"On the Existence and Positivity of a Mass Structured Cell Population Model"

In this paper, we address the problem of the existence, the uniqueness and the positivity of a class of integro-partial differential equation describing the growth of a mass structured cell population coupled with a ordinary integro-differential equation accounting for substrate consumption. We use the semi-group theory and the fixed point theorem to achieve this objective.


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Introduction
In recent years, there has been a great deal of research on modeling, control and analysis of cell population balance models. Such models describe the dynamic of cell growth and take into account the biological facts that the cell properties (e.g. mass or age) are distributed among the cell of the population. These models consist of a nonlinear partial integro-differential equation with non linear boundary condition coupled with an ordinary integro-differential equation [4,5,6]. In addition to the complex structure of such models, another difficulty is due to the intrinsic physiological functions, namely the growth rate function, the cell division probability density function, and the partitioning probability density function, whose selection may appear to be a complex task in many instances. For these reasons, the existing results on the analysis of such systems, including the analysis of the solution, the study of the existence, multiplicity and stability of equilibrium profile or on the control have only been obtained numerically. Mantzaris et al. [4]- [9] have presented several finite difference, spectral and finite element algorithms for solving cell population balance models. Several controllers have been proposed for this class of systems to control the different moments by using a nonlinear linearizing controller [4]. The analysis study of the existence of multiple equilibrium profiles and the stability analysis has been the object of several research. However, this analysis was performed always for the nonlinear ordinary differential equations of the first moment of cell population number and substrate concentration but not for the original mathematical model consisting of the partial and ordinary integro-differential equations. Our primary objective in this paper is to present results about the existence, uniqueness and positivity of the solution of the original mathematical model, this analysis is primordial to the control design for cell population models. For these considerations, we discuss the existence and uniqueness of nonnegative state trajectories of such a model by using the semi-group theory and the fixed point theorem. The paper is organized as follows: In the second section we present the basic dynamical model, and use a new formulation of the problem within the framework of the non-linear systems such asẋ(t) = A(x)x, where A(x) is a linear unbounded operator in a Banach space for a fixed x. The sufficient conditions of existence, uniqueness and positivity of this class of systems is given in section 3. Section 4 contains our main results.

Mathematical model
Let us consider a cell population growing in a continuous stirred tank reactor. The cells are distinguishable from each other in terms of their mass or any other property of the cell, which obeys the conservation law. Let N (m, t) be the number of cells which have a mass between m and m + dm at time t. The cells are considered to grow with a rate r(m, S) that depends on their mass and on the concentration of the limiting substrate S. We also assume that the value of the mass is standardized and that m ∈ [0, 1]. The cell division and the birth processes of the cell population are described by the division rate Γ(m, S) defined as follows (see [4]): where f (m) is the division probability density function which is assumed to depend only on the cell mass, and is taken to be a left hand side truncated Gaussian distribution with the mean of µ f and standard deviation of σ f . The probability p that a mother cell of mass will give birth to a daughter cell of mass m is assumed to be independent of substrate concentration. This function should further satisfy the following normalization conditions: We finally assume that the probability function is a symmetrical beta distribution with a parameter of q defined by the following equation: We assume also that no cell death occurs and that cells grow in a batch bioreactor. Under these assumptions, the cell population dynamics are described by the following integro-differential equations, see e.g. [4]: subject to the initial condition : System (3)-(4) is completed by the following boundary conditions: The cell population equation (3) consists of four terms: the accumulation term, the growth term, the division term and the birth term.
The behavior of the cell population depends on the substrate concentration, the source of nutrient for its growth. The mass balance equation for the substrate expresses in particular, that the substrate consumption is proportional to the total biomass production +∞ 0 r(m, S)N (m, t)dm with a yield coefficient Y which is the ratio of the biomass production rate over the rate of substrate consumption, assumed to be constant. The substrate concentration mass balance will be read as follows: subject to the initial condition: In this work, we are interested in the existence of the solution for this class of models in the case where: We consider the Banach space is endowed with usual norm: Then the evolution of x(t) is described by the following system: We can rewrite the equation (9)-(10) in the abstract form: Let D be defined as: The nonlinear (unbounded) operator A(x) is defined on its domain D by: where: Remark 2.1 By this formulation and semi-group theory we are looking for the existence, uniqueness and the positivity of the solution, Contrary to the previous work where this model was solved only numerically see [7,8,9].

Basic concepts and results
Let X be a Banach space endowed with the norm denoted by . , u 0 ∈ X and let B u 0 := {u ∈ X, u − u 0 ≤ r} the closed ball centred at u 0 with radius r. This section is concerned with the existence and uniqueness of global solution of the following class of non linear infinite dimensional systems: for all fixed u ∈ B u 0 , A(u) is an unbounded linear operator defined on the subspace D ⊂ X. Assume that D is endowed with a norm denoted by . D . Recall that if A is a bounded linear operator from D to X, then: We suppose that for all u ∈ B u 0 and T > 0, the operators A(u) satisfies the following assumptions: (H 2 )− There exist two constants M ≥ 1 and w such that for any finite sequence u 1 , u 2 , . . . , u k , with k ∈ N, we have: (H 4 )− There is a positive constant L such that: The following result, which is an equivalent version of Theorem 4.3, pp.202 of [12], gives sufficient conditions for the existence and the uniqueness of the solution of system (17) on the interval [0, T ].

Proof
We define firstly the set S, for 0 < T ≤ T , by: Assumptions (H 1 ), (H 2 ) and (H 3 ) ensure that for all v ∈ S, the system: has a unique solution U v (t)u 0 such that U v (t) ≤ M e wt (see Theorem 4.2, pp.140 of [12]).
By the continuity of U u 0 , there exist a t 1 > 0 such that For T > 0, we define the function F in S by: and we show that the function F has a fixed point in S by showing that F is a contraction function.
In the both cases, there exist t 2 such that: then for T = min{t 1 , t 2 }.
If u 1 , and u 2 ∈ S we have: So that F is a contraction. From the contraction mapping theorem it follows that F has a unique fixed point u ∈ S which is the desired solution on [0, T ] of (17). The presence of integral term in the cell population model complicates the determination of the resolvent operators. In theorem 3.1, we replace the assumption on the resolvent operators in Theorem 4.3 of [ [12],p.202] by (H 2 ) given in terms of the semi-groups. If for all u ∈ B u 0 , A(u) is a generator of a contraction C 0 -semigroup, then assumption (H 2 ) is satisfied.
Assume that X is a real Banach lattice endowed with a partial order which is denoted here by ≤. The associated positive cone is given by X + = {x ∈ X : 0 ≤ x} is supposed to be a closed set of X (see [11]) and for all x, y ∈ X, x ≤ y if and only if y − x ∈ X + . Recall that a linear operator Λ on X is said to be a positive if and only if for all 0 ≤ x, 0 ≤ Λx and a C 0 semigroup Λ(t) is said to be a positive if Λ(t) is a positive operator for all t ≥ 0. A necessary and sufficient condition for the positivity of a C 0 semigroup (Λ(t)) t≥0 is given in the following proposition: By this proposition we obtain the following results on the positivity of the solution of (17) and the proof of Theorem 4.2, pp.140 of [12]: Theorem 3.4 If X is a real Banach lattice space, the operators A(u), u ∈ B u 0 satisfies the assumptions (H 1 ), (H 2 ), (H 3 ) and (H 4 ) and if for all u ∈ B u 0 ∩ X + , T u is a positive C 0 semi group on X, then the system (11) has a unique positive solution for all positive initial condition.

Application to the cell population model
This section is concerned with the existence, the uniqueness and the positivity of the global solution of (3)-(6). Let us define the linear operator: defined in its domain: where k is a nonzero constant. In order to apply the result given in Theorem 3.4, we need the following lemmas.

Lemma 4.1
The operator A defined on ∆ is generator of a C 0 semi-group in L 1 (0, +∞).

Proof
To show that the operator A generate a C 0 semi-group in L 1 (0, +∞), we apply the Hille-Yosida theorem [1].
-A is a closed and densely defined.
-We shall now show that λI − A is invertible for all λ > k.
-We have: and for all n ∈ N, we have: Then A is infinitesimal generator of C 0 semi-group (T (t)) t≥0 such that T (t) ≤ e kt for all t ≥ 0.
To discuss the positivity of the C 0 semi-group, we define respectively two partial orders in L 1 (0, ∞) and H: for all f and g ∈ L 1 (0, ∞), f ≤ g if and only if f (z) ≤ g(z) for all z ∈ [0, +∞), and for all (f, a) and (g, b) ∈ H, (f, a) < (g, b) if and only if f (z) ≤ g(z) for all z ∈ [0, +∞) and a ≤ b.
For the first order we have the following Lemma: For α 1 > 0, let us define Our focus in the sequel is to prove that the family of the operators A(x), x ∈ B ∩ H + satisfies the assumptions (H 1 )-(H 4 ). To do so, we have: 2) For all x ∈ B, A(x) satisfies (H 2 ).
3) For all x ∈ B, A(x) is a bounded operator from D = ∆ × R to H with D is endowed with the following norm: and for all (a 1 , a 2 ) and (b 1 , b 2 ) in B: Recall that the operator A(x) defined as: for all x ∈ B, the operator A 1 (x) has the same structure as the operator A giving in (21), where k = r(x 2 ).
-A 2 (X) and A 3 (X) are linear bounded operators on H.
-The family of the operators A(x) depend only on x 2 ∈ R.
-The function r(.), the non-linear term in the operators A(X), is an upper bounded and Lipshitz function i.e, there exist two positive constants c 1 and c 2 such that for all (a, b) ∈ R 2 we have So, it is easy to show that if A 1 (x) satisfies the properties (H 1 )-(H 4 ) then it is the same for the operator A(x).

Lemma 4.4
The family of the operators A 1 (x) are generators of a C 0 semigroups (T x (t)) t≥0 for all x ∈ B and there exist w such that T x (t) ≤ e wt for all T ∈ R and all t ∈ [0, T ]. If x ∈ B ∩ H + , (T x (t)) t≥0 is positive.

Proof
From Lemma 4.1, we deduce that the operator A 1 (x) defined on ∆ by: We have also for all x ∈ B for all λ > c 1 = ω: Then for all x ∈ B, (T x (t)) t≥0 ≤ e wt for all t ∈ [0, T ] and for all T ∈ R + . In other words, taking into account the Lemma 3.3, if x ∈ B ∩ H + , the C 0semigroup (T x (t)) t≥0 is positive.

Lemma 4.5
For all x ∈ B, the operator A 1 (x) is a bounded operator from D to H and for all (a 1 , a 2 ) and (b 1 , b 2 ) in B ∩H + there exist a positive constant L such that: Proof For (a 1 , a 2 ) and (b 1 , b 2 ) in B∩ and b ∈ ∆ we have: A 1 (a 1 , a 2 )b 1 = r(a 2 )m ∂b ∂m + (r(a 2 ) + r(a 2 )mγ(m))b 1 ≤ c 1 ( m ∂b ∂m This complete the proof of the lemma.

Conclusion
In this paper, we have studied the existence and uniqueness of the solution of the mass structured cell population balance model. By using the semi-group theory and a fixed point theorem, it has been proved the existence of the cell numbers N (., t) in L 1 (0, +∞) and the substrate S(t) in , R + for all t > 0. It has also been proved that N (., t) and S(t) remain positive for all t > 0 and for all m ∈ [0, +∞] if the initial conditions are positive. The existence, multiplicity and stability of equilibrium points of this model is under investigation.