Structured Models for Cell Populations: Direct and Inverse Problems

Vinicius Albani,Jorge P Zubelli,V Volpert,J Clairambault

doi:10.1051/itmconf/20150500016

Vinicius Albani, Jorge P Zubelli + Show 2 more

Open Access

https://doi.org/10.1051/itmconf/20150500016

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Abstract

Structured population models in biology lead to integro-differential equations that describe the evolution in time of the population density taking into account a given feature such as the age, the size, or the volume. These models possess interesting ana- lytic properties and have been used extensively in a number of areas. In this article, we consider a size-structured model for cell division and revisit 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. The focus is to compare from a computational view point the perfor- mance of different algorithms. Finally, we will discuss real data reconstructions with E. coli data.

Highlights

Structured population models in biology lead to integro-differential equations that describe the evolution in time of the population density taking into account a given feature such as the age, the size, or the volume
We present some numerical examples of the calibration of the division-rate B from the stable distribution N using Tikhonov-type regularization and statistical inverse-problem techniques, with synthetic as well as real E. coli data
The reconstructions are obtained with Maximum a Posteriori (MAP) and Conditional Mean (CM) point estimators

Summary

Structured Population Models

Where g(x) is the microscopic growth rate of individuals at size x, and k(x, x ) is the proportion of cells of size x that divide into two cells, one of size x, and the other of size x − x Under this generality, the model is hard to calibrate and to make predictions. We remark that the choice of g ≡ 1 was made and that a natural alternative would be that of an affine function It is well-known [4,5,6] that there exist unique λ0 and N = N(x), forming an eigenpair, such that, after a suitable time re-normalization, the solutions of (2) satisfy the limit n(t, x)e−λ0t −→ ρN(x), as t → ∞,. Theorem 1 (Perthame-Zubelli [8]) Under assumption (5), the map B → (λ0, N), from L∞(R+) into [Bm, BM] × L1 ∩ L∞(R+) is continuous under the weak-∗ topology of L∞(R+), locally Lipschitz continuous under the strong topology of L2(R+) into L2(R+), and of class C1 in L2(R+)

The Inverse Problem

Numerical Results

Discusion