DYNAMICS OF HAEMATOPOIETIC STEM CELLS MODEL DYNAMICS OF HAEMATOPOIETIC STEM CELLS MODEL

stem cells, equilibrium, stability, differential system, delay argument.


Introduction
The population of haematopoietic stem cells gives rise to all of the differentiated elements of the blood (see [1,2]): the white blood cells, red blood cells and platelets; which may be either actively proliferating or in a resting phase. After entering the proliferating phase, a cell is committed to undergo cell division at a fixed time τ later. The generation time τ is assumed to consist of four phases: the pre-synthesis phase, the DNA synthesis phase, the post-synthesis phase and the mitotic phase. Just after the division, both daughter cells go into the resting phase. Once in this phase, they can either return to the proliferating phase and complete the cycle or die before ending the cycle. The dynamics of the model is governed by the nonlinear differential system with delay argument: (1) where β(N) is a monotone decreasing function and has the explicit form of a Hill function: n ϭ 1, 2, … .
The symbols in (1) have a following interpretation: N is the number of cells in nonproliferative phase, P is the number of cycling proliferating cells, γ is the rate of cells loss from proliferative phase, δ is the rate of cells loss from nonproliferative phase, τ is the time spent in the proliferative phase, β is the feedback function which represents the rate of recruitment from nonproliferative phase, β 0 is the maximum recruitment rate and θ, n control the shape of the feedback function. We assume that δ, γ, τ, β 0 , θ ʦ (0,∞).

Equilibria of System (1) Without Delay
For τ ϭ 0 the system (1) has a form , .
The equilibrium points of (2) we obtain from the system ϪγP ϭ 0.
Proof. The matrix of the linearized system (2) has a form and the matrix of the linearized system around E 0 ϭ (0, 0) is given by .
Then the characteristic equation of the linearized system of (2) around E 0 is This equation has two roots given by Thus the point E 0 ϭ (0, 0) is asymptotically stable equilibrium of the system (2).

Model with Delay
.
We set i.e t ϭ sτ and we get , .
Using the relations above the system (3) is transformed into the next one In the next consideration we will use the root τ of the equation We get , , . We will use the notation .
Proof. We have and implies that τ Ͼ 0.
The first equation of (5) is satisfied for For u ϭ 0 from the second equation of (5) we get that v ϭ 0.
So it cannot be μ Ն 0. Thus μ Ͻ 0 and E* is asymptotically stable equilibrium.

Conclusion
Since the differentiation and interaction in a population of haematopoietic stem cells is very complex and complicated, it is difficult to comprehend the large scale dynamics of this process without the formal structure of a mathematical model.

Acknowledgement
The research was supported by the Grant 1/1260/12 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic.