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

A numerical framework for the simulation of molecular diffusion in the micro-vascular system

Although direct simulations of the whole cerebral microcirculation are computationally cumbersome, we will show that they can be carried out on a smaller scale (about 50 to 10000 vessels). For that purpose, we introduce a numerical framework able to solve the fully coupled problem: tissue-diffusion and vessels- advection/diffusion

Structural and hemodynamic comparison of anatomical and synthetic cerebral capillary networks

A computational method is presented for generating synthetic, random 3D capillary networks which match the topological, geometrical and functional properties of the cerebral microcirculation. These networks, which can be generated in volumes larger than can currently be extracted by high-resolution imaging, can then be coupled to lower-resolution data sets of whole-brain vasculature to model blood flow and mass transport, and to validate equivalent continuum/hybrid models. Another motivation is to reveal the dominant structural features of cerebral capillary networks, which can then be tuned to model different brain regions or pathological states such as Alzheimer’s disease. Previous works [1, 2] lacked physiological basis, and although resulting networks conformed to expected global morphometric properties, were not subjected to thorough topological or functional analysis. In contrast, our approach is based on the physiological assumption that the maximum separation of tissue cells from the nearest capillary is limited by the diffusion distance of oxygen [3]. Previously, synthetic, space-filling 2D networks were constructed by placing one point randomly in each cell of an n × n grid; from this set of points, Voronoi diagrams were extracted with the edges producing a 2D network with mainly three capillaries per vertex, a characteristic feature of cerebral capillary networks. Here, we extend this approach to 3D. In 3D, Voronoi diagrams produce polyhedrons with many capillaries per vertex. To derive a network with only bifurcations, clusters of vertices were systematically merged and capillaries then removed randomly. Geometrical metrics such as the mean/S.D. of lengths and edge/length/vertex densities were compared to those of capillary regions extracted from mouse cerebral anatomical data sets [5, 6]. Capillary loops were studied to measure the interconnected network topology, while the distribution of extravascular distances allowed comparison of the spatial arrangement of capillaries. Finally, hemodynamic properties were captured through the network permeability. Overall, synthetic networks showed excellent agreement with the anatomical data. This work was supported by ERC BrainMicroFlow GA615102. We acknowledge D. Kleinfeld, P. Tsai and P. Blinder for kindly sharing their anatomical data with us

Structural and hemodynamic comparison of synthetic and anatomical cerebral capillary networks

A computational method is presented for generating 3D synthetic, random capillary networks which match the topological, geometrical and functional properties of the cerebral microcirculation. This enables production of larger capillary networks than can currently be extracted using high-resolution imaging modalities. These networks can then be coupled to lower-resolution data sets of whole-brain vasculature (capillaries unresolved) to model blood flow and mass transport, and to validate equivalent continuum/hybrid models. Another motivation is to reveal the dominant structural features of cerebral capillary networks, enabling us to tune these features to model different brain regions or pathological states such as Alzheimer's disease. Previous works (Linninger et al, Ann Biomed Eng, 2013; Su et al, Microcirc, 2012) lacked physiological basis, and although resulting networks conformed to expected global morphometric properties, they were not subjected to thorough topological or functional analysis. In contrast, our approach is based on the physiological assumption that the maximum separation of tissue cells from the nearest capillary is limited by the diffusion distance of oxygen (Lorthois & Cassot, J Theor Biol, 2010). Previously, synthetic, space-filling 2D networks were constructed by placing one point randomly in each cell of an n × n grid; from this set of points, Voronoi diagrams were extracted with the edges producing a 2D capillary network with mainly three capillaries per vertex, a characteristic feature of cerebral capillary networks. Here, we present a 3D extension of this approach and compare the resulting structural and hemodynamic properties to those of anatomical cerebral capillary networks. In 3D, Voronoi diagrams produce polyhedrons with many capillaries at each vertex. To derive a network with only bifurcations, clusters of vertices were systematically merged and capillaries were then randomly removed. The resulting network structures were compared to capillary regions extracted from human and mouse anatomical data sets (Cassot et al, Microcirc, 2006; Tsai et al, J NeuroSci, 2009; Blinder et al, Nat Neurosci, 2013), showing excellent agreement. Geometrical metrics included the mean/S.D. of capillary lengths and edge/length/vertex densities. To measure the interconnected network topology, capillary loops were identified and the mean number of edges per loop, loop length, and number of loops per edge were compared. The spatial arrangement of capillaries was compared by studying the distribution of extravascular distances. Finally, the permeability was computed as a hemodynamic measure of blood flow conductivity

Modelling solute transport in the brain microcirculation: is it really well mixed inside the blood vessels?

Most network models describing solute transport in the brain microcirculation use the well-mixed hypothesis and assume that radial gradients inside the blood vessels are negligible. Recent experimental data suggest that these gradients, which may result from heterogeneities in the velocity field or consumption in the tissue, may in fact be important. Here, we study the validity of the well-mixed hypothesis in network models of solute transport using theoretical and computational approaches. We focus on regimes of weak coupling where the transport problem inside the vasculature is independent of the concentration field in the tissue. In these regimes, the boundary condition between vessels and tissue can be modelled by a Robin boundary condition. For this boundary condition and for a single cylindrical capillary, we derive a one-dimensional cross-section average transport problem with dispersion and exchange coefficients capturing the effects of radial gradients. We then extend this model to a network of connected tubes and solve the problem in a complex anatomical network. By comparing with results based on the well-mixed hypothesis, we find that dispersive effects are a fundamental component of transport in transient situations with relatively rapid injections, i.e. frequencies above one Hertz. For slowly varying signals and steady states, radial gradients also significantly impact the spatial distribution of vessel/tissue exchange for molecules that easily cross the blood brain barrier. This suggests that radial gradients cannot be systematically neglected and that there is a crucial need to determine the impact of spatio-temporal heterogeneities on transport in the brain microcirculation

Upscaling mass transfer in brain capillary networks

Brain perfusion imaging techniques rely on the measurement of spatio-temporal concentration fields of various endogenous or exogenous tracers in the brain tissue. Their resolution is typically between 1 mm3 Magnetic Resonance Imaging) and (10 mm)3 Positron Emission Tomography). This is much coarser than the diameters of most arterioles and venules, which are typically below 100 μm, and, of course, of capillaries, whose diameters are tenfold smaller. This implies that methods to deduce the regional blood flow rate out of these large-scale concentration fields should rely on upscaled models, i.e. models describing the macro-scale behavior of the vascular system with effective properties taking into account its microstructure. To derive such models, the Volume Averaging Technique, which has been previously developed for upscaling mass transfer in heterogeneous porous media, can be applied to the advection-diffusion equations. Capillary networks indeed exhibit a space filling mesh-like structure, for which a Representative Elementary Volume (REV), can be extracted: a 3D network of capillaries with diameters ranging from 1 to 10 μm embedded in tissue, with volume about (150 to 300 μm)3. In this technique, closure equations must be solved in REVs to deduce effective coefficients, representing its macro-scale behavior. Being able to solve closure equations on any 3D network geometry taking into account individual vessels is a computational challenge. Here, we developed a numerical framework to solve partial differential equations on anatomically accurate capillary networks using the finite element library Feel++. This framework is used to 1) solve the closure equations on a REV and deduce its effective coefficients and 2) perform direct simulations of mass transfers problems as references to validate the upscaling procedure

A new model for molecule exchange in the brain microvascular system: consequences of capillary occlusions in Alzheimer's disease

The brain microvascular system is a key actor in Alzheimer’s disease (AD) development. Indeed, a significant decrease of cerebral blood flow is the earliest biomarker of AD. In vivo TPLSM of cortical vasculature in APP/PS1 mice suggests the mechanism underlying the blood flow reduction is capillary occlusions. Leucocytes adhere to inflamed vessel walls and limit the flow. The impact of capillary occlusions on blood flow has been quantified numerically in large (>10000 vessels) anatomical networks in humans and mice. The regional blood flow has been found to depend linearly with no threshold effect on the fraction of capillary occlusions, so that a small fraction of stalls (2-4%) yields a significant decrease in blood flow (5-12%). Such flow decrease has a strong impact on nutrient delivery and waste clearance. That is why we devised a new model to study the effect of capillary stalling on molecule transport. The geometry of anatomical networks is too complex to use classic numerical approaches like finite elements. Instead, our model, inspired by pore-network approaches, reduces computational costs while capturing most of the underlying physics. To derive this model, we apply upscaling methods to the 3D transport equations within each vessel to obtain 1D average equations along the axis. Contrary to previous models, this new formulation describes accurately radial concentration gradients, capturing effects like longitudinal dispersion. We further use a Green’s function formulation to calculate the concentration fields inside the tissue where diffusion and reaction occur. The coupling between vessels and tissues is modelled using a membrane condition representing the blood brain barrier. This new molecule transport model is coupled with our previously validated blood flow model to examine the effects of capillary stalling on molecular exchange in transient and stationary regimes in anatomical networks. In particular, in stationnary regimes, we demonstrate an increase of the extraction coefficient with the proportion of stalled capillaries, which does not compensate for the associated blood flow reduction

VITAE : VIrTual brAin pErfusion

VITAE is an ERC funded software project aimed at providing full brain simulations of cerebral blood flow and solute exchange between blood and the neural tissue. The endgoal is to understand fine scale interactions between the architecture of the microvascular network in the brain and its functions (blood supply, oxygen and nutrient delivery, waste removal). This may indeed help unveil potential causes of cerebral disease like Alzheimer’s Disease. In the actual state of the art, full scale brain simulations are something new. First, acquiring input anatomical data of the blood vessel network is difficult and is an active domain of research. Next, simulation by itself is a CPU intensive Computational Fluid Dynamic problem requiring both inversion of large matrices and manipulation of large amounts of data. The current milestone is capable of running pressure resolution in a full mouse brain composed of about 5 millions of microvessels in one second on 1024 processor cores. The software written in C++ fully supports parallelized IO and graph partitioning to optimize the placement of vertices and reduce computing times. The next challenge is to run simulations taking the complex behavior of blood into account, which requires to run the pressure solver from one hundred to several thousand times. This will require to improve significantly the convergence time. Acknowledgements: ERC Funded Project: Proof of Concept (PoC), ERC-2018-PoC A. Sauvé, J.-D. Julien, M. Berg, M. Peyrounette, P. Elyakime, Y. Davit, M. Pigou, S. Lorthoi

Brain capillary networks across species : a few simple organizational requirements are sufficient to reproduce both structure and function

Despite the key role of the capillaries in neurovascular function, a thorough characterization of cerebral capillary network properties is currently lacking. Here, we define a range of metrics (geometrical, topological, flow, mass transfer, and robustness) for quantification of structural differences between brain areas, organs, species, or patient populations and, in parallel, digitally generate synthetic networks that replicate the key organizational features of anatomical networks (isotropy, connectedness, space-filling nature, convexity of tissue domains, characteristic size). To reach these objectives, we first construct a database of the defined metrics for healthy capillary networks obtained from imaging of mouse and human brains. Results show that anatomical networks are topologically equivalent between the two species and that geometrical metrics only differ in scaling. Based on these results, we then devise a method which employs constrained Voronoi diagrams to generate 3D model synthetic cerebral capillary networks that are locally randomized but homogeneous at the network-scale. With appropriate choice of scaling, these networks have equivalent properties to the anatomical data, demonstrated by comparison of the defined metrics. The ability to synthetically replicate cerebral capillary networks opens a broad range of applications, ranging from systematic computational studies of structure-function relationships in healthy capillary networks to detailed analysis of pathological structural degeneration, or even to the development of templates for fabrication of 3D biomimetic vascular networks embedded in tissue-engineered constructs