[Elements de theorie pour la reconstruction de la distribution spatiale du coefficient de filtration dans les tissus mous vascularisés : Une solution exacte et une solution inexacte mais simple]
Nous formulons et résolvons un problème inverse poroelastique afin de reconstruire la distribution spatiale du coefficient de filtration pour un tissu mou vascularisé ; les déplacements utilisés ici sont enregistrés pendant la relaxation du tissu. Nous présentons deux solutions pour le problème inverse, toutes deux développées en utilisant une approche directe non-itérative. La première solution est une expression explicite simple, qui fait l'hypothèse que la pression interstitielle est spatiallement homogène. La deuxième solution ne se fonde pas sur cette hypothèse et requiert la résolution d'une équation de Poisson pour obtenir la distribution de pression. L'inversion obtenue ainsi est exacte dans la limite ou la percolation est négligeable. Nous présentons des reconstructions sur des données obtenues par simulations numériques pour valider et comparer ces deux approches. La solution explicite fournit des résultats précis dans des circonstances favorables. La deuxième approche est pratique pour simuler un chargement non homogène du tissu. Il apparaît que les deux approches sont quelque peu sensibles au bruit. Nos résultats suggérent néanmoins qu'il est possible d'effectuer l'imagerie du coefficient de filtration en utilisant cette approche. Dans le futur, nous souhaitons tester davantage ces méthodes en présence de bruit puis les valider sur des données expérimentales.
We formulate and solve an inverse poroelastic problem to reconstruct the spatial distribution of the filtration coefficient for soft vascularized tissue from a collection of displacement fields obtained during its relaxation. We present two solutions for the inverse problem, both developed using direct non-iterative approach. The first is a simple closed form approximate solution. It depends upon the approximation that the interstitial pressure is spatially homogeneous. The second solution relaxes this assumption. It requires the solution of a Poisson equation to reconstruct the pressure distribution. The inversion thus obtained is exact in the limit of negligible percolation. We present inversion results from computational experiments to validate and compare the two approaches. The closed form solution provides accurate results in favorable circumstances. The exact-pressure approach accommodates inhomogeneous loading easily. Both approaches are somewhat sensitive to noise. Our results suggest that it may be possible to image the filtration coefficient using this approach. Future work would include further test with noisy data and experimental validation.
Mot clés : Biomécanique, Problème inverse poroelastique, Coefficient de filtration
@article{CRMECA_2010__338_7-8_412_0,
author = {Ricardo Leiderman and Assad A. Oberai and Paul E. Barbone},
title = {Theory of reconstructing the spatial distribution of the filtration coefficient in vascularized soft tissues: {Exact} and approximate inverse solutions},
journal = {Comptes Rendus. M\'ecanique},
pages = {412--423},
publisher = {Elsevier},
volume = {338},
number = {7-8},
year = {2010},
doi = {10.1016/j.crme.2010.07.003},
language = {en},
}
TY - JOUR AU - Ricardo Leiderman AU - Assad A. Oberai AU - Paul E. Barbone TI - Theory of reconstructing the spatial distribution of the filtration coefficient in vascularized soft tissues: Exact and approximate inverse solutions JO - Comptes Rendus. Mécanique PY - 2010 SP - 412 EP - 423 VL - 338 IS - 7-8 PB - Elsevier DO - 10.1016/j.crme.2010.07.003 LA - en ID - CRMECA_2010__338_7-8_412_0 ER -
%0 Journal Article %A Ricardo Leiderman %A Assad A. Oberai %A Paul E. Barbone %T Theory of reconstructing the spatial distribution of the filtration coefficient in vascularized soft tissues: Exact and approximate inverse solutions %J Comptes Rendus. Mécanique %D 2010 %P 412-423 %V 338 %N 7-8 %I Elsevier %R 10.1016/j.crme.2010.07.003 %G en %F CRMECA_2010__338_7-8_412_0
Ricardo Leiderman; Assad A. Oberai; Paul E. Barbone. Theory of reconstructing the spatial distribution of the filtration coefficient in vascularized soft tissues: Exact and approximate inverse solutions. Comptes Rendus. Mécanique, Volume 338 (2010) no. 7-8, pp. 412-423. doi : 10.1016/j.crme.2010.07.003. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.003/
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