Fractal-fractional modeling of chronic myelogenous leukemia immune dynamics using Laguerre wavelets method

Abstract

This investigation proposes and analyzes a novel mathematical framework for understanding the complex tumor-immune system dynamics in Chronic Myelogenous Leukemia (CML). We adapt the established Moore and Li three-population model (naive T-cells, effector T-cells, and CML cells) by reformulating it using fractal-fractional differential operators of the Atangana–Baleanu type. This advanced approach offers a significant enhancement over standard models, as it simultaneously captures the fractal geometry of tumor structures and the non-local memory effects inherent in biological immune responses. We first establish the theoretical foundations, providing rigorous proofs for the existence, uniqueness, positivity, and boundedness of solutions, ensuring biological viability. Stability analysis of the tumor-present equilibrium is then conducted. For numerical simulation, a robust and highly accurate scheme based on the Laguerre Wavelets Method (LWM) is implemented with its efficacy demonstrated via comparison and convergence analysis. Numerical experiments visualize the system’s behavior under various fractal dimensions and fractional orders, revealing their profound impact on tumor progression. A sensitivity analysis identifies key parameters for potential therapeutic intervention. This work provides a more flexible and realistic mathematical tool for predictive oncology.

Introduction

Chronic Myelogenous Leukemia (CML) is a myeloproliferative neoplasm characterized by the excessive and unregulated growth of myeloid cells in the bone marrow and their accumulation in the blood^1,2. The hallmark of CML is the Philadelphia chromosome, a cytogenetic abnormality resulting from a translocation that produces the BCR-ABL fusion protein, a constitutively active tyrosine kinase that drives uncontrolled cell proliferation³. Understanding the intricate dynamics between the burgeoning CML cell population and the body’s immune response is paramount for designing effective, long-term treatment strategies. Mathematical modeling has emerged as an indispensable tool in quantitative oncology, providing a framework to dissect complex biological interactions and predict therapeutic outcomes^4,5. One of the foundational models in this area was proposed by Moore and Li⁶, which describes the competitive and predatory interactions between CML cells, naive T-cells (), and effector T-cells (). This model, formulated as a system of ordinary differential equations (ODEs), laid the groundwork for analyzing chemo-immunotherapy strategies. However, classical ODE models operate under the Markovian assumption, implying that the future state of the system depends only on its present state. This fails to capture the significant memory and hereditary properties inherent in complex biological systems^7,8. Immune cell activation, differentiation, and the long-term effects of cellular signaling are non-local in time. To address this, fractional calculus, which employs non-integer order derivatives, was introduced. Fractional-order models (FOMs) have demonstrated superior performance in describing biological phenomena, including tumor-immune dynamics, by incorporating memory effects^9,10. Despite this advancement, standard fractional operators still do not account for the intrinsic fractal nature of many biological structures. Tumor growth is not homogenous; tumors exhibit complex, irregular, and fractal-like geometries^11,12. These fractal structures influence nutrient diffusion, cell migration, and drug delivery. To bridge this gap, a new class of operators, known as fractal–fractional (FF) derivatives, was recently introduced^13,14. These operators unify the concepts of fractal dimension (describing geometric complexity) and fractional order (describing memory) into a single mathematical expression. The Atangana–Baleanu (AB) operator, in particular, has gained prominence due to its non-singular Mittag-Leffler kernel, which adeptly captures both fractal and memory-driven processes¹⁵. The application of FF operators to biological systems is a burgeoning field of research. Recent studies have demonstrated their utility in epidemiology^16,17 and other complex dynamical systems¹⁸, showing increased flexibility and a better fit to real-world data compared to their integer-order or standard fractional counterparts. Several studies have explored CML and other tumor models using standard fractional derivatives^19–23, but the application of fractal–fractional calculus to the specific CML Immune interaction model of Moore and Li⁶ remains an open avenue of inquiry. In recent years, fractal–fractional and fractional-order derivatives have been widely applied to biological and epidemiological systems, particularly in tumor and immune modeling. For instance, models employing the Mittag-Leffler kernel have been used to investigate immune system–cancer interactions²⁴, demonstrating improved capability in capturing memory-dependent immune responses²⁵. Fractal–fractional operators have also been successfully applied to infectious disease dynamics using real clinical data²⁶, confirming their flexibility and enhanced fitting performance, A study on brain tumor dynamics in two-dimensional irregular domain with variable-order time-fractional derivative²⁷. Furthermore, several studies have analyzed tumor–immune interaction models using Caputo and other fractional derivatives²⁸, highlighting the importance of non-local effects in cancer progression. These contributions collectively indicate that fractional and fractal–fractional frameworks provide more realistic descriptions compared to classical integer-order models. However, to the best of our knowledge, the Moore and Li CML Immune model has not yet been formulated and rigorously analyzed within the Atangana–Baleanu fractal–fractional framework, which motivates the present investigation. A significant challenge in using FF operators is the development of reliable and efficient numerical methods. Standard finite difference methods can be computationally intensive and suffer from convergence issues. Wavelet-based numerical methods have proven to be powerful tools for solving complex differential equations²⁹. Among these, the Laguerre wavelets method (LWM) is particularly attractive due to its orthogonality, compact support, and ability to approximate functions on a semi-infinite interval, making it well-suited for time-domain simulations³⁰. Although several studies have applied standard fractional derivatives to tumor–immune interaction models, the combined effect of non-local memory and fractal tumor geometry has not yet been incorporated into the Moore and Li CML framework⁶. Classical and fractional models account for hereditary immune effects but assume a Euclidean interaction space, thereby neglecting the structural complexity of tumor microenvironments. The primary motivation of this study is to develop a unified mathematical framework that simultaneously captures memory effects and geometric complexity using the Atangana–Baleanu fractal–fractional (ABC) operator. In this context, the fractional order models the hereditary behavior of immune response, while the fractal dimension characterizes the irregular structure of the tumor microenvironment. The main novelty of this work lies in the reformulation of the classical CML–immune model within a fractal–fractional setting, accompanied by rigorous qualitative analysis (existence, uniqueness, positivity, boundedness, and global stability), the development of a tailored Laguerre Wavelets numerical scheme for the FF system, and a detailed investigation of how and jointly influence long-term tumor dynamics.

This manuscript is structured as follows: In “Model formulation” section, we formulate the CML Immune interaction model using the Atangana–Baleanu fractal–fractional (ABC) operator, explain the biological significance of each term, and present the model parameters. “Preliminaries” section provides the essential mathematical definitions and theorems from fractal–fractional calculus. In “Qualitative analysis” section, we conduct a rigorous qualitative analysis, proving the existence, uniqueness, positivity, and boundedness of the solutions. “ Stability analysis” section is dedicated to the stability analysis of the system’s equilibrium points. In “Numerical solution by Laguerre wavelets method (LWM)” section, we develop the numerical scheme based on the Laguerre wavelets method (LWM) to solve the FF-CML model. An algorithm for the LWM is presented in “Algorithm of the suggested method” section. “Method validation and comparison” section provides a numerical comparison of LWM with other methods and an analysis of its convergence rate. “Numerical simulation and discussion” section presents and discusses the numerical simulations, including the effects of varying fractal and fractional orders, sensitivity analysis, and 3D phase portraits. Finally, “Conclusion and future scope” section concludes the overall findings and directions for future research. A summary of key findings is tabulated in “Summary of key findings” section.

Model formulation

The foundation of our investigation is the established mathematical model of Chronic Myelogenous Leukemia (CML) and T-cell interaction proposed by Moore and Li⁶. This model captures the predator-prey-like dynamics of the immune system and the tumor. We first provide a detailed deconstruction of this classical integer-order model to elucidate the biological mechanisms it represents.

The model tracks the population densities of three distinct cell types over time t: the naive T-cells , the activated effector T-cells , and the CML cancer cells C(t).

Classical model derivation

The classical model is formulated as a system of coupled, non-linear ordinary differential equations (ODEs) that describe the rate of change for each population. The control functions and represent the efficacy of targeted therapy (e.g., imatinib) and broad cytotoxic chemotherapy, respectively.

1

2

3

Naive T-cell Dynamics(): Equation (1) governs the population. The term represents the constant source of new naive T-cells produced by the thymus, entering the compartment as a constant influx. The term models the natural death of naive T-cells, which occurs at a constant per-capita rate . This death rate is scaled by , reflecting the fact that non-specific chemotherapy is cytotoxic to healthy immune cells as well as cancer cells. The final term, , describes the primary activation process. Naive T-cells are activated into effector T-cells upon encountering CML antigen. This process is modeled to be proportional to the population itself and the availability of antigen. The antigen signal is represented by a Michaelis-Menten (or Holling Type II) function . This form is chosen because the activation response saturates: as the CML population C becomes very large, the activation stimulus reaches its maximum. The parameter is the maximum activation rate, and is the half-saturation constant, representing the CML population at which the activation response is half-maximal.

Effector T-cell Dynamics (): Equation (2) describes the population, which is the activated immune defense. The first term, , is the direct influx of new cells from the activated pool. This term is directly coupled to the loss term in the equation, where is a proliferation constant representing the number of cells produced per activated cell. The second term, , accounts for the self-proliferation (clonal expansion) of the existing population. This process is also stimulated by the CML antigen, following the same saturated Michaelis-Menten kinetics. The third term, , is the natural death of effector T-cells at rate , which is also unfortunately enhanced by cytotoxic chemotherapy . The final term, , represents the inactivation or exhaustion of cells due to direct interaction with CML cells. This is modeled using the law of mass-action, where the rate of inactivation is proportional to the frequency of encounters between and C cells, governed by the rate .

CML Cell Dynamics (C): Equation (3) defines the growth and decay of the tumor population. The first term, , models the intrinsic growth of the CML population. This is a Gompertz growth function, which is chosen over simple logistic growth because it better reflects tumor dynamics: growth is initially rapid but slows as the tumor size C approaches its carrying capacity . The parameter is the intrinsic growth rate. This entire term is multiplied by , where is the efficacy of targeted therapy (like imatinib) that specifically inhibits the BCR-ABL protein, thereby directly suppressing the tumor’s characteristic growth. The second term, , models the death of CML cells induced by broad cytotoxic chemotherapy. This death rate is scaled by the chemo dosage . The final and most crucial interaction term is . This represents the killing (predation) of CML cells by the activated effector T-cells. This interaction is also modeled using the law of mass-action, proportional to the populations of both predator () and prey (C), with being the rate of successful killing.

Model assumptions and parameters

The classical model assumes a spatially homogenous (well-mixed) environment where all cells can interact. The parameters used in this study, adopted from the original work of Moore and Li⁶, are based on clinical and experimental data for two different patient profiles (Patient A and Patient B). These values are detailed in Table 1. For the analysis of the system’s intrinsic dynamics, we typically consider the baseline case where (no targeted therapy) and (baseline chemotherapy or natural clearance).

Table 1.

Model parameter descriptions and their values for two distinct patient profiles, as established in⁶.

Parameter	Description	Patient A	Patient B
	Source rate of cells	0.29	0.071
	Natural death rate of	0.35	0.05
	Natural death rate of	0.40	0.12
	Chemo death rate of C	0.012	0.68
	Activation rate of	0.066	0.063
	Half-saturation of C	140	43
	prolif. (from )	0.39	0.56
	self-proliferation	0.65	0.53
	Carrying capacity of C	160000	190000
	Growth rate of C	0.011	0.23
	inactivation rate by C	0.079	0.077
	C killing rate by	0.058	0.047

Model interaction diagram

The complex interactions between the three cell populations, as described by the model equations, are visualized in the schematic diagram presented in Fig. 1.

Fig. 1.

A schematic diagram visualizing the interactions within the CML Immune model. Blue, green, and red nodes represent Naive T-cells, Effector T-cells, and CML cells, respectively. Solid lines indicate cell flow, transformation, or killing. Dashed lines represent stimulation or activation pathways.

Fractal–fractional model formulation

The classical ODE model (2.1) is inherently limited. It assumes that the rate of change at time t depends only on the state of the system at that exact instant. This neglects the well-established memory and hereditary properties of biological systems, where past events and cellular lineage influence current dynamics. Furthermore, the classical model assumes interactions occur in a uniform, Euclidean space. This fails to capture the complex, irregular, and often fractal-like geometry of tumor microenvironments, which governs diffusion and cell-cell interaction.

To overcome these limitations, we reformulate the model using fractal–fractional (FF) differential operators. We employ the Atangana–Baleanu derivative in the Caputo sense (ABC), denoted . This operator is defined by two parameters: the fractional order , which captures the non-local memory effects (where recovers the classical model), and the fractal dimension , which captures the geometric complexity of the underlying medium (where represents a standard, non-fractal medium).

By replacing the integer-order derivative with the FF operator, we obtain the new Fractal–Fractional CML (FF-CML) model:

4

5

6

This new system (2.2) retains the exact same biological interaction terms (the right-hand side) as the classical model. However, its dynamic evolution (the left-hand side) is now governed by fractal and fractional principles, providing a more flexible and potentially more realistic framework for describing the complex CML Immune battle.

Dimensional consistency remark

We note that the introduction of the Atangana–Baleanu fractal–fractional time operator does not alter the biological meaning of the state variables , , and C(t), which remain cell populations. To ensure dimensional consistency, we interpret the operator as a generalized time derivative of effective dimension (consistent with standard fractional modeling). Consequently, all rate-type parameters appearing on the right-hand side (e.g., , , , , ) are understood as generalized rates with units (reducing to the classical when ). The fractal dimension affects the temporal kernel weighting but does not change the physical units of populations or carrying capacity; in particular, retains units of population.

Preliminaries

We recall the fundamental definitions of the Atangana–Baleanu (AB) fractal–fractional operators based on the Mittag-Leffler kernel¹³.

Definition 3.1

(AB Fractional Derivative) Let , , . The Atangana–Baleanu derivative in the Caputo sense (ABC) is defined as:

where is a normalization function satisfying , and is the Mittag-Leffler function defined by .

Definition 3.2

(AB Fractional Integral) The corresponding AB fractional integral is defined as:

This integral and derivative pair satisfy the fundamental theorem .

Definition 3.3

(Fractal–Fractional Derivative) Let f(t) be a continuous function. The fractal–fractional derivative of f(t) of order and fractal dimension in the ABC sense are defined as¹⁴:

where . If f is differentiable, then . Thus, the operator becomes:

Definition 3.4

(Fractal–Fractional Integral) The corresponding fractal–fractional integral is¹⁴:

Applying the integral to the derivative gives the following crucial relation for converting the system (2.2) into an integral form:

This allows us to write a general FF system as a Volterra integral equation:

7

Qualitative analysis

We analyze the mathematical properties of the FF-CML model (2.2). For brevity, let and the vector field , where:

8

9

10

We assume and are constant controls and .

Existence and uniqueness

We use the fixed-point theorem approach on the integral equation (7). First, we must show that the kernels satisfy the Lipschitz condition.

Lemma 4.1

(Lipschitz Condition) The kernels for satisfy the Lipschitz condition with respect to Y in a positively invariant domain , provided C(t) is bounded away from zero.

Proof

Let and be two states in . For :

Since are bounded in (as shown later) by some , and is Lipschitz with constant , we have:

Similar, albeit more lengthy, bounds can be derived for and .

The term is Lipschitz as long as , as its derivative is , which is bounded.

Thus, is Lipschitz with constant .

Theorem 4.1

(Existence and Uniqueness) For a given , the fractal–fractional system (2.2) has a unique solution , provided the kernels satisfy the Lipschitz condition (Lemma 4.1).

Proof

We use the Banach Contraction Principle. Let be the space of continuous functions on [0, T] with the sup-norm. Define a mapping based on the integral equation (7):

Let . We show is a contraction.

Using the Lipschitz constant and :

So, .

By choosing a sufficiently small time such that , the mapping is a contraction on [0, T]. By the Banach Contraction Principle, a unique fixed point (solution) Y(t) exists on [0, T]. This solution can be extended to any arbitrary interval by partitioning the interval.

Positivity and boundedness

For the model to be biologically meaningful, solutions must remain non-negative and bounded.

Theorem 4.2

(Positivity) Let the initial conditions be . Then the solution Y(t) of the system (2.2) remains in for all .

Proof

To justify the positivity argument, we recall that the Atangana–Baleanu kernel is continuous and non-singular on [0, T]. Consequently, the operator preserves continuity of solutions. In particular, by the standard comparison lemma for non-singular AB operators (see, e.g., Tavazoei and Haeri³¹), if at a boundary point and , then y(t) cannot cross the zero level toward negative values.

Moreover, the integral representation (3.1) together with the continuity of the Mittag–Leffler kernel ensures that solutions depend continuously on initial data and remain non-negative whenever the right-hand side is non-negative at the boundary.

We examine the behavior of each variable at the boundary of the positive octant.

1. Suppose there is a first time such that and for . At , we have:

   

   The fractal–fractional derivative with a non-singular kernel (like AB) implies that the function cannot reach this point from a positive value if its derivative is strictly positive. This is a contradiction, as the function must be non-increasing to hit zero. Thus, for all .
2. Suppose there is a first time such that . At , we have:

   

   Since (from part 1) and assuming , the derivative is . By the same logic, cannot cross into the negative region. If , the derivative is 0, but this holds for all , so remains 0.
3. Suppose there is a first time such that . The C equation (6) contains . As , . At , we have:

   

   This derivative of 0 implies that if C(t) reaches 0, it can stay at 0. It does not become negative. Thus, for all .

Therefore, the domain is a positively invariant region for the system.

Theorem 4.3

(Boundedness) All solutions Y(t) of system (2.2) that start in are uniformly bounded.

Proof

The Gompertz term for C(t) provides a natural bound. From (2.2c), if , then .

If , the derivative is strictly negative. This implies C(t) cannot grow indefinitely and is bounded, i.e., for some (which is if , or otherwise, before decaying). Now consider a total population .

Assuming (baseline chemotherapy) and (assuming ),

Let . If , then

By the properties of the AB derivative and the generalized mean value theorem (or a fractional comparison lemma³²), this implies that N(t) is bounded above by as . Therefore, , , and C(t) are all uniformly bounded.

Stability analysis

This section is dedicated to a rigorous investigation of the dynamic behavior and stability of the CML Immune system. To understand the model’s intrinsic properties, we analyze the system in a baseline treatment-free scenario. This is achieved by setting the targeted therapy control and the cytotoxic chemotherapy control . The fractal–fractional model under these assumptions, which we will analyze, is given by:

11

12

13

The stability of this non-linear system is determined by the behavior of solutions near its equilibrium points, or steady states.

Local stability analysis

Local stability describes the system’s response to small perturbations near an equilibrium. We first identify the equilibrium points by setting the derivatives in system (11) to zero.

Equilibrium points

An equilibrium point is a solution to the algebraic system .

1. Tumor-Free Equilibrium (TFE): We seek a steady state where the CML population is absent, . Setting in Eq. (13) gives , which is consistent. Setting in Eq. (12) gives:

Setting and in Eq. (11) gives:

Thus, a single Tumor-Free Equilibrium (TFE) exists at .

2. Tumor-Present (Endemic) Equilibrium (TPE): We now seek a steady state where the CML population persists, . From Eq. (13), assuming :

For to be positive, we require . From Eq. (11):

This expression is always positive for . Finally, we substitute the expressions for and into the remaining equation, Eq. (12):

14

where . The TPE, , exists if and only if there is a positive solution to the highly non-linear transcendental equation (14) that also satisfies the condition . Such a solution must be found numerically.

On the basic reproduction number ()

In many epidemiological models, the stability of the disease-free equilibrium is determined by the basic reproduction number, .

However, the concept of is not applicable to this model. The reason lies in the Gompertz growth term, . The per-capita growth rate of the tumor, , approaches as .

This means that even an infinitesimally small tumor population has an infinite growth rate. Consequently, the TFE is not a hyperbolic equilibrium (its Jacobian is undefined) and is always unstable to the introduction of any cancer cells ().

Therefore, the only meaningful dynamic behavior is the convergence of the system to the TPE, . Our analysis will focus on the stability of this endemic state.

Conceptual distinction from epidemiological models.

In classical epidemiological fractional models, the basic reproduction number is well-defined because the disease-free equilibrium is hyperbolic and the infection dynamics are locally linearizable around that state. In such systems, represents the expected number of secondary infections generated by a single infected individual, and it determines whether the disease-free equilibrium is stable or unstable.

In contrast, the present tumor model contains a Gompertz growth term , whose per-capita growth rate diverges as . Consequently, the tumor-free equilibrium is non-hyperbolic and cannot be analyzed through linearization. Therefore, unlike standard epidemiological fractional systems, no threshold quantity analogous to can be meaningfully defined for this CML framework.

Stability of the tumor-present equilibrium (TPE)

The stability of is determined by the eigenvalues of the Jacobian matrix J evaluated at . For fractal–fractional systems, the following theorem holds:

Theorem 5.1

(Local Stability of FF Systems³³) An equilibrium point of a non-linear fractal–fractional system with is locally asymptotically stable if and only if all eigenvalues of the Jacobian matrix satisfy the condition .

This crucial theorem implies that the local stability conditions for this FF system are identical to its integer-order counterpart.

The Jacobian matrix for system (5.1) is:

Let and its derivative . The components of the Jacobian at are:

From the equilibrium condition for , the term in parenthesis is zero. Thus, .

The stability of depends on the roots of the characteristic equation , which expands to the cubic polynomial:

Evaluation at the tumor-present equilibrium and handling of transcendental terms.

The Jacobian is evaluated explicitly at the tumor-present equilibrium , where is obtained numerically from the transcendental equilibrium equation (14). Once is known, the remaining equilibrium components follow directly from

and the Jacobian entries become fully computable scalars involving and . Moreover, the Gompertz contribution in is handled by direct differentiation:

so that

where we used the equilibrium identity . Finally, the fractal–fractional structure affects the time-evolution operator but does not modify the linearization of the vector field; hence the characteristic polynomial in below equation is formed from exactly as in the integer-order case.

15

where:

Theorem 5.2

(Routh-Hurwitz Stability Criterion³⁴) Based on Theorem 5.1, the equilibrium is locally asymptotically stable if and only if the coefficients of its characteristic polynomial satisfy the following conditions:

1. 

2. 

3. 

Due to the complexity of the expressions, these conditions are analytically intractable. However, they are readily verified numerically for a given set of parameters. We numerically compute the equilibrium (using a root-finding algorithm on Eq. 14) and then evaluate the Jacobian and Routh-Hurwitz coefficients.

Numerical Verification:

• For Patient A: The TPE is . The coefficients are , , and . The criteria are: All conditions are met. The eigenvalues are , all of which have negative real parts.

• For Patient B: The TPE is . The coefficients are , , and . The criteria are: All conditions are met. The eigenvalues are , all of which have negative real parts.

Conclusion: For both patient profiles, the Tumor-Present Equilibrium () is locally asymptotically stable. This implies that after a small perturbation, the CML Immune dynamic will return to this endemic state of coexistence.

Global stability analysis

We now investigate the global stability of the TPE, , to determine if all biologically feasible solutions (i.e., starting in ) converge to this single equilibrium. We employ the Lyapunov direct method, adapted for fractal–fractional systems.

Theorem 5.3

(Fractional Lyapunov Stability³¹) Let be an equilibrium point for the system . If there exists a continuously differentiable, positive-definite Lyapunov function V(Y(t)) such that its fractal–fractional derivative satisfies (negative semi-definite) along the system’s trajectories, then is stable.

Theorem 5.4

(Global Stability of TPE) The Tumor-Present Equilibrium of the system (5.1) is globally asymptotically stable in the positive octant .

Proof

The proof is constructed by defining a suitable Lyapunov function and applying the fractional extension of LaSalle’s Invariance Principle. Consider the following Volterra-type Lyapunov candidate function, which is positive definite with respect to :

16

where are positive constants to be determined. The function is non-negative for all and is zero only at .

We utilize the property for fractional derivatives³¹ that . Applying this to V(t):

Substituting the system equations (11):

We now use the equilibrium conditions:

1. 

2. 

3. 

Substituting these into the derivative expression yields an analytically complex form due to the non-linear g(C) and terms. For example, the term becomes:

Using the equilibrium relations

the first and third brackets can be rewritten explicitly as

Hence,

since for . Because g(C) is increasing, the remaining mixed terms can be bounded by choosing suitable positive constants . Therefore,

with equality only at , , .

We now apply the fractional LaSalle’s Invariance Principle³¹. This principle states that all solutions Y(t) converge to the largest invariant subset M of the set .

The derivative is zero only if , , and . This is because the non-linear terms do not allow for the cancellations required for unless each component is at its equilibrium value. Therefore, the largest invariant subset M of E is the singleton .

By LaSalle’s Invariance Principle, every solution starting in the positive octant must converge to the TPE .

Final remark: The combination of local and global stability analysis confirms that, in the absence of targeted therapy, the CML Immune system will not eradicate the tumor. Instead, it will always evolve toward a unique, stable state where the tumor and the immune system coexist. The size of this tumor, , is determined by the system’s parameters.

Numerical solution by Laguerre wavelets method (LWM)

We implement the Laguerre wavelets method (LWM) to solve the FF-CML system (2.2).

Laguerre wavelets

Laguerre polynomials of degree m are defined on and are orthogonal with respect to the weight function :

Laguerre wavelets are defined on as:

where k is the resolution level, is the translation, is the degree of the polynomial, and are the normalized Laguerre polynomials.

Dilation and translation parameters

The Laguerre wavelets are constructed using a dilation parameter and a translation parameter , where denotes the resolution level and .

Thus, the wavelet basis functions can be interpreted in the standard form

where are the normalized Laguerre polynomials. The parameter a controls the time-scale (refinement), while b shifts the support of each wavelet.

Any function (after scaling) can be approximated by a truncated series:

where is a vector of coefficients and is the vector of Laguerre wavelets.

Operational matrix of FF integration

The core of the method is the operational matrix of fractal–fractional integration. We want to find a matrix such that:

The fractal–fractional integral of a wavelet vector is:

Each term in the resulting vector is projected back onto the Laguerre wavelet basis by finding its coefficients :

This yields the operational matrix .

Numerical discretization procedure

The time interval is first scaled to [0, 1] for computational convenience.

Collocation points are chosen as

or alternatively as the shifted roots of the Laguerre polynomial of degree M.

At each collocation point, the residual equations are enforced, transforming the FF system into a finite system of nonlinear algebraic equations.

Methodology

We apply the LWM to our system. Let the state variables be approximated as:

We assume the highest derivative is approximated by the wavelet series, as it is often smoother:

Applying the integral operator gives:

Substituting the wavelet approximations:

We can isolate the coefficients for :

where is the wavelet expansion of the constant . We now substitute these approximations into the original system (4). For the first equation:

The non-linear terms like are handled by expanding the product of wavelet series, which results in a known algebraic operation on the coefficient vectors. We discretize this equation by collocating at points (e.g., roots of Laguerre polynomials) in the interval . This transforms the system of FFDEs into a large system of non-linear algebraic equations for the unknown coefficients in . This algebraic system is then solved using an iterative method, such as Newton’s method. Once the vectors are found, the solutions are reconstructed.

Illustrative example (operational matrix construction)

For example, when and , the wavelet vector is

Applying the fractal–fractional integral operator to each basis function and projecting onto the same basis yields a matrix with entries

This matrix approximates the FF integral through

and larger values of k and M improve spectral accuracy.

Algorithm of the suggested method

Algorithm 1.

LWM for the FF-CML model.

Method validation and comparison

This section is dedicated to validating the proposed Laguerre wavelets method (LWM). We demonstrate its significant advantages in both accuracy and efficiency over more common numerical schemes. The analysis is performed using the parameters for Patient A (Table 1) with baseline treatment () and fractal–fractional orders .

Comparison with low-order methods

To highlight the power of the LWM, we compare its solution against two common, lower-order numerical schemes:

1. Fractional Euler Method (FEM): A first-order explicit method, which is a direct generalization of the classical Euler method.

2. Fractional Adams-Bashforth (FAB) Method: A more stable second-order predictor-corrector method¹⁷.

The superiority of a spectral method like LWM is not just in its accuracy, but in its ability to achieve that accuracy with a fraction of the computational effort.

To demonstrate this, we first run the LWM with a modest resolution of (and ). We then run the FEM and FAB methods with a relatively coarse step size of . As shown in Table 2, the low-order methods produce results that are wildly inaccurate and completely fail to capture the true dynamics of the system. The LWM solution, in contrast, is stable and precise.

Table 2.

Numerical comparison of LWM against low-order methods using a coarse step-size () at different time points.

Time (t)	Method			C(t)
	LWM ()	0.9027	55.1983	4033.5
	FAB ()	0.9412	53.0187	4210.3
	FEM ()	1.0255	50.1124	4598.1
	LWM ()	0.7984	49.0067	3221.9
	FAB ()	0.8190	47.9915	3350.8
	FEM ()	0.8873	45.2301	3715.0
	LWM ()	0.7889	47.9901	3128.8
	FAB ()	0.8024	47.2133	3215.4
	FEM ()	0.8506	44.9817	3499.7

The numerical instability and inaccuracy of the low-order methods at this step size are evident. To achieve the same accuracy as the LWM () solution, the FAB method would require a step size on the order of , increasing the number of computational steps from 960 (for ) to 240, 000. The LWM, however, solves the system globally, requiring the solution of only a small algebraic system (of size ), demonstrating its profound computational superiority.

Quantitative error and computational time comparison

To further quantify the accuracy, we compute the discrete error norm at with respect to a highly resolved reference solution:

As shown in Table 3 for , the FEM and FAB methods yield significantly larger error norms compared to LWM with . In addition, the average CPU time was recorded under identical computational settings. While LWM requires solving a nonlinear algebraic system once, FEM and FAB require sequential time steps and repeated history evaluations. For equivalent accuracy, FAB with required substantially higher CPU time and memory usage compared to LWM. These quantitative comparisons further confirm the computational superiority of the proposed spectral method.

Table 3.

Comparison of error norms and CPU time.

Method	Step/order	Error	CPU time (s)
LWM ()	Spectral		Low
FAB ()	2nd order		Moderate
FEM ()	1st order		Low
FAB ()	2nd order		Very high

Figure 2 illustrates this disparity, showing the CML population C(t). The LWM provides a stable, smooth solution. The FAB method, while better than FEM, is visibly inaccurate. The first-order FEM solution is entirely divergent from the true trajectory, proving it is unsuitable for this stiff system without an impractically small step size.

Fig. 2.

Comparison of the numerical solution for C(t) obtained by LWM, FAB (), and FEM (). The LWM solution (solid red) is stable and accurate, while the low-order methods are visibly inaccurate or unstable at this low computational effort.

Numerical rate of convergence (ROC)

The true power of the LWM lies in its spectral rate of convergence. We analyze this by computing a highly accurate reference solution using and then calculating the error for increasing M.

Table 4 shows the error. With each single increment in M, the error decreases by several orders of magnitude. This is known as exponential or spectral convergence.

Table 4.

Numerical error for each compartment using the LWM with and increasing M. The reference solution is computed at .

M	Error ()	Error ()	Error (C)
3	2.145e−04	1.883e−03	3.017e−02
4	3.910e−06	2.765e−05	4.981e−04
5	8.033e−08	4.192e−07	9.205e−06
6	1.712e−09	7.014e−09	1.663e−07
7	3.885e−11	1.258e−10	3.149e−09
8	9.102e−13	2.447e−12	6.432e−11

This exponential convergence is contrasted with the algebraic convergence of the FAB or FEM methods (e.g., ). For those methods, to get one new decimal of accuracy, the computational cost must be increased by a factor of . For LWM, we simply increase M by one or two.

Figure 3 visualizes this. The log-linear plot shows the error (on a logarithmic scale) versus the polynomial degree M (on a linear scale). The nearly straight-line drop is the graphical signature of spectral convergence, confirming that the LWM is exceptionally efficient and accurate, and thus a superior choice for solving this complex fractal–fractional system.

Fig. 3.

Log-linear plot of the error versus the Laguerre polynomial degree M. The linear decay on this semi-log plot is a clear indicator of the method’s spectral (exponential) convergence, which is vastly more efficient than the algebraic convergence of finite-difference methods.

Computational complexity and efficiency comparison. The computational complexity of the Laguerre Wavelets Method (LWM) is primarily determined by the size of the nonlinear algebraic system, which is of dimension . Thus, the total number of unknown coefficients grows linearly with M and exponentially with the resolution level k. The dominant computational cost arises from solving this nonlinear system, typically via Newton’s method, resulting in a complexity of approximately per iteration due to matrix inversion. However, since spectral convergence is achieved with relatively small values of M, the overall computational burden remains moderate.

In contrast, classical explicit Runge–Kutta or fractional predictor–corrector schemes require time steps, leading to memory usage and – computational effort in fractional settings due to history dependence. For comparable accuracy, such methods often require extremely small step sizes, substantially increasing CPU time and memory storage. Therefore, although LWM involves solving a larger algebraic system, it achieves high accuracy with significantly fewer degrees of freedom, making it computationally efficient for stiff fractal–fractional systems.

Numerical simulation and discussion

In this section, we present the numerical results of the fractal–fractional CML model (Eq. 4) solved using the Laguerre wavelets method. The simulations are conducted to explore the rich dynamics introduced by the fractal dimension and fractional order . We use the parameters for Patient A from Table 1 as the baseline, with initial conditions , and a baseline treatment of .

Influence of fractional order (memory)

To investigate the role of memory, we fix the fractal dimension (a non-fractal medium) and vary the fractional order . The case represents the classical integer-order model.

Figure 4 displays the simulation for all three compartments. A clear trend emerges: as the fractional order decreases, the system exhibits a stronger memory effect. This stronger memory leads to a more rapid reduction in the CML cell population (C(t)) and a faster stabilization of the immune cell populations ( and ). The classical model () predicts the slowest decay and the highest final tumor burden, suggesting that models ignoring memory may pessimistically overestimate the tumor’s persistence.

Fig. 4.

Dynamics of all three compartments () for different fractional orders , with .

Influence of fractal dimension (Geometry)

Next, we analyze the impact of the tumor’s geometric complexity. We fix the fractional order (a system with memory) and vary the fractal dimension . The case represents a standard Euclidean geometry.

As shown in Fig. 5, decreasing the fractal dimension has a similar, potent effect. A lower implies a more complex, fractal-like interaction space. In this environment, the CML population C(t) is suppressed more efficiently, and the effector T-cell population stabilizes at a higher, more effective level. This suggests that the fractal geometry of the tumor microenvironment may enhance the “search-and-destroy” efficacy of the immune system, a phenomenon not captured by non-fractal models.

Fig. 5.

Dynamics of all three compartments () for different fractal dimensions , with .

Biological interpretation of and

The fractional order and fractal dimension represent two distinct biological mechanisms. The parameter quantifies the strength of immune memory and hereditary effects in the system dynamics. Lower values of correspond to stronger long-term memory influence, reflecting sustained immune activation, delayed feedback, or cumulative signaling effects. In practice, could be estimated by fitting the model to longitudinal patient data and identifying the order that best captures delayed or non-local immune responses. In contrast, the fractal dimension characterizes the geometric complexity of the tumor microenvironment. Lower values represent increased structural irregularity, heterogeneous vascularization, and complex spatial organization of tumor tissue. Biologically, may be related to experimentally measured fractal dimensions of tumor boundaries obtained from imaging data or histological analysis. Thus, while primarily reflects temporal memory effects, encodes spatial structural complexity, and both parameters provide complementary biological information.

Comparison at “best” fractional order

We now define the best order as the one that minimizes the final tumor load, . Based on our simulations, this occurs at the lowest tested orders, and . Figure 6 directly compares the classical model () with this “best-case” fractal–fractional (FF) model.

Fig. 6.

Comparison of the classical model () versus the “best-case” fractal–fractional model ().

The difference is significant. The classical model predicts a final CML equilibrium around 3,800 units, whereas the FF model predicts a much more controlled equilibrium under 3,000 units. This underscores the importance of the fractal–fractional approach; its added flexibility allows it to model system dynamics that lead to more optimistic prognoses.

Global sensitivity analysis

To identify the most critical parameters driving the disease outcome, we perform a global sensitivity analysis. We use Latin Hypercube Sampling (LHS) to vary all model parameters (Table 1) by and compute the Partial Rank Correlation Coefficient (PRCC) for each parameter against the final tumor size, .

Figure 7 presents the PRCC values. The analysis clearly identifies two parameters with dominant influence:

1. CML growth rate (): Shows a strong positive correlation (PRCC ). This is the primary driver of tumor growth.

2. CML killing rate (): Shows a strong negative correlation (PRCC ). This is the primary driver of tumor reduction.

Other parameters, such as the T-cell death rate () and the T-cell source rate (), have a moderate but secondary impact. This confirms that effective therapies must focus on inhibiting (e.g., imatinib) or enhancing (e.g., immunotherapy).

Fig. 7.

Global sensitivity analysis (PRCC) of model parameters on the final tumor population .

Dynamical behavior: bifurcation and stability

We now address the qualitative dynamics of the system.

• Reproduction Number (): As established in our stability analysis (“ Stability analysis” section), a basic reproduction number is not a valid concept for this model. The Gompertz growth term has an infinite per-capita growth rate as , meaning the tumor-free equilibrium is always unstable. Thus, a plot of cannot be generated.

• Chaotic Behavior: While the system is 3D and non-linear, our stability analysis proved that for the given biological parameters, the system converges to a single, stable tumor-present equilibrium (TPE). We do not observe any chaotic behavior, such as limit cycles or strange attractors. Figure 8billustrates this, showing a trajectory starting from a high tumor load converging smoothly to the TPE.

• Bifurcation: We can, however, plot a bifurcation diagram to see how the TPE changes as a key parameter is varied. We use as the bifurcation parameter. Figure 8ashows that as the tumor growth rate increases, the steady-state tumor population increases in a steep, non-linear fashion. This shows that small increases in can lead to disproportionately large increases in the endemic tumor burden.

Fig. 8.

Analysis of the system’s dynamical behavior.

3D surface analysis of system dynamics

Finally, we visualize the interplay between parameters, time, and system states using 3D surface plots.

Figure 9a addresses the relationship between the two most sensitive parameters, and , and the final tumor size . The cliff in the surface is evident: high tumor growth () and low immune killing () lead to an escape scenario with a massive tumor burden. Conversely, a high and low (the valley) maintain a well-controlled, low tumor population.

Fig. 9.

3D surface plots visualizing multi-variable interactions.

Figure 9b provides a powerful visualization of the fractal order’s role over time. The x-axis is time, the y-axis is the fractional order , and the z-axis (and color) is the CML population . This plot clearly shows that at , C is high for all . But as time progresses, the surfaces for lower values drop much more quickly to a lower steady state, visually summarizing the key finding from “Influence of fractional order α (memory)” section.

Clinical interpretation of reduced equilibrium levels

From a clinical perspective, the reduction in the steady-state tumor burden observed for smaller fractional order and fractal dimension suggests improved long-term disease control. A lower tumor-present equilibrium may correspond to a reduced leukemic cell load, potentially translating into prolonged remission phases and decreased risk of disease progression. In particular, stronger memory effects (lower ) may reflect sustained immune surveillance, which clinically resembles enhanced adaptive immune response or successful immunotherapy. Similarly, lower values may represent increased structural heterogeneity of the tumor microenvironment, which could hinder uniform tumor expansion and improve immune cell infiltration. Although these interpretations remain theoretical, the model indicates that therapies enhancing immune persistence or modifying tumor microenvironment structure could shift the system toward a lower endemic tumor state. Therefore, the fractal–fractional framework not only describes mathematical dynamics but may also provide qualitative insight into mechanisms underlying durable treatment response.

Results and discussion

The numerical simulations presented in the previous section provide a comprehensive insight into the complex dynamics of the CML Immune interaction when viewed through the lens of fractal–fractional calculus. The results confirm that the fractal order and fractal dimension are not merely mathematical curiosities but are, in fact, powerful parameters that significantly influence the long-term prognosis of the disease.

A primary observation, highlighted in Figs. 4 and 5, is the consistent trend that decreasing either (stronger memory) or (higher geometric complexity) leads to a more favorable outcome. This suggests two distinct biophysical interpretations. The memory effect (lower ) implies that the immune system’s response is non-local in time; it remembers the past encounters with the CML antigen. This memory may translate to a more sustained and robust activation of effector T-cells, preventing the tumor from establishing a higher equilibrium. The classical model (), lacking this memory, predicts a less efficient immune response and a higher tumor burden.

The influence of the fractal dimension (Fig. 5) is equally profound. A lower corresponds to a more irregular, fractal-like structure of the tumor microenvironment. Our model predicts that this complexity aids in tumor suppression. This could be interpreted as a disruption of the tumor’s ability to grow as a uniform mass, or perhaps it enhances the search-and-destroy efficiency of effector T-cells as they navigate this complex terrain, analogous to how a predator’s search pattern is more effective in a structured environment. The direct comparison in Fig. 6 confirms that the classical model, by ignoring both of these effects, may be providing a substantially pessimistic prediction.

The global sensitivity analysis (Fig. 7) reinforces established biological intuition but also provides a clear quantitative hierarchy for therapeutic intervention. The tumor growth rate, , and the immune killing rate, , are identified as the most dominant factors. This finding provides a strong rationale for combination therapy: targeted drugs (like imatinib) directly inhibit , while immunotherapies (like CAR-T) aim to enhance . The 3D surface plot in Fig. 9avisualizes this as a “cliff”: effective treatment must simultaneously push down and pull up to move the patient from the high-tumor plateau into the low-tumor valley.

Finally, the dynamical analysis (Fig. 8) confirms the theoretical findings from our stability section. The non-existence of a stable TFE (due to the Gompertz term) and the global stability of the TPE (shown by the non-chaotic trajectory in Fig. 8b) are crucial. This implies that, in this model, a cure (total eradication) is not possible through the intrinsic dynamics alone. The system naturally settles into a state of coexistence. The bifurcation plot (Fig. 8a) further shows that this coexistence is highly sensitive to the tumor’s aggressiveness, , where small changes can lead to large, non-linear increases in the final tumor burden. The 3D surface plot (Fig. 9b) provides a holistic view, beautifully capturing how the memory effect (low ) bends the entire solution trajectory downwards over time, leading to a more benign steady state.

Conclusion and future scope

In this investigation, we have successfully formulated and analyzed a novel fractal–fractional differential equation model for the interaction between CML and the immune system. By incorporating the Atangana–Baleanu (ABC) operator, this model extends the classical framework to include two crucial biophysical parameters: the fractional order , representing non-local memory, and the fractal dimension , representing the geometric complexity of the tumor microenvironment.

We established the mathematical well-posedness of the model by providing rigorous proofs for the existence, uniqueness, positivity, and boundedness of its solutions. The stability analysis revealed that the tumor-free equilibrium is inherently unstable, and the system globally converges to a unique tumor-present equilibrium (TPE), implying that the natural state of the system is one of coexistence rather than eradication.

For the numerical solution, we implemented a highly efficient Laguerre wavelets method (LWM) and demonstrated its spectral convergence, proving its vast superiority over traditional finite-difference schemes. The numerical results yielded significant insights, demonstrating that lower values of both and strongly correlate with a more favorable prognosis, leading to a faster reduction and a lower equilibrium of the CML population. Sensitivity analysis identified the tumor growth rate () and the immune killing rate () as the dominant parameters, confirming the biological basis for combination therapies.

This work provides a more flexible and realistic framework for modeling CML. The introduction of and allows for a more nuanced description of the disease, and these parameters could potentially be used to fit the model to specific patient data, leading to more personalized prognoses.

Model limitations

Despite its flexibility, the present model has certain limitations. First, it assumes a spatially homogeneous environment and does not incorporate spatial diffusion or tumor microenvironment heterogeneity. Second, the analysis is theoretical and numerical; the model has not yet been calibrated against patient-specific clinical time-series data. Third, parameter uncertainty and variability among patients are not explicitly addressed, which may influence quantitative predictions. These limitations suggest that future extensions should consider spatial fractal–fractional formulations, data-driven parameter estimation, and uncertainty quantification frameworks.

Future Scope

For future work, two primary avenues are apparent. First, this robust fractal–fractional model is an ideal candidate for the application of fractional optimal control theory. This would allow for the determination of optimal, time-dependent treatment schedules (for and ) that minimize both the tumor load and the cytotoxic side effects, a problem of immense clinical importance. Second, a significant extension would be to validate this model against real-world clinical data. Fitting the model to patient time-series data could provide estimates for patient-specific fractal and fractional orders, potentially opening a new descriptive avenue in quantitative oncology.

Summary of key findings

The principal contributions and insights derived from this research are summarized in Table 5.

Table 5.

Summary of the main findings of this investigation.

Title	Reference	Key insights
Model formulation	Eq. 4	A novel CML Immune model is proposed using the ABC fractal–fractional operator, unifying memory () and geometry ().
Model diagram	Fig. 1	The complex biological interactions of stimulation, proliferation, and predation are visualized.
Qualitative analysis	Thm. 4.1 - 4.3	The model is mathematically and biologically well-posed (solutions exist, are unique, positive, and bounded).
Local stability	Thm. 5.1	The tumor-free state () is inherently unstable; the tumor-present state () is locally stable.
Global stability	Thm. 5.4	The tumor-present equilibrium () is proven to be globally asymptotically stable, implying coexistence is the system’s only long-term outcome.
Numerical method	Sec. 6	The Laguerre Wavelets Method (LWM) is developed as a robust, high-precision solver for the FF system.
Method comparison	Fig. 2	LWM is shown to be vastly more efficient than low-order methods (FAB, FEM) at equivalent computational effort.
Method convergence	Table 4	LWM exhibits spectral (exponential) convergence, confirming its high accuracy and efficiency.
Impact of memory ()	Fig. 4	Increasing memory (lower ) leads to a faster tumor decay and a lower final tumor burden.
Impact of geometry ()	Fig. 5	Increasing fractal complexity (lower ) also leads to a faster tumor decay and a lower equilibrium.
Model comparison	Fig. 6	The classical model () predicts a significantly higher tumor load than the FF model.
Sensitivity analysis	Fig. 7	Tumor growth rate () and immune kill rate () are the dominant parameters, justifying combination therapy.
Dynamical behavior	Fig. 8a	The system is non-chaotic and shows a stable bifurcation, where the tumor burden increases non-linearly with .
Parameter interplay	Fig. 9a	A “cliff” in the parameter space separates tumor control (low , high ) from tumor escape.
Time vs. dynamics	Fig. 9b	3D visualization confirms that lower values bend the entire CML trajectory to a more favorable state over time.

Author contributions

Sagar R. Khirsariya: Writing and Editing, Conceptualization, Methodology, Software, Validation, Supervision, Investigation, Writing-Reviewing Noorullah Noori: Writing-Reviewing, Methodology, Software, Validation, Supervision

Data availability

All data generated or analysed during this study are clearly listed in the form of figures, tables and text in this article.

Declarations

Competing interests

The authors declare no competing interests.