## Section: New Results

### Numerical methods for biological flows

Participants : Chloé Audebert, Benoit Fabrèges, Miguel Ángel Fernández Varela, Jean-Frédéric Gerbeau, Céline Grandmont, Sanjay Pant, Marc Thiriet, Irène Vignon-Clementel.

In [37] we present a closed-loop global lumped parameter model for pre stage-II single-ventricle physiology. This model, which is built on a fibre mechanics based description of the heart chambers, benefits from a novel method to describe regurgitant valves. As many as 33 model parameters are estimated from uncertain clinical measurements in two patients—with and without atrioventricular valve regurgitation—through the method of data assimilation. Results are validated qualitatively through measurements and clinical estimates that were not included in the parameter estimation procedure. The methods are shown to successfully capture patient-specific clinical observations such as double peaked nature of valvular flows and abnormalities in electrocardiogram readings.

In [39] we propose a methodology for full propagation of uncertainty from clinical data to model results that enables estimation of the confidence associated with model predictions. We illustrate this problem in a pre stage-II single-ventricle physiology, for which coherence of simulations and clinical data indicated that the flow split to the right lung was highly uncertain. We want to assess here how such uncertainty translates into surgical planning of removing the stenosis or not. Taking into account the effect of the rest of the circulation is also studied in the uncertainty propagation.

In [21] 3D blood flow simulations are carried out for the design of a stented valve reducer in enlarged ventricular outflow tracts. Different device designs are built and compared with the initial device-free state, or with the reducer alone. Results suggest that pressure loss is higher for the reducer alone than for the full device, and that the latter successfully restores hemodynamics to a healthy state. Pressure forces on the reducer and on the valve have the same magnitudes. Migration would occur towards the right ventricle rather than the pulmonary arteries.

In [44] we aim at developing a mathematical model in order to reproduce hemodynamics changes due to liver ablation surgeries. First, a 0D closed-loop model is developed, to simulate hepatectomy and compute post-operative average values. Due to the closed loop, the surgery impact both on and from the whole circulation can be captured, including bleeding and infusion. Then, a one-dimensional artery model is implemented to improve the closed-loop model and simulate better the changes in arterial waveforms due to surgery.

In [54] we investigate the spatial and time discretization of the transient Oseen equations. Finite elements with symmetric stabilization in space are combined with several time-stepping schemes (monolithic and fractional-step). Quasi-optimal (in space) and optimal (in time) error estimates are established for smooth solutions in all flow regimes. We first analyze monolithic time discretizations using the Backward Differentation Formulas of order 1 and 2 (BDF1 and BDF2). We derive a new estimate on the time-average of the pressure error featuring the same robustness with respect to the Reynolds number as the velocity estimate. Then, we analyze fractional-step pressure-projection methods using BDF1. The stabilization of velocities and pressures can be treated either implicitly or explicitly. Numerical results illustrate the main theoretical findings.

In [26] we study the effects of inserted needle on the subcutaneous interstitial flow. A goal is to describe the physical stress affecting cells during acupuncture treatment. The model consists of the convective Brinkman equations to describe the flow through a fibrous medium. Numerical studies in FreeFem++ are performed to illustrate the acute physical stress developed by the implantation of a needle that triggers the physiological reactions of acupuncture. We emphasize the importance of numerical experiments for advancing in modeling in acupuncture. In [40] we show that the acupoint must contain a highly concentrated population of mastocytes (e.g., very-high–amplitude, small-width Gaussian distribution) to get an initial proper response. Permanent signaling is provided by chemotaxis and continuous recruitment of mastocytes. Therefore, the density and distribution of mastocytes are crucial factors for efficient acupuncture as well as availability of circulating and neighboring pools of mastocytes.

In [61] we carry out a three-dimensional blood flow simulation through a complete macrovascular circuit, the cerebral venous network, rather than using reduced order simulation and partial vascular network. The bio-mechanical modeling step is carefully performed and leads to the description of the flow governed by the Navier-Stokes equations for an incompressible viscous fluid. We then numerically solve the equations with a free finite element software in five meshes of a realistic geometry obtained from medical images to prove the feasibility of the pipeline. Some particularities of the venous network, as asymmetry for example, are discussed.