According to the Routh-Hurwitz criterion, the roots of Eq. (14) will have negative real parts if the conditions , , and are satisfied.
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
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.
(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
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 exist. 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.
If be the s with corresponding states , , , and , then there exists adjoint variables , satisfying the following
Furthermore, the control variables are defined using the Pontryagin Maximum Principle as
and
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.