Abstract
ObjectiveThe goal of this study was to explore conceptual benefits of characterizing delineated target volumes based on surface area and to utilize the concept for assessing risk of therapeutic toxicity in radiosurgery.Methods and materialsFour computer-generated targets, a sphere, a cylinder, an ellipsoid and a box, were designed for two distinct scenarios. In the first scenario, all targets had identical volumes, and in the second one, all targets had identical surface areas. High quality stereotactic radiosurgery plans with at least 95% target coverage and selectivity were created for each target in both scenarios. Normal brain volumes V12Gy, V14Gy and V16Gy corresponding to received dose of 12 Gy, 14 Gy and 16 Gy, respectively, were computed and analyzed. Additionally, V12Gy and V14Gy volumes and values for seven prospective toxicity variables were recorded for 100 meningioma patients after Gamma Knife radiosurgery. Multivariable stepwise linear regression and best subset linear regression analyses were performed in two statistical software packages, SAS/STAT and R, respectively.ResultsIn a phantom study, for the constant volume targets, the volumes of 12 Gy, 14 Gy and 16 Gy isodose clouds were the lowest for the spherical target as an expected corollary of the isoperimetric inequality. For the constant surface area targets, a conventional wisdom is confirmed, as the target volume increases the corresponding volumes V12Gy, V14Gy and V16Gy also increase. In the 100-meningioma patient cohort, the best univariate model featured tumor surface area as the most significantly associated variable with both V12Gy and V14Gy volumes, corresponding to the adjusted R2 values of 0.82 and 0.77, respectively. Two statistical methods converged to matching multivariable models.ConclusionsIn a univariate model, target surface area is a better predictor of spilled dose to normal tissue than target largest dimension or target volume itself. In complex multivariate models, target surface area is an independent variable for modeling radiosurgical normal tissue toxicity risk.
Highlights
Surface area is an essential building block for numerous theoretical and technological concepts in mathematics, physics, chemistry, biology, cosmology, and other natural sciences
Target surface area is a better predictor of spilled dose to normal tissue than target largest dimension or target volume itself
Target surface area is an independent variable for modeling radiosurgical normal tissue toxicity risk
Summary
Surface area is an essential building block for numerous theoretical and technological concepts in mathematics, physics, chemistry, biology, cosmology, and other natural sciences. An alternate way of characterizing target volume is the volume itself, with the corresponding prescription specified as a percent volume coverage[5] There is yet another fundamental alternative, volumes of three-dimensional objects can be characterized in terms of two-dimensional surface area. This is linked to a mathematics problem spanning to the origins of geometry, i.e., the classical isoperimetric problem[16]. The goal is to determine a closed plane curve of a given perimeter which encloses the greatest area. The solution to this problem is a circle. In Euclidian spaces, the isoperimetric inequality states that a sphere has the smallest surface area per given volume
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