Topology and hemodynamics of the cortical cerebrovascular system

Abstract

The cerebrovascular system continuously delivers oxygen and energy substrates to the brain, which is one of the organs with the highest basal energy requirement in mammals. Discontinuities in the delivery lead to fatal consequences for the brain tissue. A detailed understanding of the structure of the cerebrovascular system is important for a multitude of (patho-)physiological cerebral processes and many noninvasive functional imaging methods rely on a signal that originates from the vasculature. Furthermore, neurodegenerative diseases often involve the cerebrovascular system and could contribute to neuronal loss. In this review, we focus on the cortical vascular system. In the first part, we present the current knowledge of the vascular anatomy. This is followed by a theory of topology and its application to vascular biology. We then discuss possible interactions between cerebral blood flow and vascular topology, before summarizing the existing body of the literature on quantitative cerebrovascular topology.

Figure 1

Reproduction of Cohnheim's seminal experimental work on the embolic process in the frog tongue (fig. 1) from 1872. He introduced the concept of the end-artery, which does not form anastomoses with any other artery (figs. 4 to 6).

Figure 2

(A) Scanning electron micrograph of a vascular corrosion cast from the monkey visual cortex (primary visual cortex). Arteries are shaded in red and veins are blue. Bar=1 mm. (B) The red box shows the precise location of the imaged area (horizontal schematic section taken from Saleem and Logothetis, 2007).

Figure 3

Vascular density of a microvascular system across the cortical layers and cortical depths (μm from cortical surface) in the macaque striate cortex (modified from Weber et al, 2008). (A) Nissl stain, with easily identifiable laminae of striate cortex (wm=white matter); (B) cytochrome oxidase (COX) stain; (C) filtered and thresholded binary vessel image based on anticollagen (type IV) immunohistochemistry; (D) the black trace shows the overall mean length density (±s.d. indicated with gray shaded area), the blue trace shows the values for the capillaries and the red for the noncapillary vessels, and the green trace represents the COX activity in arbitrary units.

Figure 4

Three continuous objects: (A) simple disk, (B) a more complex tree-like structure, and (C) ring. Objects (A) and (B) are connected objects with one outside boundary and are therefore topologically identical. Object (C) contains a hole and is topologically different from the other two.

Figure 5

Two choices for the definition of neighborhood on the 2D image raster. The central pixel of interest is black and the neighbors are dark gray. (A) 4-neighborhood (4-NB), where all pixels sharing an edge are neighbors and (B) 8-neighborhood (8-NB), where all pixels with common corners and sides are considered as neighbors. (C, D) Topology is sensitive to the neighborhood definition. (C) Contains two 4-connected objects or only one 8-connected object. (D) The object is an 8-connected loop but a 4-connected open arc. (E–H) Topological skeletons (denoted by black dots) of four raster objects. The topological skeleton is not unique, for (E) and (F) any point in the object will equally well present a topological skeleton. (G) The skeleton contains a loop, e.g., due to lumen effect. (H) Skeletonization is inherently sensitive to small boundary perturbations; single pixels will lead to stubs in the skeleton. (I, J) Two raster midline representations of a line (I) and a curve (J) on an 8-NB topology. The piecewise linear approximate (solid line) is longer or equal to the actual distance (dashed line).

Figure 6

Two graphs. (A) Network graph featuring loops. (B) Graph of a binary tree.

Figure 7

Extracted vessels from the rat somatosensory cortex retrieved from synchrotron radiation-based micro-CT with a resolution of 700 nm³. The network has been extracted by a standard procedure with skeletonization and subsequent distance transform to estimate the local radius. The vessels are color coded (log scale) according to their radius. (A) Feeding vessels down to 10 μm diameter and (B) capillary bed below 10 μm only. CT, computed tomography.

Figure 8

(A) Graph with two loops and one bridge. (B) Graph containing two generating loops. The choice of the generating loops is arbitrary. (C) The Strahler number is an interesting taxonomy to characterize the hierarchy of a binary tree. Leaf nodes are of order 1 and orders increase upstream (see text).