Extracting the gradient component of the gamma index using the Lie derivative method

Objective. The gamma index (γ) has been extensively investigated in the medical physics and applied in clinical practice. However,γhas a significant limitation when used to evaluate the dose-gradient region, leading to inconveniences, particularly in stereotactic radiotherapy (SRT). This study proposes a novel evaluation method combined withγto extract clinically problematic dose-gradient regions caused by irradiation including certain errors.Approach. A flow-vector field in the dose distribution is obtained when the dose is considered a scalar potential. Using the Lie derivative from differential geometry, we definedL,S, andUto evaluate the intensity, vorticity, and flow amount of deviation between two dose distributions, respectively. These metrics multiplied byγ(γL,γS,γU), along with the threshold valueσ, were verified in the ideal SRT case and in a clinical case of irradiation near the brainstem region using radiochromic films. Moreover, Moran's gradient index (MGI), Bakai's χ factor, and the structural similarity index (SSIM) were investigated for comparisons.Main results. A highL-metric value mainly extracted high-dose-gradient induced deviations, which was supported by highSandUmetrics observed as a robust deviation and an influence of the dose-gradient, respectively. TheS-metric also denotes the measured similarity between the compared dose distributions. In theγdistribution,γLsensitively detected the dose-gradient region in the film measurement, despite the presence of noise. The thresholdσsuccessfully extracted the gradient-error region whereγ> 1 analysis underestimated, andσ= 0.1 (plan) andσ= 0.001 (film measurement) were obtained according to the compared resolutions. However, the MGI, χ, and SSIM failed to detect the clinically interested region.Significance. Although further studies are required to clarify the error details, this study demonstrated that the Lie derivative method provided a novel perspective for the identifying gradient-induced error regions and enabled enhanced and clinically significant evaluations ofγ.