Mathematical model of optimal control for endometrial cancer with treatments dostarlimab and chemotherapy using Caputo derivative

Abstract

We develop a mathematical model to investigate endometrial cancer progression and treatment response under dostarlimab and chemotherapy using nonlinear ordinary differential equations (s), and extend the framework to fractional differential equations (s) with the Caputo derivative to capture memory effects. Existence and uniqueness of solutions are established via the Banach fixed-point theorem, and stability analysis is performed using the Routh–Hurwitz criteria. An optimal control framework is formulated to evaluate single and combined therapies under both and settings. Numerical simulations are carried out in MATLAB, employing the ode45 solver for systems and the fde12 solver for systems. Results indicate that dostarlimab monotherapy is more effective than chemotherapy alone, while combined therapy achieves the greatest reduction in cancer cells and the strongest activation of CD8+ T-cells. The model provides faster tumor reduction and higher immune activation, whereas the model achieves lower overall therapeutic cost by balancing tumor reduction with reduced drug usage. These findings highlight the potential of optimal control strategies particularly combined therapy for improving treatment outcomes in endometrial cancer management.

Introduction

Cancer, a complex and heterogeneous group of diseases, poses a significant challenge in modern medicine due to its diverse origins, aggressive progression, and variable treatment responses. It is marked by the uncontrolled growth and spread of abnormal cells, forming malignant tumors in various tissues and organs. Despite substantial progress in cancer research and treatment, effective management remains challenging, underscoring the need for innovative strategies to understand its mechanisms and develop targeted therapies.

In 2007, Hagemann et al. investigated the significant role of inflammation in promoting cancer, while also recognizing the immune system’s crucial protective role against it¹. The global incidence and mortality patterns for common cancers and the opportunities for prevention in developing countries are discussed by Ahmedian et al.². Studies in Ref.³ aims to refine the estimates of the global and regional burden of infection-attributable cancers, thereby guiding research priorities and enhancing prevention efforts. Additionally, an approach for quantifying cancer mortality and incidence using existing data sources was described by Mathers et al.⁴. One of the most prevalent gynecologic cancers in the world, endometrial cancer (EC) develops in the endometrium, the inner lining of the uterus. It primarily affects postmenopausal women, although occurrences among younger individuals are also documented^5,6. Early detection benefits from symptoms like abnormal vaginal bleeding, enabling timely intervention and improving prognosis⁷. Nonetheless, addressing advanced or recurrent cases poses substantial challenges, underscoring the need for innovative treatment approaches to optimize patient outcomes^8–11.

Mathematical modeling is a crucial tool in cancer research, facilitating the quantitative analysis of tumor growth dynamics, treatment responses, and disease progression. Modeling and simulations are essential for uncovering the behavior of tumor cells^12–14. The primary treatment modalities include surgery, radiotherapy, and chemotherapy. A mathematical model for the cytotoxic T lymphocyte response to the development of an immunogenic tumor was described¹². Kirschner and Panetta used a mathematical modeling to investigate the dynamics between immune-effector cells, tumor cells, and IL-2¹³. Furthermore, the authors in Ref.¹⁴, offer a model of cancer-immune surveillance that takes into consideration different immunological responses from natural killer cells and CD8+ T-cells, for more details, see^15,16. Chemotherapy, a fundamental component of cancer treatment, involves the use of cytotoxic drugs to target and destroy rapidly dividing cancer cells. Although chemotherapy has proven effective in controlling tumor growth and improving survival rates, its non-specific action often results in substantial toxicity and adverse effects on healthy tissues. On the other hand, immunotherapy offers a more focused and maybe less dangerous treatment alternative by using the body’s immune system to identify and eradicate cancer cells^17–21.

A prospective treatment option for people with EC is dostarlimab, a programmed death-1 (PD-1) receptor antagonist that is a promising immunotherapy medication that strengthens the immune response against cancer cells. This article outlines the development of dostarlimab, a humanized monoclonal antibody acting as a PD-1 receptor antagonist, for the treatment of various cancers. Recent approvals in both countries EU and USA have been granted for its use in adult patients with recurrent or advanced EC lacking mismatch repair, following promising results from the GARNET trial. The developmental journey of dostarlimab, leading to these initial approvals, is detailed herein^22–25.

Fractional calculus, particularly through the use of Caputo derivatives, offers a flexible method for modeling complex biological systems that exhibit memory effects²⁶. By integrating fractional derivatives into our mathematical model, we can effectively represent non-local interactions and the long-term memory effects intrinsic to tumor growth dynamics^27–29. This methodology has been successfully employed in numerous biomedical applications, including cancer modeling, for example, the models in Figs. 1, 2, 3 were investigated in^30–32, by Samei et al., Rezapour et al. and Mohammadaliee et al., respectively.

Fig. 1.

Model for Covid-19 introduce by authors in Ref.³⁰.

Fig. 2.

model of Covid-19 introduce by authors in Ref.³¹.

Fig. 3.

Proposed model for introduce by authors in Ref.³².

The theory of optimal control , provides a robust mathematical framework for developing treatment protocols aimed at optimizing specific objectives, such as reducing tumor burden while minimizing treatment-related toxicity^33,34. By formulating and solving optimization problems, theory facilitates the creation of personalized treatment strategies that are tailored to the unique characteristics of each patient and the biology of their tumor^35–38. Fractional-order models are increasingly used because they can capture memory and hereditary properties in real-world systems better than classical models. They have been applied in studying Hepatitis (B) and (C)^39,40, cancer treatment with radiotherapy⁴¹, and the spread of the Ebola virus⁴². These models also help in analyzing COVID-19 panic spreading⁴³ and carbon capture in forest systems⁴⁴. In control applications, fractional-order models have been used for plant-locust surveillance⁴⁵, controlling Hepatitis (B) transmission⁴⁶, and managing helminth infections⁴⁷. These recent studies motivate us to apply fractional-order derivatives in our work to improve the accuracy and effectiveness of cancer chemo-immunotherapy modeling and control.

In this paper, we present both ordinary and fractional mathematical models with a singular kernel for a cancer treatment framework, as follows

1

in which the parameters and variables are explained later. The primary objective is to minimize tumor cells by employing fractional optimal control problems (s) within the proposed model. The MATLAB toolboxes for ode45 and fde12 are used to perform numerical simulations. To the best of our knowledge, the integration of ordinary and with a singular kernel for cancer treatment, leveraging the synergy between the dostarlimab model and chemotherapy, has not been previously explored.

The novelty and originality of this article are highlighted as follows:

• Development of a new mathematical model describing the combined effects of dostarlimab and chemotherapy in the treatment of EC.

• Comparative analysis between the classical integer-order () model and the fractional-order () model to capture memory effects in cancer dynamics.

• Establishment of the existence and uniqueness of solutions using the Banach fixed-point theorem.

• Stability analysis of the proposed model using the Routh–Hurwitz criteria.

• Investigation of the optimal control strategies for the chemo-immunotherapy treatment framework.

• Numerical simulations performed using the ode45 solver for the model and the fde12 solver for the model to illustrate the effectiveness of the proposed strategies.

The rest of this article is organized as follows: The necessary definitions and introduction are given in “Definitions and preliminaries” section. In “Proposed model of dostarlimab-chemotherapy” section, we build a mathematical model using a system of s that include chemotherapy and dostarlimab treatment variables. Dostarlimab inhibits the interaction between PD-1 and Programmed Death-Ligand 1 (PD-L1) receptors. This model is subsequently extended to encompass s. “Uniqueness and existence of the model” section focuses on the uniqueness and existence of the solutions to the model. Further, we examine the stability of drug-free stable states, taking into account conditions in which dostarlimab, chemotherapy, or both treatments are absent in “Stability analysis” section. “Optimal control analysis” section formulates and solves the for both s and s, utilizing two control variables, and discusses the existence and optimality conditions of the solutions. “Numerical analysis for s” section provides numerical simulations for both s and s, along with a comparative analysis. The paper is finally concluded in “Conclusion” section with a review of the results and any possible ramifications.

Definitions and preliminaries

This section delves into the foundational concepts of fractional derivatives alongside their pertinent definitions. Fractional calculus encompasses various formulations for derivatives of arbitrary order, generalizing the classical integer-order derivatives. Among these, the Riemann–Liouville (RL) and Caputo definitions are prevalent. Notably, the Caputo derivative presents a significant advantage in its use of integer-order derivatives for initial conditions. This alignment with well-established concepts facilitates the interpretation of physical systems and translates to greater applicability in real-world scenarios. Throughout the paper, denotes the time variable.

Definition 2.1

²⁶ The fractional integral of order with the lower limit for a function is defined as follows

Definition 2.2

²⁶ The RL derivative of order for a function is defined as

Definition 2.3

²⁶ The Caputo fractional order derivative is defined by

Theorem 2.4

²⁸ Consider the system

and Jacobian matrix , of the system evaluated at the equilibrium point . For ,

(i)

The equilibrium point is locally asymptotically stable iff all the eigenvalues of satisfy

(ii)

The equilibrium point is stable if all the eigenvalues of satisfy and eigenvalues with

have the same geometric and algebraic multiplicity.

(iii)

The equilibrium point is unstable if and only if there exist eigenvalues of satisfy .

Proposed model of dostarlimab-chemotherapy

The programmed cell death protein 1 (PD-1) and its ligand (PD-L1) play a pivotal role in the mechanism by which cancer cells evade immune surveillance. The checkpoint receptor PD-1 is predominantly expressed on immune cells, notably on T-lymphocytes, which are essential for recognizing and eliminating abnormal and malignant cells in the body. Conversely, many tumor cells overexpress the ligand PD-L1 on their surface, enabling them to interact with and suppress the immune system by engaging with T-cells.

When PD-L1 binds to the PD-1 receptor on T-cells, it sends inhibitory signals that inactivate these immune cells, impairing their ability to detect and destroy cancer cells. This interaction in Fig. 4, allows tumor cells to escape immune-mediated destruction, promoting uncontrolled tumor growth and the spread of cancer within the body.

Fig. 4.

Immune evasion by tumor cells through PD-1/PD-L1 interaction in the absence of dostarlimab.

Through this mechanism, cancer cells gain a significant survival advantage by actively suppressing the immune system’s ability to mount an effective response against them. This immune checkpoint pathway is one of the major reasons why many cancers can persist and progress despite the presence of functional immune cells in the tumor microenvironment.

To counteract this immune evasion strategy, dostarlimab, a humanized monoclonal antibody used in immunotherapy, has been developed to block the interaction between PD-1 and PD-L1. Dostarlimab specifically binds to the PD-1 receptors on T-cells, preventing PD-L1 expressed on tumor cells from binding and deactivating the T-cells. This blockade effectively reactivates the T-cells, allowing them to recover their ability to recognize, target, and destroy cancer cells. By restoring the functionality of the immune system in Fig. 5, dostarlimab enhances the body’s capacity to fight cancer.

Fig. 5.

Restoration of T-cell activity and immune-mediated tumor destruction in the presence of dostarlimab.

In this study, we propose a mathematical model that incorporates the mechanism of action of dostarlimab in combination with chemotherapy for the treatment of EC. The rationale for using dostarlimab alongside chemotherapy is based on the synergistic potential of these therapies: while chemotherapy directly targets and kills rapidly dividing cancer cells, dostarlimab reactivates the immune system to sustain and enhance the elimination of residual cancer cells and prevent recurrence. By disrupting the PD-1/PD-L1 immune checkpoint pathway, dostarlimab ensures that T-cells remain in an active state, facilitating continuous immune surveillance and targeted destruction of cancer cells even after the administration of chemotherapy.

This integrated approach of chemo-immunotherapy aims to improve treatment outcomes for EC patients by leveraging the combined strengths of direct cytotoxic effects from chemotherapy and the immune reactivation provided by dostarlimab, forming the foundation for our proposed fractional-order mathematical modeling and control analysis framework in this study.

This mathematical model is intended to be used for the analysis of immune system reactions, tumor formation, and the impacts of chemotherapy and dostarlimab, among other medicines. The following populations and parameters are monitored by the model

• Cell populations

  • *
    : Concentration of cytotoxic T-lymphocytes (CD8+) in the bloodstream (cells/L).
  • *
    : Concentration of activated cytotoxic T-lymphocytes (CD8+) in the bloodstream (cells/L).
  • *
    : Total count of cancer cells (cells).
• Medications

  • *
    : Concentration of dostarlimab in the blood (mg/L).
  • *
    : Concentration of chemotherapy drugs in the blood (mg/L).
• Treatments

  • *
    : Dostarlimab injection dose per liter of blood, administered daily (mg/L).
  • *
    : Chemotherapy medication dose administered daily in milligrams per liter of blood (mg/L per day).

The full system of s of the model is given below

2

The first equation describes the dynamics of CD8+ T-cells through several terms. The term models the logistic growth of CD8+ T-cells, where represents the proliferation rate and is the carrying capacity coefficient. The stimulation and conversion of CD8+ T-cells into activated T-cells, driven by their interaction with active T-cells or cytokines, is represented by the term , with being the conversion rate. The natural mortality rate of CD8+ T-cells is modeled by , where is the mortality rate coefficient. The term represents the reversion of activated T-cells back to normal CD8+ T-cells, with as the reversion rate. Finally, the term

accounts for chemotherapy-induced death of CD8+ T-cells, where is a scaling factor, is the chemotherapy effectiveness parameter, and is the chemotherapy dosage.

The second equation characterizes the expansion of activated CD8+ T-cells as a result of the direct presence of cancer cells through the term , where indicates the antigenicity of the tumor. To convert T-cells into activated T-cells, use the formula . The stimulatory effect of dostarlimab on activated T-cells is captured by the term

The term indicates the death of activated T-cells due to exhaustion from killing cancer cells. represents the spontaneous turnover of activated T-cells. The death of activated T-cells as a result of medicine toxicity is finally described by

In the third equation, the term represents the logistic growth of cancer cells, indicating their proliferation constrained by carrying capacity. The loss of cancer cells due to immune cell interaction is represented by

where is the interaction rate and is the half saturation constant, modeled using Michaelis–Menten kinetics to show the limited immune response. The combined effect of chemotherapy and dostarlimab on cancer cell death is given by

Lastly, the term accounts for cancer cell death directly due to interaction with dostarlimab. Since dostarlimab is not created by the body naturally, the quantity in the fourth equation represents dostarlimab therapies; no other growth terms are present. The natural breakdown of the dostarlimab protein in the body is represented by the phrase . The loss of available dostarlimab when it attaches to cancer cells is represented by the term

Due to dostarlimab’s strong binding affinity for target growth-factor receptors, which are found in large quantities on all cells, it is believed that a significant number of dostarlimab molecules are lost with every cancer cell. Additionally, when the concentration of dostarlimab is noticeably higher than the concentration of growth-factor receptors, it is presumed that the growth-factor receptors are fully saturated. Thus, as long as the dostarlimab concentration is not near zero, the number of growth-factor receptors on a given cancer cell may be used to estimate the number of dostarlimab molecules lost along with it. The quantity of doses delivered into the system is represented by the term in the fifth equation. The excretion and removal of medication toxicity is explained by the phrase .

The fractional modeling

The mathematical model of for EC, including treatments with dostarlimab and chemotherapy, is detailed in Eq. (1). This model examines the dynamics between cancer cell populations and the immune system, mediated by CD8+ T-cells and activated CD8+ T-cells. However, this model does not account for memory effects, which are crucial in many biological processes. Fractional derivatives inherently incorporate memory effects, making it relevant to extend Eq. (1) to a fractional model.

To achieve this, all integer-order derivatives in Eq. (1) are substituted with their corresponding -order fractional derivatives, where ranges between 0 and 1. Additionally, to ensure that both sides of the fractional model maintain consistent dimensionality of , biological parameters with dimensions of in the original equations are adjusted by a factor of . Based on these modifications, the generalized fractional model is proposed by (1). And the following, Table 1, are the specifics of the parameters.

Table 1.

Parameters with their definitions, values, and sources.

Parameter	Definition	Value	Units/Source
	Rate of T-cell growth	0.5	day15,16,18,20
	T-cell inverse carrying capacity	1/20	day15,16,18,20
	Level of T-cell activation	0.201	15,16,18,20
	T-cell mortality rate	0.07	day15,16,18,20
	Rate of T-cell deactivation	0.05	15,16,18,20
	T-cell death rate from chemotherapy	0.65	day18,21,34
	Medicine efficiency coefficient	1.8328	18,21,34
	Antigenicity of cancer cells	0.103	day18,27,34
	The percentage of T-lymphocytes that are activated		cell day15,16,18,20
	Activated T-cell growth rate		cell day15,16,18,20
	Half-saturation constant		volume18,34
	Activated T-cell death rate naturally		day15,16,18,20
	Chemotherapy’s effectiveness in killing activated T-cells		day18,21,34
	Medicine toxicity coefficient	1.8328	day18,21,34
p	Rate of activated T-cells death due to tumor interaction		cell day18,34
	Rate of cancer cell proliferation		day15,16,18,20
	Cancer cells’ inverse carrying capacity		day15,16,18,20
	Rate at which activated T-lymphocytes demolish malignant cells		cell day    (15,16,18,20
	Half-saturation constant		volume18,34
	Chemotherapy’s ability to eradicate cancer cells		day18,21,34
	Rate at which dostarlimab kills cancer cells		day    Assumed
	Medicine efficiency coefficient	1.8328	day18,21,34
	Rate of dostarlimab-induced cancer cell death		day    Assumed
	Decay rate of dostarlimab		day18
	Rate of dostarlimab-cancer cell complex formation		day    Assumed
	Concentration for half-maximal EGFR binding		cell day    Assumed
	Decay rate of chemotherapy drugs		day18,21,34

Uniqueness and existence of the model

Consider the system (1) with the initial conditions , , , , and . System is able to be expressed as follows:

3

where

and

Definition 4.1

Let be the class of continuous column vector whose components , , , , in , with the norm is given for by

Definition 4.2

belongs to a solution of the system (3) if

• , belongs to the interval [0, s] where ,

  

  where s, x, y, z, m and n are positive constants.

• satisfy the system (3).

Theorem 4.3

The system (3) has a unique solution .

Proof

Applying on both sides, we get

4

Let be defined by

5

Then

6

Using substitution method by choosing , then we obtain

If we choose s.t. . This implies . Thanks to the Banach fixed point theorem, the operator has a unique fixed point. As a result, there is only one solution to system (3), . From (4), we have

7

Differentiate with respect to , we get

8

This implies . Now, applying from (3) on both sides, we get

9

Thus (4) is equivalent to the system (3).

Stability analysis

Consider

That is,

The Stability analysis of four cases:

Case i If Dostarlimab and chemotherapy drugs are absent, we set and .

(i)

In a steady-state, tumor-free condition, CD8+ T-cells persist, and the number of cancer cells is zero. Consequently, CD8+ T-cells are not activated in the absence of cancer cells,

(ii)

In a persistent-tumor steady state, tumor cells are present, but CD8+ T-cells are not activated,

(iii)

In a persistent-tumor steady state, tumor cells are present, leading to the activation of CD8+ T-cells,

Case ii Dostarlimab is administered, but chemotherapy is not used,

Case iii Chemotherapy is administered, but Dostarlimab is not administered,

Case iv Both Dostarlimab and chemotherapy are used,

Theorem 5.1

Let be the equilibrium points of system (3). Then is stable only if and .

Proof

The Jacobian matrix at the equilibrium point , denoted as , is expressed as:

The eigen values of are as follows:

leads to .

if , thus .

if , implying .

, therefore .

, hence .

According to Theorem 2.4, which states that if all eigenvalues have arguments satisfying for , the system at is declared stable. Therefore, based on this criterion, the system at is stable only if and . Otherwise it is unstable.

Remark 5.2

The parameter represents the proliferation rate of cancer cells. At the equilibrium point , the cancer cell population is absent, which means cancer cells cannot persist in this state. Thus, the local stability of requires . In contrast, if , cancer cells proliferate with a positive growth rate, and the equilibrium point becomes unstable.

Theorem 5.3

Let be the equilibrium point of system (3). Then is unstable only if

Proof

At the equilibrium point , the Jacobian matrix is provided by

with

The eigen values of are as follows

because , leading to ;

because , thus ;

Clearly, for , implying .

According to Theorem 2.4, if all eigenvalues have arguments satisfying , the system at is declared unstable. Therefore, based on this criterion, the system at is unstable.

Theorem 5.4

Let be the equilibrium points of system (3). Then is locally asymptotically stable.

Proof

The substituted Jacobian matrix is given by

with

The eigen values of are as follows and . The characteristic equation of the reduced matrix is

10

where

11

According to Routh–Hurwitz criterion, the roots of Eq. (10), will possess negative roots when , and . Consequently, , the equilibrium point, is locally asymptotically stable.

Theorem 5.5

Let represent the system’s equilibrium points (3). In such case, is asymptotically stable locally.

Proof

The substituted Jacobian matrix is given by:

where ,

The eigen values of are as follows . The characteristic equation of the reduced matrix is

12

where ,

According to Routh–Hurwitz criterion, the roots of Eq. (12) will possess negative roots when , , , and . Therefore, the equilibrium point is locally asymptotically stable.

Theorem 5.6

Let represent the equilibrium points of system (3). Then, is locally asymptotically stable.

Proof

The substituted Jacobian matrix is given by

where

The eigen values of are as follows and . The characteristic equation of the reduced matrix is

13

where

By the Routh–Hurwitz criteria, if , , and are met, then the roots of Eq. (13) will have negative real parts. As a result, the equilibrium point is steadily increasing locally asymptotically.

Theorem 5.7

Let represent the system’s equilibrium points (3). In such case, is asymptotically stable locally.

Proof

The Jacobian matrix is given by

with

The eigen values of are as follows: The characteristic equation of the reduced matrix is

14

where and

According to the Routh–Hurwitz criterion, the roots of Eq. (14) will have negative real parts if the conditions , , and are satisfied.

Optimal control analysis

Consider a state system in , with the admissible control set defined as and are Lebesgue measurable, , , , where is the final time. The objective functional is given by

Here, the weight factors and quantify the patient’s tolerance to dostarlimab and external chemotherapy, respectively. The objective is to minimize the following functional

This minimization is subject to the following constraints

where

with initial conditions , , , , . The Hamiltonian is defined as follows

In other words,

Given that the functional is defined by

Thus, the modified objective functional is

For the fractional order in the sense of Caputo derivatives, the necessary conditions are given by

and , ,

and , . Here, the Lagrange multipliers (for ) are utilized.

Boundedness and existence of an

Theorem 6.1

For the , there exists an pair such that

if each of the below circumstances is true:

(1) The admissible control set is nonempty.

(2) is a closed, convex, and nonempty admissible control set.

(3) The right-hand side of the state system is bounded by a linear combination of the state and control variables.

(4) On , the integrand of the objective functional is convex.

(5) There exist constants such that

15

Proof

We must first demonstrate that the system’s solution exists before we can validate the aforementioned requirements. Considering that

Thus the following inequality results from ignoring the negative elements in the model

The vector form of the aforementioned system may be rewritten as follows:

This system is linear over a finite time horizon, its solutions are uniformly bounded, and its coefficients are bounded. As thus, the nonlinear system’s solutions are both bounded and exist¹¹. The first requirement is therefore met. The second criterion is obviously satisfied by the definition of . The control variables and indicate that the system is bilinear, and it can be stated as

where is the vector-valued function of . The solutions are bounded, and we have

where depends on the coefficients of the system and

We also observe that the system’s integrand, , is convex. Finally, we have

where the lower bounds of determine . Moreover, . Thus, we conclude that and are the variables.

Necessary conditions for s

Theorem 6.2

If be the s with corresponding states , , , and , then there exists adjoint variables , satisfying the following

1. Adjoint Equation:

   

2. Tranversality condition: ,

3. Optimality condition:

   

Furthermore, the control variables are defined using the Pontryagin Maximum Principle as

and

Numerical analysis for s

To numerically solve the optimality system described in Eq. (1) under the specified initial conditions, we employ the MATLAB fde12 toolbox for fractional optimal control problems (s) in the Caputo sense, and the ode45 toolbox for classical integer-order optimal control problems (s). The transversality conditions for the adjoint variables are imposed as for , ensuring that the necessary optimality conditions are satisfied at the final time.

The initial conditions for the state variables are set as:

while the simulation horizon is set to days to capture the complete dynamics of the dostarlimab-chemotherapy treatment and immune response.

The parameters used in the model are derived from Table 1, reflecting biologically realistic rates and interactions within the tumor-immune-drug dynamics. The fractional order is considered within the range , allowing us to investigate how varying the memory effect influences treatment outcomes.

The fde12 toolbox is specifically designed for solving s and provides accurate numerical approximations for systems involving fractional derivatives, thereby preserving the non-local memory characteristics intrinsic to the system under study. This is particularly important for modeling cancer chemo-immunotherapy, where the system’s response depends not only on the current state but also on its history.

For comparison, the ode45 toolbox, which implements an adaptive Runge–Kutta method, is utilized to solve the classical -based , serving as a benchmark to evaluate the impact of fractional-order modeling. The classical case captures immediate system interactions without memory, allowing us to contrast with fractional-order dynamics where .

By leveraging these numerical solvers, we can accurately solve the optimality system while ensuring that the transversality conditions are met at the final time . This approach enables a comprehensive investigation into the optimal control of dostarlimab-chemotherapy treatment, providing insights into the differences between classical and fractional-order models and their implications on treatment optimization and system stability.

Our objective is to investigate how cancer cells and immune system cells respond to changes in the fractional order within the proposed chemo-immunotherapy model. By adjusting the fractional order parameter , we aim to understand the dynamic behavior and interactions of these cells under the influence of dostarlimab and chemotherapy. Variations in the fractional order can reveal insights into the complex mechanisms governing tumor growth, immune activation, and the effectiveness of treatment strategies, capturing memory and hereditary effects that are not addressed in classical integer-order models. Furthermore, these dynamics are effectively illustrated through graphical representations. By plotting the time evolution of cancer cells, activated immune cells, and drug concentrations under different fractional orders and treatment strategies, we can observe clear trends and patterns that highlight the interplay between tumor suppression and immune response. This visual approach facilitates a systematic comparison of treatment impacts across varying fractional orders, ultimately contributing to the development of more effective and personalized therapeutic strategies for endometrial cancer management.

Applying without controls

Figure  6 illustrates the cell population dynamics in the absence of control interventions under varying fractional orders and the classical integer-order case. The results indicate a continuous increase in cancer cell populations in this uncontrolled scenario. Although immune cells, including T-cells and activated T-cells, also increase due to the immune system’s natural response to the presence of cancer cells, this immune activation alone is insufficient to suppress or eliminate tumor growth effectively. Consequently, the persistent rise in cancer cells underlines the need for therapeutic interventions, such as dostarlimab and chemotherapy, to control tumor proliferation and enhance immune system efficacy.

Fig. 6.

Cell populations across different values in the absence of controls.

Applying for dostarlimab

Figure 7 illustrates the cell population dynamics under dostarlimab therapy across various fractional orders and the classical integer-order case. Compared to the uncontrolled scenario, the application of dostarlimab results in a significant enhancement in the populations of T-cells and activated T-cells, reflecting a strengthened immune response. Furthermore, the therapy effectively reduces the cancer cell population, demonstrating the ability of dostarlimab to block immune checkpoint pathways and enable immune cells to target and eliminate cancer cells more efficiently. These findings highlight the potential of dostarlimab therapy in improving immune-mediated control of tumor growth.

Fig. 7.

Cell populations under dostarlimab treatment across different values.

Applying for chemotherapy

Figure 8 illustrates the cell population dynamics under chemotherapy treatment across various fractional orders as well as the classical integer-order case. The results indicate that chemotherapy does not lead to a significant increase in the populations of T-cells and activated T-cells, suggesting a limited enhancement of the immune response in this context. Additionally, the reduction in the cancer cell population is not substantial, reflecting the limited effectiveness of chemotherapy alone in reducing tumor burden within the modeled system. These findings suggest that while chemotherapy may contribute to cancer control, it is insufficient by itself to elicit a robust immune-mediated tumor reduction.

Fig. 8.

Cell populations under chemotherapy treatment across different values.

Applying for dostarlimab and chemo drugs

Figure 9 presents the populations of T-cells, activated T-cells, and cancer cells under the combined administration of dostarlimab and chemotherapy across various fractional orders . The results demonstrate that the populations of T-cells and activated T-cells increase, while the cancer cell population decreases significantly under the combined treatment. Furthermore, as the value of increases, the populations of T-cells and activated T-cells continue to rise, while the number of cancer cells correspondingly declines. These observations indicate that the combined therapy of dostarlimab and chemotherapy yields superior outcomes compared to either therapy alone, effectively enhancing the immune response and reducing the cancer cell burden. This highlights the potential of the combined treatment approach in achieving improved therapeutic efficacy in the chemo-immunotherapy management of EC.

Fig. 9.

Cell populations under chemotherapy treatment across various values.

Comparison between s and s:

Figure 10 compares the cell populations in the (integer-order) and (fractional-order, ) frameworks without any control interventions. The results indicate that although there is an increase in the population of activated T-cells due to the immune system’s natural response, this increase is insufficient to eliminate the continuously growing cancer cell population. Consequently, the cancer cells continue to proliferate despite the immune activation. This comparison highlights the limitations of the immune response in controlling cancer growth in the absence of treatment and underscores the necessity of therapeutic interventions to effectively target and reduce cancer cells for successful management of EC.

Fig. 10.

Cell populations without controls in and .

Figure 11 illustrates the cell populations under dostarlimab treatment in both and () frameworks. We observe a significant increase in the population of activated T-cells due to the enhancement of the immune response. In the case, the activated T-cells begin to increase rapidly after approximately 20 days, reaching a count of cells. In the case, a similar cell count is achieved after around 30 days, with both models maintaining consistent growth beyond this point. Regarding cancer cells, a clear reduction is observed in both and . In the model, cancer cell populations start to decrease gradually after around 10 days, with a significant decline observed after 40 days, reducing the population to approximately cells and maintaining this level without further proliferation. In the model, a similar trend is observed; however, the reduction in cancer cells occurs slightly later, and the final cancer cell count remains marginally higher compared to the case. Overall, dostarlimab treatment proves effective in reducing cancer cell populations and enhancing the immune response in both models, with the framework exhibiting a quicker and slightly more pronounced reduction in cancer cells compared to the framework.

Fig. 11.

Cell populations with dostarlimab in and .

Figure 12 illustrates the cell populations under chemotherapy treatment in both the and () frameworks. We observe a clear reduction in the cancer cell population, reflecting the direct cytotoxic effects of chemotherapy drugs, which effectively kill or reduce the size of cancer cells. Additionally, the number of activated T-cells increases during treatment; however, the extent of this increase varies depending on the parameter values used in the simulations. This observation indicates that while chemotherapy efficiently targets and reduces cancer cells in both and models, its impact on enhancing the immune response, as reflected by the increase in activated T-cells, is parameter-dependent. Overall, the results demonstrate that chemotherapy contributes to cancer cell reduction while modestly influencing the immune response within both modeling frameworks.

Fig. 12.

Cell populations with chemotherapy in and .

Figure 13 specifically compares the changes in immune cell and cancer cell populations under control strategies within the and frameworks. Figure 13a displays the growth of T-cells, Fig. 13b illustrates the increase in activated T-cells, and Fig. 13c shows the dynamics of cancer cell populations. We observe that in both modeling approaches, T-cells and activated T-cells increase over time, while cancer cells decrease under treatment application. These results indicate that the model demonstrates a more rapid and pronounced enhancement in immune cell populations and a faster reduction in cancer cell populations compared to the model. Figure 13d,e present the administration and management of treatment drugs, specifically dostarlimab and chemotherapy, over time in both frameworks, highlighting how these treatments influence cell populations during therapy.

Fig. 13.

Cell populations with controls in and .

Overall, the comparison across all cases shows that the model exhibits superior performance in effectively managing immune and cancer cell populations, providing clearer treatment dynamics and faster therapeutic outcomes. This suggests that, within the current parameter settings and treatment protocols, the framework may offer more accurate predictions and efficient outcomes in the design and application of cancer chemo-immunotherapy strategies.

Overall combined cases

Figure 14 provides a comprehensive overview of the findings discussed in this study. Figure 14a–c illustrate the populations of T-cells, activated T-cells, and cancer cells under various scenarios: without controls, with controls, with dostarlimab treatment alone, and with chemotherapy treatment alone. It is evident that dostarlimab treatment demonstrates superior efficacy in reducing cancer cell populations compared to chemotherapy. Additionally, the implementation of controlled conditions yields results that surpass those achieved with dostarlimab alone, highlighting the effectiveness of optimized control strategies in reducing cancer cells and enhancing immune cell responses.

Fig. 14.

Cell populations across various combinations.

Figure 14d,e further depict the comparison between the and frameworks under these treatment conditions. In both modeling approaches, there is a clear increase in activated T-cells accompanied by a corresponding decrease in cancer cells over time, reflecting the positive impact of treatment interventions. In summary, the application of controlled conditions yields the most favorable therapeutic outcomes. For effective management of EC, combined therapy under optimal control strategies is recommended to achieve a significant reduction in cancer cell populations while simultaneously enhancing the levels of activated T-cells. Furthermore, within the current parameter settings, the model demonstrates superior performance over the model across various scenarios, suggesting its potential for providing more accurate predictions and improved therapeutic outcomes in the context of cancer chemo-immunotherapy.

Figure 15 together with Table 2 illustrates the behavior of the total cost functional under different scenarios. Figure 15a shows that as the fractional order increases from 0.5 to 1, the cost functional also increases, with the integer-order case () yielding the highest cost. This trend is supported numerically in Table 2, where the cost functional rises from at to at . These results clearly demonstrate the advantage of fractional-order dynamics in achieving cost-effective control by reducing tumor burden with lower treatment cost compared to the classical integer-order model. Figure 15b further presents the convergence behavior of the cost functional across optimization iterations. Starting from an initially high value, decreases monotonically and stabilizes at a minimum, directly confirming that the proposed optimal control framework effectively minimizes the treatment cost while fulfilling therapeutic objectives.

Fig. 15.

Behavior of the total cost functional .

Table 2.

Cost functional for different fractional orders .

Fractional order	Cost functional
0.5
0.6
0.7
0.8
0.9
1.0

Observation of this model

• In single therapy, dostarlimab demonstrates superior efficacy compared to chemotherapy in reducing cancer cells.

• Combined therapy proves to be the most effective approach in our model, as it leads to optimal outcomes.

• modeling shows better performance than modeling in terms of faster tumor reduction.

In conclusion, leveraging dostarlimab in combination with chemotherapy under controlled conditions presents the best strategy for managing EC in this model. This approach not only maximizes therapeutic efficacy by reducing cancer cell proliferation but also enhances immune responses, thereby potentially improving patient outcomes.

Although the integer-order model () demonstrates a faster reduction in tumor cells compared to the fractional-order model, the cost functional analysis indicates that the fractional-order system provides a lower overall cost. This is because the cost functional penalizes both tumor load and drug administration. The model achieves stronger tumor reduction but at the expense of higher drug usage, which increases the total cost. In contrast, the fractional-order model achieves a more balanced trade-off, resulting in a minimized cost functional.

Conclusion

In this paper, a new mathematical model for EC progression was developed to analyze cancer cell proliferation under dostarlimab and chemotherapy while enhancing the activation of CD8+ T-cells. The model was formulated and analyzed under both and frameworks, with existence, uniqueness, and stability established using the Banach fixed-point theorem and Routh–Hurwitz criteria around six equilibrium points. Optimal control strategies were designed and implemented for both frameworks, with numerical simulations conducted in the MATLAB environment using the ode45 solver for models and the fde12 solver for models.

The numerical results demonstrate that dostarlimab monotherapy is more effective than chemotherapy alone in reducing cancer cell populations while increasing activated CD8+ T-cell levels. Combined therapy under optimal control yields the best outcomes, achieving the highest reduction in cancer cells and the greatest increase in activated T-cells. Furthermore, the -based optimal control framework shows faster convergence and greater tumor reduction, sustaining higher levels of activated T-cells compared to the framework. However, cost functional analysis indicates that the framework achieves a lower overall therapeutic cost by balancing tumor reduction with reduced drug usage, thereby highlighting a trade-off between treatment intensity and cost-effectiveness.

Overall, the study concludes that implementing suitable optimal control strategies particularly combined dostarlimab and chemotherapy can significantly enhance immune response, reduce cancer progression, and improve treatment outcomes in EC management. As a future direction, it is proposed to develop and analyze mathematical models that focus on the conversion of cancer cells to normal cells under suitable therapeutic interventions. This would involve formulating new state variables and control structures to represent reprogramming mechanisms, immune modulation, and targeted treatment strategies that mathematically characterize the pathways by which malignant cells may revert to normal phenotypes. Such an analysis would provide deeper insights into cancer eradication strategies beyond suppression, opening new avenues for advanced optimal control applications in personalized cancer treatment.

Abbreviations

Ordinary differential equation

Fractional differential equation

EC

Endometrial cancer

PD-1

Programmed death-1

Optimal control

Fractional optimal control problem

Author contributions

KR: Actualization, formal analysis, validation, investigation, initial draft and was a major contributor in writing the manuscript. MES: Formal analysis, validation, methodology, software, simulation and was a major contributor in writing the revision manuscript. All authors read and approved the final manuscript.

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.