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Healthcare Analytics 5 (2024) 100335
Available online 16 April 2024
2772-4425/© 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-
nc/4.0/).
Contents lists available at ScienceDirect
Healthcare Analytics
journal homepage: www.elsevier.com/locate/health
A multi-objective optimization framework for determining optimal
chemotherapy dosing and treatment duration
Ismail Abdulrashid a,∗,1, Dursun Delen b,c, Basiru Usman d, Mark Izuchukwu Uzochukwu e,
Idris Ahmed f
aSchool of Finance and Operations Management, Collins College of Business, The University of Tulsa, Tulsa, OK 74104, USA
bDepartment of Management Science and Information Systems, Spears School of Business, Oklahoma State University, Stillwater, 74078, OK, USA
cDepartment of Industrial Engineering, Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey
dDepartment of Business Management, Poole College of Management, North Carolina State University, Raleigh, NC 27695, USA
eDepartment of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36104, USA
fDepartment of Mathematics, Faculty of Natural and Applied Sciences, Sule Lamido University, Kafin Hausa, Jigawa 700271, Nigeria
ARTICLE INFO
Keywords:
Prescriptive analytics
Healthcare analytics
Drug scheduling
Chemotherapy
Heuristics
Simulation
ABSTRACT
Traditional randomized clinical trials are regarded as the gold standard for assessing the efficacy of chemother-
apy. However, this procedure has drawbacks such as high cost, time consumption, and limited patient
exploration of treatment regimens. We develop a multi-objective optimization-based framework to address
these limitations and determine the best chemotherapy dosing and treatment duration. The proposed frame-
work uses patient-specific biological parameters to create a mathematical model of cell population dynamics in
the patient’s body. The framework employs evolutionary heuristic search methods (simulated annealing and
genetic algorithms) and a prescriptive analytics approach to optimize therapy sessions that transition from
treatment to relaxation. We carefully adjust the chemotherapy dose during treatment to reduce tumor cells
while preserving host cells (such as effector-immune cells). We strategically time the relaxation sessions to aid
recovery, considering the ability of tumors and healthy cells to regenerate. We use a combined optimization
method to determine the length of the session and the amount of drug to be administered. We compare
quadratic and linear optimal control solvers for drug administration while genetic algorithms and simulated
annealing are used to optimize session length. This approach is especially important in limited healthcare
resources, ensuring efficient allocation while accurately identifying high-risk patients to optimize resource
allocation and utilization.
1. Introduction
Cancer, currently the primary contributor to mortality in economi-
cally developed nations [1], is projected to become a significant source
of illness and death across all global regions in the coming decades [2].
During cancer treatment with chemotherapy, timely decision-making is
critical for efficient resource allocation. Notably, this echoes the ethos
of the United Nations (UN) Sustainable Development Goals (SDGs),
specifically Goal 3: Good Health and Well-Being, which seeks to reduce
premature mortality from communicable [3–6] and noncommunicable
diseases [7,8] by one-third by 2030 through prevention and treatment,
as well as promote mental health and well-being [9,10]. In the case
of cancer treatment, knowing a potential patient’s length of stay dur-
ing chemotherapy and proper treatment scheduling can assist medical
professionals in designing and optimizing treatment protocols, lowering
financial costs and treatment burden [11,12].
∗Correspondence to: Collins College of Business, The University of Tulsa, 800 South Tucker Drive, Helmerich Hall 118A, Tulsa, 74104 OK, USA.
E-mail addresses: ismail-abdulrashid@utulsa.edu (I. Abdulrashid), dursun.delen@okstate.edu (D. Delen), busman@ncsu.edu (B. Usman),
muzochukwu@alasu.edu (M.I. Uzochukwu), idris.ahmed@slu.edu.ng (I. Ahmed).
1Assistant Professor of Data Analytics.
Randomized clinical trials have traditionally been used as the stan-
dard approach to treatment for evaluating chemotherapy efficacy and
toxicity levels in most cancers and neoplastic diseases [13,14]. How-
ever, they are deemed inefficient in several ways. First, these trials are
frequently expensive; second, they are time-consuming and difficult to
test different treatment regimens. Finally, these studies are frequently
based on empirical evidence and clinical data from drug development.
Mathematical modeling, analysis, and simulation have been widely
used to address the limitations of human intervention in protocol
selection and treatment scheduling [15,16].
The capability to accurately design a treatment protocol for cancer
chemotherapy is paramount in the realm of optimizing drug admin-
istration, tumor cells population, and toxicity level reduction. This
https://doi.org/10.1016/j.health.2024.100335
Received 9 February 2024; Received in revised form 1 April 2024; Accepted 8 April 2024
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
critical endeavor not only enhances the efficacy of the treatment but
also ensures the strategic allocation of medical resources. For this
reason, numerous studies have been developed utilizing prescriptive
analytics techniques in an attempt to enhance the efficacy of the
cancer chemotherapy treatment [17–19]. In particular, to minimize the
toxicity of the drug and predicting the growth of the tumor, many of
the studies have investigated chemotherapy treatment from dynami-
cal [20], optimization [21], and compartmental point of view [22]. For
instance, Abdulrashid and Han [23] conducted a study on a nonau-
tonomous dynamical system that was designed to simulate the dy-
namics of tumor cells, normal cells, and a chemotherapy agent in
the presence of time-varying environmental conditions. While the de-
veloped strategy may demonstrate the efficacy of treatments with
time-varying infusions of the chemotherapy drug in reducing tumor
size, it fails to account for stratifying total therapy length and allowing
for relaxation between each treatment session. Furthermore, no drug
administration optimization is considered during the course of the
therapy. In a recent study by Esmaili et al. [24], a novel optimal
control problem for a stochastic model of tumor growth with drug
application was proposed. The study utilized both stochastic and deter-
ministic cost functions to ascertain an optimal control strategy for the
diffusion of nutrients and the administration of drugs in the context of
tumor growth. Their primary goal, however, was to demonstrate the
existence and uniqueness of optimal control for the stochastic model
and its associated stochastic adjoint equations. Despite advancements
in prescriptive analytics techniques that have shown promise in refining
cancer chemotherapy to optimize drug doses and treatment schedul-
ing [25], the comprehension of vital patient-specific health indicators
that augment successful chemotherapy planning is still lacking. None
of the previously mentioned studies took into account current clinical
practice, which is weight-based dose calculation, as well as expert
opinions in determining the optimal treatment and relaxation duration.
To address some of the shortcomings of previous continuous optimal
control studies, the current study proposes a universally applicable
multi-objective optimization-based structure that can be used as a
decision-making tool for determining the appropriate amount of drug
to be administered and the duration of treatment. The contribution
of the present study to the domains of medical decision-making and
chemotherapy treatment planning is threefold. Firstly, our framework’s
modeling component uses patient biological parameters to generate a
mathematical model that simulates interactions between cell popula-
tions within the patient’s body. Secondly, we used a combination of
evolutionary heuristic search methods and prescriptive analytics tech-
niques to optimize our framework to propose a sequential alternation
between treatment and relaxation sessions during therapy. We admin-
istered an optimal amount of chemotherapy drug during the treatment
session to reduce the number of tumor cells without overly harming
the effector-immune cells, while during the relaxation session, we set
a time limit to allow the body to recover. Tumors and other natural
cells have the potential to grow again during this phase. Thirdly, we
use a joint optimization approach to determine the duration of each
therapy session and the amount of drug to be administered. The optimal
drug control at each treatment session is determined by comparing
both quadratic and linear optimal control solvers, while the duration of
each session is optimized using the genetic algorithm (GA) [26,27] and
simulated annealing (SA) [28,29]. This is especially important given
the limited resources available during therapy and the importance of
accurately identifying high-risk patients so that healthcare resources
can be allocated optimally.
The remainder of this article is organized as follows. The related
studies from the literature are reviewed in Section 2. Next, we describe
the proposed framework and the specific experimental methods used
to validate it. Following that, the result from comparing two optimal
control solvers using different heuristic algorithm are demonstrated.
In Section 5, we discussed the sensitivity analysis, which used partial
rank correlation coefficients to identify the model’s most important
parameters. Finally, we discuss our findings and limitations, as well as
future research directions.
2. Background
Chemotherapy, known as the "warrior of healing’’, is a remarkable
example of human ingenuity and resilience in the face of one of
life’s most formidable adversaries: cancer. This extraordinary med-
ical treatment uses the power of science to combat the relentless
growth of malignant cells, bringing hope where there was once despair.
Chemotherapy, like a skilled archer, targets and destroys rogue cells
with incredible precision while sparing healthy ones in its quest to
restore balance within the body.
It is a symphony of chemical agents, each with a distinct role, all
working together to weaken and eventually defeat cancer’s relentless
advance. Chemotherapy exemplifies the human spirit’s unwavering
determination to overcome adversity and serves as a beacon of hope
for many people and their loved ones. Its journey from the labo-
ratory bench to the patient’s bedside demonstrates the unwavering
commitment of medical professionals, scientists, and researchers who
are constantly refining and innovating this powerful therapy. When
administered by a skilled oncologist, chemotherapy can be a lifeline,
promising remission, recovery, and renewal. It is a beacon of hope,
illuminating the path to a better, cancer-free future. In the face of
adversity, chemotherapy stands out as a symbol of our unwavering
commitment to healing and a testament to the limitless potential
of human compassion and innovation. Researchers used a variety of
methods to demonstrate their approaches to modeling chemotherapy
treatments.
Chemotherapy is an influential modality in the treatment of cancer.
Chemotherapy, as opposed to more localized therapeutic approaches
like radiation therapy [30] and surgery [31], operates on a systemic
level and specifically targets malignant cells across the entire body
[32,33]. As a result, chemotherapy is extensively administered to indi-
viduals diagnosed with advanced stages of cancer; in 2016, chemother-
apy was utilized by over 60% of patients in the United States who had
been diagnosed with cancers of the breast, colon, rectal, lung, testicu-
lar, urinary bladder, and uterine corpus at stages III or IV [19]. More-
over, the efficacy of the chemotherapy can be greatly enhanced through
precise dosing and control mechanisms. Linear control models [34]
have emerged as a promising approach to optimizing chemotherapy
administration and minimizing its side effects. These models rely on
mathematical representations of the pharmacokinetics and pharmaco-
dynamics of chemotherapy drugs within the patient’s body [35,36].
By using linear control theory [37], researchers can design algorithms
that continuously adjust drug infusion rates to maintain therapeutic
drug concentrations while avoiding toxic levels. This real-time control
allows for a more personalized and adaptable treatment approach as
it accounts for individual patient variations and dynamic changes in
drug metabolism over time. For example, in [38], Shi at el summarize
the mathematical models applied to the optimal design of the cancer
chemotherapy; in [39], Sabir et al. aim at minimizing the drug dose
and the tumor using linear optimal control but have not considered
quadratic optimal control.
Chemotherapy models incorporating quadratic optimal control
[40,41] represent a sophisticated approach to optimizing cancer treat-
ment strategies. These models leverage mathematical frameworks, such
as optimal control theory, to determine the most effective drug dosage
regimens by considering not only the desired therapeutic effects but
also minimizing undesirable side effects and toxicity. By formulating
the problem as a quadratic optimization problem, researchers can
incorporate complex factors, such as drug pharmacokinetics, patient-
specific characteristics, and tumor dynamics, into the model. This
approach allows for the calculation of optimal drug infusion rates
that seek to strike a balance between maximizing tumor cell kill and
minimizing harm to healthy tissues, resulting in a more precise and
tailored chemotherapy regimen. Furthermore, quadratic optimal con-
trol models can account for time-varying parameters and uncertainties,
making them adaptable to the dynamic nature of cancer progression
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
Fig. 1. Graphical depiction of the proposed methodology.
and treatment response, ultimately offering the potential for improved
treatment outcomes and reduced adverse effects for cancer patients.
As presented by Sabir et al. [42], which proposed a quadratic optimal
control study to minimize both the tumor density and the treatment
dose. In this paper, we did not only consider quadratic or linear, but we
considered both and highlighted the results similarities and differences.
In addition, we consider the treatment and relaxation phases. The
relaxation phase will allow the normal cells to grow without the
hindrance of the chemotherapy drug.
Continuous optimal control models capture the biological dynamics
of chemotherapy processes well [19,43,44]; however, cancer treatment
involves important discrete components and operational constraints.
For instance, some cytotoxic drugs are available in pill form and are
administered orally. For these drugs, an administration dose must be
a multiple of the pill size, and any deviation from this regimen can
lead to an under-dose or overdose. Often, doctors prescribe oral drugs
with food, and certain drugs may require mandatory relaxation periods
due to metabolic processes. Continuous optimal control models fail to
capture the discrete nature of drug administration scheduling due to
these factors. Modeling such operational constraints for chemotherapy
planning requires discretized control variables.
In order to address the aforementioned disparities between current
chemotherapy optimization models and clinical practice, we propose a
framework based on multi-objective optimization that aims to ascertain
the most effective chemotherapy dosing and treatment duration. By
minimizing the tumor cell populations at the conclusion of a treat-
ment, this model attempts to identify the optimal administration dose
and schedule for drugs. By comparing linear and quadratic optimal
control solvers and employing discretization, we reframe our ordinary
differential equations (ODEs) that represent biological and pharmaco-
logical processes within a framework of multi-objective optimization.
This framework’s adaptability permits the simulation of the intricate
operational constraints of chemotherapy. In particular, we incorporate
discrete administration dose and schedule, as well as clinically man-
dated relaxation periods, into our model. We use the host cell count as
an explicit indicator of treatment toxicity. To address the uncertainty
associated with various types of tumors, we propose chance constraints
and demonstrate a chemotherapy optimization model for treatment
planning. Before implementing the heuristics methods, we compare the
simulation results for linear and quadratic solvers. Then we presented
a detailed comparison of the effectiveness of the simulated annealing
and genetic algorithms in determining therapy duration. To calibrate
our model parameters and perform sensitivity analysis, we use some
clinical literature and published data from recently conducted studies
for cancer patients.
3. Problem description
A dynamic model simulating the tumor-immune interaction, an
optimization model for patient-level outcome, and a combination of
evolutionary heuristic search methods are incorporated into the pro-
posed system’s conceptual design to consider drug combination, dose
adjustment, and therapy duration. Three development steps (or phases)
are followed in order to create an efficient framework for develop-
ing a multi-objective optimization-based approach, which includes:
(1) combining patient-level biological parameters to create a mathe-
matical model that simulates the interaction between chemotherapy
drug, tumor, and host cells inside the patient’s body; (2) developing
different optimization models to optimize the tumor and overcome tox-
icity levels during each treatment session; and (3) allotting relaxation
sessions between any two consecutive treatment sessions so that the
body can partially recover from the adverse effects of the drugs. Fig. 1
graphically depicts the proposed methodology. Each of these phases is
described in detail in the sections that follow.
3.1. Problem setting
To design a multi-objective optimization-based problem [6] that
analyzes the impact of different objective functions on a dynamic
chemotherapy scheduling problem, we first present the chemotherapy
model, which is based on the dynamical models detailing the interac-
tions between the drugs, the host cells, and the tumor cells at a single
site for treatment in the body (see, e.g., [46,47]), with modifications to
correspond to the practical situation encountered in hospitals, where
chemotherapy drugs are frequently combined to treat patients and
physicians frequently adjust the length of treatment and drug dosages.
Our model describes the proliferation of both tumor cells and host cells
within the body.
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
Table 1
Dynamical description of state variables and parameters of model (1)–(3).
Variable Description
𝑄ℎHost cells concentration
𝑄𝑐Tumor cells concentration
𝑄𝑚Chemotherapeutic drug concentration in the body
𝜓Infusion rate at the treatment site
Parameter Interpretation/Description
𝛽𝑐Per capita growth rate of tumor cells
𝛽ℎPer capita growth rate of host cells
𝜆𝑐Intraspecific competition coefficient of host on tumor cells
𝜆ℎIntraspecific competition coefficient of tumor on host cells
𝛿𝑐Half-saturation constant for tumor cells
𝛿ℎHalf-saturation constant for the host cells
𝜅ℎCarrying capacity of host cells
𝜅𝑐Carrying capacity of tumor cells
𝛼𝑐Tumor cell fractions killed by chemotherapeutic drug
𝛼ℎHost cell fraction killed by chemotherapeutic drug
𝜁Toxicity (tolerance level) of the host cells
𝜏Patients’ acceptance level of chemotherapy agent
𝜙Injection rate of the chemotherapeutic drug
𝛾Concentration of the chemotherapeutic drug
𝜆Pareto weight
We denote the concentration of the chemotherapy drug by 𝑄𝑚(𝑡),
and the concentrations of host and tumor cells at the treatment site
by 𝑄ℎ(𝑡)and 𝑄𝑐(𝑡), respectively. The following equations describe the
population dynamics resulting from chemotherapy treatment:
𝑑𝑄ℎ
𝑑𝑡 = −𝛼ℎ
𝑄𝑚(𝑡)𝑄ℎ(𝑡)
𝛿ℎ+𝑄ℎ(𝑡)+𝛽ℎ𝑄ℎ(𝑡)1 − 𝑄ℎ(𝑡)
𝜅ℎ
−𝜆ℎ𝑄ℎ(𝑡)𝑄𝑐(𝑡),(1)
𝑑𝑄𝑐
𝑑𝑡 = −𝛼𝑐
𝑄𝑐(𝑡)𝑄𝑚(𝑡)
𝛿𝑐+𝑄𝑐(𝑡)+𝛽𝑐𝑄𝑐(𝑡)1 − 𝑄𝑐(𝑡)
𝜅𝑐
−𝜆𝑐𝑄ℎ(𝑡)𝑄𝑐(𝑡),(2)
𝑑𝑄𝑚
𝑑𝑡 =𝜙(𝑡)𝛾(𝑡) − 𝑄𝑚(𝑡) − 𝑄ℎ(𝑡)𝑄𝑚(𝑡)
𝛿ℎ+𝑄ℎ(𝑡)−𝑄𝑐(𝑡)𝑄𝑚(𝑡)
𝛿𝑐+𝑄𝑐(𝑡),(3)
where 𝜙and 𝛾represent the injection and removal rates, respectively,
of the chemotherapy drug. 𝜓(𝑡) = 𝜙(𝑡)𝛾(𝑡)is then used to determine
the infusion rate at the treatment site. To more accurately represent
chemotherapy, we assumed that the host and tumor cells followed a
logistic growth model, with 𝛽ℎand 𝛽𝑐representing the specific birth
rates for small concentrations and 𝜅ℎand 𝜅𝑐representing the carrying
capacities of the host and tumor cells at the treatment site, respec-
tively [46,48]. In addition, in the absence of competition and predation,
the half saturation constants 𝜆ℎand 𝜆𝑐determine the rates at which the
host or tumor cell reaches its maximum sustainable population size. The
quantities of interest used in our current study are supported by patient-
specific biological parameters to track the cell population dynamics
inside the patient’s body, as described below:
•It is well known that see, for example, [17,18], chemotherapy
drug when injected into the patient’s body, it has a negative
effect. For this reason, in Equations (1)–(3), the term −𝛼ℎ
𝑄𝑚(𝑡)𝑄ℎ(𝑡)
𝛿ℎ+𝑄ℎ(𝑡)
and −𝛼𝑐
𝑄𝑐(𝑡)𝑄𝑚(𝑡)
𝛿𝑐+𝑄𝑐(𝑡)represents the negative effect of the chemother-
apy drug on host and tumor cell populations, where 𝛼ℎand 𝛼𝑐rep-
resent the fractional host and tumor cells killed by the chemother-
apy drugs and 𝛿ℎand 𝛿𝑐are the half saturation constants for host
and tumor cells, respectively.
•The terms 𝛽ℎ𝑄ℎ(𝑡)1 − 𝑄ℎ(𝑡)
𝜅ℎand 𝛽𝑐𝑄𝑐(𝑡)1 − 𝑄𝑐(𝑡)
𝜅𝑐found in
Eqs. (1) and (2) represent the growth of the host and tumor cells
inside patient’s body, which we assumed to be logistic, with 𝛽ℎ
and 𝛽𝑐indicating the per capita growth rates while 𝜅ℎand 𝜅𝑐are
the environmental carrying capacity of host and tumor cells at the
treatment site, respectively. Other recently published studies on
chemotherapy that have utilized this assumption include [19,40].
•Competition (for available resources) does exist between the host
and tumor cells; see, for example, [46], for this reason, 𝜆ℎand 𝜆𝑐
are the competition coefficients that represent the effect of host
cells on tumor cells and tumor cells on host cells, respectively. The
product terms −𝜆ℎ𝑄ℎ(𝑡)𝑄𝑐(𝑡)and −𝜆𝑐𝑄ℎ(𝑡)𝑄𝑐(𝑡)then represent
the effect of an equivalent number of host and tumor cells,
respectively. It is worth noting that when 𝜆ℎ<1the effect of
tumor cells on host cells is less than the effect of host cells on
tumor cells, On the other hand, when 𝜆𝑐>1the effect of tumor
cells on host cells is greater than the effect of host cells on tumor
cells.
•The chemotherapy drug dosage (infusion rate of the chemother-
apy drugs at the treatment site) 𝜓(𝑡) = 𝜙(𝑡)𝛾(𝑡), where 𝜙and 𝛾
are the rates at which the chemotherapy agent is injected and
washout, respectively.
The model’s parameters and their estimated values are described in
detail in Tables 1 and 2, respectively.
To meet the operational requirement of cancer chemotherapy, we
design a chemotherapy scheduling protocol that minimizes the tumor
population inside the patient body with minimum therapy duration
while maintaining low toxicity levels during 𝐺𝑞cycles. A cycles 𝑝of
duration 𝛺𝑝, where 𝑝= 1,2,3,…, 𝐺𝑞, corresponds to two consecutive
sessions, i.e., one treatment and one relaxation session with duration
𝑇𝑝and 𝑅𝑝, respectively (that is, 𝛺𝑝=𝑇𝑝+𝑅𝑝). We assume that a
cycle 𝑝starts at the time instant 𝑡𝑝and ends exactly at the beginning
of the next cycle 𝑡𝑝+1, that is, 𝑡𝑝+1 =𝑡𝑝+𝛺𝑝. Next, we formulate the
optimization problem aimed at minimizing the tumor concentration 𝑄𝑐
and drug dosage 𝜓. Then two evolutionary heuristic algorithms (genetic
algorithm and simulated annealing) are employed during this phase to
optimizes the duration of each session. These two techniques are used
to compare and determine an optimized chemotherapy schedule and
duration while simultaneously minimizing both the size of the tumor
and the amount of chemotherapy that will be required.
In this regard, we define the following optimization problem to be
solved during each 𝑇𝑝treatment session:
(𝑊) ∶ 𝐻(𝜓𝑝(𝑡)) = ∫𝑡𝑝+𝑇𝑝
𝑡𝑝𝑄𝑝
𝑐(𝑡) + 𝜏𝜓
2𝜓𝑝(𝑡)𝑟𝑑𝑡,
where 𝑟∈ {1,2},(4)
𝑆.𝑡. ∶𝑂𝐷𝐸 𝑠𝑦𝑠𝑡𝑒𝑚 (1)–(3), for 𝑎𝑙𝑙 𝑡 ∈ [𝑡𝑝, 𝑡𝑝+𝑇𝑝],(5)
𝑄𝑝
ℎ≥𝜁𝑄1
ℎ(𝑡1),∀𝑡∈ [𝑡𝑝, 𝑡𝑝+𝑇𝑝],(6)
0≤𝜓𝑝(𝑡)≤1,∀𝑡∈ [𝑡𝑝, 𝑡𝑝+𝑇𝑝],(7)
𝑄𝑝
ℎ(𝑡𝑝) = 𝑄𝑝−1
ℎ(𝑡𝑝),(8)
𝑄𝑛
𝑐(𝑡𝑝) = 𝑄𝑝−1
𝑐(𝑡𝑝),(9)
𝑄𝑝
𝑚(𝑡𝑝) = 𝑄𝑝−1
𝑚(𝑡𝑝),(10)
where 𝑝= 1,2,3,…, 𝐺𝑞. Note that if 𝑟= 1 or 𝑟= 2 in Eq. (4),
the problem is reduced to linear or quadratic optimal control. The
purpose of Eq. (4) is to minimize the overall tumor volume 𝑄𝑝
𝑐and the
volume of the chemotherapeutic drug 𝜓𝑝in the body of the patient. The
patient’s acceptance level of chemotherapy is described by the factor
𝜏𝜓. Constraint (5) describes the dynamics of the cells at every therapy
session of 𝑇𝑝, while constraint (6) describes the minimum threshold of
the host cells. This is critical for the patient’s survival; therefore, the
normal cell population is controlled by the toxicity level tolerance pa-
rameter 𝜁1in ∈ [0,1]. Constraint (7) enables flexible drug admission at
every therapy session 𝑇𝑝, with a continuous piece-wise control function
𝜓𝑝(𝑡) ∈ [0,1] representing the amount of the chemotherapeutic drug
administered. For example, in a treatment 𝑝of duration 𝑇𝑝,𝜓𝑝(𝑡)=1
implies a maximum chemotherapy drug presence in the body, whereas
𝜓𝑝(𝑡) = 0 means no chemotherapy drug presence in the body. Finally,
the constraints (8),(9) and (9) represent the patient’s body’s condition
prior to administering the dosage of the agents, and at each cycle 𝑡𝑝of
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
Table 2
Summary table for parameter values of model (1)–(3).
Parameter Baseline value Unit Source
𝛽𝑐4.31 × 10−3 day−1 Dhieb et al. [13] and de Pillis et al. [45]
𝛽ℎ1.02 × 10−14 day−1 de Pillis et al. [45] and Dhieb et al. [13]
𝜆𝑐0.5 day−1 Estimated
𝜆ℎ0.9 day−1 Estimated
𝜅𝑐5.00 Dimensionless Han [46]
𝜅ℎ5.00 Dimensionless Han [46]
𝛼𝑐6.00 × 10−1 day−1 Abdulrashid et al. [15]
𝛼ℎ6.00 × 10−1 day−1 Abdulrashid et al. [15]
𝛿ℎ2.02 cell−1 day−1 Han [46]
𝛿𝑐2.02 cell−1 day−1 Han [46]
𝜁0.7 [0 − 1] Dimensionless Dhieb et al. [13]
𝜏0.01 Dimensionless Dhieb et al. [13]
𝜆0.8 [0 − 1] Dimensionless Dhieb et al. [13]
Fig. 2. A comparison of linear and quadratic optimal control solutions.
a treatment, these parameters are set to the values they had towards
the final step of the preceding cycle 𝑝− 1. In the initial cycle denoted
as 𝑡1, it is assumed that the value of 𝑄𝑚is zero. Additionally, the
values of 𝑄ℎand 𝑄𝑐are initialized with the patient’s initial values
obtained prior to the chemotherapy sessions. The paper by de Pillis
et al. [45] and Ehrgott et al. [49] demonstrates the existence of an
optimal control solution 𝜓∗
𝑝(𝑡)for analogous situations by the use of
Pontryagin’s maximum principle. Furthermore, this solution may be
obtained numerically using readily available software.
4. Results and discussion
4.1. Comparison of the quadratic and linear controls
In this section, we will investigate the effect of using both quadratic
and linear controls on normal and tumor cells. At the same time, we
want to observe the drug infusion rate over time and the concentration
of the chemotherapy drug at the end of the therapy for these two
controls. The mathematical formulation of both linear and quadratic
controls are highlighted in Eq. (4) when 𝑟= 1 and 𝑟= 2, respectively.
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
Fig. 3. Times, % of recovered cells and the consumed chemo for a 5 cycles treatments and relaxation periods.
First of all, we use the same patient’s parameters and tolerated toxicity
level of 50 percent for both of the controls. The tumor deterioration
follows almost the same curve for both controls, achieving almost total
tumor elimination in about seven days (see, Fig. 2(a)). As expected,
when chemotherapy starts, there will be an occasional loss of the
normal cells, but the percentage of normal cells stabilizes around about
two days, retaining the majority of the cells intact, as observed in
Fig. 2(b). Over all, linear control has a higher percentage of normal
cells retained. In Fig. 2(c), we see that, at about 1 to 3 days of the
therapy, the quadratic control has a slightly higher concentration of the
drug, but by the time therapy reaches 4 days, the concentration levels
off and is negligible in both controls. Finally, as observed in Fig. 2(d),
the infusion rate for both controls was similar for almost a day, and
then the linear control started having more before the quadratic took
over in about a day and a half.
In order to effectively observe the differences between the linear
and quadratic control, we recommend using the same patient’s param-
eter and the same model parameters, as that will give the observer pure
comparative results.
4.2. Comparison of the heuristics methods
We started by comparing the five cycles of the genetic algorithm
(GA) and simulated annealing (SA). For the GA, we use an initial
population of 50, a crossover rate of 0.9, and a mutation rate of 0.00625
(which is 1/(bit length ×number of variables)) in 50 iterations. Simi-
larly, for the SA, we use an initial temperature of 10 and a maximum
step size of 0.3 in 100 total iterations. We have also considered different
levels of tolerated toxicity, ranging from 50% to 80% in both of the
heuristic methods.
As in Fig. 3(a), we observe that for the scheduling times, GA has
a scheduling time ranging from 15 to 16 days for all the levels of
toxicity we have considered. However, when it comes to SA, it requires
significantly more days for the therapy to be quite effective; the number
of days required by the SA therapy session ranges from 56 to 75 days.
Fig. 3(b) shows that GA was able to achieve total tumor eradication
after the specified therapy days; similarly, with SA, we have a negligible
portion of tumor left after the therapy days. In Fig. 3(c) we see that
more normal cells have been recovered using GA than using SA; the
average percentage of normal cells recovered using GA is about 85%
which is slightly higher than the average of 82% that was recovered
using SA. Finally, when it comes to the total drug consumed during the
therapy sessions, we expect the amount consumed using SA to be higher
than the amount consumed using GA due to SA having more therapy
days. That is exactly what we observed, as depicted in Fig. 3(d).
In the second step of the comparison, we consider three cycles of GA
and SA. As with the first case, the therapy time required using SA ranges
from 48 to 56 days, whereas for GA it is about 9 days for all levels of
toxicity; see Fig. 4(a). In Fig. 4(b), we see that similar to the analysis of
5 cycles, using GA, a zero tumor has been achieved and a significantly
negligible tumor portion was left at the end of SA therapy sessions.
When it comes to the percentage of normal cells, using SA, we get an
average of about 93% which is significantly higher than the average
of 90% that occurred while using GA. Finally, since the therapy days
are higher while using SA than GA, we expect the drug consumption
to be higher in SA than in GA, as depicted in Fig. 4(d). Pareto weight
is an important parameter in both heuristic methods; it controls total
therapy time while considering the percentage of tumor in the patient’s
body. So, we consider different levels of Pareto weight, from 0to 1with
an increment of 0.1.
For the 3cycle treatment, while using SA and a Pareto weight
ranging from 0.9 to 0, we observed changes in the final tumor size
ranging from 2𝐸−13 to 2𝐸− 10 percent. For the 5cycle treatment while
using SA and a Pareto weight ranging from 0.9 to 0, we observed the
change in the final tumor size to be from 2𝐸− 19 to 9𝐸− 11 percent.
When the Pareto weight is 1, there is total tumor eradication in both
cycles; see Fig. 5(b). For the SA scheduling times, it is more of a rise
and fall behavior; see Fig. 5(a). For the 3cycle treatment, the therapy
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
Fig. 4. Times, percentage of recovered cells,percentage of tumor cells and the consumed chemo for a 3 cycles treatments and relaxation periods.
Fig. 5. Scheduling time and percentage of tumor cells is affected by Pareto weight for both GA and SA.
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
Fig. 6. The plot of PRCC of tumor severity for the sensitivity analysis against the model parameters.
days range from 27 days to 52 days. Likewise, for the 5cycle treatment,
the days range from 60 days to 93 days.
Similarly, for both 3and 5cycle treatments, while using GA, the
Pareto weight does not have a significant effect on the eradication of
the tumor. We then observed the GA scheduling time, and we notice
that, for a 5cycle treatment and a Pareto weight ranging from 0to 0.9
the Pareto weight has a few hours difference ranging from 19 h to 5h,
thus the total number of days for the therapy ranges from 15 days and
19 h to 15 days and 5h, see Fig. 5(c). For the 3cycle treatment and a
Pareto weight ranging from 0to 0.9, the therapy days are about 9 days.
When the Pareto weight is 1we observe a spike in the therapy days;
for a 3cycle, it spikes to about 27 days, and for the 5cycle, it spikes to
about 63 days and 17 h.
5. Sensitivity analysis
When dealing with numerous parameters in a model it is ideal to
perform sensitivity analysis. In our sensitivity analysis, we want to
check the parameters that are crucial in making the condition severe.
To that end we define a tumor severity function which is the tumor
growth rate minus chemotherapy effect and used it as our reaction
function. The tumor severity function is given by
tumor_severity =tumor_growth ∗ (1 − tumor_effect)
−host_growth ∗ (1 − host_effect)
where
tumor_growth =𝛽𝑐∗ (1 − (𝜅𝑐∕(𝜅𝑐+𝛿𝑐))) ∗ (1 − (𝜆𝑐∗𝛽ℎ∕𝛽𝑐))
host_growth =𝛽ℎ∗ (1 − (𝜅ℎ∕(𝜅ℎ+𝛿ℎ))) ∗ (1 − (𝜆ℎ∗𝛽𝑐∕𝛽ℎ))
tumor_effect =𝛼𝑐∗ (1 − 𝜁) ∗ drug_sensitivity_ratio
host_effect =𝛼ℎ∗ (1 − 𝜁)
the meaning of the parameters is indicated in Table 1. The result of
our sensitivity analysis is depicted in Fig. 6. 1000 random samples
were generated using Sobol sequences for each parameter and then the
model was simulated for each random parameter to compute the tumor
severity. Finally, PRCCs were computed between each parameter and
tumor severity.
Our sensitivity analysis result in Fig. 6 indicates that, 𝛽𝑐is the
top-ranked biological parameter that help in tumor persistence then
followed by 𝛼ℎ, 𝜆ℎ, 𝜅ℎ, and 𝛿ℎ.
6. Conclusion
This paper presents a multi-objective optimization model for com-
bination chemotherapy optimization that aims to find the best admin-
istration dose, schedule, and relaxation for chemotherapeutic drug ad-
ministration while minimizing the tumor cell population at the end of a
treatment period. Unlike previous works, which frequently ignored op-
erational considerations (such as not including relaxation periods) [50–
52], we incorporate these constraints by employing metaheuristic tech-
niques (simulated annealing and genetic algorithms), which provide us
with the optimal length of treatment and relaxation sessions required
for the model to achieve the desired result. We calibrate our model
parameters using the literature and randomly generated data, and
then present the results of our numerical study. We use sensitivity
analyses to determine which parameters have the greatest influence on
the model outcomes. Our mathematical models offer a framework for
investigating new, personalized dose recommendations.
Specifically, we embarked on a comprehensive exploration of
chemotherapy scheduling models, pitting linear optimal control against
quadratic optimal control. Unlike the majority of research studies,
where either the linear or quadratic optimal control is studied alone
[53–55], or treatment is not incorporated with relaxation [56–58].
The primary objective was to discern the advantages and limitations
of these control strategies in the context of optimizing chemotherapy
treatments. Our findings revealed that, for a considerable portion of
patient parameters, the results obtained from both linear and quadratic
control strategies exhibited remarkable similarity. However, it is crucial
to note that distinctions emerged when we implemented optimization
algorithms, specifically comparing the outcomes of genetic algorithms
and simulated annealing. This nuanced variation underscores the im-
portance of considering algorithmic approaches in addition to control
strategies when devising chemotherapy scheduling models.
A distinctive feature of our study, which sets it apart from previ-
ous research, is the incorporation of relaxation with treatment time
within the optimization problem. Additionally, we introduced a novel
approach by implementing both linear and quadratic control strategies
within a single problem. These innovations open up new avenues
for more flexible and patient-centric treatment protocols. However, it
is important to note that biological systems are extremely complex,
Healthcare Analytics 5 (2024) 100335
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I. Abdulrashid et al.
and thus a careful consideration including medical personal inspection
is required before clinical applications. As for the limitations of our
study, the primary constraint lies in the requirement for patient-specific
data as initial inputs for our algorithms. Nonetheless, the potential
benefits are substantial. By providing a superior scheduling protocol
for chemotherapy treatments, our research has the potential to sig-
nificantly reduce the financial burden on patients while enhancing
their emotional and financial satisfaction. This outcome aligns with
the broader goal of improving the overall quality of cancer care and
underscores the potential impact of optimal control strategies in the
field of medical treatment.
The model has many biological parameters to be estimated. In our
sensitivity analysis, we employ partial rank correlation coefficients,
with tumor severity as our input. Our findings reveal that the primary
contributor to tumor persistence is the per capita growth rate of the
tumor. Of particular interest, the second most influential parameter
affecting tumor severity is the fraction of host cells killed by chemother-
apeutic drugs. Conversely, the per capita growth rate of host cells
emerges as the least significant factor influencing tumor severity.
Finally, our study offers valuable insights into the optimization of
chemotherapy scheduling models. It demonstrates that the choice of
control strategy and optimization algorithm can have a substantial
impact on treatment outcomes, paving the way for more personalized
and cost-effective cancer therapies.
6.1. Limitations and future work
This study is not perfect, but it does lead researchers down some
promising new paths. The use of simplified assumptions about com-
plex biological and chemical processes is a necessary first step in our
modeling-based approach. On the other hand, this approach has the
potential to address some of the most pressing issues in customiz-
ing treatment plans for each patient by providing a framework for
investigating alternative approaches using simulation trials.
Treatment costs were not included in the optimization models’
objectives, which significantly limits the results presented in this ar-
ticle. Our optimal scheduling methods allow us to assess the cost-
effectiveness of treatment based on drug dosages and chemotherapy
frequency. Second, once resistance is detected, drugs must be replaced
immediately. Decisions will need to be made dynamically and sequen-
tially as more information about the tumor’s characteristics is revealed
during the procedure. We have not yet updated our model to allow for
parameter changes as new data becomes available during treatment.
We believe that using the proposed prescriptive model framework
could significantly improve the quality of chemotherapy regimens used
in clinical trials. Nonetheless, significant and intriguing directions for
future research exist, which could help to expand on the findings of
this study. The use of in vitro tests and simulation studies, which are
examples of pre-clinical models, may help us optimize our process.
Placing constraints on the integration process may rule out the possibil-
ity of drug combinations that have been shown to interact negatively
or cooperatively in preclinical wildlife.
Summary table
What was Already Known on this topic
•Continuous therapy scheduling without relaxation session.
•Optimizing drug administration at a single session.
•No relaxation sessions are considered.
•No optimal duration of therapy and relaxation sessions are deter-
mined.
•No comparison of the effectiveness of using quadratic and/or
linear terms in the optimization model.
What this study adds to our knowledge
•It is more effective to design discrete treatment and relaxation
sessions for chemotherapy.
•Controlling the adverse effect of the drug on normal cells by
optimizing drug administration during each phase of therapy is
more effective.
•Stratifying therapy into treatment and relaxation phases reduces
tumor volume with fewer side effects on normal cells.
•Determining treatment length and relaxation sessions reduces
tumor recurrence and drug side effects.
•Compared the model’s effectiveness in eradicating tumors when a
quadratic or linear term is used during the optimization process.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Dursun Delen is an Associate Editor for Healthcare Analytics journal
and was not involved in the editorial review or the decision to publish
this article.
Data availability
No data was used for the research described in the article.
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