**3. Results**

This paper focuses on the image-based circulatory network of the BBB and shows the versatility of the presented methodology for analyzing up to five cortical regions of the murine brain vasculature. The main purpose of the framework is to introduce a consistent methodology for elucidating the murine microvascular hemodynamics and other functions related to the BBB. The presented synthetic anatomical networks are easy to be treated using 1D hemodynamics. In this section, we illustrate some computational results in terms of flow, pressure, hematocrit and endothelial shear stress. The results took a few CPU minutes to be obtained and required around 20 GB of memory in serial execution on a HP Z440 Intel Xeon computer.

The network geometries are represented in Figure 5 and are colored by the values of the vessels diameter. These regions considered one inlet each but a different number of outlets, different vessel densities, mean diameters, curvatures and tortuosities and the number of bifurcations among other morphological differences. As visible from the figure, the morphology of the five regions is widely different. Of course, these topologies strongly influence the flow patterns and the associated nutrient transport in the surrounding tissue. For these reasons, it is relevant to show the different statistics associated to the flow simulations of the five cerebral regions. Frequency distributions of vessel diameters, length, surface area and volume that characterize the five networks reconstructed using the presented algorithm are shown in Figure 6. The geometries show a very good correlation in terms of segmen<sup>t</sup> diameters (in (μm)) and lengths (in (μm)), surface area (in (μm2)) and

total vascular volume (in (μm3)) distributions, as shown in Figure 6 where the cumulative distribution function (CDF) of these variables is depicted. The presented curves match well the shape and order of magnitude of those presented by Linninger and coworkers [9] which were obtained using a mathematical synthesis of the cortical circulation for a whole mouse brain. The obtained relations between the frequency of appearance and diameters, lengths, surfaces and volumes are in agreemen<sup>t</sup> also with those found by other authors, showing that the used geometries are suitable for further use in the numerical simulations of the microvascular blood flow. This comparison ensures that our image-based modeling presents anatomically consistent microvasculature.

**Figure 5.** Morphology of the 5 cerebrovascular regions of the murine cortex considered in this study. The heat map represents by color the distribution of the value of the diameters within the microcirculatory synthetic network.

Additionally, the framework is capable of controlling the number of arteries and bifurcations and all the associated geometrical features that are quantified in the image data and included in the synthetic model. Previously published capillary networks use only straight segments with cylindrical shape for describing the microvasculature. However, real networks present curvature and tortuosity. Both variables were measured here directly from the images. In particular, the tortuosity was computed using the metric SOAM described by Bullit et al. [80]. We imposed the tortuosity measured directly from the images for mimicking imaged networks. Its CDF is depicted in the Figure 6e). Moreover, we provided a venous connection between arteriolar and capillary regions thanks to the double staining. Previous studies habitually neglected curvature and tortuosity, presenting straight vessel instead, and only a few consider venous drainage [9].

The five different geometries that have been analyzed in this work and depicted in Figure 5 are defined by the parameters summarized in Table 1.

**Table 1.** Morphological properties of the 5 considered cortical regions.

**Figure 6.** Statistical analysis of the considered regions: cumulative distribution functions of diameters (**a**), lengths (**b**), surface areas (**c**), volumes (**d**) and tortuosity (**e**).

The blood flow distributions of the five regions is depicted in Figure 7 in logarithmic scale for enhancing differences within the vessel segments. As the regions are of different size and present important morphological differences, the maximum and minimum of the scale is different for each geometry. The results of the simulations showed that a peak blood flow of 437.33 nL/min was found in Geometry #4, while the minimum blood flow was 99.65 nL/min and belonged to Geometry #3. Summarizing, we found a mean blood flow of 268.49 ± 168.84 nL/min. This value differed from the values by Hurtung and coworkers [24]. However, even though they have found a maximum blood flow of around 780 nLmin, they considered wider regions and scales than the ones used in this work. Of course, the comparison can be only performed qualitatively because it is about different samples with variable morphologies. The important variability of the blood flows found in the present work can be explained by the geometrical differences presented by the 5 regions. In some of them, the feeding arteriolar branch present 'shortcuts' to the outlets, having a preferential flow path of little resistance and increasing the blood flow. This happens, for example, in Geometries #4 and #5, indicated in Figure 7. Furthermore, there are slight differences in the feeding arteriolar trunk diameters, varying from 17.24 μm in Geometry #1 to 22.62 μm in Geometry #5, for instance. This causes less flow resistance for the same pressure loss between inflow and outflow, leading again to an increase of the blood flow.

**Figure 7.** Computed blood flow (in [nL/min]) within the 5 cerebrovascular regions of the murine cortex. The heat map (in logarithmic scale) represents by colors the average blood flow distribution within the microcirculatory synthetic network.

Figure 8 shows the hematocrit distribution within the five regions. Initially, a maximum hematocrit of 80% has been set for any segment. The obtained distributions, as visible in the figure tend to be chaotic in all the regions. This happens because of the used geometries, as this distribution mostly depends on the asymmetry of the bifurcations inside the models due to the nature of the blood. As can be found in the literature, the hematocrit distribution differs from realistic to synthetic, symmetric, binary tree geometries. In these geometries, the hematocrit distribution is in fact mostly homogeneous [24]. The variations seen in Figure 8 are the result of the morphology of the five considered regions. The position of the venous drainage in the geometries can affect hugely the hematocrit distribution, as this is the location where the flow exits and where convergen<sup>t</sup> bifurcations appear. Additionally, there are also some locations in the geometries where divergent

bifurcations appear, leading to a decrease of the hematocrit in the segments until it reduces even to 0%, as seen for instance in Geometry #1. On the other hand, in all geometries, some vessels with slightly high hematocrit values can be seen. These increases of hematocrit depends both on the segmen<sup>t</sup> diameter and on the feeding segmen<sup>t</sup> hematocrit, being usually convergen<sup>t</sup> bifurcations.

**Figure 8.** Computed hematocrit (in [%]) within the 5 cerebrovascular regions of the murine cortex. The heat map represents by colors the hematocrit percentage per segmen<sup>t</sup> within the microcirculatory synthetic network.
