Study of the numerical method for an inverse problem of a simplified intestinal crypt

Marie Haghebaert (1, 2) , Béatrice Laroche (3, 2) , Mauricio A. Sepulveda Cortes (4)

1 INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
2 MUSCA - Dynamiques de populations multi-échelles pour des systèmes physiologiques
3 MaIAGE - Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas]
4 Universidad de Concepción - University of Concepcion [Chile]

In this work we consider the study of an inverse problem for an intestinal crypt model. The original model is based on the interaction of epithelial cells with microbiota-derived chemicals diffusing in the crypt from the gut lumen. The 5 types of cells considered in the original model were reduced in this work to 3 types of cells for simplifications of the inverse problem. The inverse problem consists of determining the shape of the secretory cells of the deep crypt from observations of the stem cells and progenitor cells at a fixed time. The method used is the calculation of the adjoint state associated with the second-order BGK numerical scheme considered in [3], which allows calculating the critical points of the Lagrangian associated with the inverse problem, and applying a gradient method in order to minimize the cost function. The algorithm is described, and some numerical examples are given.

Mots clés: Inverse problems approach; parabolic degenerate equation; EDK scheme

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Marie Haghebaert, Béatrice Laroche, Mauricio A. Sepulveda Cortes. Study of the numerical method for an inverse problem of a simplified intestinal crypt. 2023. ⟨hal-04332907⟩