Modeling oxygen transport in the brain: An efficient coarse-grid approach to capture perivascular gradients in the parenchyma.

Recent progresses in intravital imaging have enabled highly-resolved measurements of periarteriolar oxygen gradients (POGs) within the brain parenchyma. POGs are increasingly used as proxies to estimate the local baseline oxygen consumption, which is a hallmark of cell activity. However, the oxygen profile around a given arteriole arises from an interplay between oxygen consumption and delivery, not only by this arteriole but also by distant capillaries. Integrating such interactions across scales while accounting for the complex architecture of the microvascular network remains a challenge from a modelling perspective. This limits our ability to interpret the experimental oxygen maps and constitutes a key bottleneck toward the inverse determination of metabolic rates of oxygen. We revisit the problem of parenchymal oxygen transport and metabolism and introduce a simple, conservative, accurate and scalable direct numerical method going beyond canonical Krogh-type models and their associated geometrical simplifications. We focus on a two-dimensional formulation, and introduce the concepts needed to combine an operator-splitting and a Green's function approach. Oxygen concentration is decomposed into a slowly-varying contribution, discretized by Finite Volumes over a coarse cartesian grid, and a rapidly-varying contribution, approximated analytically in grid-cells surrounding each vessel. Starting with simple test cases, we thoroughly analyze the resulting errors by comparison with highly-resolved simulations of the original transport problem, showing considerable improvement of the computational-cost/accuracy balance compared to previous work. We then demonstrate the model ability to flexibly generate synthetic data reproducing the spatial dynamics of oxygen in the brain parenchyma, with sub-grid resolution. Based on these synthetic data, we show that capillaries distant from the arteriole cannot be overlooked when interpreting POGs, thus reconciling recent measurements of POGs across cortical layers with the fundamental idea that variations of vascular density within the depth of the cortex may reveal underlying differences in neuronal organization and metabolic load.

Multiscale modelling of blood flow in cerebral microcirculation: Details at capillary scale control accuracy at the level of the cortex.

Aging or cerebral diseases may induce architectural modifications in human brain microvascular networks, such as capillary rarefaction. Such modifications limit blood and oxygen supply to the cortex, possibly resulting in energy failure and neuronal death. Modelling is key in understanding how these architectural modifications affect blood flow and mass transfers in such complex networks. However, the huge number of vessels in the human brain-tens of billions-prevents any modelling approach with an explicit architectural representation down to the scale of the capillaries. Here, we introduce a hybrid approach to model blood flow at larger scale in the brain microcirculation, based on its multiscale architecture. The capillary bed, which is a space-filling network, is treated as a porous medium and modelled using a homogenized continuum approach. The larger arteriolar and venular trees, which cannot be homogenized because of their fractal-like nature, are treated as a network of interconnected tubes with a detailed representation of their spatial organization. The main contribution of this work is to devise a proper coupling model at the interface between these two components. This model is based on analytical approximations of the pressure field that capture the strong pressure gradients building up in the capillaries connected to arterioles or venules. We evaluate the accuracy of this model for both very simple architectures with one arteriole and/or one venule and for more complex ones, with anatomically realistic tree-like vessels displaying a large number of coupling sites. We show that the hybrid model is very accurate in describing blood flow at large scales and further yields a significant computational gain by comparison with a classical network approach. It is therefore an important step towards large scale simulations of cerebral blood flow and lays the groundwork for introducing additional levels of complexity in the future.

Numerical Investigation of Thermal Counterflow of He II Past Cylinders.

We investigate numerically, for the first time, the thermal counterflow of superfluid helium past a cylinder by solving with a finite volume method the complete so-called two-fluid model. In agreement with existing experimental results, we obtain symmetrical eddies both up- and downstream of the obstacle. The generation of these eddies is a complex transient phenomenon that involves the friction of the normal fluid component with the solid walls and the mutual friction between the superfluid and normal components. Implications for flow in a more realistic porous medium are also investigated.

Determination of Chlorinated Solvent Sorption by Porous Material-Application to Trichloroethene Vapor on Cement Mortar.

Experiments have been performed to investigate the sorption of trichloroethene (TCE) vapor by concrete material or, more specifically, the cement mortar component. Gas-flow experiments were conducted using columns packed with small pieces of cement mortar obtained from the grinding of typical concrete material. Transport and retardation of TCE at high vapor concentrations (500 mg L-1) was compared to that of a non-reactive gas tracer (Sulfur Hexafluoride, SF6). The results show a large magnitude of retardation (retardation factor = 23) and sorption (sorption coefficient = 10.6 cm3 g-1) for TCE, compared to negligible sorption for SF6. This magnitude of sorption obtained with pollutant vapor is much bigger than the one obtained for aqueous-flow experiments conducted for water-saturated systems. The considerable sorption exhibited for TCE under vapor-flow conditions is attributed to some combination of accumulation at the air-water interface and vapor-phase adsorption, both of which are anticipated to be significant for this system given the large surface area associated with the cement mortar. Transport of both SF6 and TCE was simulated successfully with a two-region physical non-equilibrium model, consistent with the dual-medium structure of the crushed cement mortar. This work emphasizes the importance of taking into account sorption phenomena when modeling transport of volatile organic compounds through concrete material, especially in regard to assessing vapor intrusion.

Kinetic modeling in the context of cerebral blood flow quantification by H2(15)O positron emission tomography: the meaning of the permeability coefficient in Renkin-Crone׳s model revisited at capillary scale.

One the one hand, capillary permeability to water is a well-defined concept in microvascular physiology, and linearly relates the net convective or diffusive mass fluxes (by unit area) to the differences in pressure or concentration, respectively, that drive them through the vessel wall. On the other hand, the permeability coefficient is a central parameter introduced when modeling diffusible tracers transfer from blood vessels to tissue in the framework of compartmental models, in such a way that it is implicitly considered as being identical to the capillary permeability. Despite their simplifying assumptions, such models are at the basis of blood flow quantification by H2(15)O Positron Emission Tomgraphy. In the present paper, we use fluid dynamic modeling to compute the transfers of H2(15)O between the blood and brain parenchyma at capillary scale. The analysis of the so-obtained kinetic data by the Renkin-Crone model, the archetypal compartmental model, demonstrates that, in this framework, the permeability coefficient is highly dependent on both flow rate and capillary radius, contrarily to the central hypothesis of the model which states that it is a physiological constant. Thus, the permeability coefficient in Renkin-Crone׳s model is not conceptually identical to the physiologic permeability as implicitly stated in the model. If a permeability coefficient is nevertheless arbitrarily chosen in the computed range, the flow rate determined by the Renkin-Crone model can take highly inaccurate quantitative values. The reasons for this failure of compartmental approaches in the framework of brain blood flow quantification are discussed, highlighting the need for a novel approach enabling to fully exploit the wealth of information available from PET data.

Comment on "Frequency-dependent dispersion in porous media".

In a recent paper, Valdès-Parada and Alvarez-Ramirez [Phys. Rev. E 84, 031201 (2011)] used the technique of volume averaging to derive a "frequency-dependent" dispersion tensor, D(γ)(*), the goal of which is to describe solute transport in porous media undergoing periodic processes. We describe two issues related to this dispersion tensor. First, we demonstrate that the definition of D(γ)(*) is erroneous and derive a corrected version, D(γ)(*c). With this modification, the approach of Valdès-Parada and Alvarez-Ramirez becomes strictly equivalent to the one devised by Moyne [Adv. Water Res. 20, 63 (1997)]. Second, we show that the term "frequency-dependent dispersion" is misleading because D(γ)(*) and D(γ)(*c) do not depend on the process operating frequency, χ. The study carried out by Valdès-Parada and Alvarez-Ramirez represents a spectral analysis of the relaxation of D(γ)(*) towards its steady-state, independent of any periodic operation or excitation.

Calculation of effective diffusivities for biofilms and tissues.

In this study we describe a scheme for numerically calculating the effective diffusivity of cellular systems such as biofilms and tissues. This work extends previous studies in which we developed the macroscale representations of the transport equations for cellular systems based on the subcellular-scale transport and reaction processes. A finite-difference model is used to predict the effective diffusivity of a cellular system on the basis of the subcellular-scale geometry and transport parameters. The effective diffusivity is predicted for a complex three-dimensional structure that is based on laboratory observations of a biofilm, and these numerical predictions are compared with predictions from a simple analytical solution and with experimental data. Our results indicate that, under many practical circumstances, the simple analytical solution can be used to provide reasonable estimates of the effective diffusivity.