Fig 1.
Concept of the multiscale modeling of cerebrovasculature.
A hybrid approach based on image-based and mathematical models. The brain hemispheres are extracted from MRI images and the macroscale vasculature is reconstructed from CT images. The vascular subregions and signed distance function (SDF) for the brain hemispheres are defined in the preparation process as the inputs for the mathematical model. The mesoscale vasculatures at coarse and fine scales are constructed by a vascular generation (mathematical) algorithm in the multilevel region-confined manner.
Fig 2.
Reconstruction procedure of the image-based vasculature.
(A) Sequential CT images for bolus injection of contrast media in the brain. (B) Reconstructed vessel geometries at frame numbers 5 and 16, where the venous vasculature is not yet isolated from the arterial one. (C) Isolated vessels for arteries (red) and veins (blue), where the isolation is performed semi-automatically based on a Boolean operation applied to overlapped domains between the vessel geometries at frames 5 and 16. (D) Vascular centerlines (solid line) with radii and terminal points (square symbol) obtained through a thinning process applied to the vessel geometries extracted from (C).
Fig 3.
Schematic of the MRC algorithm.
(A) Vascular generation in the multilevel region-confined manner. In the LV1 reconstruction, two arterial structures (red) and one venous (blue) structure are generated from their root edges, where the terminal vertices enter into respective vascular subregions Di. The following arterial and venous structures are then generated while being confined within each subregion Di in the LV2 reconstruction. (B) Definitions of the vascular structure. A single graph is shown with vertices vm and edges en. Also, a schematic of the edge subdivision for en is shown, where the edge is divided by the line segments ln,k with subpoints pn,k. In the representation, the edgewise line segment is given as a straight cylinder with the radius an,k and length Ln,k. (C) Description of the generation of new branches. Four bifurcation patterns (P1, P2, P3, P4) are considered for solving the optimization problem (7) when adding a new terminal vertex vM+1 to the graph with vertices vm and edges en. In the schematic, the edge en is selected as the closest edge of the new vertex, and a branching vertex vM+2 and edges eN+1, eN+2 are newly added. (D) Schematic of the relocation of an edge. Here, the vascular subregions are given on a curved surface Γ, and the reconstruction domain Ω is defined as the extended domain within the lower limit surface Γlow and upper limit surface Γup. As the edge passes across the boundary of Ω (outside of the lower limit Γlow), the edgewise subpoints outside the domain are relocated on the curved surface Γ.
Fig 4.
Flowchart of vascular generation in the MRC algorithm.
The coarse-scale vasculatures for both arteries and veins are constructed in the LV1 reconstruction, where the location and geometries of the root edges are inherited from the terminal ends of the LV0 model. The fine-scale vasculatures are then generated in the LV2 reconstruction, where the generation continues from the terminal ends of the arteries and veins in the former level (LV1). The vascular subregions and SDF for the brain surface are used for the reconstructions at each level. The vascular generation algorithm at each level follows the geometry-prioritized CCO model described in this paper.
Table 1.
Mathematical model parameters.
Fig 5.
The reconstructed model for arteries (red) and veins (blue).
(A) Superimposed views of the image-based reconstruction (LV0) and mathematical generation (LV1 and LV2). (B) Respective vascular structures in the different levels.
Fig 6.
Vascular pathways to/in a single vascular subregion.
(A) Configuration of the vascular subregion on the brain surface in LV1, with the color indicating the subregion number (ID). (B) Vascular structures in the LV1 reconstruction. Here, there are two arterial pathways and one venous pathway entering an arbitrary vascular subregion Di′. (C) LV2 reconstruction within the above subregion Di′. The vascular pathways have continued from previous terminal vertices. In each figure for an artery/vein, the terminal ends of another vessel are shown as dots. (D) Configurations of terminal ends for arterial (red) and venous (blue) pathways to the introduced subregions on the brain surface. Note that, owing to modeling constraints, all subregions in LV2 have two arterial ends and one venous end.
Fig 7.
Comparison of superficial cortical vessels.
(A) Overview of the vascular structure in the present model for the LV1 and LV2 processes and enlarged view of the present model around a gyral surface. (B) Distributions of the vessel diameter for the terminal edges in the present LV2 model and human pial vessel measurements [7]. Note that the measurement data are characterized by the minimum and maximum diameters of the major intracortical vessels, which can be regarded as the terminal points of the pial vessels before penetration into the cerebral cortex.
Fig 8.
Morphological features along vascular pathways.
(A) Example of the arterial pathway from a single root with a diameter-defined Strahler order, where the color denotes the order number. (B, C) Morphological features of the diameter, length, and number of elements with respect to the element order for arteries (B) and veins (C). The symbols denote the evaluated values and the solids denote regression lines of the form log10 q = a + bn, where two regression lines are fitted for the data from n ∈ [1, 5] and n ≥ 6 excepting the last order number. The values of the intercept a, slope b, and determination coefficient R2 are shown in the graphs. In (B), the fitting curve given by the measurement of rat vessels [33] is also plotted by ∝nb.
Table 2.
The comparison of slope b in the regression line log10 q = a + bn (n ∈ [1, 5]) for morphological features (diameter d, length l and number of elements Nelem) along arterial pathways.
Fig 9.
Vascular territories occupied by major cerebral arteries.
(A) Superimposed views of all the arterial territories. (B) Respective territories with the image-based vascular structures. The vascular territory is classified into anterior cerebral artery (ACA), middle cerebral artery (MCA), and posterior cerebral artery (PCA) for the left(L) and right(R) hemispheres.
Table 3.
Relative sizes of arterial territories (ACA and MCA to PCA).
Table 4.
The numbers of root arteries with respect to each arterial territory (ACA, MCA and PCA) and their ratios to the PCA.
Here, the total number of terminal edges is 129, as has been already listed in Table 1.
Fig 10.
A relationship between vascular pathways and geometries and the number of coarse-scale vascular subregions.
(A) An example of arterial structures from a single root. The color differentiates the reconstruction level of the vascular generation. (B) The relative frequency of the path length C, the summation of length for all the edges consisting of the whole pathway from the root to terminal edges. (C) The distribution of terminal edge diameters for the LV1 model and comparison with the measurement data for the human cortical vessels [7]. The measurement data are referred to as the minimum and maximum diameters of central and peripheral vessels.
Table 5.
Comparison of execution time for the reconstruction at different Nu.
Fig 11.
Comparisons of statistical features for vascular structures of whole-pathways in the models reconstructed with actual and simplified brain hemisphere shapes.
(A) Hemisphere shapes for the actual model (orange) and simplified model (white), in which the major folds (cerebral sulcus) were artificially eliminated. (B) Reconstructed vessels in the simplified model for arteries (red) and veins (blue). (C) Relationships between the Euclidean distance L and path length C of the whole pathways between the root vertex and terminal ends. A linear regression line is also plotted for each model. (D,E) The relative frequencies of the path length C (D) and Euclidean distance (E) for the whole pathways. (F) Relative frequencies of the number of bifurcations on the whole pathways. (G) Relative frequencies of the index of whole-pathway tortuosity given as C/L − 1, where curves fitted using logarithmic normal distributions are shown.