Stability estimates for a Robin coefficient in the two-dimensional Stokes system

In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim and under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed

Airflow and Particle Deposition Simulations in Health and Emphysema: From In Vivo to In Silico Animal Experiments

Image-based in silico modeling tools provide detailed velocity and particle deposition data. However, care must be taken when prescribing boundary conditions to model lung physiology in health or disease, such as in emphysema. In this study, the respiratory resistance and compliance were obtained by solving an inverse problem; a 0D global model based on healthy and emphysematous rat experimental data. Multi-scale CFD simulations were performed by solving the 3D Navier-Stokes equations in an MRI-derived rat geometry coupled to a 0D model. Particles with 0.95 μm diameter were tracked and their distribution in the lung was assessed. Seven 3D-0D simulations were performed: healthy, homogeneous, and five heterogeneous emphysema cases. Compliance (C) was significantly higher (p = 0.04) in the emphysematous rats (C = 0.37 ± 0.14 cm(3)/cmH2O) compared to the healthy rats (C = 0.25 ± 0.04 cm(3)/cmH2O), while the resistance remained unchanged (p = 0.83). There were increases in airflow, particle deposition in the 3D model, and particle delivery to the diseased regions for the heterogeneous cases compared to the homogeneous cases. The results highlight the importance of multi-scale numerical simulations to study airflow and particle distribution in healthy and diseased lungs. The effect of particle size and gravity were studied. Once available, these in silico predictions may be compared to experimental deposition data