The cortical angiome: an interconnected vascular network with noncolumnar patterns of blood flow

Abstract

What is the nature of the vascular architecture in the cortex that allows the brain to meet the energy demands of neuronal computations? We used high-throughput histology to reconstruct the complete angioarchitecture and the positions of all neuronal somata of multiple cubic millimeter regions of vibrissa primary sensory cortex in mouse. Vascular networks were derived from the reconstruction. In contrast with the standard model of cortical columns that are tightly linked with the vascular network, graph-theoretical analyses revealed that the subsurface microvasculature formed interconnected loops with a topology that was invariant to the position and boundary of columns. Furthermore, the calculated patterns of blood flow in the networks were unrelated to location of columns. Rather, blood sourced by penetrating arterioles was effectively drained by the penetrating venules to limit lateral perfusion. This analysis provides the underpinning to understand functional imaging and the effect of penetrating vessels strokes on brain viability.

Figure 1

Examples of the vectorized data sets. (a,b) Example of data obtained throughout the full depth of cortex and extending into the white matter. Surface and penetrating arterioles are colored red, venules blue and the borders of cortical columns are denoted by a golden band. A selected slice from this data set is shown to illustrate the extent of penetrating vessels (b). (c) Example of data obtained through the upper half of cortex from mice used for transcranial imaging of intrinsic optical signals; see Figure 6. (d) Schematic of the make-up of edges in terms of individual centerlines, each with length $l_{mn}^{\left(k\right)}$, where m and n label the vertices and k labels the consecutive centerlines between vertices, and radius $r_{mn}^{\left(k\right)}$, computed as the average between the measured radii at vertices m and n. (e) Schematic of labeling of edges (E_nm) and vertices (V_m) used for topological analyses.

Figure 2

Analysis of the local geometry and topology of the microvasculature. (a) Scatter plot as a function of the total length of each vessel, defined as $L_{mn}={\sum }_k{l}_{mn}^{\left(k\right)}$, and the median radius, denoted as R_mn. The lines are plots of constant fluid resistance (equation in Edge resistance of Online Methods). Data are from 101,992 edges across four brains. The line plots are probability distribution functions (PDFs) for different brains, found by projecting the data across all lengths (right) or radii (top). (b) Plot of the mean radius across all segments in an axial slice as a function of depth. (c) Plot of the density of the microvasculature and neuronal density as a function of depth. The vascular density in an axial slice was defined as the fractional length of all edges in an axial slice. WM, white matter. (d) PDF of the number of branches in different microvascular loops. Data are from 59,909 loops across four brains. The bars denote 1 s.d. For comparison, the distribution of branches in the surface pial network is reproduced. The inset is a close up of a section of a vectorized network showing only the microvascualture. The colored edges highlight a loop that consists of eight branches, each with a distinct color. (e) Plot of the flow resistance per unit length as a function of vessel radius; the total resistance is found by multiplying by the length, in micrometers. Note the marked increase in resistance for radii below ~5 μm, where the Hagen-Poiseuille law (dashed line) no longer holds. The concurrent histogram shows the distribution of vessel radii for all vessels. (f) Schematic of the numerical probe of total resistance between two vertices in the network. (g) Scatter plot of the resistance between pairs of vertices (1,000 pairs per brain across four brains) as a function of the Euclidian distance between the vertices. The asymptote highlights the constant value averaged across all data sets, indicative of a three dimensional lattice. The slope, (3.0 ± 4.4) × 10⁻⁵ (mean ± 95% confidence interval, found with robust linear regression), was not significant.

Figure 3

Community analysis of the global topology of the microvasculature. The analysis makes use of the adjacency matrix for the branches. (a) Views of the complete vectorized vasculature that show different communities, each labeled by an individual color, as well as columnar boundaries (golden bands). (b–d) Representative communities. (e) The distribution of separate communities as a function of depth (362 communities across 4 mice; different colors represent data from different mice). The boxes highlight the extent of layer 4. The area of the dot scales as the size of the community relative to that of each angiome. The PDF on the right was computed by averaging over all examples. (f) Scatter plot of the number of edges between each pair of communities versus the number of branches in the community. The gray region is bounded by power law slopes of 2/3 and 1, and the dashed line is a fit of a power law to the data, with exponent 0.83 ± 0.04 (mean ± 95% confidence interval, found with robust linear regression). (g) Scatter plot of the number of communities that are either encompassed or pass through a cortical column as a function of the size of the column. Size is measured in terms of the number of branches that are either encompassed by or pass through the column. The horizontal line is the expected result when there is one community per column and the dashed line is a linear fit to the data, with slope 0.041 ± 0.004 (mean ± 95% confidence interval). The size of the dot is the fraction of the community that has the largest overlap with a given cortical column.

Figure 4

Calculated fluid flow and domains of common input in complete vectorized networks. (a) Vectorized vasculature in which the blood flow through a given penetrating arteriole is numerically labeled to determine the vessels that receive at least half of their flow from the chosen penetrating arteriole. This exercise is repeated for all penetrating arterioles and each territory is labeled with a different color. (b,c) Examples of different flow domains relative the cortical columns. (d) Overlap of multiple flow domains with cortical columns in layer 4. (e) Scatter plot of the number of flow domains that are either encompassed or pass through a cortical column. The size of the dot is the fraction of the flow domain that has the largest overlap with a given cortical column. The horizontal line is the expected result when there is one domain per column and the dashed line is a linear fit to the data, with slope 0.028 ± 0.003 (mean ± 95% confidence interval).

Figure 5

Relation of penetrating vessels to cortical columns. (a) Example data set from a flattened cortex. The location of all penetration arterioles (red squares) and all penetrating venules (blue squares) are superimposed on an axial projection of the upper 150 μm of cortex. The cortical columns are based on imaging data taken with a flattened cortex. (b) Summary statistics on the location of penetrating vessels relative to the centroid of the cortical columns. The numbers of vessels in each bin, beginning at the center of the column and heading toward the midline of the septum (insert), were 15, 55, 47, 31 and 36 for the penetrating arterioles and 30, 84, 102, 63 and 68 for the penetrating venules. The locations of cortical columns boundaries were deduced from the cell density in layer 4. We plotted the fraction of pixels covered by arterioles or venues in each of five bins relative to the fraction expected for uniform coverage. (c) Probability density function for the distance between pairs of nearest penetrating arterioles (red) and between pairs of nearest venules (blue). The two distributions were significantly different (Kolmogorov-Smirnov test, P < 0.0001). (d) Examples of primary branches (green) from penetrating arterioles (red) and venules (blue). (e) Probability density function of the arteriole (red) and venule (blue) primary branches as a function of depth below the pia. The two distributions were significantly different (Kolmogorov-Smirnov test, P < 0.0001); the number of arteriole branches peaked near layer 4, whereas that for venules peaked at the surface.

Figure 6

Relation of images of the intrinsic optical signal (IOS) to the centroids of the cortical columns. A thinned-skull window was prepared above vS1 cortex and individual vibrissae were deflected at 10 Hz for 4 s. The mice were anesthetized with isoflurane so that only a net deoxyhemoglobin signal was observed by reflectance of light with a center wavelength of 625 nm. (a) Selected frames for four different vibrissa from the same mouse. Each frame is 0.5 s in duration and represents an average over ten trials. M, medial; R, rostral. (b) Complete time dependence for the spatial location of maximal change for the data in a. Shaded areas represent s.d. (c) Responses from all columns, normalized in amplitude and thresholded to avoid spatial overlap, are superimposed on a map of cortical columns obtained from flattened tissues optically sectioned with two-photon microscopy. Part of the mismatch at the lateral side results from an incomplete correction for the curvature of the brain. (d) Example of relatively small cortical columns in which the optical signal is centered (×) on the columnar centroid (+) as opposed to nearby penetrating arterioles (red dots) and penetrating venules (blue dots). Smoothed by convolution with a σ = 50 μm Gaussian filter. (e–g) Amplitude of the optical signal as a function of the distance to the columnar centroid (e), to the nearest penetrating arteriole (f) and to the nearest penetrating venule (g). The dashed lines are the null hypotheses, formed from a random distribution of signal centroids. (h) Lateral extent of the column-centered IOS. Maps of 94 individual whiskers from four mice were aligned on the spatial location of maximal change and averaged across time (frames). These column-centered maps were symmetric in all directions. Shaded areas represent s.d.

Figure 7

Calculated loss of lateral flow under numerically imposed pathological conditions in comparison with experimental observations. (a) Cartoon of local occlusion and representative necrotic cyst formed after occlusion of a rat penetrating arteriole along with thin section stained with the pan-neuronal marker αNeuN. Reproduced from ref. . (b) Computed vascular perfusion domains and their estimated parenchymal volume were consistent with measured cyst volumes formed after single artery occlusion. The vascular volume of 115 domains in 4 data sets (colored circles) was computed following numerical dye tracing (Fig. 4) as a function of the perfusion current. The parenchymal volume perfused by each domain was computed as 1/0.0074 of the vascular volume following previous measurements. The linear fit to the data holds for a >98.5% confidence limit. We further plotted the cyst volumes (yellow and green diamonds), found from targeted photothrombotic occlusion of rat penetrating arterioles in rat cortex^,, as a function of the initial flux in the arteriole. (c) Schematic of the occlusion of an individual penetrating arteriole with scheme for labeling the order of downstream edges. (d) We simulated the occlusion of selected, individual penetrating arterioles and calculated redistribution of flow in microvessels up to 15 edges downstream from the occlusion site. The reduction in vascular flux is plotted as a function of each vessel's topological distance, that is, in terms of vertices, from the occluded plunging arteriole. Shown are the results of 100 simulations per order of the downstream edge (red circles) along with the median reduction in flux (red diamonds). We compared these results with the published in vivo data of downstream flux measurement before and after penetrating arteriole occlusion in rat neocortex (green points are data from 175 vessels with median values shown as yellow and green diamonds). (e) Schematic of the occlusion of an individual penetrating venule with scheme for labeling upstream edges. (f) We simulated the occlusion of selected, individual penetrating venules and calculated redistribution of flow in microvessels up to 15 edges upstream from the occlusion site. The reduction in vascular flux is plotted as a function of each vessel's topological distance from the occluded venule. The distribution of vascular responses is shown for 100 simulations per order of edge (blue circles) along with the median reduction in flux (blue diamonds). The results were compared with the published in vivo data of upstream flux measurement before and after penetrating arteriole occlusion (green points are data from 170 vessels).

Figure 8

Lattice models of the angiome. The sources are penetrating arterioles (red, PAs), sinks are penetrating venules (blue, PVs), network resistors represent the asymptotic value of the microvasculature (Fig. 2e), and the source and sink resistances are about half the value of the network resistances (Fig. 2e). (a) Linear circuit, the directions of current flow indicated for normal conditions (top) and after a block of a penetrating arteriole (bottom). (b) Planar circuit with a rhombic lattice and two penetrating venules for each penetrating arteriole. (c) Blockage of a penetrating arteriole leads to a region of no flow with an effective radius of $\sqrt{3\sqrt{3}∕\left(2\pi \right)}=0.91$ in units of median spacing between penetrating venules, whereas blockage of a penetrating venule leads to a region of no flow with an effective radius of $\sqrt{3}∕\left(2\sqrt{\pi }\right)=0.49$ in the same units. (d) Comparison of the prediction from the lattice model and data for penetrating arterioles and venules. The number of vertices between a pair of penetrating venules, which sets the distance scale, was estimated from our analysis (Figs. 2e and 5c) as ${(100\mu \mathrm{m}∕(50\mu \mathrm{m}∕\sqrt{2}))}^2=10$ vertices.