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, γ has 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 with γ to 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 defined L , S , and U to evaluate the intensity, vorticity, and flow amount of deviation between two dose distributions, respectively. These metrics multiplied by γ ( γL , γS , γU ), along with the threshold value σ , 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 high L- metric value mainly extracted high-dose-gradient induced deviations, which was supported by high S and U metrics observed as a robust deviation and an influence of the dose-gradient, respectively. The S- metric also denotes the measured similarity between the compared dose distributions. In the γ distribution, γL sensitively detected the dose-gradient region in the film measurement, despite the presence of noise. The threshold σ successfully extracted the gradient-error region where γ > 1 analysis underestimated, and σ = 0.1 (plan) and σ = 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 of γ .
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