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# Asymptotic behavior of solutions of nonlinear models from erythropoiesis

- Xiao Liang
^{1}Email author

**2013**:207

https://doi.org/10.1186/1687-1847-2013-207

© Liang; licensee Springer 2013

**Received: **17 February 2013

**Accepted: **27 June 2013

**Published: **11 July 2013

## Abstract

We consider a class of nonlinear models associated with erythropoiesis and establish a global asymptotic stability result for the trivial steady state, which extends essentially some previous results. Moreover, we give numerical simulations to illustrate this theoretical result.

## Keywords

- nonlinear models
- asymptotic behavior of solutions
- erythropoiesis

## 1 Introduction

Erythropoiesis is the process by which red blood cells (erythrocytes) are formed. It is a complex process, stimulated by decreased O_{2} in circulation. Detecting this decrease, the kidneys then secrete the hormone erythropoietin. This hormone stimulates proliferation and differentiation of red cell precursors, which activates increased erythropoiesis in the hemopoietic tissues, ultimately producing red blood cells. This process (erythropoiesis) is based upon the differentiation of the hematopoietic stem cells. The hematopoietic stem cells are undifferentiated cells in a self-maintained stem cell compartment (located in the bone morrow), which are either proliferating or nonproliferating cells and have unique capacities of producing cells committed to one of the three blood cell types: red blood cells, white cells or platelets, and self-renewal. Erythrocytes (red blood cells) carry out the exchange of oxygen and carbon dioxide between the lungs and the body tissues. To effectively combine with oxygen, the erythrocytes must contain a normal amount of the red protein pigment hemoglobin (*cf.* [1]), the amount of which in turn depends on the iron level in the body. Erythrocytes are biconcave in shape, which increases the cell’s surface area and facilitates the diffusion of oxygen and carbon dioxide. From [2] and the related literature, one can get to know the following information on erythrocytes. Erythrocytes are produced primarily from the CD34^{+} pluripotent hematopoietic stem cells of bone marrow. CD34^{+} progenitors are isolated from adult peripheral blood or cord blood and grown in liquid medium in fibronectin-coated wells. These progenitor stem cells constitute approximately 0.1% of nucleated cells in the bone marrow, only about 5% of which are in cycle at any one time. Cell differentiation along the erythroid lineage occurs over a two-week span in humans. The earliest erythroid progenitor, the BFU-E (burst forming unit-erythroid), is small and without distinguishing histologic characteristics. BFU-Es express the cell surface antigen, CD34, as do all other early hematopoietic progenitors. The stage after the BFU-E is the CFU-E (colony forming unit-erythroid), which is larger and is the stage right before hemoglobin production begins. Immature erythroblasts, which start producing hemoglobin, also start condensing their nuclei. Mature erythroblasts are smaller with tightly compacted nuclei which are expelled as the cells become reticulocytes. Cell division ceases with the formation of the orthochromatic erythroblast. Division rate, death rate, and maturation rate are influenced by the level of erythropoietin. The erythrocyte lineage shares the precursor CFU-GEMM (granulocyte, erythrocyte, macrophage, megakaryocyte) with other types of blood cells (white blood cells, platelets, *etc.*). The process of erythropoiesis has been modeled in many physiological scenarios. For more information about erythropoiesis and the differentiation of the hematopoietic stem cells, we refer the reader to [1–6] and references therein.

*cf.*[3, 5–10]). Stimulated and inspired by these works, especially by [7], we investigate the following general nonlinear system with time delay

*τ*, corresponding to the cell cycle duration where the cells in cycle are divided only in two groups: proliferating and nonproliferating cells,

*t*respectively, $\delta >0$ is the rate of differentiation of nonproliferating cells,

*τ*is the average duration of the proliferating phase and the term ${e}^{-\delta \tau}$ then describes the survival rate of proliferating cells, $\beta (\cdot )$ is a continuously differentiable, positive, and decreasing function with

*L*is a positive constant) describing the nonlinear change of the number of nonproliferating cell compartments. This nonlinear system is more general than those in the previous literature since we take the change (being nonlinear in many cases) of the number of nonproliferating cell compartments into account. A typical example of

*f*is the following function:

where $\mu >0$, $k>0$, and $r>1$ are constants. This is a single-humped function of *x*, which was first considered for modeling the hematopoietic stem cells’ dynamics in [6] by Mackey and Glass. With such a nonlinear term, the study of system (1.1)-(1.2), which is a generalized Mackey-Glass-type model, is more complicated than that of previous models. The purpose of the paper is to establish a global asymptotic stability result for the trivial steady state of this class of systems, which shows that in many cases, the hematopoietic stem cell population is definitely extinct. The next section is devoted to proving this criterion. In the last section, we give numerical simulations to illustrate this phenomenon.

## 2 Asymptotic behavior of solutions

It follows from many books on delay equations (*e.g.*, [9]) that for each continuous initial condition, system (1.1)-(1.2) has a unique continuous solution $(S(t),N(t))$ for $t\ge 0$. For more information on delay equations, see, *e.g.*, [7, 11–15].

**Theorem 2.1** *Let* $(S(t),N(t))$ *be a solution of* (1.1)-(1.2) *for a positive initial datum*. *Then* $S(t)$ *and* $N(t)$ *are positive for* $t\ge 0$.

*Proof*To obtain the required conclusion, we will show that if $(S(t),N(t))$ is a solution of (1.1)-(1.2) for a positive initial datum, then the following situations:

- (i)
neither $S(t)$ nor $N(t)$ is positive;

- (ii)
$N(t)$ is positive but not $S(t)$;

- (iii)
$S(t)$ is positive but not $N(t)$;

do not appear definitely.

Step 1: We show that it is impossible that neither $S(t)$ nor $N(t)$ is positive.

Suppose that neither $S(t)$ nor $N(t)$ is positive. Then $S(t)$ has one zero point in $[0,+\mathrm{\infty})$ at least, and so does $N(t)$. Let ${t}_{0}$ and ${t}_{1}$ be the first zero point of $S(t)$ and $N(t)$ in $[0,+\mathrm{\infty})$, respectively. Then, by $S(0)>0$ and $N(0)>0$, we know that ${t}_{0}>0$ and ${t}_{1}>0$.

Case 1: ${t}_{0}\le {t}_{1}$.

*i.e.*, $S(t)>0$ and $N(t)>0$ for $t\in [-\tau ,0]$) that

This contradicts (2.1).

Case 2: ${t}_{1}\le {t}_{0}$.

*i.e.*, $S(t)>0$ and $N(t)>0$ for $t\in [-\tau ,0]$) imply that

This contradicts (2.3).

Consequently, it is impossible that neither $S(t)$ nor $N(t)$ is positive.

Step 2: We prove that it is impossible that $N(t)$ is positive but not $S(t)$.

So, (2.1) is true. Moreover, by the positivity of the initial datum and $N(t)$, we get (2.2) from (1.2)-(1.4), which contradicts (2.1). Hence, it is impossible that $N(t)$ is positive but not $S(t)$.

Step 3: We prove that it is impossible that $S(t)$ is positive but not $N(t)$.

Thus, we get (2.3). On the other hand, by the positivity of the initial datum and $S(t)$, we have (2.2) from (1.2)-(1.4), which contradicts (2.4). Therefore, it is impossible that $S(t)$ is positive but not $N(t)$.

In conclusion, if $(S(t),N(t))$ is a solution of (1.1)-(1.2) for a positive initial datum, then $S(t)$ and $N(t)$ are positive. □

**Theorem 2.2** *Let* $(S(t),N(t))$ *be a solution of* (1.1)-(1.2) *for a positive initial datum*. *If* $N(t)$ *is bounded*, *then so is* $S(t)$.

*Proof* By Theorem 2.1, we know that $(S(t),N(t))$ is positive.

since *β* is decreasing. This means that $S(t)$ is bounded. □

**Theorem 2.3** *Let* $(S(t),N(t))$ *be a solution of* (1.1)-(1.2) *for a positive initial datum*, *and* ${lim}_{t\to +\mathrm{\infty}}N(t)=0$. *Then* ${lim}_{t\to +\mathrm{\infty}}S(t)=0$.

*Proof* From ${lim}_{t\to +\mathrm{\infty}}N(t)=0$, it follows that (2.6) holds for some positive constant *M*.

where $\xi \in [\frac{t+\tau}{2},t]$.

□

**Theorem 2.4**

*Let*$(S(t),N(t))$

*be a solution of*(1.1)-(1.2)

*for a positive initial datum*,

*and*

*Assume that*

*Then*

*and*

*Proof*By Theorem 2.1, we see that

Hence, (2.9) is true.

Case 1: $2{e}^{-\delta \tau}-1\le 0$.

*β*, (2.11), (1.4) and (2.12), we get

Case 2: $2{e}^{-\delta \tau}-1>0$.

*β*, (2.11), (1.4), (2.12) and (2.8), we obtain

□

**Theorem 2.5**

*Let*$(S(t),N(t))$

*be a solution of*(1.1)-(1.2)

*for a positive initial datum*,

*and*$Y(t)$

*be given by*(2.7).

*Assume that*(2.8)

*holds*.

*Then*

*Proof* It follows from Theorem 2.4 that $Y(t)$ is decreasing and lower bounded by 0 for $t\ge \tau $. So, ${inf}_{t\ge \tau}Y(t)$ exists.

Hence, $Y(t)$ is bounded on $[\tau ,+\mathrm{\infty})$. Consequently, $N(t)$ is bounded on $[0,+\mathrm{\infty})$, and Theorem 2.2 shows that $S(t)$ is also bounded on $[0,+\mathrm{\infty})$.

*β*and

*f*, we know that

By virtue of (2.16), we know that (2.15) is true, owing to the following known result by Barbălat (see Gopalsamy [15]):

*Let*$g:[a,+\mathrm{\infty})\to R$

*,*$a\in R$

*, be a differentiable function. If*${lim}_{t\to +\mathrm{\infty}}g(t)$

*exists and*${g}^{\prime}(t)$

*is uniformly continuous on*$(a,+\mathrm{\infty})$

*, then*

□

**Theorem 2.6** *Let* $(S(t),N(t))$ *be a solution of* (1.1)-(1.2) *for a positive initial datum*, *and let* (2.8) *hold*. *Then* $(S(t),N(t))$ *tends to* $(0,0)$.

*Proof*First, we prove that

*n*, there is ${t}_{n}>n$ such that

where *α* is a constant. Therefore, we have the following observations.

Case 1: $2{e}^{-\delta \tau}-1\le 0$.

which contradicts (2.15).

Case 2: $2{e}^{-\delta \tau}-1>0$.

*β*, (2.11), (1.4), (2.12) and (2.8), we have

which also contradicts (2.15).

□

## 3 Numerical simulations

With Matlab software we give two numerical simulations of the main result, Theorem 2.6.

Both numerical simulations illustrate our result very well and show that for these cases, the hematopoietic stem cell population is definitely extinct.

In this paper, we investigated the asymptotic behavior of solutions of some nonlinear delay models of hematopoietic stem cell dynamics. The nonlinearity depends upon the entire hematopoietic stem cell population as well as the nonlinear change of the number of nonproliferating cell compartments, which is different from the models considered in the previous literature on blood cell production models (*cf.* [7] and references therein). Moreover, this class of models covers essentially those in the previous works. By using arguments different from [7], we proved Theorems 2.1 and 2.3. By employing a new analysis process, we overcame the difficulty caused by the nonlinear term *f* and proved Theorem 2.6. As special cases, we can deduce the corresponding results given in these works from Theorems 2.1-2.6. Finally, by giving a numerical investigation, we illustrated efficiently the asymptotic stability of the solutions to some models of the cell population dynamics. For further analysis, the stability of the models with appropriate feedback controls is a good issue.

## Declarations

### Acknowledgements

I would like to thank the anonymous referees and Prof. Z Lei for their valuable comments and suggestions.

## Authors’ Affiliations

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