Murray's Law in Elastin Haploinsufficient (Eln+/−) and Wild-Type (WT) Mice

Abstract

Using either the principle of minimum energy or constant shear stress, a relation can be derived that predicts the diameters of branching vessels at a bifurcation. This relation, known as Murray's Law, has been shown to predict vessel diameters in a variety of cardiovascular systems from adult humans to developing chicks. The goal of this study is to investigate Murray's Law in vessels from mice that are haploinsufficient for the elastin protein (Eln+/−). Elastin is one of the major proteins in the blood vessel wall and is organized in concentric rings, known as lamellae, with smooth muscle cells (SMCs) around the vessel lumen. Eln+/− mice have an increased number of lamellae, as well as smaller, thinner vessels. It is possible that, due to decreased amounts of elastin available for vessel wall remodeling during development and in adulthood, Eln+/− vessels would not follow Murray's Law. We examined vessel bifurcations in six different physiologic regions, including the brain, heart, epidermis, ceocum (or cecum), testes, and intestines, in Eln+/− mice and wild-type (WT) littermates. All vessels were between 40 – 300 μm in diameter. We found that the diameters of both Eln+/− and WT vessels have an average of 13% error from the diameters predicted by Murray's Law, with no significant differences between genotypes or physiologic regions. The data suggest that vessels are optimized to follow Murray's Law, despite limitations on the proteins available for growth and remodeling of the vessel wall.

Consistent Biomechanical Phenotyping of Common Carotid Arteries from Seven Genetic, Pharmacological, and Surgical Mouse Models

“…Comparing these geometry-based estimates of pressure with the adjusted measured systolic pressures in Table 1 revealed that the values corresponded well, in general (percent error < 15%). This finding suggests a possible normal, or adaptive, growth and remodeling in response to changes in either genotype or pharmacologic treatment for most of the cases studied, which is consistent with the report by Sather et al 44 that arterial caliber was well adapted in an elastin haploinsufficient mouse model ( Eln +/− ). A similar analysis of the Aortic Banding model revealed values of ε ranging from 0.83 (0 day) to 1.91 (10–14 day) and γ ranging from 0.94 (0 day) to 2.59 (35–56 day); the actual data suggest a value of ε ~ 2 at the 7–10 day period but only a maximum value of γ ~1.28 at 10–14 days 12 .…”

“…Note, therefore, that the Acta2 −/− carotids were the only ones that were hypotensive (~81 mmHg) and the 35–56 day banding carotids were the only ones to have experienced an abrupt increase (from 120 to 154 mmHg at 7–14 days), then decrease (to 124 mmHg) in systolic blood pressure. In other words, it appears that “reverse” growth and remodeling (e.g., thinning of the wall) in response to decreases in pressure was much less adaptive than “positive” growth and remodeling in response to increases in blood pressure see, for example, previous results for Eln +/− carotids 44 as well as the Fbn1 mgR/mgR , Fbln5 −/− , Ang-II ApoE −/− , and Aortic Banding models at 7–10 and 10–14 days, all of which only experienced elevated pressures. That central arteries tend to adapt better, or quicker, to increases rather than decreases in pressure has been known since the important work of Wolinsky 53 and may have important implications regarding the clinical goal to reverse effects of established hypertension.…”

A Modification of Murray's Law for Shear-Thinning Rheology

“…(1) and have also found that the data are in agreement with Murray's law [9,10]. Nevertheless, some of these studies [5,7,8,10] have reported that the scaling exponent, as determined by a least-squares fit to the data, differs slightly from 3 with the reported exponent being 2.7-3.0. How ever, considerable scatter exists in the data sets.…”

“…However, the Newto nian fluid assumption used in Murray's analysis is relaxed. Sev eral previous studies [8,[11][12][13] have suggested that discrepancies between experimental data and Murray's theory may be due to the rheological properties of blood. The goal of this work, therefore, is to test the hypothesis that deviations from the cubic scaling exponent may be in part due to blood's shear-thinning behavior.…”

“…With in vivo data, the vessel diameters at a bifurcation are typically compared with that predicted by Eq. (2) [3][4][5][6][7][8]. A few studies have tested the flow rate/diameter scaling of Eq.…”