Extended law of laplace for measurement of the cloverleaf anatomy of the aortic root
Authors: Ban Ehsan , Kalogerakos Paris-Dimitrios , Khosravi Ramak , Ziganshin Bulat A. , Ellauzi Hesham , Ramachandra Abhay B. , Zafar Mohammad A. , Humphrey Jay D. , Elefteriades John A. 
Published: April 2023

AbstractThe cross-sectional shape of the aortic root is cloverleaf, not circular, raising controversy regarding how best to measure its radiographic “diameter” for aortic event prediction. We mathematically extended the law of Laplace to estimate aortic wall stress within this cloverleaf region, simultaneously identifying a new metric of aortic root dimension that can be applied to clinical measurement of the aortic root and sinuses of Valsalva on clinical computerized tomographic scans. Enforcing equilibrium between blood pressure and wall stress, finite element computations were performed to evaluate the mathematical derivation. The resulting Laplace diameter was compared with existing methods of aortic root measurement across four patient groups: non-syndromic aneurysm, bicuspid aortic valve, Marfan syndrome, and non-dilated root patients (total 106 patients, 62 M, 44 F). (1) Wall stress: Mean wall stress at the depth of the sinuses followed this equation: Wall stress = BP × Circumscribing circle diameter/(2 × Aortic wall thickness). Therefore, the diameter of the circle enclosing the root cloverleaf, that is, twice the distance between the center, where the sinus-to-commissure lines coincide, and the depth of the sinuses, may replace diameter in the Laplace relation for a cloverleaf cross-section (or any shaped cross-section with two or more planes of symmetry). This mathematically derived result was verified by computational finite element analyses. (2) Diameters: CT scan measurements showed a significant difference between this new metric, the Laplace diameter, and the sinus-to-commissure, mid-sinus-to-mid-sinus, and coronal measurements in all four groups (p-value < 0.05). The average Laplace diameter measurements differed significantly from the other measurements in all patient groups. Among the various possible measurements within the aortic root, the diameter of the circumscribing circle, enclosing the cloverleaf, represents the diameter most closely related to wall stress. This diameter is larger than the other measurements, indicating an underestimation of wall stress by prior measurements, and otherwise provides an unbiased, convenient, consistent, physics-based measurement for clinical use.
Graphical abstract
“Diameter” applies to circles. Our mathematical derivation of an extension of the law of Laplace, from circular to cloverleaf cross-sectional geometries of the aortic root, has implications for measurement of aortic root “diameter.” The suggested method is as follows: (1) the “center” of the aortic root is identified by drawing three sinus-to-commissure lines. The intersection of these three lines identifies the “center” of the cloverleaf. (2) The largest radius from this center point to any of the sinuses is identified as the “radius” of the aortic root. (3) This radius is doubled to give the “diameter” of the aortic root. We find that this diameter best corresponds to maximal wall stress in the aortic root. Please note that this diameter defines the smallest circle that completely encloses the cloverleaf shape, touching the depths of all three sinuses.