Analysis and numerical simulation of an inverse problem for a structured cell population dynamics model.

Frédérique Clément,1 Inria, Université Paris-Saclay, France ,2 Lms, Ecole Polytechnique, Cnrs, Université Paris-Saclay, France ,3 Maiage, Inra, Université Paris-Saclay, France ,Béatrice Laroche,Frédérique Robin

doi:10.3934/mbe.2019150

Frédérique Clément, 1 Inria, Université Paris-Saclay, France + Show 4 more

Open Access

https://doi.org/10.3934/mbe.2019150

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Abstract

In this work, we study a multiscale inverse problem associated with a multi-type model for age structured cell populations. In the single type case, the model is a McKendrick-VonFoerster like equation with a mitosis-dependent death rate and potential migration at birth. In the multi-type case, the migration term results in an unidirectional motion from one type to the next, so that the boundary condition at age 0 contains an additional extrinsic contribution from the previous type. We consider the inverse problem of retrieving microscopic information (the division rates and migration proportions) from the knowledge of macroscopic information (total number of cells per layer), given the initial condition. We first show the well-posedness of the inverse problem in the single type case using a Fredholm integral equation derived from the characteristic curves, and we use a constructive approach to obtain the lattice division rate, considering either a synchronized or non-synchronized initial condition. We take advantage of the unidirectional motion to decompose the whole model into nested submodels corresponding to self-renewal equations with an additional extrinstic contribution. We again derive a Fredholm integral equation for each submodel and deduce the well-posedness of the multi-type inverse problem. In each situation, we illustrate numerically our theoretical results.

Highlights

Cell dynamics are classically investigated in the framework of structured populations, and especially of the McKendrick–VonFoerster model
We study and illustrate the well-posedness of a multiscale inverse problem (IP), defined from a multi-type version of the linear, age-structured formulation of the McKendrick–VonFoerster model, in the specific situation where cell death can only occur at the time of mitosis
We have analyzed an inverse problem associated with a multi-type version of the McKendrick–VonFoerster model, and consisting of retrieving the division rate functions b j, and the probability of motion p(Sj), for j ∈ 1, J − 1, from the knowledge on the total cell numbers on each layer

Summary

Model and discretized solutions

We recall the age and layer structured cell population model considered in [11]. Let J ∈ N∗. For the single layer case, note that this hypothesis is more general than the one used either in [8] where the division rate function is supposed to be non-lattice or, in the steady state approach [14] where Am1 ax = +∞. Even if an explicit solution of the PDEs (1.1) cannot be obtained, one classical way to solve the direct problem when dealing with continuous initial conditions, is to use the method of characteristics [2, 1]. Given that our problem possibly involves a non-continuous initial condition (ψ1), and lattice division rates, the method of characteristics is the most suitable one. We analyse the inverse problem (IP) enunciated in Definition 1.1 starting with the simplest case of a single layer and considering a compactly supported function as initial condition. We extend our results to the general case of several layers

Single layer case

Numerical procedure for non-synchronized populations

Multi-layer case

Conclusion