A fractal–fractional order modeling approach to understanding stem cell-chemotherapy combinations for cancer

Abstract

The main objective of this work is to study the mathematical model that combines stem cell therapy and chemotherapy for cancer cells. We study the model using the fractal fractional derivative with the Mittag-Leffler kernel. In the analytical part, we study the existence of the solution and its uniqueness, which was studied based on the fixed point theory. The equilibrium points were also studied and discussed after stem cell therapy, and the approximate solutions for the given model were obtained using the Adam Bashford method, which depends on interpolation with Lagrange polynomials. Finally, the model was simulated using the Mathematica software, and through the figures, we found that the components of the model approach the equilibrium point, which indicates the stability of the model at the equilibrium point. Also, the result of the numerical simulation and graphic for the concentration of cells over time indicate the effects of the therapies on the decay rate of tumor cells and the growth rate of effector cells to modify the cancer patient’s immune system. It is worth noting that we simulated all the model components with different fractional orders, confirming the effect of stem cell therapy and chemotherapy on the cells and the decay of cancer cells.

Introduction

Despite remarkable advancements in science and technology, cancer continues to pose a formidable challenge to human health. While conventional treatment approaches like surgery, radiation, chemotherapy, hormone therapy, and gene therapy have made progress, the core methods for tackling cancer have not transformed significantly. However, mathematical modeling offers a powerful tool to gain valuable insights into the intricate dynamics of cancer progression, immune responses, and potential therapeutic interventions^1,2. In^3,4, researchers have increasingly explored mathematical frameworks to enhance our understanding of cancer biology and develop more effective treatment strategies. Several studies have employed models to investigate the interactions between T-cells, particularly effector T-cells, and tumor cells, elucidating how these interactions influence cancer development; these quantitative models provide a comprehensive perspective that complements experimental research on the complex immune dynamics as in^5–8.

The authors in^9,10 clarified that stem cells have revolutionized the treatment of numerous diseases due to their remarkable ability to differentiate and proliferate. The researchers confirmed in^11,12 that studying stem cells mathematically has excellent importance for considering their therapeutic potential; mathematical modeling of stem cell growth dynamics can help medical scientists determine optimal screening and treatment protocols since stem cell therapies are costly. Quantitative analyses of stem cell behavior through modeling provide the key insights that complement experimental stem cell research, highlighting promising directions for medical applications of stem cells. Fractional calculus and its various applications have recently witnessed remarkable developments as many fractional differential operators have emerged, such as Riemann-Liouville,Caputo,Grunwald-Letnikov, Atangana-Baleanu, Conformable Fractional, and Opsi Hilfer. For more information about these operators, see^13–15. Furthermore, numerical and analytical methods have been formulated to meet the challenges of fractional calculus^16–19. The power of fractional or non-classical calculus has been effectively harnessed across a wide range of disciplines, including computer science, physics, biology, social sciences, economics, engineering, epidemiology, and beyond^20–26. A key advantage of employing fractional calculus with its various differential and integral operators lies in the ability to utilize fractional operators of arbitrary order, a flexibility not afforded by conventional derivatives of integer order. In recent years, there has been increased interest in mathematically modeling infectious diseases using fractional-order systems of equations. Fractional-order models are more accurate than integer-order models in representing memory effects over time and nonlocal phenomena in space. Overall, fractional calculus allows more precise modeling of infectious disease dynamics. A key benefit of fractional-order models is reducing errors from omitting parameters in natural processes see,^27–33. In³⁴, the authors apply time-fractional derivatives to model tumor growth dynamics involving cancer stem cells, generalizing an integer-order model previously analyzed. The aim is to investigate the sensitivity of the cancer growth paradox to differing mortality rates of non-stem cancer cells. It demonstrates that the paradox occurs across various parameter values in the proposed fractional-order model. A mathematical model of stem cells and chemotherapy for cancer treatment using fractional-order differential equations with the Caputo sense is used to represent the model by Özköse et al.³⁵.

In³⁴, the authors apply time-fractional derivatives to model tumor growth dynamics involving cancer stem cells, generalizing an integer-order model previously analyzed. The aim is to investigate the sensitivity of the cancer growth paradox to differing mortality rates of non-stem cancer cells. It demonstrates that the paradox occurs across various parameter values in the proposed fractional-order model. A mathematical model of stem cells and chemotherapy for cancer treatment using fractional-order differential equations with the Caputo sense is used to represent the model by Özköse et al.³⁵.

In 2017, Atangana³⁶ introduced a new class of operators called fractal–fractional operators (FFOs), combining concepts from fractional and fractal calculus. A recently developed fractal–fractional operator holds promise for addressing such mathematical challenges^37,38. Furthermore, the fractal–fractional formulation allows us to account for biological systems’ inherent complexities and memory effects, potentially leading to more accurate and realistic predictions. Over the past few years, a growing number of researchers have applied this fractal–fractional operator to model various real-world scenarios, demonstrating its potential for capturing complex system behaviors, see^39–46. Farman et al.⁴⁷ proposed a fractional order model to study the sustainable approach for cancer treatment with chemotherapy by utilizing a fractional operator. The model incorporated the fractal–fractional Mittag-Leffler operator with non-integer order derivatives to capture the complex dynamics and memory effects of cancer progression and treatment response. We developed a mathematical model of stem cells and chemotherapy for cancer treatment using a mathematical fractal–fractional model via the generalized Mittag-Leffler-type kernel. The model provides insights into how stem cells support effector cells that combat tumor cells, thereby boosting the immune system of cancer patients. At the same time, the model incorporates chemotherapy, killing infected cancer cells. By modeling the dynamics of stem cells, immune cells, and chemotherapy, it is possible to understand their complex interactions during cancer treatment. The quantitative framework enables predicting how stem cell therapies and chemotherapy could be optimized to improve outcomes for cancer patients. Overall, the mathematical modeling approach furthers comprehension of combination therapies involving immune system modulation through stem cells and direct tumor cell cytotoxicity from chemotherapy. Overall, this work highlights the potential of mathematical modeling in guiding the development of innovative and effective cancer treatment approaches, paving the way for future experimental studies and clinical applications.

This research paper consists of seven sections. The second section presents a set of fundamental definitions, while the third section discusses the mathematical model. The fourth section addresses the existence and unit of a solution to the mathematical model. The fifth section examines stability and then calculates the approximate solution in the sixth section. The seventh section is dedicated to discussing the results.

Preliminaries

Here are some fundamental ideas that will be used in this study:

Definition 1

⁴⁸ Let be continuous in (a, b) , if is fractal differentiable on (a, b) with order then, the fractal–fractional derivative of of order in the Riemann-Liouville sense using the generalized Mittag-Leffler kernel is given as

1

where ,  and  , and .

Definition 2

³⁶ If is continuous in an opened interval (a, b) , then the fractal–fractional integral of with order is defined as:

The formulation of the model

Before starting our study, we will take this model that the authors presented in¹, which includes three distinct patterns of cell division. Specifically,they assume that stem cells (denoted S) can either:

1. Undergo symmetric self-renewing divisions with probability , generating two new stem cell daughters.

2. Divide asymmetrically with probability , resulting in one stem cell and one differentiated progeny that lacks stemness.

3. Or differentiate symmetrically into two non-stem daughters with probability . We constrain the relative probabilities such that Furthermore, we define the rates of stem cell division and cell death as k and , respectively. Given that the stem cell pool can be replenished via both asymmetric and symmetric differentiating divisions, we derive an equation to capture the time evolution of stem cell numbers. This theoretical framework allows us to make quantitative predictions about how different division modalities influence net stem cell expansion or depletion. By fitting our model to experimental data, we can potentially infer the probabilities governing cell fate choices in vivo.

Initially, Alqudah et al.¹, developed a mathematical model to explore the potential of stem cell therapy in treating cancer. The model considers the interactions between three cell populations over time:

1. E(t) effector immune cells,

2. T(t) tumor cells,

3. S(t) therapeutic stem cells.

To capture the downstream impact of stem cell activity, they introduced an amplification factor A. Using these elements, we derived a simplified system of ordinary differential equations (ODEs) describing how effector cells, tumour cells, and amplified stem cell signals modulate one another. We can predict the efficacy of stem-cell-based approaches by analyzing this theoretical model under different conditions. Fitting this model to preclinical/clinical data could help refine the quantitative assumptions regarding stem cell amplification effects and immune system activation. Overall, this general modelling framework provides a means to theoretically investigate anti-cancer stem cell therapies before experimental testing.

2

In building upon the model presented in equation (2), they propose an extension incorporating combination therapy with both chemotherapy and stem cell treatment for cancer as:

3

where ,

subject to and , where U(t) is chemotherapy concentration drug. In our current work, we present the model (3) on a mathematical fractal–fractional model via the generalized Mittag-Leffler-type kernel because this operator gives accurate results for describing models and phenomena that describe real-world biological systems, such as memory effects, long-range interactions, and anomalous dynamics. So, the model (3) become as

4

The first equation in model (4) captures the interplay between the stem cell population (S) and chemotherapy drug concentration (U) over time . Furthermore, the model incorporates a decline in viable stem cells directly induced by the chemotherapeutic agent, occurring at rate .

The second equation represents the dynamics of effector cells. These cells have a constant source rate , which is the sum of a natural source rate and a rate derived from the transformation of stem cells. The equation accounts for the death rate of effector cells, their proliferation stimulated by stem cells, and their interaction with tumour cells and chemotherapy drugs at rates respectively.

The third equation shows the rate of cancer cell production minus the amount of decrease in it as a result of the interaction between the responding cells and the chemical drug with rates and respectively. The last equation represents the rate of change in the concentration of the chemical drug.

Existence and uniqueness of solution

Establishing the existence and uniqueness of solutions plays a crucial role in validating the outcomes of a study. It ensures the robustness and stability of the analyses performed and facilitates meaningful comparisons with other research endeavours. Furthermore, these results provide a solid theoretical foundation, guiding the applicability and broader generalization of the findings. Consequently, emphasizing the importance of existence and uniqueness within the study context helps reinforce the reliability and significance of the research conclusions drawn. In this section, we prove the existence of a solution for model (4) by using fixed point theory. Let be a non empty set in Banach space, s.t where with norm . For .

Now, we rewrite the R.H.S of model (4) as following

5

By Using definition 1

6

7

8

9

Then

10

become

11

So the fractal–fractional Atangana-Baleanu of (4) in the Riemann-Liouville sense is become as:

12

also, we can rewrite (12) in form of initial value problem as the following

13

where and

and

14

By definition 1 and equation (13) we get

15

By taking the fractal-fractionl Atangana- Baleanu integral on (15) we obtain

16

Now we extend the (16) as

17

We defined the oprator to derive a fixed-point problem as

18

Theorem 1

⁴⁹ Let E be a Banach space, a convex closed bounded set, an open set, and . Then for the continuous and compact map either:

1. s.t. or

2. and s.t.

Theorem 2

Let . Assume:

()

and s.t. and , .

()

19

with .

Then the solution of (4), and then for (13) exists on J.

Proof

Let and assume . Evidently, as K is continuous, so too is F. From (A1) we obtain

for . Then

20

Thus F is uniformly bounded on . Now, take s.t. and . and let , we find

Now when , we see that the expression

when . This gives the equicontinuity of F and, accordingly, the compactness of F on by the Arzela-Ascoli thoerem. Since all conditions of Theorem 1 are fulfilled on F, we have one of (1) or (2). From (), we set for some s.t.

Form (A1) and (20), we get

21

Assume that there are and such that. . Then, by the above inequality, we obtain

this contradiction with our hypthesis (A2), then F has a fixed point in P by Theorem 1. This demonstrates a solution to the model (4) exists.

Now, we prove the uniqueness of the solution of the model (4).

Lemma 1

Let and let , where .

Then the kernels and defined in (5) are satisfies the Lipschitz condition.

Proof

For , we take , and we get

we obtain

Then satisfy the Lipschitze condition w.r.t. S. Similarly we can prove that satisfy the Lipschitz condition w.r.t. U as the following:

we get

Now, we can prove that satisfy the Lipschitz condition w.r.t. E as the following:

we obtain

Finally, we can prove that satisfy the Lipschitz condition w.r.t. A as the following:

we obtain

Consequently, and are satisfying the Lipschitz condition.

Theorem 3

Assume that lemma 1be hold. Then the model (4) has exactly one solution if

22

Proof

Suppose that there is an other solution for the model (4), namely with the initial condition .

So by using the model (17)

Now, we calculate

We can say the above inequality holds if or by the same way we can prove that , and .

Then . Then the model (4) has a unique solution.

Equilibrium point

Stability theories play a significant role in understanding the long-term dynamics of our model. They provide a theoretical framework that predicts whether a system will converge to a steady state, oscillate, or diverge under small perturbations. We perform numerical simulations using the Adams-Bashforth method to validate these theoretical predictions, as detailed in Section 6. These simulations not only confirm the stability or instability of the equilibrium points but also provide insight into the behavior of the system under initial conditions. In this segment, we initiate our inquiries by conducting a local stability analysis within the framework of fractional-order derivatives. The conventional approach employed for examining local stability using the Jacobian matrix differs from that of the fractional context. The Matignon criterion⁵⁰ is utilized in fractional scenarios. The equilibrium point for the fractional differential equation is considered locally stable only when the following condition is met⁵⁰ :

23

The approach for local stability analysis in the fractional context is detailed in⁵¹. This method involves several steps, starting with calculating the Jacobian matrix using integer-order derivatives. Subsequently, the characteristic polynomial is determined through the standard process, and the eigenvalues are computed using algebraic methods. The final step involves testing all eigenvalues against the Matignon criterion (23), incorporating the fractional order. Initially, equilibrium points for the fractional (4) are derived by solving the equations.

24

Following the calculations, the obtained equilibrium points are as follows:

,

,

where

Now, we calculate the Jacobian matrix of the model (4) as the following :

25

To simplify, we directly utilize the values of the parameters as outlined in Table 1. So become , the Jacobian matrix at the point is given by

26

After performing the calculations, the characteristic polynomial of the Jacobian matrix is expressed in the following form:

27

The eigenvalues of the are given by We note that

This fulfills the Matignon criterion,then the equilibrium point is locally stable.

Table 1.

Parameters description and values.

Parameter	Description	Value/ Unit.day
	Stem cell therapy attrition rate	− 0.028251
	Effector cell turnover rate	0.171
	Attrition rate due to natural death of effecror cells	0.031
	Ability to destroy cancer cells	1
	Attrition rate of stem cells by Chemotherapy	11
r	Tumor growth rate	0.181
	Maximum effect of effector cells	0.12451
	Attrition rate due to cancer and Chemotherapy of effector cells	11
	FDeath rate due to Chemotherapy of cancer cells	0.91
	Attrition rate of cancer cells due to effector cells	0.91
	Decomposition rate of chemotherapy	6.41
V	The rate of the Chemotherapy entering the system at a specific schedule	11

, the Jacobian matrix at the point is given by

28

After performing the calculations, the characteristic polynomial of the Jacobian matrix is expressed in the following form:

29

The eigenvalues of the are given by We note that

This fulfills the Matignon criterion,then the equilibrium point is locally stable.

, the Jacobian matrix at the point is given by

30

After performing the calculations, the characteristic polynomial of the Jacobian matrix is expressed in the following form:

31

The eigenvalues of the are given by We note that

but

This not fulfills the Matignon criterion,then the equilibrium point is unstable.

Numerical treatment

In this part, we develop a numerical approach for the fractal–fractional model (4). We use the fractional Adams-Bashforth approach based on two-step Lagrange interpolation polynomials to do this. This method offers a potent combination of accuracy, efficiency, and user-friendliness. Notably, it exhibits rapid convergence to the precise solution, even when employing a substantial discretization step (h), thereby streamlining the overall^30,52. We redefine fractal fractional integral equations (17) at . In reality, these integrals (17) for are discretized as follows:

32

The formulation of the approximation of the integrals mentioned above is as follows:

After that we use two-step Lagrange interpolation polynomial to approximate on the interval with step size as the following:

33

34

35

36

where

,

.

Discussion and numerical findings

Here, the actual data for initial values , and the parameters in the table is utilized. Table 1 assumed by¹ to simulate and describe the dynamic behaviors of the fractals model (4).

The study investigates the dynamic behavior of the approximate solution of the model (4) under the influence of various fractional orders, specifically , through applying the Adams-Bashforth method. When we implemented the algorithms in Mathematica, we got the solution almost as shown in the figures above where: In Fig. 1, we notice that the stem cells decrease and return to the initial value very quickly in the first moments. They decrease until they reach equilibrium, indicating the stem cells’ depletion by chemotherapy. In contrast, in Fig. 2, we notice that the immune cells responding to the treatment decrease in the first days, and then these cells begin to increase until they reach the value 0.9, representing the equilibrium point for this component. In addition, Fig. 3 shows the behavior of cancer cells, as we notice a rapid decrease in cancer cells until they disappear, which is a positive indicator of the response to therapy by chemotherapy and stem cells. In Fig. 4, we notice at the beginning the fluctuation of the concentration of chemotherapy so that it increases very quickly inside the cancer cells, leading to stability. Furthermore, in Figs. 1, 2, 3, and 4, we notice that higher values of influence how quickly the system approaches equilibrium, representing a more realistic memory.

Fig. 1.

This figure shows the approximate solution of function S(t) under the influence of different values of the fractional orders .

Fig. 2.

This figure shows the approximate solution of function E(t) under the influence of different values of the fractional orders .

Fig. 3.

This figure shows the approximate solution of function T(t) under the influence of different values of the fractional orders .

Fig. 4.

This figure shows the approximate solution of function U(t) under the influence of different values of the fractional orders .

The Fig. 1 highlights the pivotal role of stem cells in fortifying the immune system against cancer by facilitating the regeneration of healthy tissues. Additionally, the effectiveness of chemotherapy in directly targeting and eradicating cancer is visually evident in Figs. 3 and 4. It is noteworthy that chemotherapy, while instrumental in cancer eradication, initially exerts a suppressive effect on the immune system. However, immune cell counts gradually rebound, as depicted in Fig. 2. The model emphasizes the importance of achieving a delicate balance between stem cell-based therapies and chemotherapy to combat cancer while concurrently supporting immune functionality effectively. These simulations confirm the stabilization or stabilization of equilibrium points and provide insight into the system’s behavior under initial conditions.

The visual representations reveal how the stem cells aid in replenishing the effector cell population, enabling a sustained immune response against the tumour cells. Concurrently, the chemotherapy agent exerts cytotoxic effects on the tumour cells, potentially reducing their population. However, as mentioned earlier, chemotherapy can initially suppress the immune system, leading to a temporary decrease in effector cell numbers.

The study’s findings underscore the importance of carefully balancing the stem cell-based immunotherapy and chemotherapy components to maximize their synergistic effects. By leveraging stem cells’ regenerative capabilities and chemotherapy’s direct tumor-killing properties, the model suggests an optimized treatment strategy can be developed, potentially leading to improved cancer management and patient outcomes.

Conclusion

Mathematical modelling is vital in understanding, anticipating, and mitigating the devastating societal impacts of infectious diseases throughout history. Fractional-order derivatives have emerged as a more prevalent and insightful framework for capturing the intricate dynamics of real-world phenomena compared to classical integer-order models. We have presented an integer-order and fractional-order mathematical model describing the interactions between stem cells, immune effector cells, tumour cells, and chemotherapy over time. The theoretical model provides insights into how the introduction of stem cells could potentially boost anti-cancer immune responses while chemotherapy acts to kill tumour cells directly. By fitting the model to experimental data in future work, the contributions of stem cell effects on immune activation could be parsed out from the direct tumoricidal impacts of chemotherapy. Additionally, the fractional-order model enables more precise dynamics capturing than traditional integer-order modelling. Our stability analysis of the fractional system revealed that the equilibrium points are locally stable under the Matignon criterion. To facilitate numerical simulations and explore the model’s dynamics, we employed the Adams-Bashforth method, which is based on two-step Lagrange polynomials to approximate the solutions of the system. The numerical simulation was done using Mathematica. Overall, the quantitative modelling approach presented here furthers understanding of the complex interplay between combination stem cell and chemotherapy treatments, highlighting promising new therapeutic avenues for combating cancer. In the future, continued theoretical developments combined with experimental validation will aid the development of potent immuno-oncology therapies harnessing the power of stem cells. A key avenue for future research involves investigating the controllability of the cancer model, specifically by incorporating the effects of chemotherapy treatment. This would enable the design and analysis of feedback control strategies, where the therapeutic input (chemotherapy dosing) is dynamically adjusted based on the observed response of the system. Also, in the future, it would be interesting to determine whether manipulating external signals can modulate and , thereby providing a means to control stem cell self-renewal versus differentiation programmatically. . The research results emphasize the critical need to carefully balance stem cell-based immunotherapy and chemotherapy components to optimize their combined effects. The study demonstrates that by harnessing the regenerative abilities of stem cells in conjunction with the tumor-targeting efficacy of chemotherapy, an optimized treatment strategy can be achieved, potentially improving cancer management and patient outcomes.

Author contributions

E.S. and B.S.:Conceptualized the study, developed the mathematical model, and contributed to the theoretical analysis. A.H.: Performed numerical simulations and stability analysis. H.E. and A.K.: Interpreted the results and provided guidance for model validation.

Funding

No funding.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Competing interests

The authors declare no competing interests.