Dynamics of Chronic Myeloid Leukemia Under Imatinib Treatment: A Study of Resistance Development

Abstract

Chronic myeloid leukemia (CML) is a hematological disorder characterized by the abnormal proliferation of leukemic cells. This study aims to model the dynamics of leukemic and healthy cell populations in CML, considering the role of the immune system and the effects of treatment with Imatinib. The model also addresses the development of treatment resistance in cells, following the Goldie–Coldman hypothesis. We employ a system of delay differential equations to simulate the interactions between leukemic cells, healthy cells, and the immune system under treatment. The results provide insights into the dynamic balance between leukemic cells, healthy cells, and immune responses, and the impact of developing resistance on treatment outcomes.

1. Introduction

Chronic Myeloid Leukemia (CML) is a slowly progressive and clonal myeloproliferative disorder thought to be responsible for 15% to 20% of adult leukemia cases worldwide.

CML is characterized by the presence the Philadelphia chromosome, resulting in the BCR-ABL fusion gene. This genetic abnormality results in the formation of a particular gene product, a constitutively active tyrosine kinase that produces a continued proliferative signal causing the clinical manifestations of CML. Given our understanding of this mechanism, an effort has been made to develop new compounds in order to selectively inhibit the resulting abnormal tyrosine kinase. Since then, several tyrosine kinase inhibitors (TKIs) blocking the initiation of the BCR-ABL pathway have been developed and tested in patients with CML. The first TKI developed, which currently remains the first line of CML treatment, is Imatinib Mesylate (Gleevec). However, in spite of the remarkable rate of complete hematological response and complete cytogenetical remission, some cases show resistance or relapse after an initial response (secondary or acquired resistance). The methods involved in overcoming Imatinib resistance consist of the development of new TKIs and the stimulation of an immune response against the disease.

Mathematical models involving differential equations for CML dynamics have been developed mainly in the last 20 years and are based on hematopoiesis models (see, for example [1,2,3,4,5,6,7,8]). In the beginning, ordinary differential equations (ODEs) were mostly used, but as the cellular and molecular processes imply a certain time lag, the use of delay differential equations (DDEs) became more appropriate. In this framework, a recent and comprehensive review on mathematical models of leukemia and its treatments can be found in [3].

The approach of CML evolution (with or without treatment) through DDEs models can be found in many recent studies [9,10]. In [11,12], DDEs models of CML evolution with implications for the immune system are studied, while in [9], a complex model describing the competition between healthy and leukemic cell populations and the influence of T-lymphocytes is introduced and analyzed. The paper given in [13] introduces a two-dimensional system of DDEs for stem cells and mature neutrophils in CML under a periodic treatment. The existence of a strictly positive periodic solution is proved using the Krasnoselskii theorem and the stability of this solution is investigated.

One of the first studies on CML evolution in a model investigating treatment effect including Imatinib resistance through the Goldie–Coldman law was introduced in [10]. The DDE model is one-dimensional and the state variable is represented by a population of stem-like cells, as they have the property of self-renewal and acquired drug resistance is propagated by this cell population. A more elaborated model for CML dynamics with treatment and resistance, including two state variables, is considered and investigated in [14].

In this paper, a more complex DDE model for healthy and leukemic CML cell populations subjected to Imatinib treatment is introduced and its equilibrium points and stability properties are analyzed. Imatinib’s pharmacokinetics are considered here through the last two equations and treatment effects, including resistance, are introduced using a Goldie–Coldman approach. In this framework, the present model is more realistic and appropriate than other similar models.

The mathematical model is a system of delay differential equations (DDEs) with seven equations: four describing the time-evolution of healthy and CML cell populations, one describing the evolution of immune cell population and two equations involving Imatinib pharmacokinetics. The state variables of the model are healthy and leukemic cell populations with self-renewal abilities ($x_1$ and $x_3$) called stem-like cells, mature healthy and leukemia leukocytes ($x_2$ and $x_4$), which have lost their self-renewal ability, the immune system represented by the population of anti-leukemia cells, that is, CD8+ cytotoxic T cells ($x_5$), the concentration of Imatinib in the absorbtion compartment (D), and the concentration of active substance in the plasmatic compartment (P). The system that models CML under Imatinib treatment is

$$\begin{array}{cc}{\dot{x}}_1=\hfill & -{\gamma }_{1h}{x}_1-({\eta }_{1h}+{\eta }_{2h})k_h(x_2+x_4)x_1-(1-{\eta }_{1h}-{\eta }_{2h}){\beta }_h(x_1+x_3)x_1\hfill \\ \hfill & +2e^{-{\gamma }_{1h}{\tau }_1}(1-{\eta }_{1h}-{\eta }_{2h}){\beta }_h(x_{1{\tau }_1}+x_{3{\tau }_1})x_{1{\tau }_1}+{\eta }_{1h}{e}^{-{\gamma }_{1h}{\tau }_1}{k}_h(x_{2{\tau }_1}+x_{4{\tau }_1})x_{1{\tau }_1}\hfill \\ {\dot{x}}_2=\hfill & -{\gamma }_{2h}{x}_2+A_h(2{\eta }_{2h}+{\eta }_{1h})k_h(x_{2{\tau }_2}+x_{4{\tau }_2})x_{1{\tau }_2}\hfill \\ {\dot{x}}_3=\hfill & -({\gamma }_{1l}+k_1)x_3-({\eta }_{1l}+{\eta }_{2l})u_1{k}_l(x_2+x_4)x_3-(1-{\eta }_{1l}-{\eta }_{2l})u_1{\beta }_l(x_1+x_3)x_3\hfill \\ \hfill & +2e^{-\widetilde{{\gamma }_{1l}}{\tau }_3}(1-{\eta }_{1l}-{\eta }_{2l})u_1{\beta }_l(x_{1{\tau }_3}+x_{3{\tau }_3})x_{3{\tau }_3}+{\eta }_{1l}{e}^{-\widetilde{{\gamma }_{1l}}{\tau }_3}{u}_1{k}_l(x_{2{\tau }_3}+x_{4{\tau }_3})x_{3{\tau }_3}\hfill \\ \hfill & -b_1{x}_3{x}_5{l}_1(x_3+x_4)\hfill \\ {\dot{x}}_4=\hfill & -{\gamma }_{2l}{x}_4+A_l(2{\eta }_{2l}+{\eta }_{1l})u_1{k}_l(x_{2{\tau }_4}+x_{4{\tau }_4})x_{3{\tau }_4}-b_2{x}_4{x}_5{l}_1(x_3+x_4)\hfill \\ {\dot{x}}_5=\hfill & -a_2{x}_5-a_3{u}_2{x}_5{l}_2\left(x_4\right)+2^{n_1}{a}_3{e}^{-a_2{\tau }_5}{u}_2{x}_{5{\tau }_5}{l}_2\left(x_{4{\tau }_5}\right).\hfill \end{array}$$

(1)

Here, $k_1$, $u_1$, and $u_2$ are the controls, defined in what follows, depending on P, in the following system:

$$\begin{array}{ccc}\dot{D}=\hfill & -{\lambda }_{0}D+K\hfill & \\ \dot{P}=\hfill & -\nu P+\lambda D.\hfill \end{array}$$

Some assumptions are considered throughout this paper. First, we consider that the healthy and leukemic blood cell populations are in competition for resources. This is reflected in the fact that the feedback functions for self-renewal (${\beta }_{\alpha }$) and for differentiation ($k_{\alpha }$) depend on the sum of healthy and leukemic cells; hence, they are

$$\left\{\begin{array}{c}{\beta }_{\alpha }(x_1+x_3)={\beta }_{0\alpha }{\displaystyle \frac{{\theta }_{1\alpha }^{m_{\alpha }}}{{\theta }_{1\alpha }^{m_{\alpha }}+{(x_1+x_3)}^{m_{\alpha }}}}\\ k_{\alpha }(x_2+x_4)=k_{0\alpha }{\displaystyle \frac{{\theta }_{2\alpha }^{n_{\alpha }}}{{\theta }_{2\alpha }^{n_{\alpha }}+{(x_2+x_4)}^{n_{\alpha }}}}\end{array}\right.,\alpha =h,l$$

(h for healthy and l for leukemia) where ${\beta }_{0\alpha }$ and $k_{0\alpha }$ are the maximal rates of self-renewal, respectively, of differentiation, ${\theta }_{1\alpha }$ and ${\theta }_{2\alpha }$ are half of their maximal value and, and $m_{\alpha }$ and $n_{\alpha }$ control the sensitivity of ${\beta }_{\alpha }$ or $k_{\alpha }$ to changes in the size of the stem-like and mature populations, respectively.

It is assumed that T cells are able to fight leukemia only up to a certain level of leukemia population and then, after this level is exceeded, the T cell population is inhibited by the leukemic cells. Hence, the interaction between the immune system and the leukemic cell population is regulated by the following two feedback functions, describing the effect of T cells on CML cells and the effect of leukemic cells on the immune cell population, respectively:

$$\left\{\begin{array}{c}{l}_1\left(z\right)={\displaystyle \frac{1}{b_3+z^2}}\\ l_2\left(z\right)={\displaystyle \frac{z}{b_4+z^2}}.\end{array}\right.$$

The evolution of the T cells is modeled starting with a moment when they are already mature, thus skipping all the intermediate levels. Such an approach is similar with that of [15]. It is assumed that a stem-like cell is capable of undergoing three types of division when it enters the cell cycle; hence, in this model, it is considered that a fraction ${\eta }_{1\alpha }$, $\alpha =h,l$, of stem-like cells is susceptible to asymmetric division: one daughter cell proceeds to differentiate and the other re-enters the stem cell compartment, while another fraction ${\eta }_{2\alpha }$, $\alpha =h,l$, is susceptible to differentiate symmetrically with both cells that result following a phase of maturation, and the fraction $1-{\eta }_{1\alpha }-{\eta }_{2\alpha }$, $\alpha =h,l$, is susceptible to self-renewal so both cells that result after mitosis are stem-like cells.

In view of [16], in the equations for Imatinib pharmacokinetics, K is the amount of drug administrated, considered as constant. According to [17], the standard dose of 400 mg is $K=6.74$ μM. Also, ${\lambda }_0$ is the first order absorption rate and $\nu$ is the total plasma clearance of drug divided by the volume of distribution of the drug.

It is known that the treatment with Imatinib, targeting the BCR-ABL gene, affects the apoptosis and the proliferation rate of leukemia cells. Its action is considered constant in time, corresponding to a period when, after an initial transitory stage, the effect at the cell level can be seen as a steady one. Also, we consider that Imatinib acts on the cells of the immune system, affecting its maturation and correct functioning. The treatment effects are introduced in the model through the factor $u_1$ in terms of controlling the multiplication of cells, through the factor $u_2$ in the equation of CD8+ T cytotoxic cells and through the mortality rate of stem-like CML cells $k_1$.

In order to choose the functions $u_1$ and $u_2$, we used the data in [18]. In a Goldie–Coldman framework, let $x_0$ represent the initial number of leukemic stem-like cells, $R_0$ the number of cells resistant to treatment, and p the probability of mutation where $q=-p\in (-1,0]$. Because the standard dose of Imatinib reduces the rate of differentiation 100 times, it follows that

$$u_1\left(P\right)={\displaystyle \frac{1}{1+(99/3.2^2)P^2}}.$$

Also, as a dose of 10 μM leads to a 75% decrease in the activation of T cells,

$$u_2\left(P\right)={\displaystyle \frac{100/3}{100/3+P^2}}.$$

The function modeling the treatment’s effect on the mortality of leukemic stem-like cells, including resistance, can be taken as

$$k_1=r_1\left(P\right)x_3^q,$$

with

$$r_1\left(P\right)={\displaystyle \frac{P^{\mu }}{P^{\mu }+P_0^{\mu }}}{\displaystyle \frac{x_0-R_0}{x_0^{-p+1}}},$$

where $P_0$ is the half maximum activity concentration, $\mu$ is a Hill coefficient, and $R_0$ is the initial number of resistant cells. We take $R_0=r{x}_0$ with $r\in [0,1].$

The duration of the cell cycle for healthy and leukemia stem-like cells are ${\tau }_1$ and ${\tau }_3$, which are supposed to be independent of the type of division; the time necessary for differentiation into mature leukocytes for healthy and leukemia cells is ${\tau }_2$ and ${\tau }_4$, respectively. ${\tau }_T$ is the duration of the cell cycle for cytotoxic T cells and ${\tau }_5=n{\tau }_T$ where n is the number of antigens depending on divisions. With $\alpha =h,l,{\gamma }_{1\alpha }$ is the natural apoptosis and $k_l$ is the supplementary one provided by treatment. $A_{\alpha }$ is an amplification factor, $a_2$ indicates the apoptosis rate of CD8+ cytotoxic T cell population, and the last terms in the T cell population equation give the rate at which naive T cells leave and re-enter the effector state after finishing the minimal developmental program of $n_1$ cell divisions (due to antigen stimulation) combined with positive growth and suppressive signals at different rates for self-development or regulatory mechanisms. The time delay ${\tau }_5=n_1{\tau }_T$, where ${\tau }_T$ is the duration of the cell cycle for T cell division, is the duration of this program.

Leukemia cells suppress the anti-leukemia immune response. The precise mechanism is unknown. It is assumed that the level of downregulation depends on the current leukemia population. This competition between T cells and leukemia cells is expressed by the presence of the mature population of leukemic cells in the denominators of the second and the third terms, $-a_3{x}_5{l}_2\left(x_4\right)$ and $2^{n_1}{a}_3{x}_{5{\tau }_6}{l}_2\left(x_{4{\tau }_6}\right)$. These terms, respectively, represent the loss and production of T cells due to competition with leukemic cells.

It is not difficult to see that if the initial conditions are positive, the solution will be positive for $t>0$.

2. Equilibrium Points and Medical Interpretations

System (1.1) has four possible types of equilibrium points: (0, 0, 0, 0, 0) (a death state), $(x_1^{*},x_2^{*},0,0,0)$ (a healthy state), $(0,0,{\overline{x}}_3,{\overline{x}}_4,{\overline{x}}_5)$ (an acute state), and $({\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3,{\widehat{x}}_4,{\widehat{x}}_5)$ (a chronic state).

From a medical point of view, we are interested only in equilibrium points with non-negative components.

As state variables represent cell populations (of blood in this case), the normal evolution of a healthy state is characterized by constant behavior, eventually displaying small fluctuations. Mathematically, this translates into a stable non-trivial steady-state or a stable limit cycle. Suppose now that an incipient leukemic process is initiated. Two situations can occur: the defense mechanisms of the organism can defeat the leukemic process at the very beginning or the cancer cell population develops and invades the healthy cell population. The term “invasion” means that, starting from a small number of cells, the number of leukemic cells growths and outnumbers the healthy ones. The invasion happens when, although in a neighborhood of zero, the number of leukemic cells is large enough to have already escaped the defense mechanisms. This kind of invasion is taken from the ecology of populations and does not have the same meaning with the term used in oncology.

CML cell populations are characterized by lower mortality and higher proliferation or differentiation, features that progress in time (with the accumulation of more mutations) and confer more “strength” to these cancer cells (see [19]). It follows that, in a hypothetical system of solely cancer cell lines, the features just mentioned ensure an increasing growth. This and the invasive character of leukemia imply that, in a theoretical case of only CML cell populations, one would expect the trivial equilibrium to be unstable and an eventual limit cycle that is also unstable.

In a real system of coexisting healthy and CML cell populations, although CML, as a chronic disease, advances slowly, in a couple of years, if no treatment is present, the cancer cell populations will overwhelm the organism. A parallel can be made with invasive populations from ecology (see [20,21]). It follows from this reasoning that leukemia develops when, in the real system of healthy and CML cell populations, the equilibria with nonzero components for healthy populations and zero ones for leukemic populations are not stable.

An interesting equilibrium point is $(0,0,{\overline{x}}_3,{\overline{x}}_4,{\overline{x}}_5)$, which corresponds to the extinction of the healthy cell populations and the establishment of the CML cell populations. This would correspond to the fateful case of a blast crisis.

The equilibria $({\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3,{\widehat{x}}_4,{\widehat{x}}_5)$ is equivalent to a chronic phase of the disease, in which the healthy and leukemic cells constantly fight for resources.

3. Stability of the Steady States

The stability analysis focuses on the equilibrium points $E_3=(0,0,{\overline{x}}_3,{\overline{x}}_4,{\overline{x}}_5)$ and $E_4=({\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3,{\widehat{x}}_4,{\widehat{x}}_5)$, as the first two equilibrium points (representing patient death and a healthy state) are not medically relevant to the study of treatment resistance. For these points, linear stability analysis cannot be applied due to the singularity in the derivative with respect to $x_3$ at zero.

To analyze stability at the relevant equilibria, we linearize the system around points $E_3$ and $E_4$. This is carried out by constructing the matrices A, B, C, D, E, and F (see Appendix A), which represent the partial derivatives of the system with respect to the state variables x and their corresponding time delays.

3.1. Equilibrium $E_3$

For the equilibrium $E_3$, the matrix for which we need to calculate the determinant has the following form:

$$G=\left(\begin{array}{ccccc}{g}_{11}& 0& 0& 0& 0\\ g_{21}& g_{22}& 0& 0& 0\\ g_{31}& g_{32}& g_{33}& g_{34}& g_{35}\\ 0& g_{42}& g_{43}& g_{44}& g_{45}\\ 0& 0& 0& g_{54}& g_{55}\end{array}\right).$$

The complexity of solving this equation arises from the elements $g_{35}$ and $g_{45}$, which are defined as follows:

$$g_{35}=-a_{35}=b_1{\overline{x}}_3{l}_1({\overline{x}}_3+{\overline{x}}_4)and{g}_{45}=-a_{45}=b_2{\overline{x}}_4{l}_1({\overline{x}}_3+{\overline{x}}_4).$$

To handle this, we use the approach from [22], and we begin by introducing a perturbation matrix ${\mathsf{\Delta }}_0$, which has all zero entries except for the elements $g_{35}$ and $g_{45}.$

We then adjust the matrix G by defining

$$G_1=G-{\mathsf{\Delta }}_0.$$

This leads to a simplified determinant expression:

$$detG=d_1{g}_{11}{g}_{22}{g}_{55},$$

(2)

where $d_1$ is the determinant of a submatrix formed by the interactions between $g_{33}$, $g_{34}$, $g_{43}$, and $g_{44}:$

$$d_1=\left|\begin{array}{cc}{g}_{33}& g_{34}\\ g_{43}& g_{44}\end{array}\right|.$$

(3)

The elements $g_{11}$, $g_{22}$, and $g_{55}$ are defined as follows:

$$\begin{array}{cc}\hfill g_{11}=& \lambda -a_{11}-b_{11}{e}^{-\lambda {\tau }_1}\hfill \\ \hfill g_{22}=& \lambda -a_{22}-c_{22}{e}^{-\lambda {\tau }_2}\hfill \\ \hfill g_{55}=& \lambda -a_{55}-f_{55}{e}^{-\lambda {\tau }_5}.\hfill \end{array}$$

(4)

Theorem 3.12 from [23] is applied to assess the asymptotic stability of the equilibrium, $E_3$, utilizing a linearized approach and stability results from the first approximation theorem. Asymptotic stability is guaranteed if the exponential stability condition is met, meaning that all solutions decay exponentially to zero over time.

We calculate the norm of the perturbation matrix ${\mathsf{\Delta }}_0$, which yields

$$\left|\right|{\mathsf{\Delta }}_0\left|\right|\le \sqrt{(g_{35}^2+g_{45}^2)}$$

and defines the spectral radius

$$\rho =\sqrt{(g_{35}^2+g_{45}^2)}.$$

(5)

The following result is obtained:

Theorem 1

([24], Th1, [23], Th3.12). Suppose that for $d_1$, $g_{11}$, $g_{22}$, and $g_{55}$ defined in (3) and (4), the equations $d_1=0$, $g_{11}=0$, $g_{22}=0$, and $g_{55}=0$ all have roots λ with $Re\lambda <0$. Then, if ρ defined in (5) verifies the condition for ρ from Theorem ([23], Th3.12), the equilibrium point $E_3$ is asymptotically stable.

The equation that can be used following Theorem 1 in order to study the stability of the linearized system corresponding to the equilibrium point $E_3$ is as follows:

$$\begin{array}{cc}\hfill & \left[\left(\lambda -a_{33}-d_{33}{e}^{-\lambda {\tau }_3}\right)\left(\lambda -a_{44}-e_{44}{e}^{-\lambda {\tau }_4}\right)-\left(a_{33}+d_{34}{e}^{-\lambda {\tau }_3}\right)\left(a_{43}+e_{43}{e}^{-\lambda {\tau }_4}\right)\right]\hfill \\ \hfill & \left(\lambda -a_{11}-b_{11}{e}^{-\lambda {\tau }_1}\right)\left(\lambda -a_{22}-c_{22}{e}^{-\lambda {\tau }_2}\right)\left(\lambda -a_{55}-f_{55}{e}^{-\lambda {\tau }_5}\right)=0\hfill \end{array}$$

(6)

To study the stability of this equation, we apply the theory from [25,26,27]. Knowing that $a_{11}<0$, $a_{22}<0$, and $a_{55}<0$ and $b_{11}>0$, $c_{22}>0$, and $f_{55}>0$, Equation (6) is stable if the following conditions are respected:

a

$a_{11}+b_{11}<0$

b

$a_{22}+c_{22}<0$

c

$a_{55}+f_{55}<0$

d

The equation

$$\left[\left(\lambda -a_{33}-d_{33}{e}^{-\lambda {\tau }_3}\right)\left(\lambda -a_{44}-e_{44}{e}^{-\lambda {\tau }_4}\right)-\left(a_{34}+d_{34}{e}^{-\lambda {\tau }_3}\right)\left(a_{43}+e_{43}{e}^{-\lambda {\tau }_4}\right)\right]=0$$

has roots with a negative real part.

We have to study the roots of the following equation:

$$E\left(\lambda \right)=P_0\left(\lambda \right)+P_1\left(\lambda \right)e^{-\lambda {\tau }_3}+P_2\left(\lambda \right)e^{-\lambda {\tau }_4}+P_3\left(\lambda \right)e^{-\lambda ({\tau }_3+{\tau }_4)}=0$$

(7)

where

$$\begin{array}{cc}\hfill P_0\left(\lambda \right)& ={\lambda }^2-(a_{44}+a_{33})\lambda +a_{33}{a}_{44}-a_{43}{a}_{43}\hfill \\ \hfill P_1\left(\lambda \right)& =a_{44}{d}_{33}-a_{43}{d}_{34}-d_{33}\lambda \hfill \\ \hfill P_2\left(\lambda \right)& =a_{33}{e}_{44}-a_{34}{e}_{43}-e_{44}\lambda \hfill \\ \hfill P_3\left(\lambda \right)& =e_{44}{d}_{33}-d_{34}{e}_{43}.\hfill \end{array}$$

(8)

For the case when ${\tau }_3={\tau }_4=0$, we have

$$P_0\left(\lambda \right)+P_1\left(\lambda \right)+P_2\left(\lambda \right)+P_3\left(\lambda \right)=0$$

with the explicit form

$${\lambda }^2+q_1\lambda +q_2=0$$

(9)

where

$$q_1=-(a_{33}+a_{44}+d_{33}+e_{44}),q_2=a_{33}{a}_{44}+a_{33}{e}_{44}+a_{44}{d}_{33}+e_{44}{d}_{33}-a_{34}{a}_{43}-a_{43}{d}_{34}-a_{34}{e}_{43}-d_{34}{e}_{43}.$$

(10)

Thus, Equation (7) is stable for ${\tau }_3={\tau }_4=0$ if, and only if, $q_1>0$ and $q_2>0$.

Assume that when ${\tau }_3={\tau }_4=0$, the conditions are met and Equation (7) is stable. Stability may be lost if and only if the roots of Equation (7) move across the imaginary axis from left to right as the delay parameters change. This can occur only if purely imaginary roots exist, which leads us to consider the equation $E\left(i\omega \right)=0$, where $\omega \in \mathbb{R}$. Under these conditions, Equation (7) transforms into

$$\begin{array}{cc}\hfill E\left(i\omega \right)=& -{\omega }^2-(a_{44}+a_{33})i\omega +a_{33}{a}_{44}-a_{34}{a}_{43}-i\omega d_{33}{e}^{-i\omega {\tau }_3}-i\omega e_{44}{e}^{-i\omega {\tau }_4}+(a_{44}{d}_{44}-a_{43}{d}_{43})e^{-i\omega {\tau }_3}\hfill \\ \hfill & +(a_{33}{e}_{44}-a_{34}{e}_{43})e^{-i\omega {\tau }_4}+(e_{44}{d}_{33}+d_{34}{e}_{43})e^{-i\omega ({\tau }_3+{\tau }_4)}=0.\hfill \end{array}$$

(11)

We now seek the conditions under which $E\left(i\omega \right)=0$ has no real roots. For this, we will examine the equation to determine whether any specific constraints on the coefficients and parameters can prevent the existence of real roots.

Furthermore, following [28], we denote the real part of $E\left(i\omega \right)$ as $R\left(\omega \right)$ and the imaginary part as $I\left(\omega \right)$ and then square each of them for further analysis.

$$\begin{array}{cc}\hfill R^2\left(\omega \right)=& [-{\omega }^2+(a_{33}{e}_{44}-a_{34}{e}_{43})cos\omega {\tau }_4+e_{44}\omega sin\omega {\tau }_4+(a_{44}{d}_{33}-a_{43}{d}_{34})cos\omega {\tau }_3\hfill \\ \hfill & -d_{33}\omega sin\omega {\tau }_3+(e_{44}{d}_{33}-d_{34}{e}_{43})cos\omega ({\tau }_3+{\tau }_4)+a_{33}{a}_{44}-a_{34}{a}_{43}{]}^2\hfill \\ \hfill I^2\left(\omega \right)=& [-(a_{44}+a_{33})\omega +(a_{34}{e}_{43}-a_{33}{e}_{44})sin\omega {\tau }_4-\omega e_{44}cos\omega {\tau }_4+(a_{43}{d}_{34}-a_{44}{d}_{44})sin\omega {\tau }_3\hfill \\ \hfill & -d_{33}\omega cos\omega {\tau }_3-(e_{44}{d}_{33}+d_{34}{e}_{43})sin\omega ({\tau }_3+{\tau }_4){]}^2\hfill \end{array}$$

(12)

So, Equation (7) only has roots with a negative real part if the equation $R^2\left(\omega \right)+I^2\left(\omega \right)=0$ has no positive roots.

3.2. Equilibrium $({\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3,{\widehat{x}}_4,{\widehat{x}}_5)$

The analysis of the equilibrium point $({\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3,{\widehat{x}}_4,{\widehat{x}}_5)$ using the characteristic equation appears difficult; therefore, we will explore a Lyapunov–Krasovskii functional to assess its stability.

The first step involves translating the system to the origin by defining

$$y_i=x_i-{\widehat{x}}_i,i=\overline{1,5}.$$

This yields the following system in terms of deviations:

$$\begin{array}{c}\begin{array}{cc}\hfill {\dot{y}}_1=& {\displaystyle -{\gamma }_{1h}(y_1+{\widehat{x}}_1)-({\eta }_{1h}+{\eta }_{2h})k_h[(y_2+{\widehat{x}}_2)+(y_4+{\widehat{x}}_4)](y_1+{\widehat{x}}_1)}\hfill \\ & {\displaystyle -(1-{\eta }_{1h}-{\eta }_{2h}){\beta }_h[(y_1+{\widehat{x}}_1)+(y_3+{\widehat{x}}_3)](y_1+{\widehat{x}}_1)}\hfill \\ & {\displaystyle +2e^{-{\gamma }_{1h}{\tau }_1}(1-{\eta }_{1h}-{\eta }_{2h}){\beta }_h[(y_{1{\tau }_1}+{\widehat{x}}_1)+(y_{3{\tau }_1}+{\widehat{x}}_3)](y_{1{\tau }_1}+{\widehat{x}}_1)}\hfill \\ \hfill & {\displaystyle +{\eta }_{1h}{e}^{-{\gamma }_{1h}{\tau }_1}{k}_h[(y_{2{\tau }_1}+{\widehat{x}}_2)+(y_{4{\tau }_1}+{\widehat{x}}_4)](y_{1{\tau }_1}+{\widehat{x}}_1)}\hfill \\ {\dot{y}}_2=\hfill & {\displaystyle -{\gamma }_{2h}(y_2+{\widehat{x}}_2)+A_h(2{\eta }_{2h}+{\eta }_{1h})k_h[(y_{2{\tau }_2}+{\widehat{x}}_2)+(y_{4{\tau }_2}+{\widehat{x}}_4)](y_{1{\tau }_2}+{\widehat{x}}_1)}\hfill \\ {\dot{y}}_3=\hfill & {\displaystyle -({\gamma }_{1l}+k_1)(y_3+{\widehat{x}}_3)-({\eta }_{1l}+{\eta }_{2l})u_1{k}_l[(y_2+{\widehat{x}}_2)+(y_4+{\widehat{x}}_4)](y_3+{\widehat{x}}_3)}\hfill \\ \hfill & {\displaystyle -(1-{\eta }_{1l}-{\eta }_{2l})u_1{\beta }_l[(y_1+{\widehat{x}}_1)+(y_3+{\widehat{x}}_3)](y_3+{\widehat{x}}_3)}\hfill \\ \hfill & {\displaystyle +2e^{-{\gamma }_{1l}{\tau }_3}(1-{\eta }_{1l}-{\eta }_{2l})u_1{\beta }_l[(y_{1{\tau }_3}+{\widehat{x}}_1)+(y_{3{\tau }_3}+{\widehat{x}}_3)](y_{3{\tau }_3}+{\widehat{x}}_3)}\hfill \\ \hfill & {\displaystyle +{\eta }_{1l}{e}^{-{\gamma }_{1l}{\tau }_3}{u}_1{k}_l[(y_{2{\tau }_3}+{\widehat{x}}_2)+(y_{4{\tau }_3}+{\widehat{x}}_4)](y_{3{\tau }_3}+{\widehat{x}}_3)}\hfill \\ \hfill & {\displaystyle -b_1(y_3+{\widehat{x}}_3)(y_5+{\widehat{x}}_5)l_1[(y_3+{\widehat{x}}_3)+(y_4+{\widehat{x}}_4)]}\hfill \\ \hfill \\ {\dot{y}}_4=\hfill & {\displaystyle -{\gamma }_{2l}(y_4+{\widehat{x}}_4)+A_l(2{\eta }_{2l}+{\eta }_{1l})u_1{k}_l[(y_{2{\tau }_4}+{\widehat{x}}_2)+(y_{4{\tau }_4}+{\widehat{x}}_4)](y_{3{\tau }_4}+{\widehat{x}}_3)}\hfill \\ \hfill & {\displaystyle -b_2(y_4+{\widehat{x}}_4)(y_5+{\widehat{x}}_5)l_1[(y_3+{\widehat{x}}_3)+(y_4+{\widehat{x}}_4)]}\hfill \\ {\dot{y}}_5=\hfill & {\displaystyle -a_2(y_5+{\widehat{x}}_5)-a_3{u}_2(y_5+{\widehat{x}}_5)l_2(y_4+{\widehat{x}}_4)}\hfill \\ \hfill & {\displaystyle +2^{n_1}{a}_3{e}^{a_2{\tau }_T}{u}_2(y_{5{\tau }_5}+{\widehat{x}}_5)l_2(y_{4{\tau }_5}+{\widehat{x}}_4)}\hfill \end{array}\hfill \end{array}$$

For our analysis, we will consider the following form of a Lyapunov–Krasovskii functional:

$$V=\sum _{i=1}^5{\alpha }_i{y}_i^2+\sum _{j=1}^5{\beta }_j\underset{t-{\tau }_j}{\overset{t}{\int }}{y}_j^2\left(s\right)ds+\sum _{i\ne j}{\delta }_{ij}\underset{t-{\tau }_j}{\overset{t}{\int }}{y}_i^2\left(s\right)ds$$

where ${\alpha }_i>0,{\beta }_j\ge 0,\forall i=\overline{1,5},j=\overline{1,5}$, and ${\delta }_{ij}\ge 0$ for $i\ne j.$

As we are working in the framework of the first approximation method, we consider the linearized system

$$\dot{y_i}=g_i\left(y\right),i=\overline{1,5},$$

where

$${\displaystyle g_i\left(y\right)=\sum _{k=1}^5\frac{\partial f_i}{\partial y_k}\left(\widehat{x}\right)y_k+\sum _{k,j}\frac{\partial f_i}{\partial y_{k{\tau }_j}}\left(\widehat{x}\right)y_{k{\tau }_j}}$$

and we denote the coefficients of the functions $g_i$, $i=\overline{1,5}$ as follows:

$$\begin{array}{cc}\hfill g_1\left(y\right)=& s_{11}{y}_1+s_{12}{y}_2+s_{13}{y}_3+s_{12}{y}_4+s_{14}{y}_{1{\tau }_1}+s_{15}{y}_{2{\tau }_1}+s_{16}{y}_{3{\tau }_1}+s_{15}{y}_{4{\tau }_1}\hfill \\ \hfill g_2\left(y\right)=& s_{21}{y}_2+s_{22}{y}_{1{\tau }_2}+s_{23}{y}_{2{\tau }_2}+s_{23}{y}_{4{\tau }_2}\hfill \\ \hfill g_3\left(y\right)=& (s_1+u_1{s}_{31})y_3+u_1{s}_{32}{y}_1+u_1{s}_{33}{y}_2+(s_2+u_1{s}_{34})y_4+s_{35}{y}_5+u_1{s}_{36}{y}_{3{\tau }_3}\hfill \\ \hfill & +u_1{s}_{37}{y}_{1{\tau }_3}+u_1{s}_{38}{y}_{2{\tau }_3}+u_1{s}_{38}{y}_{4{\tau }_3}\hfill \\ \hfill g_4\left(y\right)=& s_{41}{y}_4+s_{42}{y}_3+s_{43}{y}_5+u_1{s}_{44}{y}_{4{\tau }_4}+u_1{s}_{44}{y}_{2{\tau }_4}+u_1{s}_{45}{y}_{3{\tau }_4}\hfill \\ \hfill g_5\left(y\right)=& (s_3+u_2{s}_{51})y_5+u_2{s}_{52}{y}_4+u_2{s}_{53}{y}_{5{\tau }_5}+u_2{s}_{54}{y}_{4{\tau }_5}.\hfill \end{array}$$

Proposition 1.

Assume the following conditions hold:

$$\left[{\displaystyle \frac{s_{14}^2}{{\beta }_1}+\frac{s_{15}^2}{{\gamma }_{21}}+\frac{s_{16}^2}{{\gamma }_{31}}+\frac{s_{15}^2}{{\gamma }_{41}}+2s_{12}^2+s_{13}^2}\right]{\alpha }_1^2+2s_{11}{\alpha }_1+\left({\beta }_1+{\gamma }_{12}+{\gamma }_{13}+1\right)<0$$

$$\left[{\displaystyle \frac{s_{23}^2}{{\beta }_2}+\frac{s_{22}^2}{{\gamma }_{12}}+\frac{s_{23}^2}{{\gamma }_{42}}}\right]{\alpha }_2^2+2s_{21}{\alpha }_2+\left({\beta }_2+{\gamma }_{21}+{\gamma }_{23}+{\gamma }_{24}+2\right)<0$$

$$\left[{\displaystyle \frac{u_1^2{s}_{36}^2}{{\beta }_3}+\frac{u_1^2{s}_{37}^2}{{\gamma }_{13}}+\frac{u_1^2{s}_{38}^2}{{\gamma }_{23}}+\frac{u_1^2{s}_{38}^2}{{\gamma }_{43}}+u_1^2{s}_{32}^2+u_1^2{s}_{33}^2+{(u_1{s}_{34}+s_2)}^2+s_{35}^2}\right]{\alpha }_3^2$$

$$+2{\alpha }_3(u_1{s}_{31}+s_1)+\left({\beta }_3+{\gamma }_{31}+{\gamma }_{34}+2\right)<0$$

$$\left[{\displaystyle \frac{u_1^2{s}_{44}^2}{{\beta }_4}+\frac{u_1^2{s}_{44}^2}{{\gamma }_{24}}+\frac{u_1^2{s}_{45}^2}{{\gamma }_{34}}+s_{42}^2+s_{43}^2}\right]{\alpha }_4^2+2s_{41}{\alpha }_4+\left({\beta }_4+{\gamma }_{41}+{\gamma }_{42}+{\gamma }_{43}+{\gamma }_{45}+3\right)<0$$

$$\left[{\displaystyle \frac{u_1^2{s}_{53}^2}{{\beta }_5}+\frac{u_1^2{s}_{54}^2}{{\gamma }_{45}}+u_1^2{s}_{52}^2}\right]{\alpha }_5^2+2(u_2{s}_{51}+s_3){\alpha }_5+\left({\beta }_5+1\right)<0.$$

Then, the equilibrium point $({\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3,{\widehat{x}}_4,{\widehat{x}}_5)$ is stable, independent of delays.

Proof. 

We begin the proof by calculating the derivative of V:

$${\displaystyle \frac{dV}{dt}=\sum _{i=1}^{9}2{\alpha }_i{y}_i{g}_i\left(y\right)+\sum _{j=1}^9\left({\beta }_j\left[y_j^2\left(t\right)-y_j^2(t-{\tau }_j)\right]+\sum _{i\ne j}{\gamma }_{ij}\left[y_i^2\left(t\right)-y_i^2(t-{\tau }_j)\right]\right).}$$

To derive sufficient stability conditions, we need to ensure that ${\displaystyle \frac{dV}{dt}}$ is negative, which involves analyzing the terms related to $g_i\left(y\right),i=\overline{1,5}$ and constructing perfect squares to confirm that the expressions are negative.

By applying this to all complicated terms, we obtain the following conditions:

$$\left[{\displaystyle \frac{s_{14}^2}{{\beta }_1}+\frac{s_{15}^2}{{\gamma }_{21}}+\frac{s_{16}^2}{{\gamma }_{31}}+\frac{s_{15}^2}{{\gamma }_{41}}+2s_{12}^2+s_{13}^2}\right]{\alpha }_1^2+2s_{11}{\alpha }_1+\left({\beta }_1+{\gamma }_{12}+{\gamma }_{13}+1\right)<0$$

$$\left[{\displaystyle \frac{s_{23}^2}{{\beta }_2}+\frac{s_{22}^2}{{\gamma }_{12}}+\frac{s_{23}^2}{{\gamma }_{42}}}\right]{\alpha }_2^2+2s_{21}{\alpha }_2+\left({\beta }_2+{\gamma }_{21}+{\gamma }_{23}+{\gamma }_{24}+2\right)<0$$

$$\left[{\displaystyle \frac{u_1^2{s}_{36}^2}{{\beta }_3}+\frac{u_1^2{s}_{37}^2}{{\gamma }_{13}}+\frac{u_1^2{s}_{38}^2}{{\gamma }_{23}}+\frac{u_1^2{s}_{38}^2}{{\gamma }_{43}}+u_1^2{s}_{32}^2+u_1^2{s}_{33}^2+{(u_1{s}_{34}+s_2)}^2+s_{35}^2}\right]{\alpha }_3^2$$

$$+2{\alpha }_3(u_1{s}_{31}+s_1)+\left({\beta }_3+{\gamma }_{31}+{\gamma }_{34}+2\right)<0$$

$$\left[{\displaystyle \frac{u_1^2{s}_{44}^2}{{\beta }_4}+\frac{u_1^2{s}_{44}^2}{{\gamma }_{24}}+\frac{u_1^2{s}_{45}^2}{{\gamma }_{34}}+s_{42}^2+s_{43}^2}\right]{\alpha }_4^2+2s_{41}{\alpha }_4+\left({\beta }_4+{\gamma }_{41}+{\gamma }_{42}+{\gamma }_{43}+{\gamma }_{45}+3\right)<0$$

$$\left[{\displaystyle \frac{u_1^2{s}_{53}^2}{{\beta }_5}+\frac{u_1^2{s}_{54}^2}{{\gamma }_{45}}+u_1^2{s}_{52}^2}\right]{\alpha }_5^2+2(u_2{s}_{51}+s_3){\alpha }_5+\left({\beta }_5+1\right)<0.$$

By following the steps outlined in [9], we arrive at the following conditions:

$$s_{11}^2-\left[{\displaystyle \frac{s_{14}^2}{{\beta }_1}+\frac{s_{15}^2}{{\gamma }_{21}}+\frac{s_{16}^2}{{\gamma }_{31}}+\frac{s_{15}^2}{{\gamma }_{41}}+2s_{12}^2+s_{13}^2}\right]\left({\beta }_1+{\gamma }_{12}+{\gamma }_{13}+1\right)>0,s_{11}<0,{\alpha }_1\in ({\alpha }_{11},{\alpha }_{12})$$

$$s_{21}^2-\left[{\displaystyle \frac{s_{23}^2}{{\beta }_2}+\frac{s_{22}^2}{{\gamma }_{12}}+\frac{s_{23}^2}{{\gamma }_{42}}}\right]\left({\beta }_2+{\gamma }_{21}+{\gamma }_{23}+{\gamma }_{24}+2\right)>0,s_{21}<0,{\alpha }_2\in ({\alpha }_{21},{\alpha }_{22})$$

$${(u_1{s}_{31}+s_1)}^2-\left[{\displaystyle \frac{u_1^2{s}_{36}^2}{{\beta }_3}+\frac{u_1^2{s}_{37}^2}{{\gamma }_{13}}+\frac{u_1^2{s}_{38}^2}{{\gamma }_{23}}+\frac{u_1^2{s}_{38}^2}{{\gamma }_{43}}+u_1^2{s}_{32}^2+u_1^2{s}_{33}^2+{(u_1{s}_{34}+s_2)}^2+s_{35}^2}\right]$$

$$·\left({\beta }_4+{\gamma }_{41}+{\gamma }_{42}+{\gamma }_{43}+{\gamma }_{45}+3\right)>0,u_1{s}_{31}+s_1<0,{\alpha }_3\in ({\alpha }_{31},\alpha 32)$$

$$s_{41}^2-\left[{\displaystyle \frac{u_1^2{s}_{44}^2}{{\beta }_4}+\frac{u_1^2{s}_{44}^2}{{\gamma }_{24}}+\frac{u_1^2{s}_{45}^2}{{\gamma }_{34}}+s_{42}^2+s_{43}^2}\right]\left({\beta }_4+{\gamma }_{41}+{\gamma }_{42}+{\gamma }_{43}+{\gamma }_{45}+3\right)>0,$$

$$s_{41}<0,{\alpha }_4\in ({\alpha }_{41},{\alpha }_{42})$$

$${(u_2{s}_{51}+s_3)}^2-\left[{\displaystyle \frac{u_1^2{s}_{53}^2}{{\beta }_5}+\frac{u_1^2{s}_{54}^2}{{\gamma }_{45}}+u_1^2{s}_{52}^2}\right]\left({\beta }_5+1\right)>0,u_2{s}_{51}+s_3<0,{\alpha }_5\in ({\alpha }_{51},{\alpha }_{52}).$$

□

Remark 1.

The above parameter conditions are only sufficient; they are not necessary.

This study has demonstrated that using a Lyapunov–Krasovskii functional provides a viable method for assessing the stability of the equilibrium point $({\widehat{x}}_1,{\widehat{x}}_2,{\widehat{x}}_3,{\widehat{x}}_4,{\widehat{x}}_5)$. By constructing appropriate stability conditions through the first approximation method, we ensured that the derivative of the Lyapunov–Krasovskii functional is negative. Although the derived conditions are sufficient and guarantee stability, they are not necessary, meaning there could be other parameter values that also lead to stability.

4. Numerical Results and Simulations

For the numerical study, we used the parameter values found in

Table 1. Description of parameters.

Parameter Significance	Parameter	Parameter Value
Maximal value of the ${\beta }_h$ function	${\beta }_{0h}$	$1.77$
Maximal value of the ${\beta }_l$ function	${\beta }_{0l}$	$1.87$
Maximal value of the function $k_h$	$k_{0h}$	0.1
Maximal value of the function $k_l$	$k_{0l}$	$0.4$
Parameter for the ${\theta }_h$ function	${\theta }_{1h}$	0.5
Parameter for the ${\theta }_l$ function	${\theta }_{1l}$	0.5
Parameter for the function $k_h$	${\theta }_{2h}$	36
Parameter for the function $k_l$	${\theta }_{2l}$	36
Loss of stem cells due to mortality for healthy cells	${\gamma }_{1h}$	0.1
Loss of stem cells due to mortality for leukemic cells	${\gamma }_{1l}$	$0.04$
Rate of asymmetric division for healthy cells	${\eta }_{1h}$	0.7
Rate of asymmetric division for leukemic cells	${\eta }_{1l}$	$0.1$
Rate of symmetric division for healthy cells	${\eta }_{2h}$	0.1
Rate of symmetric division for leukemic cells	${\eta }_{2l}$	$0.2$
Instant mortality of mature normal leukocytes	${\gamma }_{2h}$	2.4
Instant mortality of mature leukemic leukocytes	${\gamma }_{2l}$	$0.15$
Amplification factor for normal leukocytes	$A_h$	500
Amplification factor for leukemic leukocytes	$A_l$	1500
Loss of leukemic stem cells due to cytotoxic T cells	$b_1$	0.06
Loss of mature leukemic leukocytes due to cytotoxic T cells	$b_2$	0.6
Coefficient of the feedback function $l_1$	$b_3$	2
Coefficient of the feedback function $l_2$	$b_4$	0.5
Anti-leukemia T cell death rate	$a_2$	0.4
Coefficient of influence of the regulatory mechanism	$a_3$	0.4
The number of antigen-depending divisions	$n_1$	4
Duration of cell cycle for normal stem cells	${\tau }_1$	2.8
Duration of cell cycle for normal leukocytes	${\tau }_2$	3.5
Duration of cell cycle for leukemic stem cells	${\tau }_3$	2
Duration of cell cycle for leukemic leukocytes	${\tau }_4$	$2.8$
Duration of one T cell division	${\tau }_5$	2.13

and

Table 2. Description of parameters related to treatment.

Parameter Significance	Parameter	Parameter Value
Constant dose of administrated drug (Imatinib)	K	400
Total plasma of clearance of Imatinib/volume of distribution of Imatinib	$\nu$	$0.412$
Initial number of LSC	$x_0$	$0.449$
Probability of mutation of stem cells and of developing resistance to Imatinib	p	$0.5$

.

Based on the numerical calculations, $q_1=-4.8241<0$, indicating that $E_3$ is unstable. Figure 1 addresses the equilibrium point $E_3$, where the number of healthy cells is zero. The persistence of leukemic stem cells ($x_3$), even with treatment, indicates an unstable equilibrium. The resistant population prevents full suppression, meaning that any slight perturbation (e.g., a reduction in immune response or a slight change in conditions) could allow these cells to repopulate rapidly, moving the system further from equilibrium. The low but persistent level of mature leukemic cells ($x_4$) contributes to instability because it reflects ongoing activity from resistant leukemic stem cells. This creates a feedback loop where resistant stem cells produce mature cells, keeping the leukemic cell population from reaching zero and maintaining an unstable state. The immune system ($x_5$) appears to be unable to mount an effective response against the resistant leukemic cells. This may be due to the resistant cells evading immune detection or due to the immune system being suppressed by the high burden of leukemic cells. An ineffective immune response allows resistant leukemic cells to persist and proliferate, contributing to instability. This simulation demonstrates that even with treatment, the presence of resistance in leukemic cells drives the system toward instability.

Figure 2 shows that in the absence of treatment resistance, the population of leukemic stem cells $\left(x_3\right)$ and mature leukemic cells $\left(x_4\right)$ declines significantly over time. As the leukemic cells decrease, the healthy stem cells $\left(x_1\right)$ and mature healthy cells $\left(x_2\right)$ begin to recover. The immune system $\left(x_5\right)$ remains stable, suggesting that the treatment effectively controls the leukemia and allows the healthy cell populations to stabilize.

When resistance to treatment is introduced, a stark contrast appears. Both leukemic stem cells $\left(x_3\right)$ and mature leukemic cells $\left(x_4\right)$ begin to oscillate heavily, indicating unstable dynamics and the persistence of the leukemic cell population. These oscillations suggest that the treatment becomes ineffective over time, and leukemic cells continue to proliferate. The healthy stem cells $\left(x_1\right)$ and mature healthy cells $\left(x_2\right)$ remain suppressed, unable to recover due to the continued dominance of leukemic cells.

Figure 3 demonstrates that, despite the initially high population of healthy mature cells ($x_2$), resistance allows leukemic cells to quickly dominate, driving both healthy stem and mature cell populations to near-zero. The presence of resistance (black) results in an unstable equilibrium, where leukemic cells proliferate aggressively and suppress healthy cells. Without resistance (red), treatment is effective, allowing for a more stable state with low but steady healthy cell populations. Mature cells are replenished by stem cells through differentiation. However, if there is an initially high number of mature cells ($x_2$) and low number of stem cells ($x_1$), the system may experience a delayed adjustment period as it tries to balance the mature cell population.

Due to the fact that the immune system is inhibited by a large CML cell population, it is straightforward that the plot for $x_5$ will show a decreasing evolution once the leukemic cell population reaches a certain burden.

5. Conclusions

In this study, we conducted a stability analysis for two key equilibrium points, $E_3$ and $E_4$, within a mathematical model of leukemia dynamics under treatment. The equilibrium point $E_3$ was analyzed using a rank-one perturbation approach, which provided insights into the sensitivity of this equilibrium to small disturbances. For equilibrium $E_4$, sufficient conditions for stability were established through the construction of a Lyapunov–Krasovskii functional, a method well suited to handling the delays inherent in biological systems. These analytic techniques provided a robust framework for understanding the stability properties of the system under various scenarios, including the presence and absence of resistance to treatment.

Simulations confirmed the theoretical findings, demonstrating that the presence of treatment resistance significantly impacts healing. Specifically, resistance to treatment results in the dominance of leukemic cell populations despite initial low levels. These results highlight the critical role that resistance plays in the effectiveness of treatment and underscore the challenges associated with achieving long-term control of leukemic cell populations in resistant cases.

Overall, this work provides a comprehensive analysis of leukemia dynamics under treatment with resistance, combining analytic and numerical approaches to reveal the undesired effects of resistance.