ASYMPTOTIC BEHAVIOR OF A SINGULAR TRANSPORT EQUATION MODELLING CELL DIVISION

This paper analyses the behavior of the solutions of a model of cells that are capable of simultaneous proliferation and maturation. This model is described by a first-order singular partial differential system with a retardation of the maturation variable and a time delay. Both delays are due to cell replication. We prove that uniqueness and asymptotic behavior of solutions depend only on cells with small maturity (stem cells).

1. Introduction.Mathematical models of biological populations structured in age or maturity appeared in various contexts more than forty years ago.In particular, there exists a vast literature on models for progress through the cell cycle and its various phases.See for example the references given in Metz and Diekmann (1986), and in Gyllenberg and Heijmans (1987).More recently Mackey andRudnicki (1994 and1999) studied a particular time-and-age maturation structured model of biological process of hematological cell development in bone marrow.This model is a generalization of those that have been introduced previously both in the absence of maturation by Mackey (1978 and1979) and presence of maturation with only one phase by Rey andMackey (1992 and1993).They considered a mathematical model for the dynamics of a population of cells which can be distinguished from each other according to their maturity and their position in the cell cycle phase.Mackey andRudnicki (1994 and1999) assumed that the cell cycle consists of two distinct phases.The cells in the first phase (resting phase) cannot divide, they mature, and provided they do not die, they eventually enter the second phase.In the second phase (proliferating phase) the cells are committed to undergo cell division a time τ later.The position of a cell in each phase is denoted by a (cell age) which is assumed to range in the proliferating phase from a = 0 (the point of commitment) to a = τ (the point of cytokinesis), and in the resting phase from a = 0, when the cell enters, to a = +∞.The maturity variable m represents the concentration of what composes a cell such as proteins, or other elements one can measure experimentally.The maturity variable m can range, without loss of generality, from 0 to 1. Cells can be lost in the proliferating phase with a rate γ(m) and in the resting phase with a rate δ(m).At the point of cytokinesis, a cell with maturity m divides into two daughter cells with maturity g(m) ≤ m, that enter directly the resting phase.In the resting phase, cells can return to the proliferating phase with a rate β and complete the cycle.The cells of both types mature with the same velocity V (m).In the model considered by Mackey andRudnicki (1994 and1999), the main results concern the global stability for the following first order partial differential equation where ∆ : [0, 1] → [0, 1] is a continuous function such that ∆(0) = 0 and ∆(m) < m, for m ∈ (0, 1].These authors gave in 1999 a criterion for global stability of Equation (1).However, they considered only the case when the term f (u, v) in Equation ( 1) does not depend on the maturity variable.This restriction implies that the rates of mortality in the two phases, the rate of return from the resting to the proliferating phase and V (m) are independent of the maturity variable.This allowed Mackey and Rudnicki to introduce the following delay differential equation and to establish the connection between the global solution behavior of the equation ( 2) and the local and global solution behavior of the partial differential equations (1).More exactly, they proved that local stability of Equation ( 1) and global stability of Equation (2) yield to global stability of (1).This model has also been analyzed by Dyson et al. (1996a and1996b) in the particular case when the maturity X(−τ, g −1 (m)) is independent of τ and is equal to αm, with 0 < α < 1.In these studies, it is shown that the behavior of solutions is dependent upon the stem cells.Numerical studies performed by Rey and Mackey (1993) suggest such a result.For other recent developments concerning properties of similar equations see for example Dyson et al. (2000a and2000b) and Bernard et al. (2001) The model investigated in the present paper is more general than those considered by Mackey andRudnicki in (1994 and1999).It takes into account the maturity variable.In particular, we consider a general velocity of maturation V (m) and a general function g(m) which is the maturity of the two daughter cells if m is the maturity of their mother.Moreover, the rates of mortality and the rate of return from the resting to the proliferating phase are, in our paper, depending on the maturity variable.We prove, under local conditions on the rates of mortality and the rate of return from the resting to the proliferating phase, exponential stability of the trivial solution of our system.This result is obtained under the assumption that the proliferating phase is long enough, We also obtained under Condition (3), that if the initial population of cells ϕ(t, m) is 0 for the maturity m in [0, b], where b > 0 can be chosen as small as we want (stem cells), then the population goes to extinction at a finite time.Moreover, if ϕ(t, 0) = 0 (primitive cells), then N (t, m) → 0 as t → ∞.In the interpretation of these results the case of a sufficient supply of stem cells implies a normal blood production and the case of injury or destruction of stem cells or primitive cells corresponds to an abnormal production (the aplastic anemia).
2. Presentation of the model.
The conservation equation for the proliferating phase is with the initial condition where Here We assume that the rate The conservation equation for the resting phase is We suppose that δ and β are continuous functions.

Boundary conditions.
The cellular flux between the two phases is given by the system where g(m) represents the maturity of the daughter cells when m is the maturity of their mother.
The first boundary condition describes the flux of daughter cells into the resting phase just after the division of their mother.Note that the maturity m of the daughter cells just after division is smaller than g (1).This fact is expressed by the following boundary condition n(t, m, 0) = 0, for m > g( 1).
(7) The second condition in ( 6) represents the re-entry of resting cells into the proliferating phase.We assume that g : 3. Equations for N and P .We define the characteristic curve s → X(s, m) through (0, m), m ∈ [0, 1], in the two phases by the ordinary differential equation We have X(0, m) = m and X(s, 0) = 0, for s ∈ R and m ∈ [0, 1].Note that the expression X(s, m) appears in our model only for s ≤ 0. Furthermore, one can prove that X(s, .) is given explicitly, for s ≤ 0 and m ∈ [0, 1] , by Using the method of characteristics, one can derive evolution equations for N (m, t) and P (m, t) : If m ∈ [g(1), 1] and t ≥ 0, Since our objective in this work is to prove that uniqueness and asymptotic behavior of the solutions depend only on the cells with small maturity (stem cells), we will focus our study on the production process given by Equation ( 14) on the maturity interval [0, g (1)].
The first term on the right-hand side of Equation ( 14) describes the loss due to death and transition to the proliferating phase.The second term describes the birth of cells with maturity m from mother cells completing their proliferating phase who entered into the proliferating phase a time τ ago with a maturity X(−τ, g −1 (m)).

Existence and uniqueness.
The integrated version (variation of constants formula) of Equation ( 14) for t ≥ τ and m ∈ [0, g (1)] is given by where and Note that the initial datum ϕ is in fact the solution of Equation ( 11) where k is a positive constant.If ϕ ∈ C ([0, τ] × [0, g(1)]), then there exists a unique solution We set and Let (N n ) n∈N be a sequence of continuous functions given, for t ∈ [0, 2τ ] and m ∈ [0, g (1)], by Due to the continuity of X, δ, V , ϕ and f 2 , there exist α > 0 and M 0 ≥ 0 such that Then, we obtain for t ∈ [τ, 2τ ] and m ∈ [0, g (1)] that and thus In the general case, we have By induction, we obtain So, the limit N := lim n→∞ N n exists uniformly on [0, 2τ ] and N is continuous on [0, 2τ ].
To prove uniqueness, we suppose that Z is also a solution of Equation (20).Then, We suppose that t ∈ [τ, 2τ ] is fixed and we consider the function w for σ ∈ [τ, t] and m ∈ [0, g(1)], defined by By the Gronwall's inequality, it follows that w = 0.Then, N = Z on [τ, 2τ ].By the method of steps, we deduce the existence and uniqueness for all t ≥ 0.

Remark 1. If x → β(x, m)x is only locally Lipschitz continuous, one can prove using Gronwall's inequality that Equation (20) has at most one solution.
We will now study the transformation of maturity from one generation of cells to another.We put for m ≤ g (1).
(25) Recall that ∆(m) represents the maturity of a mother cell at commitment when m is the maturity of its two daughter cells.The function ∆ : [0, g (1)] → [0, g (1)] is continuously differentiable and satisfies Let 0 < m 0 ≤ g(1) be fixed.We set then m 0 can be chosen equal to zero, and which in this case means that τ 1 = +∞.As in the linear case, to prove our main results, we need the following lemma.1), (iv) τ > τ 0 if and only if ∆(m) < m, for all m ∈ [m 0 , g (1)] .
The condition τ > τ 0 means that the duration of the proliferating phase is sufficiently long to let a future mother cell, whose maturity at the point of commitment was ∆(m), increase sufficiently its maturity during this phase such that, just after its division the maturity m of its daughter cells will be greater than the maturity ∆(m).We assume in the sequel of this paper that and τ > τ 0 , with m 0 = 0.The results hold with obvious modifications, for any m 0 ∈ (0, g (1)] .We give now an example for which the condition (26) is satisfied.< +∞ if and only if g (0) > 0.
Note that a cell in the proliferating phase whose maturity at the point of commitment is h −1 (e −τ ) will give birth to two daughter cells with maturity g (1), which is the maximal maturity of cells just after division.So, we can consider the function Λ : [0, g (1)] → [0, 1] defined by .
Note that the function Λ is continuously differentiable on 0, h −1 (e −τ ) and satisfies The condition τ > τ 0 implies that the sequence (b n ) n∈N is increasing and that there exists (1).
(28) Now we give the first of our main results, which emphasizes the strong link between the process of production of cells and cells with small maturity (stem cells).The following result has been proved by Adimy and Pujo-Menjouet (2001) in the linear case (i.e.β = β(m)).It has also been proved by Dyson, Villella-Bressan and Webb (1996b) in the special case when the maturity X(−τ, g −1 (m)) is independent of τ and equal to αm, 0 < α < 1.It is also a first step to study the asymptotic behavior and instability of our model.
] and t ≥ t, where t can be chosen equal to Proof.Recall that h −1 (e −τ ) < g (1).
The proof is based on our proof in the linear case and the use of the Gronwall's lemma.First, we establish by induction that Hence, Consequently, By the Gronwall's inequality, it follows that By a method of steps, we deduce that Secondly, we reconsider the sequence (b n ) n≥0 given by ( 27), and the sequence (t n ) n∈N defined by This sequence (t n ) n∈N is increasing.Thus, we prove by induction the following result So, by induction and using (28), we obtain According to the proof above, we also have the following existence result.It is believed that the pathology of aplastic anemia is due to injury or destruction of a common pluripotential stem cell.Theorem 4.1 proves that the production of cells depends strongly on the state of cells with small maturity.In particular, it describes the destruction of a cell population when its starting value is defective.This is the first step to study the asymptotic behavior of our model that we present in the following section.We set We define the sequence of continuous functions (N n ) n∈N , for t ≥ 0 and m ∈ [0, b], by and and that the function x → β(x, m)x satisfies the following Lipschitz condition where k := k( ) is a positive constant such that and Similarly, we have and As g (m) > 0 for m ∈ [0, 1], we have On the other hand, where Proof.Condition (35) and the continuous property of ϕ imply that there exists b such that Consequently, Proposition 5.1 gives the result.
We also obtain the following global stability result.