An analytical formalism for the assessment of dose uncertainties due to positioning uncertainties

PURPOSE: To present an analytical formalism for the in depth assessment of uncertainties of field output factors in small fields related to detector positioning based on dose profile measurements. Additionally, a procedure for the propagation of these uncertainties was developed. METHODS: Based on the assumption that one dimensional and two dimensional second‐order polynomial functions can be fitted to dose profiles of small photon beams, equations for the calculation of the expectation value, the variance, and the standard deviation were developed. The following fitting procedures of the dose profiles were considered: A one‐dimensional case (1D), a quasi two‐dimensional case (2Dq) based on independently measured line profiles and a full 2D case (2Df) which also considers cross‐correlations in a two‐dimensional dose distribution. A rectangular and a Gaussian probability density function (PDF) characterizing the probability of possible positions of the detector relative to the maximum dose were used. Uncertainty components such as the finite resolution of the scanning water phantom, the reproducibility of the determination of the position of the maximum dose, and the reproducibility of the collimator system were investigated. This formalism was tested in a 0.5 x 0.5 cm(2) photon field where dose profiles were measured using a radiochromic film, a synthetic diamond detector, and an unshielded diode detector. Additionally, the dose distribution measured with the radiochromic film was convoluted with a convolution kernel mimicking the active volume of the unshielded diode. RESULTS: Analytic expressions for the calculation of uncertainties on field output factors were found for the 1D, the 2Dq, and the 2Df case. The uncertainty of the field output factor related to the relative position of the detector to the maximum dose increased quadratically with increasing limits of possible detector positions. Analysis of the radiochromic film showed that the 2Dq case gave a more conservative assessment of the uncertainty compared to the 2Df case with a difference of < 0.1%. The 2Dq case applied to the film measurements agreed well with the same approach as was applied to the unshielded diode. The investigated uncertainty components propagated to an uncertainty of the field output factors of 0.5% and 0.4% for the synthetic diamond and the unshielded diode, respectively. Additionally, the expectation value was lower than the maximum dose. The difference was 0.4% and 0.3% for the synthetic diamond and the unshielded diode, respectively. CONCLUSIONS: The assessment of uncertainties of field output factors related to detector positioning is feasible using the proposed formalism. The 2Dq case is applicable when using online detectors. Accurate positioning in small fields is essential for accurate dosimetry as its related uncertainty increases quadratically. The observed drop of the expectation value needs to be considered in small field dosimetry.