percentage eectiveness of the drug.
Several dierent approaches have been used in the past to model
the fate of a drug within the human body [118]; several of them
are summarized in this paper. First, we address the mechanistic
compartmental models, the one-compartment model and the two-
compartment model. e one-compartment model is based on the
schematization of the human body as a single reactor; in this case, the
evolution of the drug concentration in the tumor tissues over time
[119] is given by:
0
γ
=−+
du uu
dt
(35)
Here,
0
u
is a control function characterizing the rate at which the
drug is introduced into the tumor, and
γ
is the dissipation coecient.
e generalizations of this model (25) were formulated in [51,120] to
demonstrate its exibility.
Another form is used to describe the PK in a one or two
compartmental model [121]; this form is explained as follows:
In the case of the one-compartment model, the human body
is represented as a single reactor, as shown in Figure 3. e two-
compartment model, shown in Figure 4, is based on the representation
of the body by two reactors, the rst of which is a central compartment,
involved in absorption, distribution, metabolism and elimination,
which represents the plasma, and the second represents tissues,
involved only in distribution.
In these two models,
()gt
represents the mass rate with which the
drug reaches the reactor and depends on the route of administration,
P
V
and
T
V
are the plasma and the tissues volumes, respectively,
PT
k
and
TP
k
are the rate constants that describe the transport between the
plasma and the tissue, respectively, EL
k
is the elimination rate constant,
and
()
P
ut
and
()
T
ut
are the time functions that express the evolutions
of the drug concentration in the plasma and tissue, respectively. All of
the rate constants are assumed to be rst-order kinetic constants.
e mathematical statements of the one- and two-compartment
models are given by the mass balances. ey are:
0In constant rate intravenous injection
In constant rate intravenous injection
()
() () () ()
() () ()
()
( 0) 0
=+− −
= −
=
= =
P
P TP T T PT P P EL P P
T
T PT P P TP T T
P
du t
V gt k Vu t k Vu t k Vu t
dt
du t
V k Vu t k Vu t
dt
gt k
ut
(36)
Where
()
P
P
V
du t dt
is the drug mass balance in the central
compartment (plasma),
()
T
T
V
du t dt
is the drug mass balance in the
peripheral compartment (tissue), and
0
k
is the constant-mass rate.
Another approach is used to distinguish the plasma and the
active drug concentrations [122]; most mathematical models assume
that the drug is instantaneously delivered to the cancer site. We
avoided this undesirable simplication by considering the dynamic
relationship between the kinetic behavior of the drug administered
and its corresponding concentration prole at the cellular level. We
propose a compartment model, as shown in Figure 5, to express the
Figure 4: Representation of human body in two-compartments model.
Figure 5: Compartmental model expressing pharmacokinetics of
plasmatic
1
()ut
and active
2
()ut
drug concentrations on the tumor site.
Figure 3: Representation of human body in one-compartment model.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 85
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
pharmacokinetics of the plasmatic
()
p
ut
and active
2
()ut
drug
concentrations on the tumor site [119].
P
V
and
2
V
are the volumes of
distribution, and
1E
k
,
12
k
and
2E
k
are the rate constants.
Let
()
p
ut
denote the concentration of the administered anticancer
drug in the plasma. e second compartment with a volume of
distribution
2
V
is the eect compartment of the active concentrations
2
()ut
[118,123]. From the plasma compartment, the drug reaches the
eect compartment, and then it is eliminated from it:
1 12
2
12 2 2
2
11
1
2
() ()
( ) ()
() () ()
0
() , 2,...,
(0) 0
(0) 0
−
=−+ +
= −
≤≤
=≤≤ =
=
=
p
Ep
p
p
pE
i ii
p
du t ut
k kut
dt V
V
du t k ut kut
dt V
d for t t
ut d for t t t i N
u
u
(37)
()ut
represents the drug-input function, which is the intermittent or
continuous drug delivery by intravenous infusion, the amount
i
d
,
denoting infused function between the start- and the end-of- infusion
times
1i
t
−
and
i
t
, respectively.
Another approach to determine the drug concentration in the body
is described below [124]:
e distribution of the drug in a tumor is represented in terms of its
concentrations in three compartments: intracellular,
()
i
ut
; extracellular,
()
e
ut
; and vascular (plasma),
()
v
ut
. e drug binds extensively to
proteins in both the plasma and the tissue [125];
()
e
ut
and
()
v
ut
therefore represent only protein-unbound drug. e following system
of ordinary dierential equations is solved over time
t
:
max
max
()
φ
φ
= −
++
= −− −
++
i ei
ee ii
e ei
tv e c
ee ii
du u u
V
dt u K u K
du u u
PS u u d V
dt u K u K
(38)
e plasma concentration for an infusion time
T
is, for
tT<
[126]:
( ) (1 ) (1 ) (1 )
αβγ
αβγ
−−−
= −+− +−
t tt
v
uA B C
ut e e e
T
(39)
And for t>T:
( ) ( 1) ( 1) ( 1)
αα β β γγ
αβγ
−−−
= −+ −+ −
tt t t tt
v
uA B C
ut e e e e e e
T
(40)
en, the plasma concentration
()
v
ut
for bolus injection is given
by:
()
α
−
=
t
v
u t uAe
(41)
Here,
u
is the total dose injected,
12
0.693 t
α
α
=
;
12
t
α
is the initial
plasma half-life of drug and the parameter
A
is the inverse volume
of distribution in plasma. Values for the other pharmacokinetic
parameters
,,B
βγ
and
C
are given in [124].
e equation used for a liposomal drug in the plasma is a
biexponential t made by Gabizon et al. [128] for their clinical data:
( )
2
1
12
()
−−
= +
kt kt
Lv
G
u
u t Ae A e
D
(42)
For the liposomal drug in the extracellular space:
( )
τ
= −−
Le Le
L t Lv Le
re
du u
PS u u
dt
(43)
erefore, the plasma concentration of the free drug
v
u
is governed
by:
( )
α
τ
=− −+ −
v t Lv
t v e B cv
B rv
du V u
PS u u AV
dt V
(44)
e parameters appearing in these equations are dened in Table
1 of reference [124].
e above model for liposomes has been adapted for thermo
sensitive liposomes [128,129], which are designed to release their
contents rapidly upon heating. First,
re
τ
the time constant for drug
release from the liposomes in the tumor extracellular space, is replaced
by the following time function:
0
()
τ
ττ
< >+
=<< +
re h h d
re h
re h h d
if t t or t t t
tif t t t t
(45)
Here,
h
t
is the time at which hyperthermia begins (aer the initial
drug injection), and
d
t
is the duration of the hyperthermia. A new
constant,
h
re
τ
is the time of drug release from the heated liposomes
[130]. e permeability of the vessel to liposomes is assumed to increase
during heating, so the parameter
L
P
is replaced by a function of time:
0
0
() < >+
=<< +
L h hd
L
L h hd
P if t t or t t t
Pt EP if t t t t
(46)
E is an enhancement factor for
L
P
at 45°C. Gaber et al. [130]
observed a 76-fold increase in the liposome extravasation upon heating
to 45°C. It is dicult to deduce from this the increase in the value of
L
P
itself. e range of E=1 to 100 is considered here.
bolus drug injection: e injection of a drug (or drugs) in a
high quantity is called a bolus [12]. It is a relatively large dose of
medication administered into a vein in a short period, usually within
1 to 30 minutes. e intravenous (IV) bolus is commonly used
when rapid administration of a medication is needed, such as in an
emergency; when drugs that cannot be diluted, such as many cancer
chemotherapeutic drugs, are administered; and when the therapeutic
purpose is to achieve a peak drug level in the bloodstream of the patient.
e IV bolus is not used when the medication must be diluted in a
large-volume parenteral uid before entering the bloodstream or when
the rapid administration of a medication, such as potassium chloride,
may be life threatening. Bolus injections allow medication to become
useful to a patient faster, which can be the dierence between life and
death in some situations.
e mathematical model used to describe the diusion throughout
the body of drugs that are administered via bolus drug injection is
similar to the model (Eq. (36)). However, in this case,
()gt
is equal to
zero because of the instantaneous (limited to the initial instant) drug
absorption.
To describe the evolution in the time of the drug concentration
with the one- and two-compartment models, the initial conditions that
are required to solve the balanced equations can be found in the recent
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 86
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
article [121] and are observed below:
In bolus intravenous injection
( 0)= =
P
P
u
ut u
(47)
Orally Administered Drugs: Oral chemotherapy doses are set up
so that the patients will have constant levels of the drugs in their bodies
to kill the cancer cells. Not taking chemotherapy drugs as they should
be taken can aect how well the treatment works, and it can even allow
the cancer to grow, so sometimes changes may be needed. Even aer
starting to feel better, they may still have cancer cells in the body that
must be kept under control with chemotherapy [12].
Oral chemotherapy drugs may be taken every day, every week, or
once or twice a month; sometimes, the frequency is as much as several
times a day. It may necessary to use chemotherapy for several months
or longer. Chemotherapy is oen given in cycles. is means that the
chemotherapy will be used for a period of time, and then there will be
a break. is allows the body of the patient to grow new, healthy cells.
e mathematical model used to describe the diusion throughout
the body of drugs that are administered via the oral method is similar
to the model (Eq. (36)). However, in this case,
()gt
is equal to
()
A BODY
kA t
, where
()
BODY
At
is the drug mass available at the time
t
at the
site of absorption, the prole of which is dened by the pharmaceutical
system-dissolution characteristics and by the drug solubility.
For the oral method, the drug mass balance at the site of absorption
is [121,131]:
() () ()= −
BODY
A BODY
dA t df t kA t
dt dt
(48)
e function
()ft
corresponds to immediate availability of the
entire dose
u
, and we dene
()ft
as follows:
constant (case a)
()
( ) (case b)
=
ft
fur t
(49)
For case a, the ingestion is of a system with a high solubility, or a
high-permeability drug; in this case, the drug’s mass that lies at the site
of absorption changes over time only because of absorption proceeding.
For case b, the system ingested is a low solubility or low permeability
drug; in this case, the drug’s mass that lies at the site of absorption
additionally changes between the instant
t
and the following instant
because of the change in the fraction of the drug that is actually available
for absorption.
A
k
is the absorption constant, and the function
()rt
expresses the in vitro release. e relationship between in vitro and in
vivo release is
() ()f t fur t=
, where
fu
is the maximum fraction of the
pharmacological dose (
u
) that could actually be absorbed, taking into
account the liver metabolism.
To describe the evolution in the time of drug concentration for
the one- and two-compartment models, the initial conditions that are
required to solve the balanced equations are mentioned in the recent
article [122] and are observed below:
In oral assumption( 0) 0
(case a)
( 0) 0 (case b)
= =
= =
P
BODY
ut
fu
At
(50)
In these three modes of administration, the drug can be taken
alone with dilution or in a liposome or a thermo liposomal capsule.
Liposomes are articially prepared vesicles made of a lipid bilayer,
which can be lled with drugs and used to deliver drugs for cancer and
other diseases. ey are composite structures made of phospholipids
and may contain small amounts of other molecules [132]. ermo
liposomal capsules have the same denition as liposomal capsules with
the dierence that the thermo liposomal capsule can release its drug
content within minutes of heating [124].
Drug combinations and treatment protocols
Chemotherapy was rst introduced in the 1940s [133]. For the
next 20 years, it was considered an investigational treatment. In the
last 30 years, chemotherapy information has evolved, and many
more eective drugs have been developed.During this time, doctors
have documented responses and conducted clinical trials comparing
standard treatments to new treatments. is process of gathering
chemotherapy information has helped to establish specic protocols
regarding the types, doses and dosing schedule of drugs that are based
on the type, stage, and other specics of a person’s cancer. ere is
no one correct choice in chemotherapy treatment. Each treatment
protocol has advantages and disadvantages, and there may be more
than one good option. In addition, treatment choices can change over
time. A good chemotherapytreatment choice at one time may not be
the right choice later.
e development of drug resistance is one reason that drugs are
oen given in combination. It is thought that this may reduce the
incidence of a resistance developing to any one drug. Oen, if a cancer
becomes resistant to one drug or group of drugs, it is more likely that
the cancer will also be resistant to other drugs. us, it is very important
to select the best possible treatment protocol at the outset.
Table 1 presents the dierent combinations of treatment used
for some cancers. For some of the combinations, we cited treatment
protocols:
• CAF treatment consists of the cyclic administration of [134]
drug “C” orally for 14 days, while “A” and “F” are given together,
intravenously (i.v.) into the hand or arm, on days 1 and 8 of that 2-week
period. is schedule will repeat four to six times, once every 4 weeks.
e entire process takes a total of approximately 4 to 5 months, barring
any complications that slow it down. Another option is that C, A, and
F are all given simultaneously, via a drip into the arm or hand. is
treatment is repeated every three weeks, four to six times, barring any
complications. e standard dosages of CAF are as follows: 600 mg/
m2 cyclophosphamide (19.9 mg/kg), 60 mg/m2 adriamycin (1.9 mg/kg)
and 600 mg/m2 uorouracil.
Doxorubicin, bleomycin, vinblastine, and dacarbazine regimen
(ABVD) [135] : Combined modality consisting of a doxorubicin 25-
mg/m2 IV plusa bleomycin 10-IU/m2 IV plusa vinblastine 6-mg/m2 IV
plusa dacarbazine 375-mg/m2 IV on days 1 and 15; every 28ad for 2-4
cycles; followed by involved eld radiation therapy (IFRT) at a dose of
approximately 20 Gy.
• Methotrexate, vinblastine, adriamycin, and cisplatin, a
regimen for bladder cancer [136-141]: Patients on the MVAC regimen
received methotrexate 30 mg/m2 on days 1, 15, and 22; vinblastine 3
mg/m2 on days 2, 15, and 22; doxorubicin 30 mg/m2 on day 2; and
cisplatin 70 mg/m2 as a 1- to 8-hour infusion on day 2. Cisplatin was
administered with adequate pre- and posthydration. e cycles were
repeated every 28 days.
Review of the Objective Function
Optimal control, which has been widely used in the cancer literature,
is the process of determining the control and state trajectories for a
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 87
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
dynamic system over a period of time to minimize (or maximize) the
nal value of a single variable
J
. is can be written mathematically
as, Eq. (1):
( )
( )
(), () , ,
Tf
f
Ti
f ff
Min J F dt f
ut wt u w T
= +
∫
e objective function may be composed of any number of
functions, combined to produce a single number, usually through a
weighted sum. It is dicult to convey the knowledge and experience
of a clinician to an optimizer through one or more simple functions.
e use of the drug is subjected to a constraint represented by the
limit of toxicity. erefore, numerical solutions are found, considering
various objective functions. It is assumed that the drug is administered
by continuous infusion and that the number of tumor cells decreases
during treatment. e toxicity of the treatment causes a limitation in
the dosing [142]; then,
u
must be within the range
max
[0, ]u
for any
t
in the range
[0, ]T
, where
T
is the nal time of treatment.
In this section, we give examples of objective functions considered
in the literature on the treatment of cancers. Starting with a quadratic
function:
22
( () ())= +
∫
Tf
Ti
J C t u t dt
(51)
Where
2
()Ct
denotes the number of cancer cells sensitive to
treatment and
2
()ut
represents the nonlinear cost of the treatment.
e quadratic forms do not make much sense, as they do not represent
anything biological. e idea behind them came from control theory
and closed-loop control, in which the quadratic error is minimized and
the quadratic control terms arise from the energy interpretation.
e linear function (Eq. (52)), is used to minimize the number of
tumor cells at the end of the treatment period while limiting the side
eects of drugs [36].
( () ())= +
∫
Tf
Ti
J C t u t dt
(52)
e logarithmic function (Eq. (53)) is used to prevent, for example,
the development of new subpopulations that are resistant to drugs. On
the other hand, the logarithmic function (Eq. (52)) was mainly used
because of the logarithmic transformation of state variables, which is
especially useful in the case of Gompertz-type growth equations.
( )
( )
( )
( )
( )
ln ln= +
∫
Tf
Ti
J C t u t dt
(53)
ese functions have been used for the optimal control of the non-
cell-cycle-specic model (drugs that are eective in all of the phases
of the cell cycle) introduced by Murray [36]. Additionally, Swan [27]
provides a good review of the role of optimal control in non-cell-cycle-
specic cancer chemotherapy. However, in another study we focus on
optimal control problems as applied to cell-cycle-specic chemotherapy.
First, Eisen [143] developed a system of linear dierential equations
describing the growth dynamics of the proliferating (drug-sensitive
phase) and quiescent (drug-resistant phase) cells. In this work, the
control over a given interval reduces the cancer to a xed level while
minimizing the total drug use. Another work by Swierniak, Polanski,
and Kimmel [44] uses optimal control theory on a cell-cycle-specic
chemotherapeutic model.
In the stochastic model, dierent objectives can be used that
combine to maximize the probability of cure while minimizing
toxicity. Here, we achieve this goal by maximizing the probability of
uncomplicated control [144]:
{ } { } (1 ()) {() 0}= × =− ×=
TN N
J P no toxicity P tumor is cured CUMP t P R t
(54)
where
()
T
CUMP t
is the cumulative probability of a toxic event with:
()≤
T N tox
CUMP t U
(55)
tox
U
denotes any explicit limits on the toxicity, and
{ ( ) 0}
N
PRt =
is
the probability that the number of cancer cells equals zero.
A patient can fail treatment for two reasons. Either the treatment
may not be able to eliminate the tumor or the treatment itself may
create toxicity for a patient, to the point that the regimen cannot be
completed. Both of these results lead to treatment failure. is function
expresses the probability that neither of these circumstances occur.
It determines the probability for a given regimen that a patient can
complete treatment (i.e., that it will not be prematurely stopped due to
toxicity) and have the tumor eliminated.
Several studies using a compartmental model are able to solve their
optimal control model and nd an analytical expression for the optimal
chemotherapy treatment plan.
In most optimal control studies, the sucient and necessary
conditions for optimality, optimality conditions, can be obtained using
frameworks such as the Pontryagin Maximum Principle [102]. On the
other hand, several other studies solve their optimal control model by
using sequential quadratic programming [145]. e objective function
to be minimized is given by [120]:
( ) ( )
1
1
00
0
()
ττ
−∞
= = =
=++
∑∑∑
∫
T
lm
ii k
i il k
J CT r C T r u d
(56)
Where
m
is the number of administered drugs and
r
and
1
r
are
weighing factors. is general model even allows for the inclusion of
special cases, such as multidrug therapy. However, this model is valid
only in cases when two of the following conditions are met:
• Each drug aects cells of a dierent type (true in the basic
model of a killing and a blocking agent treatment).
• Either the molecular source of resistance to each drug is
identical (as in multidrug resistance [146]) or the innite subsystem
representing gene amplication is required for only one type of a drug
(the basis for resistance to other drugs requires only a single mutation
and there is only one level of resistance for each of them).
e objective function of the compartmental model in the case of a
single drug is described as:
( ) ( )
00
ττ
≥
= +
∑∫
T
ii
i
J rC T r u d
(57)
where
i
C
denotes the average number of cancer cells in the
i th−
compartment,
T
denotes the time at the end of therapy, and
r
and
i
r
are weighting factors. Similarly, Eq. (57) was used in the context of
phase-specic chemotherapy in a number of papers, such as [20,22,44].
For a deterministic model with the three dierential equations
described above, (Eq. (11)), we have the following objective function:
22
12
12 1 2 3 4
0
( , ) ( () () ()) ( ) ( )
22
=++ +−
∫
tf
f nf
BB
Ju u Ct u t u t dt BCt BT t
(58)
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 88
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
In the objective functional, we minimize the total cancer-cell
population over the interval
[0, ]
f
t
through the rst term in the
integrand, and at the nal time through a salvage term
( )
3f
BC t
. e
systemic costs in the body of the two drugs (
1
u
and
2
u
) are also
minimized. It is expected that the eects of the drugs are nonlinear, and
we choose the quadratic cost terms
2
1
()ut
and
2
2
()ut
to reect these
eects, as in [38]. e coecients B1 and B2 are weight constants for the
controls and include a measure of toxicity of the drugs to the body. e
salvage term
3
()
f
BC t
is included to counter the eect of using a xed
treatment time. If this term were not present, the controls could taper
o earlier and allow a rise in the cancer-cell count at the end of the
treatment period. e salvage term
4()
nf
BTC t−
is included to create a
penalty for low values of
n
T
because this aects the patient’s ability to
ght o other diseases.
For the Stepanova model [103], we have the following objective
function:
( ) ( )
( )
0
()
ε
= −+
∫
tf
ff
J u aC t bT t u t dt
(59)
Where
a
and
b
are positives coecients determined by a stable
eigenvector, and
ε
is a positive constant.
For the delay dierential equation model [83], we have the
following objective function:
0
1
( ) ( ) ( ) ( () () ())= + + + ++
∫
tf
IfMfQf IMQ
f
J Ct C t C t Ct C t C tdt
t
(60)
And:
max
0
γ
−≥TT
(61)
Another objective function is described in a recent paper [134], as
follows:
e rst objective of treating a patient with stage IIB breast cancer,
who is in adjuvant chemotherapy for four months, is to nd the optimal
value of
()ut
in a way that the number of cancer cells,
()Ct
, resulting
from this treatment, track
()
r
Ct
. us, the optimal value of
()ut
can
minimize the cost function, described as follows:
2
1
1
( ( ))
=
= −
∑
tf
ir
i
J C Ci
(62)
Where,
()
r
Ct
is a reference trajectory, and
84
f
t=
is the period of
treatment in this case. Because biological systems usually respond in a
sigmoidal fashion to inputs,
()
r
Ct
is derived by the nonlinear rescaling
[147,148].
e second objective function is used to preserving the normal
cell population
( )
Nt
in the best way by minimizing the following cost
function:
2 2 22
2
( (21 ) ) ( (42 ) ) ( (63 ) ) ( (84) )
−−−
∞ ∞ ∞∞
= −+ −+ −+ −JN NN NN NN N
(63)
where
N
∞is an asymptotic number of normal cells, and
()Ni
−
for i
= 21, 42, 63 is the normal cell population at the times just before the
second, third and fourth cycles of the chemotherapy.
(84)N
is the
normal cell population 21 days aer the fourth cycle of chemotherapy.
Dierent benecial cases can be found between these two extreme
situations by minimizing the cost function, described as follows:
11 2 2
= +J WJ W J
(64)
where, the weighting coecients
1
W
and
2
W
are selected by the
treating physician according to the particular conditions of each
patient to indicate the cases for which the cancer-cell reduction has
more priority than preserving the normal cells
12
()WW>
or vice versa.
Solution Methods
As we have already cited, optimal-control theory is usually used
to model chemotherapy-treatment planning. Optimal-control theory
uses a system of dierential equations that are solved to determine the
optimal choice of the chemotherapy-treatment plans. ese problems
are dicult to solve optimally. For these reasons, the optimal control
theory is applied with simplifying assumptions that reduces its clinical
validity.
In the rst part, this paragraph presents a simple review of the ways
in which optimal-control theory interacts with cancer chemotherapy.
In the second part, we review the optimal-control models based on the
solution methods used: the analytical solution, approximation with
analytical intuition, and heuristics [37].
Role of optimal-control theory
ere are three main areas of study: the rst studies the diverse
growth-kinetics models, the second studies cell-cycle models and the
third is a classication of ‘other models’.
Diverse growth-kinetics models: In this section, the authors in
[27] use the Gompertz equation to describe the evolution of cancer cells
under the inuence of chemotherapy treatment. e optimal-control
problem is formulated using the Hamiltonian function to determine
the positive continuous time optimal controller u that minimizes the
toxicity of the treatment subject to the perturbed growth kinetics of
the tumor.
Another model used to describe the evolution of cancer cells is
the Verhulst-Pearl equation [149]. For this model, several numerical
results to the solution are presented. e optimal control u increases,
but as time increases, its rate of increase slows down rapidly. For the
Cox-Woodbury-Meyers equation [150], there is a nonlinear algebraic
feedback relationship that connects the control and the state, and the
optimal controller is monotonically decreasing and drives the tumor
population to a more desirable level. It is hoped to see closed-loop
control.
Finally, Abulesz and Lyberatos use a model to describe the
evolution of cancer and the normal cells [151], with the introduction
of a pharmacokinetic equation. ey give an optimal solution for the
drug administered, but it is not clear what the biological signicance
of the steady state means, and it is not clear how this applies to the
dosage rate.
Cell-cycle models: e rst application of optimal-control theory
in a cancer-chemotherapy problem is to a cell-cycle model analyzed by
Bahrami and Kim [28], using a discrete-time state vector approach. e
optimal-control problem is to determine the controls u(k) such that
the size of the tumor is minimized at the end of the treatment interval
while minimizing the toxicity.
Creasey et al. [152] present a general overview of the optimal
control and cancer chemotherapy, but no particular problem is solved.
Long and Fegley [153] consider a discrete dosage program, which is cast
into the framework of dynamic programming. ey wish to determine
the doses and treatment times to maximize the surviving fraction of
normal cells at the end of the recovery period. ey do not apply their
procedure to any illustrative problem.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 89
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
Biran an dMcKinnis [154] considered a single closed loop of the
cell cycle in three phases. e time rate of change of each number of
cells in one phase is related to the ux of cells into and out of each
phase. Numerical results are presented for the case of use of the drug
Melphalan. e optimal control results suggest that cultured cells
treated with Melphalan accumulate and arrest their progress in the
mitotic phase.
In another paper, Dibrov et al. [155] examine a dynamic model
of a population of continuously dividing cells under multiple periodic
treatments with a phase-specic agent. ey introduce a model of
the cell cycle that has both deterministic and what they refer to as
“probabilistic” discrete compartments.
Other models: In this section, several models are presented that,
are not directly dependent on the growth kinetics of the tumor but
could be combined with them if desired. In the papers by Bellman and
coworkers [156] the authors study the “control aspects of the pioneering
eorts in the modeling of chemotherapy. Bischo and Dedrick [157]
described a physiological model for the anticancer drug concentration
in anatomical compartments. Swan and Alexandro [158] present the
reduction of a physiological model for the chemotherapy of brain
tumors.
Problems of the following type remain to be investigated:
determining the optimal drug inputs in regional chemotherapy
to achieve a cell kill in the tumor while minimizing the toxicity to
important anatomical compartments. Gaglio et al. [159] describe two
expert systems that are being used for the characterization of optimal
adjuvant cancer therapies. Unfortunately, no details of the optimal-
control application are provided. Maceratini et al. [160] have reviewed
expert systems and some of their applications in medicine.
For more details about the role of optimal-control theory, we refer
the reader to reference [161], in which four dierent mathematical
models of chemotherapy from the literature are investigated with
respect to the optimal control of the drug-treatment schedules. e
various models are based on two dierent sets of ordinary dierential
equations and contain either chemotherapy, immunotherapy,
antiangiogenic therapy or combinations of these. Optimal-control
problem formulations based on these models are proposed, discussed
and compared. For dierent parameter sets, scenarios, and objective
functions, optimal-control problems are solved numerically with
Bock’s direct multiple shooting methods.
Role of the Nonlinear Optimization Method
In this paragraph, we classify the optimal-control models with
respect to the solution method used.
Analytical solution: We use the analytical method when the
model is composed of linear dierential equations; in most optimal-
control studies, the sucient and necessary conditions for optimality,
the optimality conditions, can be obtained using frameworks such as
the Pontryagin Maximum Principle [99]. Zietz and Nicolini [33] use
the Pareto version of the Pontryagin Maximum Principle (Yu and
Leitmann [162]) to show that a specic type of chemotherapy plan
is optimal under certain conditions; their solution depends on the
two weighting factors in the Hamiltonian form of the model. Murray
[36] uses Pontryagin Maximum Principle to show that the optimal
chemotherapy treatment plan is a mixture of an initial bolus dose (an
immediate infusion of a single dose) drug application followed by no
drug and then continuous infusion.
Another optimal control is solved by analytical solution, for
example, by explicit formulation of the treatment plan [30,33,43], or
by explicit formulation of the treatment administration period given
the dose size [163].
Approximation with analytical intuition: In some studies, the
optimality conditions obtained using the Pontryagin’s Maximum
Principle are not simple enough to explicitly derive the optimal
chemotherapy treatment plan. e optimality conditions provide an
idea about the structure of the optimal solutions. However, optimality
conditions, which are used in the characterization of the optimal
solution, are expressed in terms of adjoint variables. ese adjoint
variables turn up in the Hamiltonian function that the optimal
solution needs to maximize for all instances. ese conditions give a
large system of dierential equations, which is dicult to solve. For
this reason, these studies use approximation techniques to determine
the optimal chemotherapy treatment plan, such as i) discretization
of the continuous decision horizon and solving of the optimal model
using control parameterization; and ii) solving the system of equations
obtained from the optimality conditions using approximate methods
such as Newton’s method [164].
Certain optimal controls are solved by approximation with
analytical intuition. Several example are by using standard methods
with hypothetical data [165], by characterizing the optimal solutions
for quadratic and linear controls [145], by using an iterative algorithm
that may converge to a local optimum [166], by transforming the
model to mixed integer linear programming (MILP) by discretizing
the decision horizon and linearizing the nonlinear constraints [15], by
characterizing analytically the form of the optimal solution and using
discretization and nonlinear programming to solve it [167], by using
a linear time-varying approximation technique, or nally, by using
an explicit formulation of the optimal treatment plan as a function of
variables from the Hamiltonian form of the model that are quantied
via approximation [38].
Heuristics: Although the optimal solution is typically
computationally intractable, we develop heuristic algorithms to solve
the optimal chemotherapy treatment. Tan et al. [168] use distributed
evolutionary computing soware that employs the resources of a
network of computers to overcome complex chemotherapy treatment
planning problems. Floares et al. [147] design an adaptive neural
network algorithm to solve this problem. Liang et al. [169] minimize
the size of the tumor population using an adaptive elitist-population-
based genetic algorithm, and they design a multimodal optimization
genetic algorithm that represents the chemotherapy treatment plans
as genes to optimize the chemotherapy plan under various cumulative
toxicity functions.
Several optimal controls are solved by heuristics. For example,
Iliadis and Barbolosi [118] use simulation to evaluate the performance
of possible drug-dose plans for given drug-administration periods.
Petrovski and McCall [170] adapted the strength Pareto evolutionary
algorithm [170] to evolve a population of treatments spread out along
the Pareto ecient frontier and nd the Pareto-optimal chemotherapy
plans. ey also develop a new heuristic algorithm in their further
research: a genetic and a particle swarm optimization (PSO) algorithm
[171]. e genetic algorithm encodes the multi-drug chemotherapy
schedules as binary strings. e PSO algorithm uses a population of
candidate-solution particles that swam around the search space. Tse et
al. [172], minimize the size of the tumor population using a customized
algorithm combining a genetic algorithm and iterative dynamic
programming.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 90
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
All of the methods cited above are used to nd a global optimum
value, knowing that there are several computational biology research
studies [142,173,174] that integrate local search algorithms into a
global research algorithm to increase the computing accuracy.
Discussion and Conclusion
e number of cancerous cells, the toxicity, and the drug resistance
are the key factors in chemotherapy-treatment planning. Most of the
studies on chemotherapy-treatment optimization problem dier from
each other according to how they model the interactions among these
key factors.
is paper provides a comprehensive review of the relevant
literature while considering several other pieces of information not
mentioned previously, such as the cancer modeling characteristics
and optimization computational methods used to solve these models.
We also provide information on the medical relevance, such as real
treatment protocols and one-drug/multi-drug involvement.
Our study is oriented toward the optimization of treatment
protocols. e optimization methods can still lead to solutions, even
if the parameters to identify are numerous. Clinically, however, to
identify the parameters, we need many experiences. e number of
parameters is large, and therefore, the number of experiences is large;
for this reason, it is preferable that the number of parameters is small.
It is clear from all of the studies presented that there is a desperate
need to create more interaction between mathematicians, clinicians
and biologists to correctly identify the model. Of course, these highly
interdisciplinary eorts are not easy to achieve. We see the role of
mathematical modeling in cancer biology as twofold. On the one
hand, mathematical models are able to verify supposed word models
suggested by experimentalists.
e eld of cancer biology is reaching a stage of maturity at which the
next step in the modeling process must be the careful parameterization
of a number of the models so that specic experimental predictions can
be made and tested in close collaboration between experimentalists and
theoreticians. Many problems in this review require the determination
of the parameters in the model by tting the model to data. Clinical
tests are performed over short durations, so we require an ecient
identication protocol to correctly identify the model parameters. e
clinical trials occur over a short duration, whereas the phenomenon is
long, occurring over several months.
In the literature, most mathematical models that describe the
diusion of drugs in the body are based on a single drug. Whereas,
in real cases, oncologist use a treatment protocol with a combination
of drugs. In some cases, these drugs work together in series or parallel
mode, or without any interaction between them. Some drugs have
physical eects and other have chemical eects. Based on these
information, it is necessary to develop these models to consider all
these aspects of administered drugs. Although there have been some
success stories in the application of mathematical models to cancer
biology with chemotherapy treatment, mathematics has much more
to oer for the use of mathematical models with real chemotherapy
treatment protocols. For example, in Eq. (9), this model describes the
use of a single drug. However, by using a real protocol, the treatment of
many drugs must be considered together, so we propose to change the
form of the equation:
1
log( ) ( ..... ) 1....
θ
αβ
= −− =
i
dC C C C fct u u i number of drugs
dt C
(65)
is form must be veried by mathematicians and biologists to
ensure its reliability. Briey, one of the advantages of mathematical
modeling is its ability to determine a delicate balance when building
any model between reliability and realism.
Research in chemotherapy should consider a more realistic
objectives function. Currently, most models aim to obtain the
chemotherapy treatment plan that minimizes the size of the tumor
population while satisfying the constraints on maximum toxicity in a
given time interval. However, this is only a part of an optimal treatment
plan. In this context, the question of how to dene the objective function
naturally becomes more important. In this paper we have reviewed all
of the objectives functions to include and cover all of the possibilities of
cancer phenomena in our optimization method.
It is important to compare the performance of all objective functions
to choose the best that can lead to the optimal solutions. Reading
various articles shows that all objective functions lead practically to
optimal solutions close to each other, without showing the specicity
of each objective function. For these reasons, comparative study is
important.
e chemotherapy process involves the interactions between the
patient’s response and the oncologist’s scheduling decision. During
the chemotherapy session, when drug resistance becomes obvious,
the drugs must be changed immediately. During this process, better
information about the characteristics of the tumor will be available,
which necessitates dynamic and sequential decision making. On the
other hand, current models consider only the decision process prior
to the initiation of chemotherapy and then calculate the optimal
plan of the entire period. For these reasons, we incorporate in our
optimization method the evolution of the normal and immune cells to
help the oncologist to make a better decision about the administration
of the treatment. In real cases, the evolution of normal and cancer cells
varies according to the treatment. e relations between the treatment
dosage and the number of normal and immune cells are clearly visible
in the mathematical models that describe the evolution of these cells.
Additionally, we can improve the optimization problem by adding
as constraints the evolution of the white blood cells (WBC) in the
body aer treatment administration. ere are mathematical models
that describes the evolution of the number of immune cells under
the inuence of cancer and drugs. We can add Eq. (66) [118] for
mathematical models that do not have an equation that describes the
evolution of the number of immune cells. In this case, the oncologist
can dene if it is necessary to stop the treatment if the number of
immune cells is less than a predened number.
( ) ( ) ( ) ( ) ( )
νµ
=−−
dWBC t r t WBC t WBC t u t
dt
(66)
where
WBC
are produced at rate
()rt
and eliminated by the rst-
order process
()WBC t
ν
. e dimensions of
()rt
,
ν
,
µ
and
WBC
are
11
(volume) (time)
−−
,
1
(time)
−
,
11
(concentration) (time)
−−
and 0
3
mm−
,
respectively. e initial condition
0
WBC
was selected as the physiological
level of
WBC
with the constant
()
c
rt r=
and without drug toxicity;
therefore, neglecting the third term in the equation, we can set
0c
WBC r
ν
=
.
e normal range for the
WBC
count is 4.3 to 10.8×109 cells per
liter. A range of 11 to 17×109/L may be considered mild-to-moderate
leukocytosis, and a range of 3.0 to 5.0×109/L may be considered mild
leucopenia [142].
In the literature, applying the optimization method to the
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 91
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
mathematical models that describe the evolution of cancer cells, the
authors do not provide any clinical applications. However, in our
further research, the usefulness of these models is shown for using real
chemotherapy treatment protocols. e major limitation of the existing
studies is the lack of clinical realism in the models. Most studies treat
chemotherapy-treatment planning as a pure optimization problem. As
a result, the practical applications of this area of research are limited.
In this paper, we provide several suggestions to improve the research
in this area.
Mathematical models must be developed to aid in our
understanding of how to implement this work and hopefully to show
how to develop new optimal treatment strategies. To accomplish this
goal, we can determine a strategy that makes it possible by following it
to apply the optimization method using real chemotherapy treatment
protocols. is strategy can be described as follows:
1) Dene the genre of the cancer treated.
2) Choose the model of cancer evolution with and without
treatment.
3) Identify the parameters of this model according to the cancer
treated.
4) Choose a real treatment protocol dened by the oncologist.
5) Choose the pharmacokinetic model that describes the evolution
of the drug concentration aer its administration.
6) Dene the model of resistance (e.g., white blood cells) added as
constraints.
7) Dene the minimal and maximal thresholds for the resistance of
treatment and white blood cells.
8) Resolve the optimization problem using one solution method
(we recommend the genetic algorithm).
By using a real treatment protocol, we suggest that the treatment
optimization is performed using genetic algorithms (GA) [175] to
avoid the problem of oversizing caused by the use of deterministic
optimization methods. GA allows us to set an initial population size
to a xed value at the beginning of the search that remains constant
throughout the run. However, making it necessary to specify this initial
parameter value is problematic in many ways. If it is too small, the GA
may not be able to reach high-quality solutions. If it is too large, the GA
spends unnecessary computational resources.
We propose the use the genetic algorithm because it is a method
that has been adopted in all cases. It allows us to oer a soware tool
ready for use by oncologists. Additionally, the deterministic models
converge according to each case, so it is not a generalized method
contrariwise for stochastic models.
Due to the complexity of cancer, chemotherapy-treatment studies
should focus on a specic cancer rather than modeling cancers in
general. It is important to biologically classify the models, based on the
cancer denition and not as a function of the mathematical description
(e.g., linear or nonlinear models).
It is evident from all of the studies presented that there is a
desperate need to have more interaction between the modelers and
either the experimentalists or the clinicians. Of course, such highly
interdisciplinary endeavors are not easily accomplished. Another
feature of importance is the need to work with models of biological
relevance that can be tested on a computer and in a laboratory situation.
At present, there are no commonly used techniques to measure
the number of tumor cells in humans when the level is below the
minimum level of a tumor that can be diagnosed, (Cd). Some clinicians
believe that one should treat the tumor only to the level Cd because, as
indicated, they cannot measure the size below Cd. is means that we
should attempt to nd a model to describe cancer appearance starting
with at least one cell. In the thesis [176], the authors demonstrate,
using stochastic models, that the cancer passes through two phases:
i) a creation phase, in which they study the probability that a cancer
will arise from a healthy population of cells under the inuence of
mutations, and ii) a phase in which the cancer appears. e rst phase
is explained by stochastic static models. ese models are divided into
three types, i) the Moran model [64,65], ii) the Wright-Fisher model
[66] and iii) the Moolgavkar, Venzon, and Knudson (MVK) model
[69]. Simulations of these models show the uncontrollable evolution
of cancer cells. e cancer cell number grows until reaching the
population size or to a nonpredictable number.
In this paper, we reviewed i) the mathematical models that describe
the evolution of cancer and biological cells; ii) the mathematical models
that describe the diusion of drugs throughout the body that are
administered via bolus injection, continuous infusion, and liposomal
and thermo liposomal drug delivery; iii) a set of objective functions that
aect the results of the optimization method; and iv) a solution method
for the optimizations models. ese studies imply signicant gaps
between theoretical research and clinical application in chemotherapy
treatments. We note that the application of the optimization method
in the real case of chemotherapy is only a tool that helps doctors to
determine the best decision for a patient’s treatment.
Finally, the question of therapeutic optimization in cancer is vast,
and it can be treated in notably dierent manners that must be adapted
for the particular clinical problem at hand. Nevertheless, modeling the
target and the means of control while considering the known clinical
issues is still a successful way to reduce the cost and time of clinical
trials. Room remains for further mathematical developments to meet
other challenges, especially for optimizing treatments for more clinical
problems. ese results will be all the more useful as more links develop
between mathematicians and clinicians.
Acknowledgment
Deepest gratitude is due to the members of the R.I.T.C.H. and L.I.S.V.
laboratories. I would also like to convey thanks to AUF for its nancial support.
Finally, best wishes go out to my beloved family, for their understanding and
endless love through the duration of my studies.
References
1. World Health Organization (2011) International Agency for Research on Cancer,
Global Cancer Facts and Figures. (2nd Edn) American Cancer Society, Atlanta.
2. Costa MI, Boldrini JL (1997) Conicting objectives in chemotherapy with drug
resistance. Bull Math Biol 59: 707-724.
3. Orme ME, Chaplain MA (1997) Two-dimensional models of tumour angiogenesis
and anti-angiogenesis strategies. IMA J Math Appl Med Biol 14: 189-205.
4. Anderson ARA, Chaplain MAJ, Newman EL, Steele RJC, Thompson AM
(2000) Mathematical modeling of tumour invasion and metastasis. Journal of
Theoretical Medecine 2: 129-154.
5. Anderson AR (2005) A hybrid mathematical model of solid tumour invasion: the
importance of cell adhesion. Math Med Biol 22: 163-186.
6. Chaplain MAJ, Lolas G (2006) Mathematical modelling of cancer invasion of
tissue: dynamic heterogeneity. Net Hetero Med 1: 399-439.
7. Painter KJ, Armstrong NJ, Sherratt JA (2010) The impact of adhesion on cellular
invasion processes in cancer and development. J Theor Biol 264: 1057-1067.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 92
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
8. Powathil GG, Gordon KE, Hill LA, Chaplain MA (2012) Modelling the effects of
cell-cycle heterogeneity on the response of a solid tumour to chemotherapy:
biological insights from a hybrid multiscale cellular automaton model. J Theor
Biol 308: 1-19.
9. Gopal R, Nair R, Saikia TK, Soman CS, Dinshaw KA, et al. (1994) Modied
MACOP-B chemotherapy for intermediate and high grade non Hodgkins
lymphomas. J Assoc Physicians India 42: 781-784.
10. Andre T, Colin P, Louvet C, Gamelin E, Bouche O, et al. (2003) Semimonthly
versus monthly regimen of uorouracil and leucovorin administered for 24 or
36 weeks as adjuvant therapy in stage II and III colon cancer: results of a
randomized trial. J Clin Oncol 21: 2896-2903.
11. André T, Boni C, Mounedji-Boudiaf L, Navarro M, Tabernero J, et al. (2004)
Oxaliplatin, uorouracil, and leucovorin as adjuvant treatment for colon cancer.
N Engl J Med 350: 2343-2351.
12. Hoff PM, Ansari R, Batist G, Cox J, Kocha W, et al. (2001) Comparison of
Oral Capecitabine Versus Intravenous Fluorouracil Plus Leucovorin as First-
Line Treatment in 605 Patients With Metastatic Colorectal Cancer: Results of a
Randomized Phase III Study. J Clin Oncol 19: 2282 - 2292.
13. Susan EC, Faina N, John CP (2000) Molecular biology of breast cancer
metastasis: The use of mathematical models to determine relapse and to
predict response to chemotherapy in breast cancer. Breast Cancer Res 2: 430-
435.
14. Retsky MW, Demicheli R, Swartzendruber DE, Bame PD, Wardwell RH, et
al. (1997) Computer simulation of a breast cancer metastasis model. Breast
Cancer Res Treat 45: 193-202.
15. Harrold JM, Parker RS (2009) Clinically relevant cancer chemotherapy
dose scheduling via mixed integer optimization. Computers and Chemical
Engineering 33: 2042-2054.
16. Hughes TP, Branford S, White DL, Reynolds J, Koelmeyer R, et al. (2008)
Impact of early dose intensity on cytogenetic and molecular responses in
chronic- phase CML patients receiving 600 mg/day of imatinib as initial therapy.
Blood 112: 3965-3973.
17. Vogel CL, Cobleigh MA, Tripathy D, Gutheil JC, Harris LN, et al. (2002) Efcacy
and safety of trastuzumab as a single agent in rst-line treatment of HER2-
overexpressing metastatic breast cancer. J Clin Oncol 20: 719-726.
18. Itik M, Metin US, Stephen PB (2009) Optimal control of drug therapy in
cancer treatment. Nonlinear Analysis: Theory, Methods and Applications 71:
e1473-e1486.
19. Fister KR, Panetta JC (2000) Optimal control applied to cell-cycle-specic
cancer chemotherapy. Siam J Appl Math 60: 1059-1072.
20. Ledzewicz U, Schättler H (2002) Analysis of a cell cycle specic model for
cancer chemotherapy. J Biol Syst 10: 183.
21. Ledzewicz U, Schättler H (2002) Optimal bang-bang controls for a two-
compartment model in cancer chemotherapy. J Opt Theory Appl 114: 609-637.
22. Swierniak A, Urszula L, Heinz S (2003) Optimal control for a class of
compartmental models in cancer chemotherapy. Int J Appl Math Comput Sci
13: 357-368.
23. Kimmel M, Swierniak A (2006) Control theory approach to cancer chemotherapy:
beneting from phase dependence and overcoming drug resistance. Tutorials
in Mathematical Biosciences III: Lecture Notes in Mathematics, Mathematical
1872: 185-221.
24. Swan GW (1975) Some strategies for harvesting a single species. Bull Math
Biol 37: 659-673.
25. Swan GW (1981) Optimal control applications in biomedical engineering-a
survey. Optim Control Appl Methods 2: 311-334.
26. Swan GW (1984) Applications of Optimal Control Theory in Biomedicine,
Marcel Dekker.
27. Swan GW (1990) Role of optimal control theory in cancer chemotherapy. Math
Biosci 101: 237-284.
28. Bahrami K, Kim M (1975) Optimal control of multiplicative control systems
arising from cancer therapy. IEEE Trans Autom Control 20: 537-542.
29. Swierniak A, Duda Z (1994) Singularity of optimal control in some problems
related to optimal chemotherapy. Math Comput Model 19: 255-262.
30. Swan GW, Vincent TL (1977) Optimal control analysis in the chemotherapy of
IgG multiple myeloma. Bull Math Biol 39: 317-337.
31. Swan GW (1977) Some Current Mathematical Topics in Cancer Research.
Published for the Society for Mathematical Biology by University Microlms
International 27: 1-289.
32. Swan GW (1988) General applications of optimal control theory in cancer
chemotherapy. IMA J Math Appl Med Biol 5: 303-316.
33. Zietz S, Nicolini C (1979) Mathematical approaches to optimization of cancer
chemotherapy. Bull Math Biol 41: 305-324.
34. Dibrov BF, Zhabotinsky AM, Neyfakh YA, Orlova MP, Churikova LI (1985)
Mathematical model of cancer chemotherapy. Periodic schedules of phase-
specic cytotoxic-agent administration increasing the selectivity of therapy.
Math Biosci 73: 1-31.
35. Shin KG, Pado R (1982) Design of optimal cancer chemotherapy using a
continuous time state model of cell kinetics. Math Biosci 59: 225-248.
36. Murray JM (1990) Optimal control for a cancer chemotherapy problem with
general growth and loss functions. Math Biosci 98: 273-287.
37. Jinghua S, Oguzhan A, Faith SE, Qiang S (2011) A survey of optimization
models on cancer chemotherapy treatment planning. Ann Oper Res 221: 331-
356.
38. Nanda S, Moore H, Lenhart S (2007) Optimal control of treatment in a
mathematical model of chronic myelogenous leukemia. Math Biosci 210: 143-
156.
39. Swierniak A, Kimmel M, Smieja J (2009) Mathematical modeling as a tool for
planning anticancer therapy. Eur J Pharmacol 625: 108-121.
40. Clairambaud J (2009) Modelling physiological and pharmacological control
on cell proliferation to optimize cancer treatments. Mathematical Modelling of
Natural Phenomena 4: 12-67.
41. Billy F, Jean C, Olivier F (2012) Optimisation of cancer drug treatments using
cell population dynamics. Mathematical Methods and Models in Biomedicine,
Lecture Notes on Mathematical Modelling in the Life Sciences 2013: 265-309.
42. Chakrabarty SP, Hanson FB (2005) Optimal control of drug delivery to brain
tumors for a distributed parameters model. Proceedings of the American
Control Conference 2: 973-978.
43. Murray JM (1990) Some optimal control problems in cancer chemotherapy with
a toxicity limit. Math Biosci 100: 49-67.
44. Swierniak A, Polanski A, Kimmel M (1996) Optimal control problems arising in
cell-cycle-specic cancer chemotherapy. Cell Prolif 29: 117-139.
45. Sethi SP, Thompson GL (1988) Optimal Control Theory. Applications to
Management Science and Economics 18: 506.
46. Cohen Y (1987) Applications of Control Theory in Ecology, Lecture notes in
Biomathematics 73: 39-56.
47. Ami D, Cartigny P, Rapaport A (2005) Can marine protected areas enhance
both economic and biological situations? C R Biol 328: 357-366.
48. Eisen M (1988) Mathematical Methods and Models in the Biological Sciences,
Nonlinear and Multidimensional Theory. Prentice Hall Advanced Reference
Series, New Jersey.
49. Martin R, Teo KL (1994) Optimal Control of Drug Administration in Cancer
Chemotherapy. World Scientic, New Jersey.
50. Swierniak A, Polanski A, Smieja J, Kimmel M (2003) Modelling growth of
drug resistant cancer populations as the system with positive feedback. Math
Comput Model 37: 1245-1252.
51. Swierniak A, Smieja J (2005) Analysis and optimization of drug resistant and
phase-specic cancer chemotherapy models. Math Biosci Eng 2: 657-670.
52. Parker RS, Doyle FJ 3rd (2001) Control-relevant modeling in drug delivery. Adv
Drug Deliv Rev 48: 211-228.
53. Tiina R, Chapman SJ, Maini KP (2007) Mathematical models of avascular
tumor growth. Siam Review 49: 179-208.
54. Norton L (1988) A Gompertzian model of human breast cancer growth. Cancer
Res 48: 7067-7071.
55. Sun X, Zhang L, Tan H, Bao J, Strouthos C, et al. (2012) Multi-scale agent-
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 93
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
based brain cancer modeling and prediction of TKI treatment response:
incorporating EGFR signaling pathway and angiogenesis. BMC Bioinformatics
13: 218.
56. Orme ME, Chaplain MA (1997) Two-dimensional models of tumour angiogenesis
and anti-angiogenesis strategies. IMA J Math Appl Med Biol 14: 189-205.
57. Alrón T (2009) Modeling tumor-induced angiogenesis: A review of individual-
based models and multi-scale approaches. Mathematics, Developmental
Biology, and Tumor Growth, Contemporary Math 492: 45-75.
58. McDougall SR, Anderson AR, Chaplain MA (2006) Mathematical modelling
of dynamic adaptive tumour-induced angiogenesis: clinical implications and
therapeutic targeting strategies. J Theor Biol 241: 564-589.
59. Dai W, Tang X, Adrian B, Zhang L, Raja N (2006) Optimal temperature
distribution in a 3D triple layered skin structure with embedded vasculature.
Journal of Applied Physics, 99: 104702-104709.
60. Dréau Didier, Dimitre S, Ted C, Mirsad H (2009) An agent-based model of solid
tumor progression. Bioinformatics and Computational Biology, Lecture Notes in
Computer Science 5462: 187-198.
61. Zhang L, Athale CA, Deisboeck TS (2007) Development of a three-dimensional
multiscale agent-based tumor model: simulating gene-protein interaction
proles, cell phenotypes and multicellular patterns in brain cancer. J Theor Biol
244: 96-107.
62. Deisboeck TS, Zhang L, Yoon J, Costa J (2009) In silico cancer modeling: is it
ready for prime time? Nat Clin Pract Oncol 6: 34-42.
63. Haeno H, Iwasa Y, Michor F (2007) The evolution of two mutations during clonal
expansion. Genetics 177: 2209-2221.
64. Haeno H, Michor F (2010) The evolution of tumor metastases during clonal
expansion. J Theor Biol 263: 30-44.
65. Nowak MA, Michor F, Iwasa Y (2003) The linear process of somatic evolution.
Proc Natl Acad Sci U S A 100: 14966-14969.
66. Beerenwinkel N, Antal T, Dingli D, Traulsen A, Kinzler KW, et al. (2007) Genetic
progression and the waiting time to cancer. PLoS Comput Biol 3: e225.
67. Serra MC, Haccou P (2007) Dynamics of escape mutants. Theor Popul Biol
72: 167-178.
68. Keinj R, Bastogne T, Vallois P (2011) Multinomial model-based formulations of
TCP and NTCP for radiotherapy treatment planning. J Theor Biol 279: 55-62.
69. Schöllnberger H, Beerenwinkel N, Hoogenveen R, Vineis P (2010) Cell
selection as driving force in lung and colon carcinogenesis. Cancer Res 70:
6797-6803.
70. Afenya EK, Bentil DE (1998) Some perspectives on modeling leukemia. Math
Biosci 150: 113-130.
71. Murray JD (2003) Mathematical Biology II. Spatial Models and Biomedical
Applications 18: 3-839.
72. Jackson TL, Byrne HM (2000) A mathematical model to study the effects of drug
resistance and vasculature on the response of solid tumors to chemotherapy.
Math Biosci 164: 17-38.
73. Papadogiorgaki M, Koliou P, Kotsiakis X, Zervakis ME (2013) Mathematical
modelling of spatio-temporal glioma evolution. Theor Biol Med Model 10: 47.
74. Wai-Yuan Tan, Leonid Hanin (2008) Mathematical Models of Cancer and Their
Relevant Insights. Handbook of cancer models with applications 14: 173-223.
75. Partt AM, Fyhrie DP (1997) Gompertzian growth curves in parathyroid
tumours: further evidence for the set-point hypothesis. Cell Prolif 30: 341-349.
76. Lairda AK (1969) Dynamics of growth in tumours and in normal organisms. Natl
Cancer Inst Monogr 30: 15-28.
77. Floares A (2003) Neural Networks Control of Drug Dosage Regimens in Cancer
Chemotherapy. Proceedings of the International Joint Conference on Neural
Networks 154-159.
78. Afenya E(1996) Acute leukemia and chemotherapy: a modeling viewpoint.
Math Biosci 138: 79-100.
79. Kozusko F, Bajzer Z (2003) Combining Gompertzian growth and cell population
dynamics. Math Biosci 185: 153-167.
80. Afenya EK (2001) Recovery of normal hemopoiesis in disseminated cancer
therapy--a model. Math Biosci 172: 15-32.
81. Moore H, Li NK (2004) A mathematical model for chronic myelogenous
leukemia (CML) and T cell interaction. J Theor Biol 227: 513-523.
82. Villasana M, Radunskaya A (2003) A delay differential equation model for tumor
growth. J Math Biol 47: 270-294.
83. Minaya Villasana, Gabriela Ochoa (2004) Heuristic design of cancer
chemotherapies. IEEE transactions on evolutionary computation 8: 513-521.
84. de Pillis LG, Gu W, Radunskaya AE (2006) Mixed immunotherapy and
chemotherapy of tumors: modeling, applications and biological interpretations.
J Theor Biol 238: 841-862.
85. Farrar JD, Katz KH, Windsor J, Thrush G, Scheuermann RH, et al. (1999)
Cancer dormancy. VII. A regulatory role for CD8+ T cells and IFN-gamma in
establishing and maintaining the tumor-dormant state. J Immunol 162: 2842-
2849.
86. O’Byrne KJ, Dalgleish AG, Browning MJ, Steward WP, Harris AL (2000) The
relationship between angiogenesis and the immune response in carcinogenesis
and the progression of malignant disease. Eur J Cancer 36: 151-169.
87. Morecki S, Pugatsch T, Levi S, Moshel Y, Slavin S (1996) Tumor-cell vaccination
induces tumor dormancy in a murine model of B-cell leukemia/lymphoma
(BCL1). Int J Cancer 65: 204-208.
88. Müller M, Gounari F, Prifti S, Hacker HJ, Schirrmacher V, et al. (1998) EblacZ
tumor dormancy in bone marrow and lymph nodes: active control of proliferating
tumor cells by CD8+ immune T cells. Cancer Res 58: 5439-5446.
89. Dalgleish AG, O’Byrne KJ (2002) Chronic immune activation and inammation
in the pathogenesis of AIDS and cancer. Adv Cancer Res 84: 231-276.
90. Dzivenu OK, Phil D, Donnel-Torney JO (2003) Cancer and the immune system:
The vital connection. A publication from: Cancer Research Institute.
91. Perelson AS, Weisbuch G (1997) Immunology for physicists. Reviews of
Moderne Physics 69: 1219-1267.
92. de Boer RJ, Hogeweg P (1986) Interactions between macrophages and
T-lymphocytes: tumor sneaking through intrinsic to helper T cell dynamics. J
Theor Biol 120: 331-351.
93. De Pillis LG, Radunskaya AE (2003) A mathematical model of immune
response to tumor invasion. Second MIT on Computational Fluid and Solid
Mechanics 2: 1661-1668.
94. Newbury G (2007) A Numerical Study of a Delay Differential Equation Model for
Breast Cancer, Master’s thesis.
95. Yaa R (2006) Dynamics Analysis and Limit Cycle in a Delayed Model for
Tumor Growth with Quiescence, Nonlinear Analysis: Modeling and Control 11:
95–110.
96. Kimmel M.et al (1998) Innite-dimensional model of evolution of drug resistance
of cancer cells. J Math Syst Estim Contr 8:1-16.
97. Swierniak A, Kimmel M, Bobrowski A, Smieja J (1999) Qualitative analysis of
controlled drug resistance model—inverse Laplace and semigroup approach.
Control Cybern 28: 61-75.
98. Harnevo LE, Agur Z (1992) Drug resistance as a dynamic process in a model
for multistep gene amplication under various levels of selection stringency.
Cancer Chemother Pharmacol 30: 469-476.
99. Panetta JC (1998) A mathematical model of drug resistance: heterogeneous
tumors. Math Biosci 147: 41-61.
100. Gyllenberg M, Webb GF (1989) Quiescence as an explanation of Gompertzian
tumor growth. Growth Dev Aging 53: 25-33.
101. S Bajzer Z, Vuk-Pavlovic S, Huzak M (1997) Mathematical modeling of tumor
growth kinetics. A Survey of Models for Tumor-Immune System Dynamics 89-
133.
102. Pontryagin LS (1964) The Mathematical Theory of Optimal Processes. Oxford,
New York International series of monographs in pure and applied mathematics
55.
103. Stepanova NV (1980) Course of the immune reaction during the development
of a malignant tumor. Biophysics 24: 917–923.
104. Ledzewicz U, Heinz Schättler, Mohammad Naghnaeian (2011) An Optimal
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 94
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
Control Approach to Cancer Treatment under Immunological Activity. Appl 38:
17-31.
105. Zhang L, Wang Z, Sagotsky JA, Deisboeck TS (2009) Multiscale agent-based
cancer modeling. J Math Biol 58: 545-559.
106. Preziosi L (2003) Cancer Modeling and Simulation. Mathematical Biology and
Medicine Series.
107. Ferreira SC Jr, Martins ML, Vilela MJ (2003) Morphology transitions induced
by chemotherapy in carcinomas in situ. Phys Rev E Stat Nonlin Soft Matter
Phys 67: 051914.
108. Ferrante L, Bompadre S, Possati L, Leone L (2000) Parameter estimation in
a Gompertzian stochastic model for tumor growth. Biometrics 56: 1076-1081.
109. Jassem J, Schoket Z, Krzakowski M (2002) Cancer clinical trials in Central
and Eastern Europe: historical review. Crit Rev Oncol Hematol 42: 157-161.
110. Kozuch P, Petryk M, Bruckner HW (2002) A comprehensive update on the use
of chemotherapy for metastatic pancreatic adenocarcinoma. Hematol Oncol
Clin North Am 16: 123-138.
111. Koukourakis MI (2002) Amifostine in clinical oncology: current use and future
applications. Anticancer Drugs 13: 181-209.
112. Greco FA (2001) Paclitaxel-based combination chemotherapy in advanced
non-small cell lung cancer. Lung Cancer 34 Suppl 4: S53-56.
113. Hudis CA, Seidman AD, Baselga J, Raptis G, Lebwohl D, et al. (1995)
Sequential adjuvant therapy with doxorubicin/paclitaxel/cyclophosphamide for
resectable breast cancer involving four or more axillary nodes. Semin Oncol
22: 18-23.
114. Bertrand N, Leroux JC (2011) The journey of a drug carrier in the body: an
anatomo-physiological perspective. J Control Release 161: 152-63.
115. Albano G, Giorno V (2006) A stochastic model in tumor growth. J Theor Biol
242: 329-336.
116. González JA, de Vladar HP, Rebolledo M (2003) New late-intensication
schedules for cancer treatments. Acta Cient Venez 54: 263-273.
117. Lo C.F (2009) Modelling Size-dependent Therapy Strategy of Tumors,
Engineering Letters 17:4.
118. Barbolosi D, Iliadis A (2001) Optimizing drug regimens in cancer chemotherapy:
a simulation study using a PK-PD model. Comput Biol Med 31: 157-172.
119. Cherruault Y (1983) Mathematical Methods in Medicine. Acta Applicandae
Mathematica 3: 319-321.
120. Smieja J, Swierniak A (2003) Different models of chemotherapy taking into
account drug resistance stemming from gene amplication. Int J Appl Math
Comput Sci 13: 297-306.
121. Muria MD, Lamberti G (2009) Modeling the pharmacokinetics of extended
release pharmaceutical systems. Heat and Mass Transfer 45: 579-589.
122. Antipov AV, Bratus AS (2009) Mathematical Model of Optimal Chemotherapy
Strategy with Allowance for Cell Population Dynamics in a Heterogeneous
Tumor. Computational mathematics and mathematical physics 49: 1825-1836.
123. Collins JM, Zaharko DS, Dedrick RL, Chabner BA (1986) Potential roles for
preclinical pharmacology in phase I clinical trials. Cancer Treat Rep 70: 73-80.
124. El-Kareh AW, Secomb TW (2000) A mathematical model for comparison of
bolus injection, continuous infusion, and liposomal delivery of doxorubicin to
tumor cells. Neoplasia 2: 325-338.
125. Doroshow JH (2006) Anthracyclines and anthracenediones. In cancer
chemotherapy and biotherapy: principles and practice. 4th edition.
126. Robert J, Illiadis A, Hoerni B, Cano JP, Durand M, et al. (1982)
Pharmacokinetics of adriamycin in patients with breast cancer: correlation
between pharmacokinetic parameters and clinical short-term response. Eur J
Cancer Clin Oncol 18: 739-745.
127. Gabizon A, Catane R, Uziely B, Kaufman B, Safra T, et al. (1994) Prolonged
circulation time and enhanced accumulation in malignant exudates of
doxorubicin encapsulated in polyethylene-glycol coated liposomes. Cancer
Res 54: 987-992.
128. Wu NZ, Braun RD, Gaber MH, Lin GM, Ong ET, et al. (1997) Simultaneous
measurement of liposome extravasation and content release in tumors.
Microcirculation 4: 83-101.
129. Zou Y, Yamagishi M, Horikoshi I, Ueno M, Gu X, et al. (1993) Enhanced
therapeutic effect against liver W256 carcinosarcoma with temperature-
sensitive liposomal adriamycin administered into the hepatic artery. Cancer
Res 53: 3046-3051.
130. Gaber MH, Wu NZ, Hong K, Huang SK, Dewhirst MW, et al. (1996)
Thermosensitive liposomes: extravasation and release of contents in tumor
microvascular networks. Int J Radiat Oncol Biol Phys 36: 1177-1187.
131. William J, Federspiel (2007) Basic transport phenomena in biomedical
engineering. ASAIO Journal 53: P e7.
132. Alberts B, Johnson A, Lewis J, Raff M, Roberts K, et al. (2002) Molecular
Biology of the Cell. Garland Science.
133. DeVita VT Jr, Chu E (2008) A history of cancer chemotherapy. Cancer Res
68: 8643-8653.
134. Khaloozadeh H, Yazdanbakhsh P, Homaei-shandiz F (2008) The optimal dose
of CAF regimen in adjuvant chemotherapy for breast cancer patients at stage
IIB. Math Biosci 213: 151-158.
135. Engert A, Schiller P, Josting A, Herrmann R, Koch P, et al. (2003) Involved-eld
radiotherapy is equally effective and less toxic compared with extended-eld
radiotherapy after four cycles of chemotherapy in patients with early-stage
unfavorable Hodgkin’s lymphoma: results of the HD8 trial of the German
Hodgkin’s Lymphoma Study Group. J Clin Oncol 21: 3601-3608.
136. Von der Maase H, Hansen SW, Roberts JT, Dogliotti L, Oliver T et al. (2000)
Gemcitabine and Cisplatin Versus Methotrexate, Vinblastine, Doxorubicin,
and Cisplatin in Advanced or Metastatic Bladder Cancer: results of a Large,
Randomized, Multinational, Multicenter, Phase III Study. J Cli Oncol 17: 3068-
3077.
137. Grobmyer SR, Maki RG, Demetri GD, Mazumdar M, Riedel E, et al. (2004)
Neo-adjuvant chemotherapy for primary high-grade extremity soft tissue
sarcoma. Ann Oncol 15: 1667-1672.
138. Anderson KC, Alsina M, Bensinger W, Biermann JS, Chanan-Khan A, et al.
(2011) Multiple myeloma. J Natl Compr Canc Netw 9: 1146-1183.
139. de Wit R, Roberts JT, Wilkinson PM, de Mulder PH, Mead GM, et al. (2001)
Equivalence of three or four cycles of bleomycin, etoposide, and cisplatin
chemotherapy and of a 3- or 5-day schedule in good-prognosis germ cell
cancer: a randomized study of the European Organization for Research and
Treatment of Cancer Genitourinary Tract Cancer Cooperative Group and the
Medical Research Council. J Clin Oncol 19: 1629-1640.
140. Booton R, Lorigan P, Anderson H, Baka S, Ashcroft L, et al. (2006) A phase
III trial of docetaxel/carboplatin versus mitomycin C/ifosfamide/cisplatin (MIC)
or mitomycin C / vinblastine / cisplatin (MVP) in patients with advanced non-
small-cell lung cancer: a randomised multicentre trial of the British Thoracic
Oncology Group (BTOG1). Ann Oncol 17: 1111-1119.
141. Mouratidou D, Gennatas C, Michalaki V, Papadimitriou A, Andreadis CH, et al.
(2007) A phase III randomized study comparing paclitaxel and cisplatin versus
cyclophosphamide and cisplatin in patients with advanced ovarian cancer.
Anticancer Res 27: 681-685.
142. Wu H, Kumar A, Miao H, Holden-Wiltse J, Mosmann TR, et al. (2011) Modeling
of inuenza-specic CD8+ T cells during the primary response indicates that
the spleen is a major source of effectors. J Immunol 187: 4474-4482.
143. Schmidt W (2007) Eisen M: Mathematical Models in Cell Biology and Cancer
Chemotherapy, Lecture Notes in Biomathematics. Biometrical Journal 23:519-
520.
144. Coldman AJ, Murray JM (2000) Optimal control for a stochastic model of
cancer chemotherapy. Math Biosci 168: 187-200.
145. de Pillis LG, Gu W, Fister KR, Head T, Maples K, et al. (2007) Chemotherapy
for tumors: an analysis of the dynamics and a study of quadratic and linear
optimal controls. Math Biosci 209: 292-315.
146. Kappelmayer J, Simon A, Kiss F, Hevessy Z (2004) Progress in dening
multidrug resistance in leukemia. Expert Rev Mol Diagn 4: 209-217.
147. Floares C, Cucu M, Lazar L (2003) Adaptive neural networks control of drug
dosage regimens in cancer chemotherapy. IEEE 1: 154-159; 2003.
148. Bojkov B (1993) Application of direct search optimization to optimal control
problems. Hung J Ind Chem 21: 177.
149. Verhulst P (1965) Correspondence Mathematique et Physique 10: 64-66.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-
095. doi:10.4172/jcsb.1000173
Volume 8(1) 074-095 (2015) - 95
J Comput Sci Syst Biol
ISSN: 0974-7230 JCSB, an open access journal
150. Cox EB, Woodbury MA, Myers LE (1980) A new model for tumor growth
analysis based on a postulated inhibitory substance. Comput Biomed Res 13:
437-445.
151. Abulesz EM, Lyberatos G, (1988) Novel approach for determining optimal
treatment regimen for cancer chemotherapy. Int J Syst Sci 19:1483-1497.
152. Creasey W (1978) Designing optimal cancer chemotherapy regimens. Proc.
9th Pittsburgh Conf. Modeling and Simulation, Instrument Society of America,
Pittsburgh 379-385.
153. Long V, Fegley KA (1979) An approach to optimal chemotherapy. Proc. 10th
Pittsburgh Conf. Modeling and Simulation, Instrument Society of America,
Pittsburgh 71-75.
154. Biran Y, McInnis B (1979) Optimal control of bilinear systems: time-varying
effects of cancer drugs. Automatica 15: 325-329.
155. Dibrov BF, Zhabotinsky AM, Neyfakh YA, Orlova MP, Churikova LI (1984)
Optimal scheduling for cell synchronization by cycle-phase-specic blockers.
Math Biosci 66:167-185.
156. Bellman R (1961) Mathematical models of chemotherapy. Proc. 4th, Berkeley
Symposium on Mathematical Statistics and Probability, Univ. of California
Press, Berkeley 57-66.
157. Bischoff KB, Dedrick RL (1970) Generalized solution to linear, to-compartment,
open model for drug distribution. J Theor Biol 29: 63-68.
158. Swan GW, Alexandro F (1980) Modeling methotrexate therapy for treatment of
brain tumors, Proc. Ilth Pittsburgh Conf Modeling and Simulation, Instrument
Society of America, Pittsburgh 289-293.
159. Gaglio S, Genovesi M, Ruggiero C, Spinelli G, Nicolini C (1987) Expert
systems for cancer chemotherapy. Comput Math Appl 14: 793-802.
160. Maceratini R, Rafanelli M, Pisanelli DM, Crollari S (1989) Expert systems
and the pancreatic cancer problem: decision support in the pre-operative
diagnosis. J Biomed Eng 11: 487-510.
161. Engelhart M, Lebiedz D, Sager S (2011) Optimal control for selected cancer
chemotherapy ODE models: a view on the potential of optimal schedules and
choice of objective function. Math Biosci 229: 123-134.
162. Yu PL, Leitmann G (1974) Nondominated decisions and cone convexity in
dynamic multicriteria decision problems. Journal of Optimization Theory and
Applications 14: 573-584.
163. Panetta JC, Adam J (1995) A mathematical model of cycle-specic
chemotherapy. Mathematical and Computer Modelling 22: 67–82.
164. Martin RB, Fisher ME, Minchin RF, Teo KL (1992) Optimal control of tumor size
used to maximize survival time when cells are resistant to chemotherapy. Math
Biosci 110: 201-219.
165. Martin RB, Fisher ME, Minchin RF, Teo KL (1990) A mathematical model of
cancer chemotherapy with an optimal selection of parameters. Math Biosci
99: 205-230.
166. Pereira FL, Pedreira CE, de Sousa JB (1995) A new optimization based
approach to experimental combination chemotherapy. Front Med Biol Eng 6:
257-268.
167. d’Onofrio A, Ledzewicz U, Maurer H, Schättler H (2009) On optimal delivery of
combination therapy for tumors. Math Biosci 222: 13-26.
168. Tan KC, Khor EF, Cai J, Heng CM, Lee TH (2002) Automating the drug
scheduling of cancer chemotherapy via evolutionary computation. Artif Intell
Med 25: 169-185.
169. Liang Y, Leung KS, Mok TS (2006) A novel evolutionary drug scheduling model
in cancer chemotherapy. IEEE Trans Inf Technol Biomed 10: 237-245.
170. Petrovski A, McCall J, (2001) Multi-objective optimisation of cancer
chemotherapy using evolutionary algorithms. Evolutionary Multi-Criterion
Optimization, Lecture Notes in Computer Science 531-545.
171. Petrovski A, McCall J (2004) Optimising cancer chemotherapy using particle
swarm optimization and genetic algorithms. Lecture Notes in Computer
Science 633-641.
172. Tse SM, Liang Y, Leung KS, Lee KH, Mok TS (2007) A memetic algorithm for
multiple-drug cancer chemotherapy schedule optimization. IEEE Trans Syst
Man Cybern B Cybern 37: 84-91.
173. Wu H, Wu L (2002) Identication of signicant host factors for HIV dynamics
modelled by non-linear mixed-effects models. Stat Med 21: 753-771.
174. Miao H, Hollenbaugh JA, Zand MS, Holden-Wiltse J, Mosmann TR, et al.
(2010) Quantifying the early immune response and adaptive immune response
kinetics in mice infected with inuenza A virus. J Virol 84: 6687-6698.
175. Musrrat Ali, Millie Pant, Ajith Abraham (2011) A Simplex Differential Evolution
Algorithm: Development And Applications. Transactions of the Institute of
Measurement and Control 34: 691-704.
176. Hoda Sbeity (2013) Optimisation sur un modèle de comportement pour la
thérapie en oncologie. Université de Versailles Saint –Quentin-en-Yvelines.
Citation: Sbeity H, Younes R (2015) Review of Optimization Methods for
Cancer Chemotherapy Treatment Planning. J Comput Sci Syst Biol 8: 074-095.
doi:10.4172/jcsb.1000173
Submit your next manuscript and get advantages of OMICS
Group submissions
Unique features:
• User friendly/feasible website-translation of your paper to 50 world’s leading languages
• Audio Version of published paper
• Digital articles to share and explore
Special features:
• 400 Open Access Journals
• 30,000 editorial team
• 21 days rapid review process
• Quality and quick editorial, review and publication processing
• Indexing at PubMed (partial), Scopus, EBSCO, Index Copernicus and Google Scholar etc
• Sharing Option: Social Networking Enabled
• Authors, Reviewers and Editors rewarded with online Scientic Credits
• Better discount for your subsequent articles
Submit your manuscript at: http://www.editorialmanager.com/systemsbiology
