## Section: New Results

### Blood flows

#### Resistive Immersed Surfaces for stent modeling

Participants : Alfonso Caiazzo, Miguel Ángel Fernández, Jean-Frédéric Gerbeau, Vincent Martin.

A. Caiazzo, M.A. Fernández, V. Martin and J.-F. Gerbeau have proposed and analyzed different finite element formulations to simulate incompressible flow through a particular type of stent, modeled as a resistive immersed surface [63] . They first analyze a variant of the monolithic approach proposed in [63] , which does not require extra pressure stabilization. Then they consider a fractional-step formulation based on the Chorin-Temam projection scheme. This requires a proper reformulation of the interface coupling conditions at each substep. They show that an appropriate Nitsche interface treatment (see [66] ) of the pressure interface conditions allows to derive uniform stability, within the whole range of resistivity r[0, + ] . The theoretical stability and convergence results are illustrated via numerical experiments. A paper is in preparation [44] .

Applications to abdominal aneurisms have been presented in [26] .

#### Inverse problems for blood flow simulation

Participants : Cristóbal Bertoglio, Laurent Dumas, Miguel Ángel Fernández, Jean-Frédéric Gerbeau.

##### Data assimilation for 3D fluid-structure problems

C. Bertoglio, D. Chapelle (MACS), M.A. Ferández, J.-F. Gerbeau and P. Moireau (MACS) have addressed some inverse problems in fluid-structure interaction. For the joint state-parameter estimation they considered a sequential approach, inspired by filtering strategies recently proposed in [68] .

The performance of the proposed approach has been investigated theoretically, using simplified models, and illustrated numerically through some inverse problems inspired from vascular mechanics [25] .

A model to account for the tissues and organs surrounding the vessels has been proposed, in collaboration with A. Figueroa, C. Taylor and N. Xian (Stanford) [52] and successfully apply to model the effect of the spine on the aorta.

##### Inverse problems for 1D blood flow models

In order to reduce the cost of complex 3D fluid-structure computations of blood flow in arteries, one dimensional model based on the averaging of the general three dimensional equations are commonly used. Under a certain number of hypotheses for the artery flow and geometry, it computes the section area A(t, z) and the volumic flux Q(t, z) at any longitudinal position z and time t .

Such model is interesting for three main reasons: first, it drastically reduces the computational time of the 3D model, secondly, its two unknowns are quantities that can be experimentally obtained by non invasive techniques, like echotracking, and finally, it allows to recover all the other hemodynamic variables like blood pressure that are more difficult to measure.

In a preliminary work, [47] , the inverse problem corresponding to the identification of the rigidity function , supposed to be piecewise constant, has been successfully solved by Laurent Dumas in the case of a straight artery. He has shown in particular that for an artery with a loss of compliance in some portion, the knowledge of only one area section profile downstream is enough to locate the exact position of this disease portion and also its associated compliance.

The next step in this problem, currently under study with a medical team of Hôpital Georges Pompidou (co-headed by Pr. Boutouyrie), is to construct the simplified numerical network of a given patient by using its own flux and section measurements obtained by echotracking.

#### Cardiac valves simulation

Participants : Matteo Astorino, Jean-Frédéric Gerbeau, Irène Vignon-Clémentel.

Important progress has been achieved in recent years in simulating the fluid-structure interaction around cardiac valves. An important step in making these computational tools useful to clinical practice is the development of postprocessing techniques to extract clinically-relevant information from these simulations. In collaboration with Shawn Shadden (Stanford University), M. Astorino, I. Vignon-Clementel and J.-F. Gerbeau showed how the concept of Lagrangian Coherent Structures (LCS) could improve insight into the transport mechanics of the flow downstream of the aortic valve [24] . In [53] , LCS were further investigated to extract from numerical simulation the Effective Orifice Area (EOA) which is a commonly used clinical index.

#### Perfusion of the myocardium

Participants : Jean-Frédéric Gerbeau, Irène Vignon-Clementel.

This activity on perfusion is done in close collaboration with the MACS project-team, in particular with D. Chapelle and J. Sainte-Marie, in the framework of the CardioSense3D INRIA project.

This work is motivated by the modeling of blood flows through the beating myocardium, namely cardiac perfusion. Perfusion is modeled here as a flow through a poroelastic medium. The main contribution of this study is the derivation of a general poroelastic model valid for a nearly incompressible medium which experiences finite deformations, illustrated by several numerical examples [45] .

A general poroelastic formulation valid for finite strains and compatible with incompressibility was introduced, as these two features are deemed to be important in the modeling of living tissues. The strategy – presented in [60] in a linear framework – of deriving the formulation from an appropriate free energy functional, which is crucial to guarantee that fundamental thermodynamics principles are satisfied, was followed.

A numerical procedure to solve the resulting system of equations was then proposed: a fixed point algorithm iteratively couples the “solid” and the “ fluid” parts of this system.

To illustrate the behavior of this poroelastic model several numerical examples were run. The first test cases consist of typical poroelastic configurations: swelling (see Figure 2 ) and complete drainage.

Finally, a simulation of cardiac perfusion was performed in an idealized left ventricle embedded with active fibers. Results showed the complex temporal and spatial interactions of the muscle and blood, reproducing several key phenomena observed in cardiac perfusion.

#### Interaction between vascularization and tumor development

Participant : Irène Vignon-Clementel.

This is a collaborative project with D. Drasdo (BANG, INRIA) and his coworkers.

In [32] , Drasdo, Vignon-Clementel and co-authors, have developed a multiscale model. Agent-based and continuum models are coupled to study the dynamic interplay of the tumor mass and its environment, including blood vessels and their remodeling, nutrients and other factors such as angiogenic growth factors. In their models the tumor cells are represented by individual agents. The models consider the competition between contact inhibition-limited and nutrient/oxygen limited growth. For these questions it turns out to be sufficient to model individual cells within a cellular automaton model where the dynamics is rule-based. The vessels are modeled explicitly as discrete objects with a simplified lumped model relating flow and pressure inside them while the diffusion of oxygen, nutrients and growth factors is represented by continuum equations. Drasdo et al. compared the growth kinetics of cell populations in 3D from a single precursor cell up to a cell population of several thousands of cells for (1) oxygen-and nutrient unlimited growth, (2) growth in a *static* vascular environment and (3) growth if the vascular environment is remodeled by angiogenesis. The model is able to explain the growth characteristics found in approximately spherically growing tumors such as multi-cellular spheroids, an experimental system used to mimic tumors in their avascular phase, and Xenografts of NIH3T3 mouse fibroblast cells which also have a spherical shape.

Work in progress includes the extension of this model to specific in-vitro and in-vivo situations, in collaboration with IGR (Institut Gustave Roussy, UPRES 4040) and the *LungSys* consortium (http://www.lungsys.de/ , Germany).