Derivative based sensitivity analysis of gamma index

Originally developed as a tool for patient-specific quality assurance in advanced treatment delivery methods to compare between measured and calculated dose distributions, the gamma index (γ) concept was later extended to compare between any two dose distributions. It takes into effect both the dose difference (DD) and distance-to-agreement (DTA) measurements in the comparison. Its strength lies in its capability to give a quantitative value for the analysis, unlike other methods. For every point on the reference curve, if there is at least one point in the evaluated curve that satisfies the pass criteria (e.g., δDD = 1%, δDTA = 1 mm), the point is included in the quantitative score as “pass.” Gamma analysis does not account for the gradient of the evaluated curve - it looks at only the minimum gamma value, and if it is <1, then the point passes, no matter what the gradient of evaluated curve is. In this work, an attempt has been made to present a derivative-based method for the identification of dose gradient. A mathematically derived reference profile (RP) representing the penumbral region of 6 MV 10 cm × 10 cm field was generated from an error function. A general test profile (GTP) was created from this RP by introducing 1 mm distance error and 1% dose error at each point. This was considered as the first of the two evaluated curves. By its nature, this curve is a smooth curve and would satisfy the pass criteria for all points in it. The second evaluated profile was generated as a sawtooth test profile (STTP) which again would satisfy the pass criteria for every point on the RP. However, being a sawtooth curve, it is not a smooth one and would be obviously poor when compared with the smooth profile. Considering the smooth GTP as an acceptable profile when it passed the gamma pass criteria (1% DD and 1 mm DTA) against the RP, the first and second order derivatives of the DDs (δD’, δD”) between these two curves were derived and used as the boundary values for evaluating the STTP against the RP. Even though the STTP passed the simple gamma pass criteria, it was found failing at many locations when the derivatives were used as the boundary values. The proposed derivative-based method can identify a noisy curve and can prove to be a useful tool for improving the sensitivity of the gamma index.