Section: New Results

Numerical methods for biological flows

Participants : Grégory Arbia, Benoit Fabrèges, Miguel Ángel Fernández Varela, Justine Fouchet-Incaux, Jean-Frédéric Gerbeau, Céline Grandmont, Sanjay Pant, Saverio Smaldone, Marc Thiriet, Irène Vignon-Clementel.

• In [19] We consider the problem of estimating the stiffness of an artery wall using a data assimilation method applied to a 3D fluid-structure interaction (FSI) model. We briefly present the FSI model, the data assimilation procedure based on a reduced order Unscented Kalman filter, and the segmentation algorithm. We then present two examples of the procedure using real data. First, we estimate the stiffness distribution of a silicon rubber tube from image data. Second, we present the estimation of aortic wall stiffness from real clinical data.

• In [29] , we propose a new approach to the loosely coupled time-marching of a fluid-fluid interaction problems involving the incompressible Navier-Stokes equations. The methods combine a specific explicit Robin-Robin treatment of the interface coupling with a weakly consistent interface pressure stabilization in time. A priori energy estimates guaranteeing stability of the splitting are obtained for a total pressure formulation of the coupled problem. The performance of the proposed schemes is illustrated on several numerical experiments related to simulation of aortic blood flow.

• In [55] we investigate the stability of numerical schemes that are classically used in the simulation of airflows and blood flows. The geometrical complexity of the networks in which air/blood flows leads to a classical decomposition of two areas: a truncated 3D geometry corresponding to the largest contribution of the domain, and a 0D part connected to the 3D part, modelling air/blood flows in smaller airways/vessels. The resulting Navier-Stokes system in the 3D truncated part may involve non-local boundary conditions, deriving from a mechanical model. For various 3D/0D coupled models, different discretization processes are presented and analyzed in terms of numerical stability, highlighting strong differences according to the regimes that are considered. In particular, two main stability issues are investigated: first the coupling between the 3D and the 0D part for which implicit or explicit strategies are studied and, second, the question of estimating the amount of kinetic energy entering the 3D domain because of the artificial boundaries. The second issue has been also the subject of a review [31] .

• In [31] we deal with numerical simulations of incompressible Navier-Stokes equations in truncated domain. In this context, the formulation of these equations has to be selected carefully in order to guarantee that their associated artificial boundary conditions are relevant for the considered problem. In this paper, we review some of the formulations proposed in the literature, and their associated boundary conditions. Some numerical results linked to each formulation are also presented. We compare different schemes, giving successful computations as well as problematic ones, in order to better understand the difference between these schemes and their behaviours dealing with systems involving Neumann boundary conditions. We also review two stabilization methods which aim at suppressing the instabilities linked to these natural boundary conditions.

• In [40] , we propose a framework for Windkessel parameter estimation in a 0D representation of the 3D fluid-flow domain. Parameters are estimated from uncertain measurements through a sequential approach, and the 0D representation is iteratively improved through 3D-CFD simulations. The application of generalized sensitivity functions to assess parameter correlation and to ascertain the measurement set needed to avoid identifiability problems is also presented through representative test cases. This method, which is capable of handling non-simultaneous measurements, is demonstrated and validated for a patient-specific case of aortic coarctation.

• In [17] we perform the first patient-specific pulmonary hemodynamics 3D-0D modeling before single ventricle stage 2 surgery. 0D parameters are automatically tuned to match flow and pressure clinical measurements that are not taken where 3D boundary conditions need to be specified. This work on six patients demonstrates how simulations can help to check the coherence of clinical data or provide insights to clinicians that are otherwise difficult to measure, such as in the presence of kinks.

• In [25] we study a case of post single ventricle stage 2 surgery with the three following aims: (i) to show how to build a patient-specific model describing the hemodynamics in the presence of collaterals, using patient-specific clinical data collected at different times; (ii) to use this model to perform virtual collateral occlusion for quantitative hemodynamics prediction; and (iii) to compare predicted hemodynamics with post-operative measurements.