Fig 1.
Multiscale architecture of microvascular networks in human brain with a representation of different length scales.
(a) Brain scale (∼63 cm2 × 300 μm): 300 μm-thick cortical section, where blood vessels have been injected with India ink for contrast enhancement [7]. (b) Macroscopic scale (∼18 mm2 × 300 μm): reconstruction of parts of the collateral sulcus by confocal laser microscopy [8]. (c) Mesoscopic scale (∼5 mm2 × 300 μm): region of interest in which vessels of more than 10 μm in diameter are colored in black and vessels of less than 10 μm in diameter are colored in red (diameters have been multiplied by 2 for visualization). In contrast with the capillary bed, the arteriolar and venular trees have a quasi-fractal structure [9]. (d) Microscopic scale (∼0.07 mm2 × 300 μm): detailed view of the capillary bed. The capillary bed is dense and space-filling over a cut-off length of ∼ 50 μm [9].
Fig 2.
Schematic diagrams of the hybrid approach and spatial correlations between coupling sites.
(a) The arteriolar (A) and venular (V) trees (black networks) plunge into the cortical tissue where their endpoints are connected to the capillary bed (light red). Because of the multiscale architecture of the human brain (Fig 1), we adopt two separate strategies to model blood flow at macroscopic scale, depending on the hierarchical position of the vessels in the microvascular network. A network approach is used in arteriolar and venular trees while the capillary bed is considered as a continuum. The continuum model is then discretized using coarse mesh cells (green) at mesoscopic scale, with size hx × hy × hz ≃ (250 μm)3. The side of a discretization cell is significantly larger than the capillary characteristic length lcap = 50 μm. The red dots correspond to the coupling points connecting arterioles (or venules) to capillary vessels, where coupling conditions must be applied. (b) We use anatomically accurate data to illustrate spatial correlations between the coupling sites. Minimum distances have been computed between all coupling points according to their types (arteriolar to arteriolar, venular to venular and arteriolar to venular). The histogram plot presents the percentages of couplings associated to the same range of minimum distance.
Fig 3.
Schematic diagrams of the derivation of the coupling condition between one of the arteriolar or venular endpoints and the continuum representing the capillary bed.
(a) The legend shows all the domains used for the development of the coupling model with a detailed nomenclature provided in Appendix D. (b) An arteriolar or venular tree ΩAV (black network) plunges into a FV grid ΩFV (green grid). The capillary vessels in Ωcap that are connected to the coupling point (highlighted by a red dot) are represented in red. The coupling condition describes the relationship between the network pressure πs at this coupling point and the pressures of the surrounding cells , which include the coupled cell. The pressure field is reconstructed in the local neighbourhoods
and
using Eq 20a and 20b. In the most distant cells
, the analytical approximation matches with the FV representation further away. To make visualizations easier, a centered coupling is illustrated. (c) Detailed view of the coupled cell. The linear and spherical approximations match at a distance lΓ from the coupling point. πΓ is the averaged weighted pressure evaluated at this distance. (d) 1D pressure profile: FV pressures (green steps) are plotted on the left-hand side while both linear (red) and spherical (blue) approximations are plotted on the right-hand side. (e) Domains for the computation of errors. Four regions have been selected to compare hybrid and CN approaches: the arteriolar and venular trees ΩAV (black), the coupling points s (red), the surrounding cells
(hatched blue) and the rest of the FV domain
(hatched green) which excludes cells where the pressure field must be analytically reconstructed.
Fig 4.
Visualization of the model three-dimensional lattices for the simplified representation of the capillary networks.
The elementary patterns of 6- and 3-regular networks are presented on the left- and right-hand sides, respectively. Each inner vertex of a 6-regular network has 6 neighbours, while an inner vertex of a 3-regular network has always 3 neighbours. These patterns can be duplicated as necessary in each direction, depending on the desired size of the capillary network.
Table 1.
Effective parameters for the continuum representation of the model capillary networks.
This table summarizes the values of effective permeability Keff and viscosity μeff used in this paper (Section 2.6). Details about the computation of these coefficients are provided in Appendix A.
Fig 5.
Schematics of the boundary conditions imposed for the validation tests.
(a) and (b) The black boxes represent either the capillary bed in CN configuration or the continuum in the hybrid approach. For comparison, the boundary conditions imposed at the boundaries of the capillary bed in the CN simulations are similar to those imposed at the faces of the continuum in the hybrid approach. The black cylinders represent the arteriolar and/or venular vessels. The in- and outflow are illustrated by arrows. The arteriolar and venular endpoints highlighted by red dots are the coupling sites, which are connected to capillaries in CN configuration or connected to the continuum using the coupling model in the hybrid approach. (a) The boundary conditions are schematized for the simplest configuration where a single arteriolar vessel is coupled with a cubic capillary bed/continuum. A pressure drop of 10 000 Pa is imposed between the arteriolar inlet and the bottom face of the capillary bed/continuum. A condition of impermeability (i.e. no flow) is imposed at the top face, and periodic conditions are imposed in the two other directions (left-right and front-back). (b) The boundary conditions are schematized for a multi-coupling configuration, where one arteriolar and one venular vessels are coupled with a parallelepipedic capillary bed/continuum. A pressure drop of 7 000 Pa is imposed between the arteriolar inlet and the venular outlet. A condition of impermeability (i.e. no flow) is imposed at the top and bottom faces, and periodic conditions are imposed in the two other directions (left-right and front-back).
Fig 6.
Comparison of hybrid and reference CN approaches for simple configurations. In the CN version, the arteriolar and venular trees are connected to a 6-regular capillary network. In the hybrid approach, they plunge into an equivalent porous medium which is discretized in FV cells of size h.
(a) Schematics of the basic tests: in the baseline configuration i, a single coupling is imposed at the center of the coupled cell. In the following configurations (ii to iv), we independently study the robustness of the model by imposing three geometrical constraints that typically occur in brain microcirculation: off-centering of the coupling in a cell (ii), effect of domain boundaries (iii), and effect of the distance to a second coupling point (iv). These effects are characterized by three distances from the coupling point to: the center of the coupled cell (ii, distance doff), the upper domain boundary (iii, distance dboundary), and another coupling point (iv, distance dAV). (b) Global pressure errors: for each configuration, global pressure errors are computed in the four domains of interest defined in Fig 3(e). The errors are normalized by the global pressure drop δP. (c) Local pressure errors: the distributions of the local pressure errors are displayed for the ratios hlcap indicated by orange dotted lines in (c). (d) Global flow rate errors: the flow rate errors are computed in the same four domains of interest. The errors are normalized by the global incoming flow rate qA.
Fig 7.
Comparison of pressure fields computed via hybrid and CN approaches.
(a) The pressure fields are displayed for two configurations of interest: (i) centered and (ii) off-centered single coupling of an arteriolar vessel in a 6-regular capillary network. The coupling point is indicated by a large orange dot. In the off-centered case, the coupling condition is either distributed among neighbouring cells or not. (b) Pressure profiles are plotted along a line of interest, which is indicated by orange dashes in (a). The reference pressure field from CN approach is represented by orange crosses. Green steps represent cell pressures of the FV representation, blue and red lines respectively represent linear and spherical approximations. Black dashed lines represent the limit of
, where pressure field is approximated by analytical solutions. The reconstructed pressure field of the hybrid approach is finally obtained by combining solid lines. The asymmetry of the field far away from the source is caused by the imposed boundary conditions at the limits of the domain. Here πref = 5 000 Pa is a reference pressure value used for nondimensionalization.
Fig 8.
Summary of errors computed for simple configurations, for h/lcap > 4.
In Section 3.1, pressure and flow rate errors have been computed for five simple configurations (Baseline, Base+off-center, etc). For h/lcap > 4, maximal errors are indicated by orange dashed lines in each individual graph of Fig 6 and S5 Fig, and combined here in the same diagram.
Fig 9.
Distribution of pressure and flow rate local errors by comparing the hybrid and CN approaches in the large realistic configuration.
(a) Pressure and flow rate error maps in arteriolar and venular trees are presented. (b) In black crosses, local errors are plotted against local pressure or flow rate reference values. Additional errors are represented as gray crosses corresponding to computations without the proposed coupling model (pressure continuity at coupling sites). Here, πref = 5 000 Pa and qref = 5 × 10−12 m3 · s−1 are reference pressure and flow rate values used for nondimensionalization.
Fig 10.
Comparison between the hybrid and CN approaches in the large realistic configuration.
(a) The cumulative flow rates are compared for both hybrid and CN approaches in the left arteriole. The same conventions as in Fig 9 are adopted. (b) The probability density functions are presented for flow rate using the hybrid approach with (black) or without (gray) the coupling model, and CN approach (orange). Here, πref = 5 000 Pa and qref = 5 × 10−12 m3 · s−1 are reference pressure and flow rate values used for nondimensionalization.
Fig 11.
Statistical study for the computation of the effective permeability in a 3-regular capillary network with uniform length (lcap = 30.62 μm) and diameter dcap = 8 μm.
The variation of the effective permeability Keff is plotted in function of the side L of the capillary network. The convergence threshold is established at L = 300 μm (orange dashed line).
Fig 12.
Statistical study for the computation of the effective permeability Keff and viscosity μeff in 6- and 3-regular capillary networks with uniform length and distributed diameters.
(a) First, we derive a reference permeability Kref by assuming a uniform diameter, thus a uniform viscosity μref, in the capillaries. The variation of Kref is plotted in function of the side L of the capillary domain. For 3-regular networks, the convergence threshold is established for L = 300 μm (orange dashed line). (b) Second, we assume the viscosity to be uniform in the capillaries and derive the effective permeability Keff from computations involving 2000 different distributions of diameters. The mean and standard deviation of the ratio Keff/Kref is plotted in function of the side L of the capillary domain. For 6-regular networks, the convergence threshold is established for L = 200 μm (orange dashed line) (c) Finally, we distribute the viscosity in the capillaries and derive the effective viscosity μeff at mesoscopic scale from computations involving the same 2000 different distributions of diameters. The mean and standard deviation of the ratio μeff/μref is plotted in function of the side L of the capillary domain.
Fig 13.
Visualization of the final matrix structure.
Using the hybrid approach, we can compute blood flow by solving a single linear system, assembled from the equations describing the finite volume discretization in the continuum (FV, in green), the network approach in the arteriolar and venular trees (AV, in black) and the coupling model at the interface of the two frameworks (coupling, in red). This yields a sparse matrix that is schematized here. Plain lines and dots represent non-zero values. The components (Pi)i, (πα)α and (πs)s are the unknowns of the linear system, corresponding to FV, AV and coupling parts, respectively.