Pressure Distribution in the Pial Arterial System of Rats Based on Morphometric Data and Mathematical Models

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Abstract
Summary: The objective of the present work was a theoretical evaluation of pial arterial pressures in normoten-sive rats and spontaneously hypertensive rats based on the geometry and topography of the pial arterial system as well as on various topological models of the vascular trees. Pial branches of the middle cerebral artery in the diameter range of 30-320 µm were selectively visualized by corrosion compound, and the diameter and length of vascular segments were measured. The vessels were classified into branching orders by the methods of Hors-field and Strahler. The steady-state pressure distribution in the pial arterial system was calculated assuming that the flow at the bifurcations was partitioned in proportion to a given power of the diameters of the daughter branches (diameter exponent). The maximum number of vascular segments along the longest branch varied between 16 and 33. The mean branching ratio was 4.14 ± 0.23 (SD). The mean diameter of vessels classified into Strahler orders 1-5 were: 50 ± 12, 71 ± 19, 106 ± 22, 168 ± 22, and 191 ± 7 µm, respectively. The calculated pressure drop in the pial trees of normotensive rats was approximately twice as large in proximal orders 3 and 4 than in distal orders 1 and 2. The mean pressure in arteries of order 1 ranged from 54.4 to 58.4 mm Hg in the normotensive rat (input pressure: 83 mm Hg), and from 77.2 to 89.0 mm Hg in the spontaneously hypertensive rat (input pressure: 110 mm Hg). The coefficient of variation of terminal pressures in vessels of order 1 increased linearly with the mean pressure drop in the system. The coefficient of variation in terminal pressure had a minimum as a function of the diameter exponent in case of each pial tree. At its minimum, it was higher in all spontaneously hypertensive rats (10.1-22.9%) than in any normotensive rats (6.0-8.5%). The corresponding diameter exponents were in most cases lower in the spontaneously hypertensive rat (1.3-2.5) than in the normotensive rat (2.5-3.0). Topologically consistent models of the pial arterial network predicted significantly less variation in intravascular pressures than was obtained by direct calculations. More idealized models suggested the dependence of coefficient of variation in terminal pressure on the total number of vascular segments contained by the tree. All models predicted the existence of the minimum of coefficient of variation in terminal pressure in function of the diameter exponent. From these, we conclude that (a) a hemodynamic configuration of the pial arterial system resulting in the smallest variation in cerebral perfusion pressure may exist, and (b) the pial vascular structure of spontaneously hypertensive rat allows less homogeneous terminal pressure distribution than does that of normotensive rats.