Section: New Results

Numerical methods for biological flows

Participants : Chloé Audebert, Jean-Frédéric Gerbeau, Céline Grandmont, Sanjay Pant, Marc Thiriet, Irene Vignon-Clementel.

In [16], we present a new approach for the outflow boundary conditions of Navier-Stokes equations in hemodynamics that consists in adding a 3D artificial part where the Navier-Stokes equations are modified to obtain an equivalent energy balance to a standard coupling with a 3-element Windkessel model. We investigate theoretically the stability of the system and compare it to previously introduced methods. We compare these coupling methods for numerical simulations of blood flow in three patient-specific models, which represent different flow regimes in the pulmonary and systemic circulations.

In [36], we highlight and present solutions to several challenges of the UKF method, a data-assimilation method, pertinent to reduced models of cardiovascular haemodynamics. These include methods to a) avoid ill-conditioning of covariance matrix; b) handle a variety of measurement types; c) include a variety of prior knowledge in the method; and d) incorporate measurements acquired at different heart-rates, a common situation in the clinic where patient-state differs between various clinical acquisitions.

In [18], we introduce a kinetic scheme to solve the 1D Euler equations of hemodynamics, which solution on several benchmark tests for both arterial and venous wall laws compares well with the literature. In particular, it is shown that it has a good behavior when the section area of a vessel is close to zero, which is an important property for collapsible or clamped vessels. The application to liver surgery shows that a closed-loop model of the global circulation, including 0D and 1D equations, is able to reproduce the change of waveforms observed after different levels of hepatectomy.

In [17], we explain with a 0D closed-loop lumped model the hemodynamics changes observed during partial hepatectomy in pigs [22]. The typical increase of portal pressure, increase of liver pressure loss, slight decrease of portal flow and major decrease in arterial flow are quantitatively captured by the model for a 75% hepatectomy. The different post-operative states, observed in experiments, are reproduced with the proposed model. Thus, an explanation for inter-subjects post-operative variability is proposed. This work needs to be translated to humans, in which liver flow modulation is a subject of surgery research [39].

In [24], we propose a computational approach for efficient design study of a reducer stent to be percutaneously implanted in enlarged right ventricular outflow tracts (RVOT) of repaired Tetralogy of Fallot. Hemodynamics of different designs are simulated in the stented RVOT via a reduce order model based on proper orthogonal decomposition on a reference device configuration. To validate the approach, forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters, and compared with the results of full CFD simulations.

Peripheral pulmonary artery stenosis (PPS) is a congenital abnormality resulting in pulmonary blood flow disparity and right ventricular hypertension, for which optimal surgical strategies remain unclear. In [38], a proof of concept study, a constant shear stress hypothesis and structured pulmonary trees are used to derive adaptive outflow boundary conditions for 3D-0D postoperative blood flow simulations. This strategy provides better predictions of pulmonary flow distribution than the conventional strategy of maintaining outflow boundary conditions.

In [26] the effect of inserted needle on the subcutaneous interstitial flow is studied. The goal is to describe the physical stress affecting cells during acupuncture needling. The convective Brinkman equations are considered to describe the flow through a fibrous medium. Three-dimensional simulations are carried out by employing an ALE finite element model. Numerical studies illustrate the acute physical stress developed by the implantation of a needle.

In [32], a fully three-dimensional blood flow simulation through a complete rigid macrovascular circuit, namely the intracranial venous network, instead of a reduced order simulation and partial vascular network is presented. The biomechanical modeling step is carefully analyzed and leads to the description of the flow governed by the dimensionless Navier-Stokes equations for an incompressible viscous fluid. The equations are then numerically solved with a free finite element software using five meshes of a realistic geometry obtained from medical images to prove the feasibility of the pipeline. Some features of the intracranial venous circuit in the supine position such as asymmetric behavior in merging regions are discussed.