**4. Discussion**

The brain is the most complex organ of humans, but despite the extensive work dedicated in recent decades, still little is known about its functionalities, including the anatomy and the hemodynamics of its vasculature in comparison with all the other organs [81]. Several studies have attempted to describe the microvasculature structure and anatomical variations in the cerebral surface region often comparing humans and rats that present many similarities but also differences [82]. We have proposed a comprehensive framework based on tissue clarification, advanced microscopy and image treatment aimed at the analysis of the murine microvasculature that is feasible to be applied to humans. Through a mathematical algorithm, specific regions or even the entire murine brain geometry can theoretically be created for the analysis of its hemodynamics. The reconstruction of the entire anatomy from image data is difficult to be obtained as patient-specific data have a limited spatial resolution [9]. For this reason, the combination of anatomical images, from the tissue clarification to the obtention of 3D geometries, and mathematical modeling using advanced algorithms that allow the analysis of a consistent circulatory network is an efficient strategy, and it is the standard methodology in the literature. The advantage of the synthetic network is that it can be used for different purposes, for example for the simulation of blood flow and nutrients transport phenomena that can mimic the 3D vasculature. These simulations can cover regions of the in vivo data sets where imaging data are not consistent, as we have discussed in the Section 2.5.1 or regions not reconstructed [9]. Alternatively, the use of mathematical networks could also be capable of complementing real anatomical data serving as boundary conditions for 3D realistic anatomical microcirculatory models. The 1D modeling can be attached to 3D models replacing the limits of the computational

vascular domains and can be used for applying the boundary conditions as elucidated by Linninger and coworkers [9]. Fractal networks have been often used in this sense for large and small arteries as well as for the cerebral vasculature [83–85].

In the past decade, synthetic vascular models have offered more and more an alternative to purely image-based approaches [65,81,86,87]. Unfortunately, binary trees can only approximate the real microvasculature because they only bifurcate in one direction and cannot take into account loops. For this reason, more recently, other authors start creating more complex vascular structures that could include anastomoses improving previous findings [9]. Our work demonstrated that the presented methodology offers such morphological structures as the obtained synthetic models faithfully represent the imaged cortical regions.

At the same time, researchers have progressively proposed improved mathematical algorithms providing increased models complexity ye<sup>t</sup> providing accurate brain databased networks. An example is the synthetic model introduced by Linninger et al., which simulates the cortical blood supply in a section of the human cortex. They provided a computational method for building realistic microcirculatory beds using Voronoi tessellation [66]. Due to the high computational costs, they later further extended this model using a single algorithm including arterial and venous trees with capillary connection [9]. Another example is the algorithm developed by Su et al. for creating a set of networks based on experimental statistics to bypass the complexities to reconstruct a cerebral microvascular network from real brain tissue data [81].

The principles of the modeling proposed in the present work are similar to those introduced by other studies in the literature [63]. The cerebral vasculature is represented by a network of bifurcating cylinders that provide a resistance to flow according to the Stokes equations. The proposed mathematical model was further used for studying the hemodynamic in the brain for showing the application of the developed methodology. Some computational results regarding the blood flow, the hematocrit and the endothelial shear stress distribution have been presented (see Figures 7–9) and demonstrate the feasibility, the utility and versatility of the presented framework. With the proposed framework, it is also possible to have a consistent quantification of the vascular morphology, providing data of the number of bifurcations, tortuosity, surface, vessel length and diameter, volume and volume density that can be used for characterizing the vascular structure and its functionality (see Figure 6). It is widely known that the neuronal tissue varies with the depth of the cerebral cortex so that the presented results may be used to help elucidate the relationship of the flow, pressure and shear stress characteristics with the depth in 1D realistic vascular networks as studied by other authors but still not ye<sup>t</sup> fully understood [77]. The computational results support the hypothesis already diffused in the literature that the flow field and the hematocrit distribution are highly heterogenous in the microvasculature, suggesting that the oxygen and nutrients brain regulations depend on the cortical layer [25,77].

The presented simulations are based on a real anatomical data so that reconstructed geometries are controllable. However, the results leads to 1D flow and average values of velocity, WSS and other variables that approximate the real cerebrovascular hemodynamics. Of course, synthetic models are based on simplified geometries and simplified hemodynamic constraints as a boundary condition so that the resulting hemodynamic features are simplified as well [9]. For this reason, the results obtained in this study were compared with published results for demonstrating the consistency and robustness of the presented tool. Unfortunately, currently, an in vitro or an experimental validation is not feasible. Nevertheless, as stated in the literature [9], simplified hemodynamic models used in combination with synthetic vascular networks do not preclude rigorous blood flow simulations. In this sense, the advantage of the presented model is that one can control all geometrical parameters and preview the results in real time. Additionally, as explained before, the presented framework is feasible to be more and more complicated adding or improving model details and additional specific conditions. In conclusion, although it includes some simplifications, the presented mathematical model which incorporates anatomic-based

morphometric properties can potentially be used for addressing open questions regarding healthy and diseased cortical blood flow in the cerebral microvasculature.

**Figure 9.** Computed endothelial shear stress of the 5 cerebrovascular regions of the murine cortex. The heat map represents by colors the average value of the shear stress per segmen<sup>t</sup> within the microcirculatory synthetic network.
