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

Frédérique Clément; Béatrice Laroche; Frédérique Robin; Frédérique Clément; Béatrice Laroche; Frédérique Robin

doi:10.3934/mbe.2019150

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 3018-3046. doi: 10.3934/mbe.2019150

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

1.
Inria, Université Paris-Saclay, France
2.
LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, France
3.
MaIAGE, INRA, Université Paris-Saclay, France

Received: 29 January 2019 Accepted: 19 March 2019 Published: 10 April 2019

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.
- age structured cell population,
- renewal model,
- McKendrick–VonFoerster equation,
- inverse problem,
- multiscale identification problem,
- Fredholm integral,
- characteristic curves,
- measure solutions
Citation: Frédérique Clément, Béatrice Laroche, Frédérique Robin. Analysis and numerical simulation of an inverse problem for a structured cell population dynamics model[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 3018-3046. doi: 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.

References

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