# A Mathematical Model of the Transition from Normal Hematopoiesis to the Chronic and Accelerated-Acute Stages in Myeloid Leukemia

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## Abstract

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## 1. Introduction

#### Medical Background

**Hematopoiesis**is the process of blood cell formation. The process starts in the intrauterine life in the mesoderm of the yolk sac and continues in the liver and the spleen during the second and seventh month. It then takes place at the level of the bone marrow, where it carries on after birth. During childhood, hematopoiesis takes place in almost all bones, gradually being replaced with growth by fat tissue. In adults, hematopoiesis occurs only in the pelvis, vertebrae, sternum (Howard et al. [43] and Young [44]), ribs, skull, proximal humerus, and femur epiphysis (Kaushansky et al. [45]).

**Leukemias**are an heterogeneous group of malignant disorders, also known as cancer, arising from one mutant hematopoietic stem cell (mHSC) (Howard et al. [43], Jilkine and Gutenkunst [35] Driessens et al. [36], Klein et al. [37], Lopez-Garcia et al. [38], and Snippert et al. [39]). In this study, we analyze the dynamics of HSCs and mHSCs by assuming that at least one mHSC can be found in the human body. Therefore, the complex biological processes on which the hematopoiesis is based are not completely involved.

**Chronic Myeloid Leukemia**(CML) is an acquired myeloproliferative disorder (Howard et al. [43] and Neiman [7]). The CML is probably the first recognized leukemia, dating back to the 1840s (Young [44]).

## 2. The Mathematical Model

#### 2.1. The Normal-Leukemic Dynamic System

**(a) Monotonicity of the solutions.**The function $x\left(t\right)$ increases during the time intervals where $dx/dt>0$, i.e., $a/(1+{b}_{1}x\left(t\right)+{b}_{2}y\left(t\right))-c>0$, or equivalently $x\left(t\right)+({b}_{2}/{b}_{1})y\left(t\right)<d$. Hence,

**(b) Steady states.**A steady state (or an equilibrium) is a constant solution, i.e., a solution for which $dx/dt=dy/dt=0$. Hence, the steady states are obtained by solving the algebraic system

**(c) Stability.**We study the stability of the steady states of the system (4)–(5) using the standard first approximation method (for details see Kaplan and Glass [60], Coddington and Levinson [61] and Jones et al. [62]). According to this method, an equilibrium $(\alpha ,\beta )$ is asymptotically stable if the Jacobian matrix $J(\alpha ,\beta )$ is a Hurwitz matrix, i.e., Re $\lambda <0$ for all its characteristic roots $\lambda $, and is unstable if Re $\lambda >0$ for at least one of its characteristic roots.

- If $D<d$, then the steady state $(d,0)$ is asymptotically stable, and the steady state $(0,D)$ is unstable.
- If $d<D<({b}_{1}/{b}_{2})d$, then the steady state $({x}^{*},{y}^{*})$ is positive and asymptotically stable, and the steady states $(d,0)$ and $(0,D)$ are unstable.
- If $D>({b}_{1}/{b}_{2})d$, then the steady state $(0,D)$ is asymptotically stable, and the steady state $(d,0)$ is unstable.

#### 2.2. The Mathematical–Biological Interpretation

## 3. Numerical Simulation of the Model

#### 3.1. Parameter Estimations

#### 3.2. Numerical Simulations

## 4. The Model Extended to Terminally Differentiated Cells

## 5. Discussion and Conclusions

- The mHSCs proliferation rate is a predictive factor for the development of the accelerated-acute state: an increased rate of proliferation of these cells in comparison to normal stem cells determines the accelerated-acute phase to occur earlier;
- The death rate of leukemic stem cells is predictive for the global evolution of the disease, influencing the shifts between the different phases of the chronic myeloid leukemia.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diagram of the transition from the normal hematopoiesis to the chronic and accelerated-acute stages in myeloid leukemia. Values of D less than d correspond to the normal hematopoietic state; values of D between d and $({b}_{1}/{b}_{2})d$ correspond to the chronic phase of leukemia; values of D larger than $({b}_{1}/{b}_{2})d$ characterize the accelerated-acute phase of the disease.

**Figure 2.**Phase portrait of the two-dimensional system (4)–(5), in the normal state (

**a**) $D<d$; in the chronic phase (

**b**) $d<D<({b}_{1}/{b}_{2})d;$ and in the accelerated-acute phase (

**c**) $({b}_{1}/{b}_{2})d<D$. The orbits $\left(x\right(t),y(t\left)\right)$ approach the unique asymptotically stable equilibrium (represented by a thickened red point): $(d,0)$, in case (

**a**); $({x}^{*},{y}^{*})$, in case (

**b**); $(0,D)$, in case (

**c**).

**Figure 3.**Behavior of the normal and abnormal (leukemic) stem cell populations in Case I. Initial conditions: (

**a**) $x\left(0\right)=1.5\times {10}^{4}$, $y\left(0\right)=5\times {10}^{3}$; (

**b**) $x\left(0\right)=2\times {10}^{4}$, $y\left(0\right)=1\times {10}^{3}$; (

**c**) $x\left(0\right)=2\times {10}^{4}$, $y\left(0\right)=1$.

**Figure 4.**Behavior of the normal and leukemic stem cell populations in Case II (accelerated-acute phase). Initial conditions: $x\left(0\right)=2\times {10}^{4}$, $y\left(0\right)=1$.

**Figure 5.**Behavior of the normal and abnormal (leukemic) stem cell populations in Case III. Initial conditions: (

**a**) $x\left(0\right)=1.5\times {10}^{4}$, $y\left(0\right)=5\times {10}^{3}$; (

**b**) $x\left(0\right)=2\times {10}^{4}$, $y\left(0\right)=5\times {10}^{3}$; (

**c**) $x\left(0\right)=2\times {10}^{4}$, $y\left(0\right)=1$.

**Figure 6.**Behavior of the normal and leukemic stem cell populations in Case IV. Initial conditions: (

**a**) $x\left(0\right)=1.5\times {10}^{4}$, $y\left(0\right)=5\times {10}^{3}$; (

**b**) $x\left(0\right)=2\times {10}^{4}$, $y\left(0\right)=5\times {10}^{3}$; (

**c**) $x\left(0\right)=2\times {10}^{4}$, $y\left(0\right)=1$.

**Figure 7.**Behavior of (

**a**) stem cell populations, (

**b**) progenitor cell populations, (

**c**) differentiated cell populations, and (

**d**) terminally differentiated cell populations for the parameter values: ${a}_{1}=0.005$, ${a}_{2}=4$, ${a}_{3}=5$, ${a}_{4}=100$, ${b}_{1}=0.75\times {10}^{-5}$, ${b}_{2}=0.38\times {10}^{-5}$, ${c}_{1}=0.002$, ${c}_{2}=0.008$, ${c}_{3}=0.05$, ${c}_{4}=1$, ${A}_{1}=0.01$, ${A}_{2}=8$, ${A}_{3}=10$, ${A}_{4}=100$, $B=0.19\times {10}^{-5}$, ${C}_{1}=0.004$, ${C}_{2}={c}_{2}$, ${C}_{3}={c}_{3}$, ${C}_{4}={c}_{4}$, and initial conditions: ${x}_{1}\left(0\right)=2\times {10}^{5}$, ${x}_{2}\left(0\right)=1\times {10}^{8}$, ${x}_{3}\left(0\right)=1\times {10}^{10}$, ${x}_{4}\left(0\right)=1\times {10}^{12}$, ${y}_{1}\left(0\right)={y}_{2}\left(0\right)={y}_{3}\left(0\right)={y}_{4}\left(0\right)=1$.

**Table 1.**The numerical simulation cases. $a,A$ = growth rates; ${b}_{1},{b}_{2},B$ = bone marrow microenvironment sensitivity; $c,C$ = death rates; $a,{b}_{1},{b}_{2},c$ = normal stem cell parameters; and $A,B,C$ = abnormal (leukemic) stem cell parameters.

Case I | Case II | Case III | Case IV |
---|---|---|---|

$a<A$ | $a<A$ | $a>A$ | $a>A$ |

$c<C$ | $c>C$ | $c<C$ | $c>C$ |

${b}_{1}>{b}_{2}>B$ | ${b}_{1}>{b}_{2}>B$ | ${b}_{1}>{b}_{2}>B$ | ${b}_{1}>{b}_{2}>B$ |

**Table 2.**Parameter values for simulations. S-S = steady state; $d=2\times {10}^{4}$ (normal); D = variable parameter (leukemic).

Figure | a | ${\mathit{b}}_{1}\times {10}^{-4}$ | ${\mathit{b}}_{2}\times {10}^{-4}$ | c | A | $\mathit{B}\times {10}^{-4}$ | C | $\mathit{S}-\mathit{S}$ |
---|---|---|---|---|---|---|---|---|

Figure 3a | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.01$ | $0.19$ | $0.009$ | $(d,0)$ |

Figure 3b | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.01$ | $0.19$ | $0.007$ | $({x}^{*},{y}^{*})$ |

Figure 3c | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.01$ | $0.19$ | $0.004$ | $(0,D)$ |

Figure 4 | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.007$ | $0.19$ | $0.001$ | $(0,D)$ |

Figure 5a | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.004$ | $0.19$ | $0.003$ | $(d,0)$ |

Figure 5b | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.0045$ | $0.19$ | $0.003$ | $({x}^{*},{y}^{*})$ |

Figure 5c | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.0045$ | $0.19$ | $0.0025$ | $(0,D)$ |

Figure 6a | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.0012$ | $0.19$ | $0.001$ | $(d,0)$ |

Figure 6b | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.0015$ | $0.19$ | $0.001$ | $({x}^{*},{y}^{*})$ |

Figure 6c | $0.005$ | $0.75$ | $0.38$ | $0.002$ | $0.0025$ | $0.19$ | $0.001$ | $(0,D)$ |

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**MDPI and ACS Style**

Parajdi, L.G.; Precup, R.; Bonci, E.A.; Tomuleasa, C.
A Mathematical Model of the Transition from Normal Hematopoiesis to the Chronic and Accelerated-Acute Stages in Myeloid Leukemia. *Mathematics* **2020**, *8*, 376.
https://doi.org/10.3390/math8030376

**AMA Style**

Parajdi LG, Precup R, Bonci EA, Tomuleasa C.
A Mathematical Model of the Transition from Normal Hematopoiesis to the Chronic and Accelerated-Acute Stages in Myeloid Leukemia. *Mathematics*. 2020; 8(3):376.
https://doi.org/10.3390/math8030376

**Chicago/Turabian Style**

Parajdi, Lorand Gabriel, Radu Precup, Eduard Alexandru Bonci, and Ciprian Tomuleasa.
2020. "A Mathematical Model of the Transition from Normal Hematopoiesis to the Chronic and Accelerated-Acute Stages in Myeloid Leukemia" *Mathematics* 8, no. 3: 376.
https://doi.org/10.3390/math8030376