# Stability Analysis of Equilibria for a Model of Maintenance Therapy in Acute Lymphoblastic Leukemia

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## Abstract

**:**

## 1. Introduction

#### 1.1. Mathematical Background

#### 1.2. Biological Background

## 2. **The Modeling of Erythropoiesis**

#### 2.1. The Mathematical Model

#### 2.2. The Equilibrium Points and Linearization

#### 2.3. Stability Analysis of the Equilibrium Point ${E}_{1}$

#### 2.3.1. The Real Solutions of the Characteristic Equation

#### 2.3.2. Analysis of the Critical Case

**Theorem**

**1.**

**.**Consider the following nonlinear system with time delays:

#### 2.3.3. The Transcendental Part of the Characteristic Equation

**Proposition**

**1.**

**Proof.**

#### 2.4. Stability Analysis of the Equilibrium Point ${E}_{2}$

**Remark**

**2.**

#### 2.5. Numerical Simulations

## 3. The Leukopoiesis Model

#### 3.1. The Mathematical Model

#### 3.2. The Equilibrium Points and Linearization

#### 3.3. Stability Analysis for Equilibrium Point ${\tilde{E}}_{1}$

**Proposition**

**2.**

#### 3.4. Stability Analysis of the Equilibrium Point ${\tilde{E}}_{2}$

#### 3.4.1. The Real Solutions of the Characteristic Equation

#### 3.4.2. The Transcendental Part of the Characteristic Equation

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Remark**

**4.**

#### 3.5. Numerical Simulations

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Abass, A.K.; Lichtman, A.H.; Pillai, S. Cellular and Molecular Immunolgy, 7th ed.; Elsevier: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Adimy, M.; Crauste, F.; Ruan, S. A mathematical study of the hematopoiesis process with application to chronic myelogenous leukemia. SIAM J. Appl. Math.
**2005**, 65, 1328–1352. [Google Scholar] [CrossRef] [Green Version] - Akushevich, I.; Veremeyeva, G.; Dimov, G.; Ukraintseva, S.; Arbeev, K.; Akleyev, A.; Yashin, A. Modeling hematopoietic system response caused by chronic exposure to ionizing radiation. Radiat. Environ. Biophys.
**2011**, 50, 299–311. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Childhood, A.; Collaborative Group. Duration and intensity of maintenance chemotherapy in acute lymphoblastic leukemia: Overview of 42 trials involving 12,000 randomized children. Lancet
**1996**, 347, 1783–1788. [Google Scholar] - Pui, C.H.; Evans, W.E. Treatment of acute lymphoblastic leukemia. N. Engl. J. Med.
**2006**, 354, 166–178. [Google Scholar] [CrossRef] [PubMed] - Lin, T.L.; Vala, M.S.; Barber, J.P.; Karp, J.E.; Smith, B.D.; Matsui, W.; Jones, R.J. Induction of acute lymphocytic leukemia differentiation by maintenance therapy. Leukemia
**2007**, 21, 1915–1920. [Google Scholar] [CrossRef] [PubMed] - Badralexi, I.; Halanay, A.; Mghames, R. A Delay Differential Equations model for maintenance therapy in acute lymphoblastic leukemia. UPB Sci. Bull. Ser. A
**2020**, 82, 13–24. [Google Scholar] - Amin, K.; Badralexi, I.; Halanay, A.; Mghames, R. A stability theorem for equilibria of delay differential equations in a critical case and some models of cell evolution. Math. Model. Nat. Phenom.
**2021**, 16, 36. [Google Scholar] [CrossRef] - Adimy, M.; Bourfia, Y.; Hbid, M.L.; Marquet, C. Age-structured model of hematopoiesis dynamics with growth factor-dependent coefficients. Electr. J. Diff. Equ.
**2016**, 2016, 1–20. [Google Scholar] - Munker, R.; Hiller, E.; Glass, J.; Paquette, R. Modern Hematology. Biology and Clinical Management, 2nd ed.; Humana Press: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Parajdi, L.G.; Precup, R.; Bonci, E.; 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. [Google Scholar] [CrossRef] [Green Version] - Greer, J.; Arber, D.; Glader, B.; List, A.; Means, R., Jr.; Paraskevas, F.; Rodgers, G. Wintrobe’s Clinical Hematology, 13th ed.; LWW: Philadelphia, PA, USA, 2013. [Google Scholar]
- Schmiegelow, K.; Nielsen, S.; Frandsen, T.; Nersting, J. Mercaptopurine/Methotrexate Maintenance Therapy of Childhood Acute Lymphoblastic Leukemia. Clinical Facts and Fiction. J. Pediatr.
**2014**, 36, 503–517. [Google Scholar] - Colijn, C.; Mackey, M.C. A Mathematical Model for Hematopoiesis: I. Periodic Chronic Myelogenous Leukemia. J. Theor. Biol.
**2005**, 237, 117–132. [Google Scholar] [CrossRef] [PubMed] - Jayachandran, D.; Rundell, A.E.; Hannemann, R.; Vik, T.A.; Ramkrishna, D. Optimal Chemotherapy for Leukemia: A model-Based Strategy for Individualized Treatment. PLoS ONE
**2014**, 9, e109623. [Google Scholar] [CrossRef] [Green Version] - Bellman, R.; Cooke, K.L. Differential-Difference Equations; Academic Press: New York, NY, USA, 1963. [Google Scholar]
- Cooke, K.; Grossman, Z. Discrete Delay, Distribution Delay and Stability Switches. J. Math. Anal. Appl.
**1982**, 86, 592–627. [Google Scholar] [CrossRef] [Green Version] - Balea, S.; Halanay, A.; Neamtu, M. A feedback model for leukemia including, cell competition and the action of the immune system. In Proceedings of the ICNPAA World Congress, Narvik, Norway, 15–18 July 2014; Volume 1637, pp. 1316–1324. [Google Scholar]
- Cooke, K.; van den Driessche, P. On Zeroes of Some Transcendental Equations. Funkcialaj Ekvacioj
**1986**, 29, 7790. [Google Scholar]

**Figure 3.**Stability of equilibrium point $\tilde{{E}_{1}}$: (

**a**) Evolution of the white blood cells’ precursors; (

**b**) Evolution of the adult leukocytes.

**Figure 4.**Stability of equilibrium point $\tilde{{E}_{2}}$: (

**a**) Evolution of the white blood cells’ precursors; (

**b**) Evolution of the adult leukocytes.

Maximal value of the function ${\mathit{\beta}}_{\mathit{e}}$ [9,14] | ${\mathit{\beta}}_{0\mathit{e}}$ | $1.5$ |

Maximal value of the function ${k}_{e}$ [9] | ${k}_{0e}$ | $0.1$ |

Parameter for the death rate [14] | $\alpha $ | $0.8$ |

Loss of stem cells due to mortality [9] | ${\gamma}_{0}$ | $0.1$ |

Rate of asymmetric/symmetric division [18] | ${\eta}_{1e},{\eta}_{2e}$ | $0.3$ |

Parameter in the Hill function [18] | m | 2 |

Standard half-saturation (estimated) | ${a}_{1}$ | 3 |

Instant mortality of mature leukocytes [9] | ${\gamma}_{2}$ | $0.025$ |

Amplification factor [9] | $\tilde{A}$ | 2400 |

Maximum effect of drug on erythrocytes [15] | ${\tilde{R}}_{m}$ | $0.0022$ |

Saturation constant for drug on erythrocytes [15] | ${\tilde{R}}_{50}$ | $82.2$ |

The supply rate of the 6-MP in the gut [15] | ${a}_{2}$ | ${3.9\times 10}^{8}$ |

6-MP absorption rate from the gut [15] | ${b}_{1}$ | $4.8$ |

6-MP elimination rate from plasma [15] | ${e}_{1}$ | 5 |

6-MP to 6-TCN conversion rate [15] | ${c}_{1}$ | $29.8$ |

Activity of TPMT enzyme [15] | ${e}_{2}$ | $0.5$ |

MM constant for 6-TGN [15] | ${c}_{2}$ | ${4.04\times 10}^{5}$ |

MeMP elimination rate from erythrocytes [15] | ${m}_{2}$ | $0.06$ |

MM constant for MeMP [15] | ${m}_{1}$ | ${3.28\times 10}^{5}$ |

Stoichiometric coefficient for 6-TGN conversion [15] | ${v}_{pt}$ | 1 |

6-TGN elimination rate from erythrocytes [15] | ${e}_{3}$ | $0.0714$ |

Self-renewal duration of erythrocytes [14] | ${\tau}_{1}$ | $2.8$ |

Differentiation duration of erythrocytes [14] | ${\tau}_{2}$ | 6 |

Maximal value of the function ${\mathit{\beta}}_{\mathit{l}}$ [9,14] | ${\mathit{\beta}}_{0\mathit{l}}$ | $1.5$ |

Maximal value of the function ${k}_{l}$ [9] | ${k}_{0l}$ | $0.1$ |

Loss of stem cells due to mortality [9] | ${\gamma}_{1l}$ | $0.1$ |

Rate of asymmetric/ symmetric division [18] | ${\eta}_{1l},{\eta}_{2l}$ | $0.3$ |

Parameter in the Hill function [18] | ${m}_{l}$ | 2 |

Standard half-saturation (estimated) | ${a}_{1}$ | 3 |

Instant mortality of mature leukocytes [9] | ${\gamma}_{2}$ | $0.025$ |

Amplification factor [9] | $\tilde{A}$ | 2400 |

Maximum effect of drug on leukocytes [15] | ${T}_{1}$ | $0.0782$ |

The supply rate of the 6-MP in the gut [15] | ${a}_{2}$ | ${3.9\times 10}^{8}$ |

6-MP absorption rate from the gut [15] | ${b}_{1}$ | $4.8$ |

6-MP elimination rate from plasma [15] | ${e}_{1}$ | 5 |

6-MP to 6-TCN conversion rate [15] | ${c}_{1}$ | $29.8$ |

Activity of TPMT enzyme [15] | ${e}_{2}$ | $0.5$ |

MM constant for 6-TGN [15] | ${c}_{2}$ | ${4.04\times 10}^{5}$ |

MeMP elimination rate from leukocytes [15] | ${m}_{2}$ | $0.06$ |

MM constant for MeMP [15] | ${m}_{1}$ | ${3.28\times 10}^{5}$ |

Stoichiometric coefficient for 6-TGN Conversion [15] | ${v}_{pt}$ | 1 |

6-TGN elimination rate from leukocytes [15] | ${e}_{3}$ | $0.1207$ |

Self-renewal duration of leukocytes [14] | ${\tau}_{1}$ | $1.4$ |

Differentiation duration of leukocytes [14] | ${\tau}_{2}$ | $3.5$ |

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**MDPI and ACS Style**

Badralexi, I.; Halanay, A.-D.; Mghames, R.
Stability Analysis of Equilibria for a Model of Maintenance Therapy in Acute Lymphoblastic Leukemia. *Mathematics* **2022**, *10*, 313.
https://doi.org/10.3390/math10030313

**AMA Style**

Badralexi I, Halanay A-D, Mghames R.
Stability Analysis of Equilibria for a Model of Maintenance Therapy in Acute Lymphoblastic Leukemia. *Mathematics*. 2022; 10(3):313.
https://doi.org/10.3390/math10030313

**Chicago/Turabian Style**

Badralexi, Irina, Andrei-Dan Halanay, and Ragheb Mghames.
2022. "Stability Analysis of Equilibria for a Model of Maintenance Therapy in Acute Lymphoblastic Leukemia" *Mathematics* 10, no. 3: 313.
https://doi.org/10.3390/math10030313