Dual therapy of cancer using optimal control supported by swarm intelligence

Abstract

BACKGROUND:

The scientific revolution in the treatment of many illnesses has been significantly aided by stem cells. This paper presents an optimal control on a mathematical model of chemotherapy and stem cell therapy for cancer treatment.

OBJECTIVE:

To develop effective hybrid techniques that combine the optimal control theory (OCT) with the evolutionary algorithm and multi-objective swarm algorithm. The developed technique is aimed to reduce the number of cancerous cells while utilizing the minimum necessary chemotherapy medications and minimizing toxicity to protect patients’ health.

METHODS:

Two hybrid techniques are proposed in this paper. Both techniques combined OCT with the evolutionary algorithm and multi-objective swarm algorithm which included MOEA/D, MOPSO, SPEA II and PESA II. This study evaluates the performance of two hybrid techniques in terms of reducing cancer cells and drug concentrations, as well as computational time consumption.

RESULTS:

In both techniques, MOEA/D emerges as the most effective algorithm due to its superior capability in minimizing tumour size and cancer drug concentration.

CONCLUSION:

This study highlights the importance of integrating OCT and evolutionary algorithms as a robust approach for optimizing cancer chemotherapy treatment.

Keywords

Hybrid optimal control particle swarm optimization evolutionary algorithms constrained optimization multi-objective optimization

1. Introduction

One of the main causes of mortality and a long-standing danger to mankind is cancer [1]. The International Agency for Research on Cancer (IARC) predicts that by 2030, there will be 21.4 million new instances of cancer worldwide, along with 13.2 million cancer-related deaths [2]. The standard therapies used to treat this condition include radiation, chemotherapy, surgery, hormone therapy, and others. The best treatment for cancer patients relies on the kind of cancer, the location of tumor, and patient’s current general health [3].

Since mathematical modeling can comprehend how illness spread and behave, it can comprehend how illnesses spread and behave, mathematical modeling is often used to address issues in public health. Ordinary differential equations (ODEs) are used in mathematical modeling to understand the development of cancer cells and the relationship between cancer growth and cancer therapy in order to assist oncologists in individually tailoring treatment for patients [4]. The link between the immune system and tumor cells was clarified using mathematical modeling, as in [5–9]. The majority of mathematical models use optimum control (OC) to optimize tumor and medication concentration. The cancer treatment model is enhanced by introducing extra constraints and transitioning to multi-objective optimization, which aims to reduce tumor size and drug concentration. The optimum chemotherapy treatment should boost effectiveness while minimizing side effects and reducing the cost of cancer treatment.

Chemotherapy, which is administering a medicine cocktail over the course of many doses, has been shown to be very effective in treating cancer [10]. Drugs used to treat cancer are used to slow down or stop the cell division cycle, which results in cell death [11]. However, chemotherapy drugs not only target cells that are dividing quickly but also inhibit the proliferation of cancer cells [12–14]. Therefore, it is essential to keep healthy cells above the minimal threshold when chemotherapy medications are administered throughout the body in order to balance the anticancer activity. However, therapy resistance and cancer recurrence are commonly caused by the conventional cancer therapeutic techniques’ poor and nonspecific target [15,16].

New potential methods to fight cancer cells and heal cancer patients have been made possible by the discovery and revolution in stem cell therapy. Stem cells have been used to rebuild immune and blood systems that have been damaged by chemotherapy side effects or cancer cells. Stem cell-based approaches for treating cancer have shown both considerable promise and difficulties via research in preclinical studies [17]. After receiving several sessions of high-dose radiation or chemotherapy, hematopoietic stem cell (HSC) transplantation has mostly been utilized as a treatment for myeloma, leukemia, and lymphomas [18]. After high dose chemotherapy, HSCs are often employed to return the immune system and bone marrow to their pre-chemotherapy levels [19]. Randomized clinical studies showed that using HSCs reduced hospital stay and increased disease-free survival rates [20–24]. Mesenchymal stem cells (MSCs) are known to speed up the healing of damaged organs and increase a patient’s tolerance for high-dose chemotherapy, which enhances the tumor-killing effects [25]. Cancer stem cells (CSCs), also known as stem-like tumor cells, are thought to be important in the genesis of cancer and provide promise for the therapy of a variety of solid tumor forms [26]. Because they are linked to increased immunological and therapeutic resistance, CSCs show a considerable improvement in therapeutic effectiveness in the fight against cancer [26–28]. Stem cell therapy may thereby lessen negative effects and shield therapeutic molecules from fast biological deterioration.

To determine the ideal medication dose for each patient, the majority of the mathematical models used in cancer treatment now use Optimal Control (OC). The optimization problem becomes increasingly challenging to provide the optimum cancer therapy as the number of variables, goals, and state restrictions rises. The multi-objective optimization problem of chemotherapy in cancer treatment is addressed in the simulated model using a linear time varying approximation strategy to kill cancer cells while reducing drug concentration [29]. The majority of research uses a technique that combines swarm intelligence (SI) with evolutionary algorithms (EA) to address the constrained multi-objective optimization problem (CMOOP). To solve a multi-objective optimization issue aimed at identifying the best control method for pharmacological cancer therapy, the author used the non-dominated sorting genetic algorithm (NSGA II) and the multi-objective optimization differential evolution algorithm (MODE). A modified multi-objective particle swarm optimization technique (M-MOPSO) is developed for addressing CMOOP issues, and it is shown that it performs better for problems with large dimension [30]. Zietz and Nicolini [31] presented a multi-objective optimal control model (OCM) to reduce the tumor population and raise the normal cell population with the constraint on intermediate drug dose and normal cell population. A maximum medication dosage was to be administered until a certain period, after which the patient was to be permitted to rest until the end of treatment, according to their Pareto-efficient chemotherapy treatment plan.

Even if there are several well-established techniques for solving mathematical models, combining various strategies may still lead to advancements. In order to manage CMOOP for the mathematical model of Alqudah, this study offers hybrid algorithms that combine Evolutionary Algorithm (EA) and Swarm Intelligence (SI) [32]. These algorithms use an indirect approach to OC theory. To demonstrate how stem cells support effector cells that fight tumor cells to strengthen the immune systems of cancer patients while chemotherapy drugs kill infected cells, the author [32] presented a study of ordinary differential equations (ODEs) model that describes stem cells and cancer chemotherapy treatment. The goal of this study is to identify the best control strategy for reducing the number of tumor cells and medicine consumption while meeting the state constraint. It might be challenging to determine the optimal chemotherapy dose that maximizes cancer cell decrease efficacy while minimizing undesirable side effects.

The present work is a novel effort to address the aforementioned problem. In this work, the mathematical model used to represent the tumour control is different because it includes stem cells. Though other papers use optimal control to schedule the treatment, dual therapy used in this work makes the treatment more challenging in terms of optimization. It is to be noted that stem cells helps in controlling tumour cells with less side effects compared to chemotherapy, however the presence of chemo-medicine in cancer patient could reduce the stem cells growth, thus making its effectiveness meaningless. Hence an optimal scheduling of chemotherapy is required to minimize the tumour while making sure the side effects are kept within certain threshold represented by effector cell constraint. Secondly, the other article fundamentally analyses the stability of the basic and widely used model where the chemotherapy level of 0.7 was identified as stable input to the system of equations used. The model stability was analysed earlier by Alqudah [32] hence in the present work, the focus is to obtain the critical treatment schedule of chemotherapy that reduces the side effects if it was combined with stem cell treatment.

2. Cancer chemotherapy model

Throughout many years of active cancer research, a number of mathematical models of cancer cell proliferation have been developed using partial differential equations (PDEs) and ordinary differential equations (ODEs). This article makes use of a mathematical model created by Alqudah [32]. With this strategy, which combines chemotherapy and stem cell treatment, cancer patients’ immune systems are meant to be strengthened. While chemotherapy is destroying the cancer cells, the effector cells need the help of the stem cells to fend off the tumor cells. This concept is crucial for planning an efficient medication regimen for cancer patients. Four ODEs make up the mathematical representation of the tumor, which represents the use of chemotherapy and stem cell therapy to cure cancer and fosters dynamic interactions between tumor cells and medication effects. This article presents the mathematical model of cancer therapy examined in [32]. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\displaystyle \frac{dS}{dt}}={\gamma}_{1}S-k_{S}MS\quad \\[12.0pt] {\displaystyle \frac{dE}{dt}}={\alpha}-{\mu}E+{\displaystyle \frac{p_{1}ES}{(S+1)}}-p_{2}(T+M)E\quad \\[12.0pt] {\displaystyle \frac{dT}{dt}}=r(1-bT)T-(p_{3}E+k_{T}M)T\quad \\[12.0pt] {\displaystyle \frac{dM}{dt}}=-{\gamma}_{2}M+V(t)\quad \end{array}\right. & & \displaystyle\end{eqnarray}$ (1) where S stands for the population of stem cells and E for the population of effector cells. T stands for the number of tumor cells, and M stands for the amount of chemotherapeutic agent present. The description of the parameters utilized in the mathematical model is shown in Table 1. The simulation settings and unit in this model were selected from [33].

Table 1

Description and value of all relevant parameters

Parameters	Description	Value
S ₀	Initial concentration of stem cells	1
E ₀	Initial concentration of effector cells	1
T ₀	Initial concentration of tumor cells	1
𝛾₁	Stem cells Decay rate	−0.02825
𝛼	Effector cells production rate	0.17
𝜇	Effector cells natural death rate	0.93
b	Tumor cells carrying capacity of	10⁻⁹
k _s	Fractional stem cell killed by chemotherapy	1
p ₁	Effector cells maximum proliferation rate	0.1245
r	Tumor growth rate	0.18
p ₂	Decay rate of the effector cells killed	1
k _T	Fractional tumor cells killed by chemotherapy	0.9
p ₃	Tumor cells decay rate killed by the effector cells	0.9
𝛾₂	Chemotherapy drug decay rate	6.4
V (t)	The time dependent external influx of chemotherapy drug	1

3. Multi-objective optimization problem

Multiple criteria decision-making uses multi-objective optimization (MOO), a branch of mathematics that deals with optimization problems involving two or more objective functions that must be maximized concurrently. The decision-making challenge in MOO enables a compromise on certain competing issues. The solutions of MOO models, which describe the optimal trade-offs between specified criteria, are represented as a collection of Pareto optima. The Pareto optimum solutions are included inside the Pareto optimal set if all feasible solutions in vector S are not dominated by other solutions. Pareto front refers to the mapping of the Pareto optimal set [33,34].

Fig. 1.

Pareto optimal front and pareto optimal solution.

The work’s goal functions are provided in (4). The objective is to preserve the patient’s health during the course of treatment while decreasing the intake of tumor cells and chemotherapy medication, respectively, within the state restriction as mentioned in (3).

The initial conditions for the ODEs in (1) are $\begin{eqnarray}\displaystyle S(t)=S_{0},E(0)=E_{0},T(0)=T_{0},M(t)=0,\quad \text{for }∼V_{0}=0. & & \displaystyle\end{eqnarray}$ (2) The dynamic process is subject to the state constraint: $\begin{eqnarray}\displaystyle E>0.4\quad \forall t{\epsilon}[0,t_{f}]. & & \displaystyle\end{eqnarray}$ (3) Denote the state variable $x=(S,E,T,M)\in \mathbb{R}^{4}$ by considering the objective: $\begin{eqnarray}\displaystyle J(x,V)=\min \int _{0}^{t_{f}}T(t)+V(t)dt. & & \displaystyle\end{eqnarray}$ (4) Which represents a weighted sum of the tumor cells and the chemotherapy drug concentration. Our optimal control problem is considered as follows:

Minimize subject to dynamic systems in (1) and the state constraint (3).

The most noteworthy feature of this study is the invention of a hybrid Swarm Intelligent (SI) and Evolutionary Algorithm (EA) model that merges with OC theory to find the best potential solution for the CMOOP. Four algorithms are used in this work: the Multi-Objective Particle Swarm Optimizer (MOPSO), which is based on SI, the Strength Pareto Evolutionary Algorithm II (SPEA II), the Pareto envelope-based selection algorithm II (PESA II), and the Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), which is based on EA.

3.1. Evolutionary algorithms (EA)

The four essential steps in EA are initialization, selection, genetic operators, and termination. The first phase of EA involves creating a sample population of solutions. The population’s members are then assessed using a fitness function. To create the next generation of the algorithm, the best members are chosen. Up till a stopping creation is reached, these operations are repeated. Since EA is less influenced by the Pareto front’s form or continuity, it enhances the likelihood of finding a nearly optimum solution throughout the optimization process [35,36].

The multi-objective problem will be divided into smaller scalar optimization problems via MOEA/D. By creating a population of solutions, all of the subproblems are resolved simultaneously. If the weight of each subproblem is suitably selected, MOEA/D confirms during decomposition that optimizing a number of subproblems may result in a collection of well-distributed Pareto optimal solutions [37]. In terms of accelerating convergence and preserving population diversity, MOEA/D has substantial benefits [38].

In recent years, SPEA has become a well-known evolutionary method for multi-objective optimization. The upgraded version of SPEA, or SPEA-II, was suggested by [39]. Strength Pareto is crucial because it demonstrates how solutions close to the top rank. A collection of non-dominated solutions, also known as a set of pareto optimum solutions, may be found and preserved using the multi-objective optimization approach SPEA-II, which has a limited number of setup parameters. Due to its speedy convergence and equally dispersed solution sets, SPEA-II seems to perform better.

The Pareto Envelope-based Selection algorithm (PESA) is a grid-based fitness assignment system that manages selection and maintenance using a hyper-grid crowding approach. Given that it features an archive in addition to the internal population, it is comparable to SPEA. They have been shown to outperform SPEA II for optimization problems with several goals in terms of their capacity to search toward the pareto front [40]. Because PESA lacks a Pareto front spread, this problem was fixed in the updated version known as PESA II.

3.2. Particle swarm algorithms (PS)

Particle swarm optimization (PSO) refers to a group of particles, such as a flock of birds, that operate and communicate similarly and move collectively in a predetermined search space in pursuit of the perfect answer. Each particle travels through the solution space, trying to get into a better position by modifying its speed and direction in accordance with its own past knowledge and data from the swarm’s best particle at the moment. The history of the best position for each particle, pbest, and the history of the best position throughout the population, gbest, serve as the particles’ primary guiding factors. The particles adjust their locations by changing their velocities [41]. When the particles migrate to a new position, the rules are modified. The process is repeated until the halting requirement is satisfied.

The MOPSO method was developed based on the PSO algorithm, however it differs from it in terms of how mutations are used, how the repository is updated, and how population growth occurs. Instead of identifying a single “best” answer, the objective of MOPSO is to identify a group of possibilities that comprise the Pareto Front. A particle in MOPSO flies in a path determined by Pareto Dominance, and the early best solutions that are saved in non-dominated vectors will be utilized by other particles to influence their own flight in order to find the best non-dominated solutions.

4. Optimization methodologies for solving CMOOP

The CMOOP may be handled while keeping a healthy balance between the objectives of objective minimization and the constraints-satisfaction by combining optimum control (OC) theory with EA and PS algorithms. Two techniques are used to address the optimization challenge.

Hybrid 1: The OC theory optimizes its single composite objective function, which is separated into multiple multi-objectives of PS and EA, to reach the Pareto optimum set with penalty strategy in PS and EA, in order to comply with the state constraint.

Hybrid 2: By using an indirect approach of OC theory and Augmented Lagrange (AL) to fulfill the restrictions to maximize its single composite objective function divided into multi-objective PS and EA, this method finds the Pareto optimum set.

Figure 2 depicts the flowcharts for the two approaches. The key difference between the two methodologies (Hybrid 1 and Hybrid 2) is how each one approaches restrictions. Multi-objective PS and EA use a penalty method to address the restrictions in Hybrid 1. The avoidance of the penalty during the optimization process is regarded as satisfying the limitations. The limitations are managed by Hybrid 2 using Augmented Lagrange of OC. This indicates that the OC manages the limitations without interfering with the OS’s and EA’s multi-objective function.

Fig. 2.

Flowchart of methodology.

The two methods for solving the optimum control theory are direct and indirect. We used an indirect way to hybridize using PS and EA since the model in [32] only has one constraint. The bang-bang control is implemented in Hybrids 1 and 2 via the indirect approach. Applying a penalty system to the multi-objective function of SI and EA algorithms allows us to satisfy the state limitations in Hybrid 1 that are not included in the OC theory. In Hybrid 2, the state constraint is included into the OC theory through AL, and only the SI and EA algorithms are employed to get the Pareto optimum set. The multi-objective problem is treated as a single objective issue using a weighting scheme, and Equation (4) is intended to provide the objective function for the optimal control theory (OCT). $\begin{eqnarray}\displaystyle \min J=w_{1}\int Tdt+w_{2}\int Vdt & & \displaystyle\end{eqnarray}$ (5) where $\begin{eqnarray}\displaystyle w_{1}+w_{2}=1 & & \displaystyle\end{eqnarray}$ (6)

The PS and EA will produce weight in Hybrids 1 and 2 that improves optimization utilizing OC. The range of the weights w ₁ and w ₂ in the search space is constrained to 0 and 1. The outcomes of OC for tumor and medication concentration were assessed using the objective function of PS and EA described in (4).

The inequality constraint in (4) becomes: $\begin{eqnarray}\displaystyle C=0.4-E<0. & & \displaystyle\end{eqnarray}$ (7)

The Hamiltonian equation for Hybrid 1 as follow $\begin{eqnarray}\displaystyle H_{\text{Hybrid}1}=w_{1}T+w_{2}V+{\lambda}_{1}{\displaystyle \frac{ds}{dt}}+{\lambda}_{2}{\displaystyle \frac{dE}{dt}}+{\lambda}_{3}{\displaystyle \frac{dT}{dt}}+{\lambda}_{4}{\displaystyle \frac{dM}{dt}}. & & \displaystyle\end{eqnarray}$ (8)

The Hamiltonian equation for Hybrid 2 will be $\begin{eqnarray}\displaystyle H_{\text{Hybrid}2}=H_{\text{Hybrid}1}+{\eta}(0.4-E). & & \displaystyle\end{eqnarray}$ (9) From the Hamiltonian, we obtain the co-states functions: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}{\lambda}_{1}=-{\displaystyle \frac{dH}{ds}}\\[12.0pt] {\lambda}_{2}=-{\displaystyle \frac{dH}{dE}}\\[12.0pt] {\lambda}_{3}=-{\displaystyle \frac{dH}{dT}}\\[12.0pt] {\lambda}_{4}=-{\displaystyle \frac{dH}{dM}}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (10)

Controllability is an essential attribute of the model with transforming into a characteristic of the system if the model is realistically accurate. Through the integration of clinical data with information about the molecular processes of drug metabolism, the utilization of mathematical models capable of precisely representing both the kinetics and dynamics of the drug, along with the dynamics of disease states, presents favorable opportunities for applying OCT. Until recently, the practical application of mathematical models for tumor growth in clinical scenarios was restricted due to challenges in obtaining input data with adequate spatial resolution in patients, even at a singular time point. This encompasses factors such as the immune infiltrate, the ratio of tumor-to-normal cells which were difficult to acquire comprehensively. The authors highlighted the exponential growth in publications on mathematical modeling of cancer. Therefore, it is imperative to promptly explore the application of control theory to assess the effectiveness of existing models. This evaluation is crucial in determining their utility as tools for optimizing dosage and scheduling of drug administration [42].

MATLAB and APMonitor are used to compute the solution to this MOO issue [43]. A free program for solving issues including linear programming, quadratic programming, nonlinear programming, and mixed integer (MILP and MINLP) is called the APMonitor Optimization Suite [44]. A very easy technique to access AP-Monitor is provided by MATLAB. In this study, the SI and EA optimization are carried out in MATLAB, whereas OCT is solved using APMonitor. The weight created by the SI or EA algorithms is input into (5), which is then resolved by the OCT algorithm. In MATLAB, the result, which is the tumor and medication concentration, is recorded and compared to produce a Pareto graph.

5. Results and discussion

As discussed earlier, the performance of the hybrid methodologies (Hybrid 1 and 2) applied in the cancer model by [32] is analyzed and compared with SI and EA methods. A set of Pareto-optimal solutions is presented in this CMOOP. It is widely accepted that a point set's quality should be determined by its proximity to the Pareto front (the closer, the better) and its diversity (the more uniformly distributed, the better).

Using the hypervolume (HV) metric indicator, the volume of the area in n-dimensional objective space between the pareto front and a dominating reference point is assessed and calculated in order to compare the results. Where n is the number of objectives, a higher HV value suggests a better collection of non-dominated solutions in terms of diversity and convergent views. Figure 3 shows that the point with the greatest distance from the initial location is the closest point to the origin in the Pareto Optimal Front. This metric is used to assess our efforts.

Fig. 3.

Calculation of the distance to the origin.

The Pareto Optimal Front findings for the two approaches shown in Fig. 2 using the SI and EA are shown in Fig. 4. The Pareto Optimal Front for the two approaches utilizing MOPSO, MOEA/D, SPEA II, and PEA II, respectively, is shown in Figs 4(a), (b), (c), and (d). The pink line displays the results of Hybrid 1, while the red line displays the findings of Hybrid 2. According to the findings, as compared to Hybrid 1, the Pareto Front curve for Hybrid 2 is nearer the origin. It demonstrates that the difficulty of handling the restriction is removed using Hybrid 2’s Augmented Lagrange approach. Due to the necessity for a larger search space by SI and EA as a result, Hybrid 1 falls short of Hybrid 2. Overall, this finding shows that the OC theory is a highly effective optimizer that works better when combined with SI and EA.

Fig. 4.

Comparison of Pareto optimal front between Hybrid 1 and Hybrid 2 using: (a) MOPSO; (b) MOEA/D; (c) SPEA-II; (d) PESA-II.

Figure 5 compares the Pareto Optimal Front approach to every one of the techniques used in Hybrids 1 and 2. This research allows us to identify the most effective algorithm for each strategy. The findings of MOPSO are shown by a red line, those of MOEA/D by a pink line, SPEA II by a turquoise line, and PESA II by a black line. In contrast to other algorithms, MOEA/D looks to be the best algorithm since it is the closest to the origin in both techniques. SPEA II is the least effective algorithm among the four for both approaches and is the most removed from the original. Due to the fact that many of their points overlap, MOPSO and PESA II function similarly.

Fig. 5.

Comparison of Pareto optimal front between all algorithms: (a) Hybrid 1; (b) Hybrid 2.

The results of the average tumor size, average medication volume, average time spent at the location closest to the origin, and the average hypervolume (HV) indicator for both Hybrid methods are shown in Tables 2 and 3. Distance from the origin defines the location that is the closest to the origin. With respect to performance, Hybrid 2 outperforms Hybrid 1. For Hybrids 2 and 1, respectively, the average distance between the four methods and the origin point is 0.877 and 0.930. Hybrid 2 with MOEA/D algorithm is the best method because of its close proximity to the origin (0.8580). Since Hybrid 1 with SPEA II has the greatest distance to the origin (0.9419), it performs the poorest. According to Table 2, Hybrid 2 requires less computational time than Hybrid 1 does. Although successful, the Hybrid 1 approach’s combination of the Penalty Method and the OC theory employs a vast search area and requires additional computing time. In OC theory, the AL technique of Hybrid 2 demonstrates that it has a better result and is computationally quicker than Hybrid 1. For both approaches, PESA II has the quickest speed of convergence, while the MOPSO algorithm is the slowest.

Table 2

Overall results of Hybrid 1 method

Algorithms	MOPSO	MOEA/D	SPEA-II	PESA-II
Distance of closest point to origin	0.893	0.8763	0.9419	0.9038
Total tumor cells	214.256	213.601	212.928	194.143
Total drug volume	363.48	355.8	380.868	388.2
Consumed time (min)	40.51	34.7	28.44	23.32
Hypervolume (HV)	0.16541	0.16701	0.08441	0.1567

Table 3

Overall results of Hybrid 2 method

Algorithms	MOPSO	MOEA/D	SPEA-II	PESA-II
Distance of closest point to origin	0.8694	0.85	0.9131	0.8703
Total tumor cells	194.88	191.62	217.01	194.4
Total drug volume	378.84	377.76	342.768	382.8
Consumed time (min)	35.88	29.1	25.69	21.45
Hypervolume (HV)	0.17639	0.1785	0.08855	0.16489

HV is calculated for the two methods with respect to the reference point (1, 1). Based on the results, Hybrid 2 has obtained larger values than Hybrid 1. Therefore, the Pareto results of Hybrid 2 are better in terms of convergence and diversity than the Pareto results of Hybrid 1. Hybrid 2 with MOEA/D performs the best with the largest HV, 0.17850, whereas Hybrid 1 with SPEA II is the least performing technique with the smallest HV, 0.08441. The overall results reveal that Hybrid 2 can reduce both tumor cell concentration and chemotherapy drug volume by using less computation effort. The AL employed in Hybrid 2’s OC freed the SI and EA from having to deal with constraint via penalty strategy. In comparison to Hybrid 1, the result allowed Hybrid 2 to enhance the closest point to the origin and hence minimize tumor cell concentration and drug volume.

Based on the previous analysis, MOEA/d has the best performance amongst all the algorithms. Figure 6 shows the results of the drug concentration and different cell concentrations against time using Hybrid 1 based on MOEA/D while Fig. 7 represents for the performance of Hybrid 2 based on MOEA/D. Figure 6 shows the outcome of the simulation with a weighting coefficient for drug therapy, w₂ of 0.058 and a weighting coefficient for tumor cells, w₁ of 0.942 for Hybrid 1 method with MOEA/D. Based on the result, the optimal length of the treatment is 1.75 days with a total drug volume used of 355 which is obtained from the area under the graph of Fig. 6(a). As seen in Fig. 6(c), the tumor cells concentration reduces over the treatment period and we obtained that the total tumor cells for the period of 10 days is 213.601 from the area under graph of Fig. 6(c). As appears in Fig. 6(c), the effector cells concentration maintains above 0.4 that satisfies the constraint introduced in ((3)). The reduction in stem cell concentration and chemotherapy concentration shows that they are being efficiently used throughout the treatment cycle.

Fig. 6.

Observation on different variable over period of treatment for Hybrid 1 with MOEA/D: (a) input control; (b) stem cells concentration; (c) effector cells concentration; (d) tumor cells concentration; (e) Chemotherapy concentration.

Figure 7 shows the outcomes of the simulation with a weighting coefficient for drug therapy, w₂ of 0.058 and a weighting coefficient for tumor cells, w₁ of 0.942 for Hybrid 2 method with MOEA/D. Based on the results, the optimal length of treatment is 1.89 days with a total drug volume used of 377.76 which is obtained from the area under the graph of Fig. 7(a). As seen in Fig. 7(d), the concentration of tumor cells reduces over the treatment period which meets the objective of this paper. The total number of tumor cells for the period of 10 days is 191.62 obtained from the area under the graph of Fig. 7(d). As appears in Fig. 7(c), the effector cells concentration maintains above 0.4 which satisfies the constraint introduced in this work. Both Hybrid 1 and Hybrid 2 methodologies achieve the main goals of this optimization, which is killing the tumor cells, minimizing the chemotherapy, and reducing the toxicity to keep the effector cells above certain safety level.

Fig. 7.

Observation on different variable over period of treatment for Hybrid 2 with MOEA/D: (a) input control; (b) stem cells concentration; (c) effector cells concentration; (d) tumor cells concentration; (e) Chemotherapy concentration.

6. Conclusions

In order to solve CMOOP utilizing a mathematical model of cancer that combines chemotherapy and stem cell treatment, a hybrid optimum control swarm intelligence optimization approach is suggested in this study. We solve optimum control issues by using Pontryagin’s Maximum/Minimum Principle. In order to calculate the ideal dose based on several optimization criteria, weighting coefficients were proposed. Our key goal is to maintain the concentration of effector cells above the safe threshold while minimizing the concentration of drug and tumor cells. In this study, two methodologies—Hybrid 1 and Hybrid 2—were created using MOPSO, MOEA/D, SPEA II, and PESA II. The findings indicate that:

Hybrid 2, which embedded the constraints in OCT, accomplished improved results than Hybrid 1 in the terms of reduction on cancer cells and drug concentration as well as the computational time consumption.

Both Hybrid 1 and Hybrid 2 cancer treatment results show proximity to chemotherapy protocol maximum tolerated dose.

MOEA/D is the best algorithm in both hybrid methods since it has the closest point to the origin compared to others which signifies that it gives the best weightage to minimize the tumor and cancer cells.

Within a more comprehensive theoretical framework, the findings of this investigation have practical significance by providing insightful information on the dynamics of chemotherapy-induced tumor development. In this paper, we show how hybrid approaches can be done for one particular cancer model previously described in [32]. The study of these theoretical results provides invaluable insights into the fundamental principles of drug resistance and tumor growth, in addition to providing predictions for the creation of novel treatment strategies. Notably, even while the researchs current focus may not directly relate to clinical problems which limit its practical applicability, but our message is broader, and we use this specific model to illustrate the hybrid optimization approach applicable to more detailed cancer evolution equations. The future of hybrid optimization works in mathematical oncology will necessitate the development of pipelines for the rapid integration of patient data such as imaging, genomics biomarkers that enhanced by optimization algorithms. While we believe that our optimization approach will have a major role in the design of future clinical trials of cancer chemotherapy, the presented version is not yet sufficiently practical, but we trust that the use of a hybrid approach for predicting optimal treatment protocols will bring us a step closer to the development of a quantitative and clinically relevant decision-making system for personalized medicine.

In general, hybrid approaches based on OCT, swarm, and evolutionary algorithms function well and might help oncologists create the most effective cancer therapy. This study aims to demonstrate how using stem cells in conjunction with chemotherapy might increase therapeutic effectiveness while minimizing chemotherapy’s negative side effects. In addition to assisting mathematicians in further analyzing the hybridization of optimum control with other optimization approaches to improve the efficiency of optimization, it is believed that this study will aid oncologists in the choice of cancer therapy for the patient.

Footnotes

Acknowledgements

This work was supported by a UM International Collaboration (grant: ST023-2022). The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Large Groups Project (grant: RGP.2/201/44).

Ethics statement

The study was conducted with due consideration of ethical principles and guidelines. No clinical data was used in the study.

Conflict of interest

The authors declare that they have no conflict of interest.

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