Introduction The principle of minimum work, alternatively known as Murray’s Law, mathematically describes the ideal radii and two-dimensional bifurcation angles which minimize the energy required to metabolically maintain and mechanically transport blood through a bifurcation. However, its application is limited by the simplification of branch points to two dimensions and the assumption of only two daughter vessels. Here, we present an extension of this law to three dimensions and an arbitrary number of daughter vessels and assess its validity by examining five middle cerebral artery (MCA) trifurcations.

Materials and Methods To derive a generalized form of the principle of minimum work, the same assumptions that Murray used in his original derivation in 1926 were applied. Specifically, Poiseuille flow and a metabolic energy cost proportional to the volume of blood were assumed. Measurements of radii and angles were made on 3D models acquired from computed tomography angiograms of MCA trifurcations. Comparison between the measured and ideal geometries was performed using a two-tailed t-test with a critical value of α = 0.05.

Results Five MCA trifurcations were analyzed. The generalized form of the principle of minimum work and accompanying definition of variables is shown in figure 1. For n > 2 daughter vessels, there are multiple solutions for ideal angles which satisfy the equation. Thus, for n =3 daughter vessels, deviation from ideality was quantified by determining how far the value of the equation with substituted measured radii and angles deviated from zero. There was no significant difference between the average observed value of the equation and the predicted value of zero, indicating good agreement between observed and predicted geometry (1.22 ± 1.57 mm² [95% CI], p = 0.20).

Conclusion This study provides an extension of a well-validated principle of physiology and tests it through application to cerebrovascular anatomy. However, this study is limited by the small sample size and by the same simplifying assumptions that were used in Murray’s original derivation. Future work may examine how well this model predicts vascular geometry at other locations to further assess its validity. Additionally, associations between deviation from the ideal geometry predicted by this model and vascular pathology, such as atherosclerosis or aneurysms, may be explored.

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Abstract E-199 Figure 1

A) Illustrated definition of variables on a mathematical diagram and application to a 3D vascular model. θ1, θj, and θn are the angles between the extended vector of the parent vessel 0 and the vectors of vessels 1, j, and n. θ1,j and θj,n are the angles between the vectors of vessels 1 and j and vessels j and n, respectively. For clarity, θ1,n is not shown. B) Generalized principle of minimum work

Disclosures S. Street: None. P. Palmisciano: None. S. Hoz: None. M. Johnson: None. A. Matur: None. J. Castiglione: None. G. Ventre: None. N. Agyeman: None. P. Shirani: None. M. Smith: None. M. Zuccarello: None. J. Forbes: None. N. Andaluz: None. C. Prestigiacomo: None.