On the inverse problem for a size-structured population model

Abstract

We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such question as an inverse problem for an integro-differential equation posed on the half line. We develop firstly a regular dependency theory for the solution in terms of the coefficients and, secondly, a novel regularization technique for tackling this inverse problem which takes into account the specific nature of the equation. Our results rely also on generalized relative entropy estimates and related Poincaré inequalities.

Estimating the division rate for the growth-fragmentation equation

“…Secondly, since the measure N ε is supposed to be in L 2 , there is no way of directly controlling ∂ ∂ x (gN ε ) even if g is known (see Section 2 of [2] for a discussion, or yet [1]). To overcome this difficulty, two regularization methods were proposed in [2,3] for the particular case of division into two equal cells, i.e. when κ(x, y) = δ x=y/2 , a third method has also been proposed in [11], and a statistical treatment to estimate the derivative in [7].…”

“…None of the three regularization methods of [2,3,11] can be directly applied here: indeed, they strongly used the fact that for the kernel κ = δ x= y 2 , the left-hand side of Equation (23) simplifies in 4BN (2x) − B(x), and can be viewed as an equation written in y = 2x. Then, a central point of the proofs in [2] as well as in [3] or [11] is the use of the Lax-Milgram theorem for the coercitive operator L : H → 4H(y) − H( y 2 ). Nothing such as that can be written here, and the main difficulty, numerically as well as theoretically, is to deal with a nonlocal kernel κ(x, y)H(y)dy.…”

“…To be able to use it as a predictive model however, it is crucial to be able to estimate quantitatively its parameters g, B and κ. A first step consists in the use of the asymptotic behavior of this equation, as first proposed in [2]. Indeed, by general relative entropy principle it is proven (see e.g [14,15,5]) that under suitable assumptions on κ, g and B one has …”

“…In [2] and [3], the problem of recovering the division rate from a measured N was addressed in the case when the growth rate is constant, i.e g(x) ≡ 1, and the daughter cells are twice smaller than their mother, i.e. when κ(x, y) = δ x=y/2 .…”

“…We first study the regularity of the direct problem, what is a necessary step for a better understanding of the inverse problem. In a second part, we investigate the inverse problem of determining B by the Quasi-reversibility and Filtering methods proposed in [2] and [3] and properly adapted to our general context. In a third part we develop new numerical approaches in order to recover the rate B following the two regularization methods ; we give some numerical illustrations of our methods.…”