A New Approach to Model Confined Suspensions Flows in Complex Networks: Application to Blood Flow

The modeling of blood flows confined in micro-channels or micro-capillary beds depends on the interactions between the cell-phase, plasma and the complex geometry of the network. In the case of capillaries or channels having a high aspect ratio (their longitudinal size is much larger than their transverse one), this modeling is much simplified from the use of a continuous description of fluid viscosity as previously proposed in the literature. Phase separation or plasma skimming effect is a supplementary mechanism responsible for the relative distribution of the red blood cell’s volume density in each branch of a given bifur- cation. Different models have already been proposed to connect this effect to the various hydrodynamics and geometrical parameters at each bifurcation. We discuss the advantages and drawbacks of these models and compare them to an alternative approach for modeling phase distribution in complex channels networks. The main novelty of this new formulation is to show that albeit all the previous approaches seek for a local origin of the phase segre- gation phenomenon, it can arise from a global non-local and nonlinear structuration of the flow inside the network. This new approach describes how elementary conservation laws are sufficient principles (rather than the complex arametric models previously proposed) to provide non local phase separation. Spatial variations of the hematocrit field thus result from the topological complexity of the network as well as nonlinearities arising from solving a new free boundary problem associated with the flux and mass conservation. This network model approach could apply to model blood flow distribution either on artificial micro-models, micro-fluidic networks, or realistic reconstruction of biological micro-vascular networks

Steady streaming confined between three-dimensional wavy surfaces

We present a theoretical and numerical study of three-dimensional pulsatile confined flow between two rigid horizontal surfaces separated by an average gap h, and having three-dimensional wavy shapes with arbitrary amplitude σ h where σ ∼ O(1), but long-wavelength variations λ, with h/λ 1. We are interested in pulsating flows with moderate inertial effect arising from the Reynolds stress due to the cavity non-parallelism. We analyse the inertial steady-streaming and the second harmonic flows in a lubrication approximation. The dependence of the three-dimensional velocity field in the transverse direction is analytically obtained for arbitrary Womersley numbers and possibly overlapping Stokes layers. The horizontal dependence of the flow is solved numerically by computing the first two pressure fields of an asymptotic expansion in the small inertial limit. We study the variations of the flow structure with the amplitude, the channel’s wavelength and the Womersley number for various families of three-dimensional channels. The steady-streaming flow field in the horizontal plane exhibits a quadrupolar vortex, the size of which is adjusted to the cavity wavelength. When increasing the wall amplitude, the wavelengths characterizing the channel or the Womersley number, we find higher-order harmonic flow structures, the origin of which can either be inertially driven or geometrically induced. When some of the channel symmetries are broken, a steady-streaming current appears which has a quadratic dependence on the pressure drop, the amplitude of which is linked to the Womersley number

Coupling and robustness of intra-cortical vascular territories

Vascular domains have been described as being coupled to neuronal functional units enabling dynamic blood supply to the cerebral cyto-architecture. Recent experiments have shown that penetrating arterioles of the grey matter are the building blocks for such units. Nevertheless, vascular territories are still poorly known, as the collection and analysis of large three-dimensional micro-vascular networks are difficult. By using an exhaustive reconstruction of the micro-vascular network in an 18 mm 3 volume of marmoset cerebral cortex, we numerically computed the blood flow in each blood vessel. We thus defined arterial and venular territories and examined their overlap. A large part of the intracortical vascular network was found to be supplied by several arteries and drained by several venules. We quantified this multiple potential to compensate for deficiencies by introducing a new robustness parameter. Robustness proved to be positively correlated with cortical depth and a systematic investigation of coupling maps indicated local patterns of overlap between neighbouring arteries and neighbouring venules. However, arterio-venular coupling did not have a spatial pattern of overlap but showed locally preferential functional coupling, especially of one artery with two venules, supporting the notion of vascular units. We concluded that intra-cortical perfusion in the primate was characterised by both very narrow functional beds and a large capacity for compensatory redistribution, far beyond the nearest neighbour collaterals

Le réseau micro-vasculaire structure la distribution de la pression sanguine

Cerebral micro-vascular networks control the blood pressure distribution when considering in vitro blood rheology models. Blood rheology is complex and non-linear. In small vessels, the effective viscosity variations are important due to red blood cells packing in capillaries, the so-called Fåhræus-Lindquist effect, whilst concomitantly phase segregation appears in bifurcations. Direct numerical simulations of different non-linear rheological models of the blood are performed on realistic three-dimensional micro-vascular networks. These simulations exhibit two significant results. First, various rheological models lead to very similar pressure distribution over the whole range of physiologically relevant hematocrits. Secondly, different models for phase segregation lead to very distinct hematocrit distributions in the micro-vacular network. Nevertheless, the hematocrit distribution very weakly affects the pressure distribution. Hence, our results suggest that the micro-vacular network structure mainly controls the pressure distribution in micro-circulation, whilst the effect of hematocrit distribution is weak

From cerebral blood flow modeling to vascular units map in primate cortex

The amazing topological and geometrical complexity of micro-vascular networks in the brain, and in other organs, has challenged many researchers for decades. Since the brain's vascular system is structured by a highly reticulated pial surface network which plunges down into a set of penetrating vessels, it is tempting to attribute a vascular unit to each penetrating arteriole. Recent experimental analysis have led to a breakthrough on the properties of the blood supply in the brain. Penetrating arterioles have been identified as the bottleneck of brain perfusion. Furthermore, it has also been realized that targeted clots of penetrating arterioles are not compensated by active changes in the diameter of their neighbor arteries. This observation suggests passive compensatory mechanisms resulting from the couplings between arteriolar territories consistent with other recent observations of active blood flow reorganization via collateral vessels (inter-arterial connections). A systematic investigation of the three-dimensional extent of compensation is not possible with experimental measurements but in silico simulations permit a systematic investigation of the spatial distribution of the brain perfusion

Analyse des écoulements micro-vasculaires cérébraux

Ces travaux s'intéressent à la micro-hydrodynamique des écoulements sanguins cérébraux dans les réseaux micro-vasculaires. A partir d'images tridimensionnelles haute résolution obtenues par micro-tomographie, les réseaux micro-vasculaires sont numérisés après différentes étapes de traitement d'images. Une résolution numérique approchée des écoulements sanguins quasi-statique de type réseau est développée. Cette approche permet à la fois de prendre en compte la structure géométrique micro-vasculaire complexe et les interactions hydrodynamiques non-locales des hématies confinées qui s'écoulent en son sein. La méthode proposée permet l'évaluation des distributions de pression, d'hématocrite et de débit sur des volumes cérébraux d'une dizaine de millimètres cube où quelques dizaines de milliers de segments vasculaires sont présents. L'analyse systématique des modèles de rhéologie sanguine existant montrent la prépondérance du choix du modèle de viscosité, à opposer au faible impact de la séparation de phase sur l'écoulement, phénomène pour lequel nous proposons aussi une modélisation alternative. De plus, nous avons analysé la perfusion sanguine cérébrale, et l'organisation générale des écoulements corticaux. Dans la situation physiologique normale, nous quantifions les territoires vasculaires qui sont les unités fonctionnelles micro-vasculaires. Nous évaluons leur évolution et leur robustesse dans des contextes pathologiques modèles et notamment pour les micro-accidents vasculaires cérébraux.This work is concerned with the hydrodynamics of blood flow in micro-vascular cerebral networks. The micro-vascular networks are reconstructed from specimens using high resolution three-dimensional micro-tomography imaging and post-processing techniques. A numerical quasi-static approximation of blood flow is then developed. This approach allows to account for both the complex geometric structure of micro-vascular networks and the non-local hydrodynamic interactions of red blood cells flowing within them. The proposed method permits the evaluation of pressure, hematocrit and flow distributions over cortical brain volumes of about ten cubic millimeters containing tens of thousands of vascular segments. A systematic analysis of blood rheology models demonstrate the importance of the choice of viscosity model, in contrast to the low impact of the phase separation on the flow, a phenomenon for which we also offer alternative model. Moreover, we analyse the the cerebral blood perfusion and the general organisation of cortical blood flow. We quantify the vascular regions that are essentially micro-vascular functional units under normal physiological conditions and assess their evolution and robustness in pathological contexts such as for cerebral micro-strokes

Benchmark of different CFL conditions for IMPES

The IMplicit Pressure Explicit Saturation (IMPES) method is a prevalent way to simulate multiphase flows in porous media. The numerical stability of this sequential method implies limitations on the time step, which may depend on the flow regime studied. In this note, three stability criteria related to the IMPES method, that differ in their construction on the observed variables, are compared on homogeneous and heterogeneous configurations for different two-phase flow regimes (viscous/capillary/gravitational). This highlights that there is no single optimal criterion always ensuring stability and efficiency. For capillary dominated flows, the Todd’s condition is the most efficient one, while the standard Coat condition should be preferred for viscous flows. When gravity effects are present, Coat’s condition must be restricted, but remains more efficient than the Todd’s condition

A comparison of various methods for the numerical evaluation of porous media permeability tensor from pore-scale geometry

In this work, several boundary value problems used to numerically evaluate the absolute permeability tensors of porous media using core-scale images are compared and discussed. The various configurations differ by the type of boundary conditions used to compute the flow at the micro-scale. The issue is the ability of the method to capture anisotropy correctly and to avoid possible percolation artifacts. This study is carried on two-dimensional synthetic, isotropic or anisotropic, porous media, that are chosen to illustrate the various difficulties mentioned above. A new method is proposed which consists in embedding the porous medium in question in a homogenized one. Using an iterative optimization procedure on the surrounding permeability, the method determines the absolute permeability tensor of the original medium. The equivalent permeability tensor that minimizes the effect on the surrounding porous medium is, unlike that of classical methods, de facto symmetrical due to the use of periodic boundary conditions and exhibits significantly lower permeabilities. The way in which non-diagonal terms of the permeability tensor are obtainedwith the various methods are thoroughly discussed

A reduced-order modeling for efficient design study of artificial valve in enlarged ventricular outflow tracts

A computational approach is proposed for efficient design study of a reducer stent to be percutaneously implanted in enlarged right ventricular outflow tracts (RVOT). The need for such a device is driven by the absence of bovine or artificial valves which could be implanted in these RVOT to replace the absent or incompetent native valve, as is often the case over time after Tetralogy of Fallot repair. Hemodynamics are simulated in the stented RVOT via a reduce order model based on proper orthogonal decomposition (POD), while the artificial valve is modeled as a thin resistive surface. The reduced order model is obtained from the numerical solution on a reference device configuration, then varying the geometrical parameters (diameter) for design purposes. 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. Such an approach could also be useful for uncertainty quantification