Tissue preparation

The mice under study (6- to 8-week-old male C57BL/6J mice [Harlan Laboratories] on a 12-h light/dark cycle with free access to food and water) were injected in vivo with retro-orbital rhodamine-red–labeled Griffonia (Bandeiraea) and Simplicifolia Lectin I (Eurobio Abcys) to achieve proper vessel labeling. At 30 min after in-vivo lectin injection, animals were anesthetized by intraperitoneal injection of a ketamine/xylazine mixture and perfused intracardially with 4% para-formaldehyde solution, then tissue was removed, oriented and post-fixed overnight at 4°C. Tissue was embedded in 1% agarose before being dehydrated by ethanol, then cleared by incubation in benzyl alcohol-benzyl benzoate solution (BABB, 1:2 vol:vol ratio) (Sigma Aldrich). A clear fat pad is illustrated in Fig 1a. All experiments complied with European Community Guidelines (2010/63/UE) and were approved by the French ethics committee.

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Fig 1. Original image and reconstructed network visualization.

(a) The cleared fat pad tissue sample rotated vertically. (b) Schematic representation of the macrocirculation: arteries (in red) and veins (in blue). Large red arrows indicate the flow directions associated with the main feeding artery inlet into which (dimensionless) pressure p = 1 pressure is applied. The nearby large blue arrow also indicates the main draining vein outlet, with (dimensionless) pressure is set to p = 0. All others secondary vein inlets and artery outlets of the network are indicated with smaller arrows. The location of the central lymph node (LN) is illustrated by the yellow ellipse. Details of boundary conditions can be found in S5 Text. (c) Maximum intensity projection of the full fat pad tissue imaged with an LSFM, the volume of which was ∼ 25 × 5 × 1 mm³. (d) The vascular network extracted from the image (c) is based on image deconvolution, segmentation and vectorization. The vascular graph has ∼ 1.7.10⁶ edges and ∼ 1.2.10⁶ nodes (cf. Fig 2a and 2b for definitions). (e) Volume rendering of a small part highlighted in (c) for a small range of slices (for a total of ∼ 250μm width) with, (f) the corresponding reconstructed vascular graph. (g) and (h) are higher zoom levels of (e) and (f), respectively, highlighting the high resolution (0.96μm voxels width) needed for micro-capillaries segmentation.

https://doi.org/10.1371/journal.pcbi.1007322.g001

Light-sheet Fluorescence Imaging Microscopy (LSFM)

Cleared samples were imaged on a custom LSFM based on cylindrical lens illumination with 488- and 561- nm lasers (Oxxius, France) and horizontal macroscope detection. The lens for the formation of the light sheet was an f = 50mm cylindrical lens coupled with a slit aperture of maximum 7mm opening to control the thickness and flatness of the illumination beam. The resulting thickness of the light-sheet at the waist varies from 5 to 50μm depending on the slit aperture. Thickness (T) was set to 35 μm to fit the largest field of view (FOV) at the lowest magnification (i.e., 1.2X). The macroscope was a Nikon AZ100M macroscope with an air lens of 2x magnification, providing an 8-fold zoom factor. The final magnification of 6.72x used here yields an estimated 1.83 μm lateral resolution against an axial lightsheet extent of about 20μm. Imaging in clearing media was through a 2 mm glass wall of a polished cuvette. Rotation of the sample was performed from below the sample, acting on the agarose block embedding the sample, with a driving rotation stage (M116-DGH, Physik Instrumente, Karlsruhe, Germany). Detection was performed with 50nm band-pass filters mounted in a filter wheel (LB10, Sutter Instruments, Novato, CA, USA) and a SCMOS camera (ORCA Flash4.0, Hamamatsu, Japan). Fig 1e illustrates the typical image of avascular network with voxel size of 0.967 × 0.967 × 2μm. The full tissue acquisition consisted of 28 tile images of 1500 × 2048 XY voxel resolution with variable depth (Z dimensions) from 400 to 1500 voxels, for a total of ∼120GB raw 16-bit data. These tiles are stitched as described in [25].

Network extraction

First, tile images were interpolated in the Z direction with cubic splines so that voxels matched an isotropic dimension of 0.967μm width and were then subdivided into 256³ voxel sized blocs that could be processed simultaneously. Image deconvolution was then applied. The Richardson-Lucy (RL) algorithm was used to remove artifacts from light microscopy (e.g., [26]) and has been found appropriate and accurate for LSFM deconvolution [27, 28] although computationally challenging. Without the use of beads, the efficient estimate of the point spread function (PSF) becomes challenging, especially when its shape varies for spatially inhomogeneous light-sheet depth. Because we chose a large light-sheet depth and FOV, we considered a spatially homogeneous PSF designed with a Gaussian shape for which its standard deviation σ(z) = 11μm is related to the light-sheet depth (cf. section Light-sheet Fluorescence Imaging Microscopy). More information about the choice of the PSF can be found in [29]. We then used the RL iteration updates with a total variation regularization. Deconvoluted images were then simply thresholded, and the resulting binary images were used for network extraction. The spatial graph representing the vascular network was built according to the vectorization of the binary image as described in [30]. Vessel center lines were extracted with an appropriate homotopy preserving skeletonization method [30]. The graph G_v = (V_v, E_v) was then created, where V_v, the set of nodes, was extracted from the branching-voxel and end-voxel, and E_v, the set of edges, was between connected node-voxels of the vessel skeleton. An edge consists of a chain of center-line voxels comprised between two nodes. Vessel radii were estimated following [31]: for each segment-element and node (illustrated in Fig 1b), a sphere was expanded until 10% of its volume was left outside the vessel shape in the binary image. Fig 1e illustrates the network extracted from Fig 1d with tubular-like visualization in Avizo (FEI) software. Once each of the 256³ voxels blocs was vectorized, networks were stitched according to global coordinates found in the previous step and merged with a simple coupling procedure to obtain the full network. Loose ends were connected by four successive Tensor Voting (TV) iterations with different parameters, trying to connect small gaps at first, with 5μm as a reference distance and 30° as reference angle (cf. [30] for more details) and up to 30μm as reference distance and 45° at fourth iteration. The connections added represented ~5% of the total vascular network edges. The quality of the skeletonization and reconnection procedures was quantitatively assessed in [32].

The above-described image processing is effective only on vessels and not macro-arteries and veins because they present different morphologies and responses to the lectin labeling. Capillary vessels (< 50μm) appear as filled near tubular structures, whereas arteries and veins appear as non tubular empty structures. A few studies tackled the particular case of automatic segmentation of such structures (e.g., [33, 34]), but we chose to manually segment these macrovessels to ensure a reliable vascular tree and geometries, which are essential for realistic flow simulations. Finally, the macrovascular network (illustrated in Fig 1) was merged with the microvascular network by adding a junction vessel by using the tensor voting method [30] as shown in Fig 1e, 1g and 1i. The resulting network was also carefully inspected along all large vessels to check and verify the quality of the reconstructed network versus the initial grey-scale image. This verified, all topological constraints needed for the modeling (Euler relation between vertex-edge number, no self-loop of segment elements, etc.).

Let us also define various notations from the graph G_v. We denote its adjacency matrix A, its degree matrix and the Laplacian matrix L = D − A.

Edge and network resistance

The aspect ratio (radius divided by length) of vessels is small, on average (≃ 1/8, on average), which justifies use of a lubrication-based discrete network approach (cf. section Edge and network resistance). The previously described vectorized network allows us to compute the hydrodynamic resistance of each vessel, defined as the ratio of the flow rate to the pressure drop applied in the vessel, formally the analog of the resistance in electric circuits. The Hagen-Poiseuille law provides the relation between the resistance R, the flow rate Q and the pressure drop Δp where the resistance is a function of the viscosity μ, the diameter D. (1)

To account for the effect on red blood cells to blood rheology, empirical models have been proposed to described blood viscosity, first in vitro [35, 36], then in vivo [37] and more recently in vivo, taking into account the effect of the endothelial surface layer [38]. In this work, we used the latter scenario because in a comparative study, it was found to have the most relative influence on resistance, pressure distribution and flow rate [13]. We also used an integrated formula of resistance because it considers the local shape variations of the vessel. Resistance in each vessel is thus defined as (2)

The expression for viscosity μ can be found in [38, 46]. Given the resistance on each edge of the vascular network, one can define the network resistance between every pair of vertices. Based on [39], the resistance R_ij between two vertices of index i and j is (3) where λ_k and ψ_k are respectively the eigenvalues and eigenvectors, respectively, of the Laplacian L of the graph G_v for which the adjacency matrix A is weighted by vessel conductance.

Graph clustering

Graph clustering, also called community partitioning, aims to regionalize networks [40]. We used a previously described method optimized for a large network [41]. The method maximizes the community features reflecting the density of edges between vertices inside communities as compared to edges between vertices in different communities. Considering vascular networks, clustering can be applied to the unweighted graph G_v, denoted G_v = (V_v, E_v, w₀) as defined in section Network extraction, as well as an edge-weighted version of G_v, denoted G_v = (V_v, E_v, w_i), where w_i is the weight function. We investigated four different edge weights, that could be of interest for vascular graph analysis: w₁ the Euclidean distance between the two vertices connected by an edge, w₂ the geodesic distance between the two vertices connected by an edge, w₃ the hydrodynamic resistance of an edge and w₄ the hydrodynamic conductance. Once the vascular graph G_v was clustered, we defined a new abstraction level of the vascular network denoted as the adjacency graph of the communities found from G_v with the weight function w_i. V_c thus represents the communities and E_c the connections between communities.

Graph features

Graph theory is widely used to characterize structured networks in technological and transportation infrastructures, social relations, or biological systems (e.g., [15]). Here we focused on characterizing the effect of communities inside graphs, , defined in section Graph clustering. We used two features: 1) the eigenvector centrality, which is a modified version of the betweenness centrality [42] and gives high scores for vertices playing the most central role (i.e., those with the smallest farness from others), and 2) the Authority/Hub features designed in [43] to discover authoritative and hub web pages on the World Wide Web concerning a topic search.

To analyze the arterial/venous vascular organization on both the network and the community scale, we computed arterial/venous domains based on generation numbers starting from the arterial and venous macro-vascular networks (Fig 4g). Both of these macro-networks were extracted from the full reconstructed network by traversing the vascular graph from every arterial boundary condition, and also from every venous boundary condition, under the condition that the mean vessel diameter is ≥ 15μm. Then, starting from the vertices of the arterial macrovascular graph, we increased by one the generation number at severy vessel connected to those nodes. The corresponding vertices of the vessels found were then looped on. Numbers were further increased by one during each loop. This procedure was continued until no further vessel was found. The very same algorithm was used starting from the venous macrovascular graph. Because of the proximity of the arterial and venous macrovascular networks, the generation numbers from both networks were very similar (see S4 Text). Finally, we used the following criterion to decide whether a vessel was part of the arterial or venous domain: if its generation number starting from the arterial macrovascular network was greater than its generation number starting from venous macrovascular network, it was considered part of the arterial domain. Otherwise, the vessel was considered part of the venous domain. At the community scale, we then computed the fraction of vessels that were part of the arterial domain. The result is displayed in Fig 4g.

Flow modeling

The adopted discrete perfusion network is based on previously proposed microcirculation models [44, 45]. These frameworks were previously used to model blood perfusion from image-based network extraction [5, 12, 13]. Here we describe the main steps and modeling issues.

The flux conservation applied at any vertex i with the set of its neighbor vertices is denoted as J (4) where Q_ij is the flow on an edge connecting vertices of i and j. From considering an integrated local lubrication approximation ([13] for more details as well as reference therein), one can provide a relation between the pressures at every vertex, which are is not associated with boundary conditions (5) where C_ij is the conductance of edge connecting vertices i and j and p_i (resp. p_j) the pressure on vertex i (resp. j). For every other vertex, a pressure Dirichlet boundary condition is imposed whose values will be discussed later.

Applying these relation to every vertex of the graph gives the following system (6) where L is the Laplacian matrix of the vascular graph G_v with hydrodynamic conductance as a weight, p is the vector of pressures to solve and is the vector of applied boundary condition, which is zero whenever the index is an internal vertex and whenever the index is a boundary condition.

The common framework for blood-flow network simulations considers also the variation of hematocrit. The hydraulic conductance (or resistance) being a function of hematocrit, this results in a coupled system of equations. For converging bifurcation, the mass conservation law gives a set of equations and for diverging bifurcations, empirical models give the fraction of hematocrit that will be convected in both daughter vessels. Reference [46] shows that hematocrit variation has a small effect on blood perfusion. Thus, we considered a constant hematocrit, with a value of 0.45 throughout the network.

A pressure Dirichlet boundary conditions is associated with any identified vertex corresponding to a cut-open macrovessel edge. A careful analysis of the network architecture allows for exhibiting a total of 26 macro-vascular vessels entering and leaving the tissue. Thus, 26 boundary conditions must be set for the flow modeling (see section Flow modeling for details). We found a total of 26 macro-vessel edges, as illustrated in Fig 1a. Expert knowledge of the macrocirculation of adipose tissue reveals that only one main inlet artery perfuses the tissue, the location of which is known in this tissue (large red arrow in Fig 1a). Similarly, because the feeding and draining vessels are associated in pairs, the corresponding vein is thus located (large blue arrow in Fig 1a). The dimensionless pressures associated with these two main inlets/outlets are defined as 1 and 0. Then every other boundary condition is considered as a secondary outlet artery or a secondary inlet vein, as shown in Fig 1a. This feature is generic to whole organ imaging because there are always various macrovessels in a given tissue, that are unavoidably cut. To reduce the problem complexity, we imposed the same inlet vein boundary condition (denoted α) and outlet artery boundary condition (denoted β) to every secondary pressure boundary condition. To reduce the impact of perfusing inlet veins, we computed the relative perfusion volume (relative to the perfusion domain) of the main inlet artery over the sum of all perfusion volumes of every inlet vein. We chose α and β values to maximize this ratio, which resulted in α = 0.35 and β = 0.3125. S5 Text summarizes the detail of perfusion volume ratios for a large range of α, β values. Perfusion domains are sub-graphs of the micro-vascular network, providing, from each entry, its (predominent)) basin of perfusion inside the entire network. In brief, it is evaluated by using a spanning-three method, starting from each entry, but restraining the sub-three extraction to predominant flux branches at each bifurcation. For more details concerning the definition and computation of perfusion domains, refer to [14].

We also investigated the effect of diameter variations that might occur due to segmentation errors on blood flow exchange between communities. Overall, a few pairs of communities ceased to exchange blood and the rest did so with a relative quadratic error up to 40%. More details can be found in S8 Text.

Tissue organization both at the macro and micro-circulatory scales

The macrovascular network (see section Network extraction for details about segmentation) extracted from the inguinal fat pad (Fig 1a and 1b) follows the known main anatomical architecture of the tissue: a principal longitudinal (apex-groin) axis, bifurcating nearby the lymph node into a (upside-down) ‘T’ shape that leads to the major arterial and venous pair [47]. The main macro-vascular artery inlet, associated with a main vein outlet, is located on the left of the tissue and is responsible for the main perfusion. Image acquisition of the whole tissue (Fig 1c) and segmentation of its entire vascular network (Fig 1d) allow for quantifying microvascular networks at close to micron-scale resolution, that is, with several voxels per diameter (Fig 1e, 1f, 1g and 1h), with their associated local parameters (diameter, segment length, orientation, density, etc.). Indeed, these structural local parameters allow for a vectorized description of the vessel (Fig 2a) and the corresponding vascular graph (Fig 2b). In contrast to many other contexts in which graphs are found in nature (e.g. metabolism pathways, gene interactions, ecology, etc.) or in practical applications (web-structure, social web, etc.), the local structure of the graph G_v of microvascular networks is very simple: almost all bifurcations result from the connection of two distinct vessels into a single one or vice-versa (Fig 1g and 1h). The vascular graph can be manipulated by associating different graph weights w_i (e.g. w₀ as illustrated in Fig 1b), that is the scalar quantity associated with each edge connecting two nodes (i.e. bifurcations) which is provided in order to consider different levels of relationships, weak or strong, between elementary nodes.

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Fig 2. Vascular network statistics.

(a) Schematic representation of the vectorized vascular network and (b) the corresponding vascular graph with common key concepts in both cases. Note that vascular segments are turned into graph edges when the vascular network is mapped onto its graph representation. (c) Data for vessel length and diameter. The average aspect ratio is found equal to D/2L ∼ 1/8. (d) Histogram distribution of bifurcation angles (cf. Fig 1b for angle conventions). (e) Local vascular density (volumetric ratio) distribution measured in the whole network according to four elementary box size widths from 100μm to 400μm.

https://doi.org/10.1371/journal.pcbi.1007322.g002

Considering this framework, one can quantify key aspects of the micro-vascular network. The aspect ratio (radius, divided by length) of vessels is small in average (≃ 1/8, Fig 2c), so as to justify the use of lubrication-based discrete network approaches (see Material and methods section Flow modeling). The segment’s orientation distribution at bifurcations given in Fig 2d displays two predominant configurations: a T-shape associated with quasi-aligned daughter and parent branches (first mode of φ₁ distribution and second shouldering of φ₂ one), and a Y-shape family one (second mode of φ₁ distribution, first mode of φ₂).

Now, considering the entire tissue, we first evaluated the vascular density inside boxes of varying sizes (from 100μm to 400μm, Fig 2e, as in [29]). This evaluation produced variable density distributions without the emergence of a homogeneous length-scale and thus revealed strong heterogeneity. Similarly, the hydraulic resistance (see Material and methods subsection Edge and network resistance and S1 Text) and the pressure distribution inside the vascular network (S1 Text) display various heterogeneous scales. Hence, various vascular characteristic scales emerge from our analysis, considering both geometrical properties (i.e. vascular density) and hydrodynamic ones (i.e. hydraulic resistance and pressure distribution), the presence of which highly suggests a meso-scale organization that we next investigated.

Structural units emerging from clustering analysis

Meso-scale vascular organization results from clusters of vessels related by a close proximity (either from structural or functional relations). Graph clustering is an established technique mostly used in computer science for applications in web networks, social networks, etc., which allows for deciphering preferential information transfer between meso-scale entities. We adapted it here for our vascular network as in [5].

At the local scale, the minimum unit of the vascular graph is a simple local binary tree (Fig 1b), but at a broader scale, this graph is much more complex, involving preferentially connected units called communities (Fig 1f and 1h). Clustering methods are based upon finding a graph partitioning (i.e. a separation of the graph into non overlapping units) that maximizes the density of (weighted) edges between vertices inside communities as compared to edges between vertices in different communities (cf. subsection Graph clustering). Hereafter we refer to the result of the clustering partitioning of the vascular graph as communities.

We consider two classes of geometry and perfusion based family for weights w_i associated with vascular segment edges: class 1, structural associated with w₀, w₁, w₂ for topological connection (e.g. Fig 2b), Euclidean distance (Fig 2a) or curvilinear (Fig 2a) respectively; and class 2, a perfusion-related relation associated with w₃ and w₄ for the vessel’s resistance and conductance respectively. For each weight a different set of communities is obtained, with one result for weight w₀ exemplified in Fig 3a, 3b and 3c. We first focus on the results obtained with the simplest weight, w₀, for which the relation between entities is binary (1 for a connection, 0 otherwise).

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Fig 3. Vascular graph clustering.

(a) Community clustering applied to the vascular network by using an unweighted vascular graph representation, w₀. Color codes for community membership. (b) and (c) Image volume rendering and the vascular graph segmentation, respectively, of a sub-volume (∼100μm width) extracted in a dense and modular area of the tissue near the central lymph node (white dotted rectangle depicted in (a)). (d) Schematic illustration of vascular communities; (e) structural community graph, with the number of vessels connecting two communities as edges weights (5 and 1 here); and (f) bi-functional community graph with in/out fluxes Q_ij as edges weights. (g) Graph clustering of the communities for geometrical based weight (w₂, curvilinear distance) and (h) perfusion based weight (hydraulic resistance w₃); the disks are located in community centers projected in the (x, y) plane with sizes coding for volumes and color for vascular density (volumic ratio from 0% to 20%) illustrated with a cold-to-warm color scale. (i) Linear regressions of the scatter plot of number of vessels between each pair of communities N_ev versus number of vessels in the community N_iv (for each community in (d), number of colored edges vs. number of black edges). Full lines are the linear regressions for each clustering weight w_i. w₀ no weight (slope s = 0.61); w₁ Euclidean distance (cf. Fig 2(a)) (s = 0.62); w₂ curvilinear distance (cf. Fig 2(a)) (s = 0.66); w₃ hydraulic resistance (Cf. S2 Text) (s = 0.25); w₄: hydraulic conductance (Cf. S2 Text) (s = 0.28). The gray curve is the result of Blinder et al. [5] with s = 0.83. Dashed lines delimit the boundaries of weak communities (therefore having strong connections between units) with slopes s = 2/3 and s = 1. Shaded areas are the standard errors associated with each regression. The linear regressions are normalized with zero offset at the origin. (j) Non-normalized scatter plot for weight w₀ with the associated linear regression in a solid line (other weights are reproduced in S2 Text).

https://doi.org/10.1371/journal.pcbi.1007322.g003

Fig 3a illustrates the spatial extent of the communities, all having vascular segments with homogeneous color. A zoom inside a sub-domain of the whole organ shows how structural entities already appear to have variable density, as visualized by maximum intensity projection (Fig 3b). The resulting graph communities spatially mapped into the vascular network with a different color for each community (Fig 3c) closely following the shape of these visible structures. This observation illustrates and validates the relevance and interest of community clustering for detecting and isolating vascular structural entities. The community of vessels (Fig 3d) can be schematically represented as a graph, with adjacent communities linked by an edge coding the number of vessels (structural community graph Fig 3e) or in/out fluxes (bi-functional community graph Fig 3f). Such a structural community graph with weight w₂ (curvilinear distance) is illustrated in Fig 3i, in which the position of center of masses (by projection in the 2D (x, y) plane), distances between communities, volumes, as well as vascular density are also coded, revealing spatial heterogeneity of the vascular density distribution inside the tissue. The large vascular density variations already revealed from the wide distribution of the vascular densities histogram of Fig 2e can now be much more precisely attributed to the presence of structural entities, the spatial distribution of which is neither regular nor patterned. A clear structural trend emerges from Fig 3g: the highest vascular densities (red and oranges circles) are found in a core region, nearby the trajectory of the main feeding artery (shown on Fig 1b). Fig 3h shows a similar trend for a perfusion based weight partitioning (w₃ hydraulic resistance) with about 10 times more communities and a much denser central region in terms of community numbers. This larger number of communities found with functional weight is robust (less than 1% variations) when taking into account the local blood flow orientation found in the blood flow modeling, and performing the clustering onto the oriented graph with weight w₃. Furthermore, the number of connected vessels associated with the edge sizes of Fig 3g and 3h reveals more heterogeneous couplings between structural communities (Fig 3g) as compared with the smaller and more homogeneous couplings for functional communities (Fig 3h). This difference between the two classes of weights suggests differences in interconnections of communities, which we next examined.

To evaluate the connectivity of communities, in Fig 3i and 3j, for each community pair, we computed the number of external connecting vessels N_ev (e.g., for the communities of Fig 3d, number of black edges) and plotted it against the number of internal vessels N_iv (e.g., for community #1 of Fig 3d, number of blue edges). The power law scaling of N_ev against N_iv, such as the representation in Fig 3d, allows for delineating a zone of weak communities (i.e., strongly connected ones) when the slope of the scaling is between 2/3 to 1 (larger than a surface to volume ratio) and strong ones when the slope is smaller than 2/3 (i.e., weakly connected nodes). Additionally, strong communities are found when the slope of the scaling is less than 2/3 (both slopes shown in Fig 3i). Considering the first class of geometry-based weights (w₀, w₁, w₂), we observe in Fig 3i that the number of vessels connected with external ones indeed scaled similar to the number of internal links N_iv of the community, in this case, . With the lower bound scaling for weakly connected units being a 2/3 power law, our results indicate that structural clustering results in already quite strongly connected units. Considering the second class of perfusion-based weights (w₃ and w₄), an even more significantly different scaling is found (cf. Fig 3d), , revealing very strongly connected perfusion communities. This very distinct behavior shows that clustering using structural weights (class 1: w₀, w₁, w₂) or functional weights (class 2: w₃ and w₄) can reveal vascular units with distinct average densities and sizes (S3 Text). We then investigated how these structural communities are functionally coupled.

Vessel community perfusion coupling map

Here we consider functional perfusion distribution between communities. We focused on the clustering results obtained from one structural weight (i.e., w₂ associated with the curvilinear distance defined in Fig 2a) because structural weights provide the most anatomic-relevant information (compared to [47]). We analyzed the total perfusion exchange fluxes between a given pair of communities (depicted in Fig 3h) denoting as the total flux from community c_i to c_j, that is, , where is the set of vessels connecting communities c_i and c_j, flux Q_k being directed from c_i to c_j ( is defined vice-versa). The robustness of the perfusion modeling and total flux exchanges between communities to the quality of the segmentation and possible errors in vessels vectorization is provided in S7 Text.

Computation of graph authority (cf. subsection Graph features for definitions) associated with functional graph features (Fig 4a) reinforces the identification of a core region previously found for structural properties (Fig 3i and S4 Text). Thus, blood perfusion confirms the spatial organization of the graph communities. Furthermore, Fig 4a, 4b, 4c and 4d shows that the core region strongly co-localizes with the largest perfusion exchanges where macrocirculation vessels illustrated in Fig 1b are also present.

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Fig 4. Functional community network analysis.

(a) Graph authority feature on the bi-functional (cf. Fig 3g) communities graph. The color code, from white to dark green indicate small to large graph authority value. (b) Identification of central area by selecting nodes with the authority graph authority feature (>0.04) core in red and (<0.04) peripheral in black. (c) Histogram distribution of vascular densities in the region depicted in (b) using the same color-code (mean vascular density is 0.06 for red, 0.02 for black), Kruskal-Wallis rank sum test: p-value < 0.0001. (d) Intercommunity flows of region depicted in (b) using the same color-code. Mean flow inside (red) is 0.0079, outside (black) is 0.0212, Kruskal-Wallis rank sum test: p-value > 0.05. (e) Bi-functional community graph (as defined in Fig 3g) with edges colored and sized by the perfusion fluxes (two arrows per edges for in/out fluxes not always visible due to fluxes asymmetry). (f) Dynamics of perfusion at the community scale. The nodes are sized according to the relative filling time of each community and are colored according to the fluid global time arrival (with a green-to-red scale) with green nodes fed first and red ones last. (g) Arterial domains computed from generation numbers from blue to red from 0 to 100% (100% means all vessels are closer to arterial macro-vascular network than to veneous one).

https://doi.org/10.1371/journal.pcbi.1007322.g004

The perfusion dynamics (cf. S6 Text for more details) emphasize the significance of core regions of the core region prerogatives, because Fig 4e and 4f shows that these central communities are fed first, but also present heterogeneous filling times, that is, the time for a community for its entire vascular bed to be perfused from the arterial entry. Also, similar heterogeneities of filling times are observed outside the core region. This observed heterogeneity is related to the distinct and variable proximity of each community with the principal arterial trunk.

Finally, one can also analyze the communities by their ability to drain perfusion from the main arterial/venous trunk. To investigate this, we compute the arterial/venous domains based on generation numbers starting from the arterial and venous macro-vascular networks (Cf S4 Text). In Fig 4g we represent the fraction of vessels that are closer (in terms of generation numbers) to the arterial than venous trunk. The core region already depicts a clear proximity to the arterial trunk. This finding is consistent with the previous observations for this section, all flow-related.

Conclusion

In conclusion, our study demonstrates the interest of new imaging and segmentation tools for analyzing structural and functional organization in tissue. We believe the presented methods as well as their technical improvements will be useful in many future biological and biomedical tissue-oriented studies. This work provides the proof of concept that in-silico all-tissue perfusion modeling can reveal new structural and functional exchanges between micro-regions in tissues, found from community clusters in the vascular graph. It also illustrates how the combination of state-of-the-art LSFM imaging, huge image data post-processing and mathematical modeling can be combined to reveal the intimate anatomy of tissues.