A random walk model for intra-fraction motion has been proposed, where at each step the prostate moves a small amount from its current position in a random direction. Online tracking data from perineal ultrasound is used to validate or reject this model against alternatives. Intra-fraction motion of a prostate was recorded by 4D ultrasound (Elekta Clarity system) during 84 fractions of external beam radiotherapy of six patients. In total, the center of the prostate was tracked for 8 h in intervals of 4 s. Maximum likelihood model parameters were fitted to the data. The null hypothesis of a random walk was tested with the Dickey-Fuller test. The null hypothesis of stationarity was tested by the Kwiatkowski-Phillips-Schmidt-Shin test. The increase of variance in prostate position over time and the variability in motility between fractions were analyzed. Intra-fraction motion of the prostate was best described as a stochastic process with an auto-correlation coefficient of ρ = 0.92  ±  0.13. The random walk hypothesis (ρ = 1) could not be rejected (p = 0.27). The static noise hypothesis (ρ = 0) was rejected (p < 0.001). The Dickey-Fuller test rejected the null hypothesis ρ = 1 in 25% to 32% of cases. On average, the Kwiatkowski-Phillips-Schmidt-Shin test rejected the null hypothesis ρ = 0 with a probability of 93% to 96%. The variance in prostate position increased linearly over time (r(2) = 0.9  ±  0.1). Variance kept increasing and did not settle at a maximum as would be expected from a stationary process. There was substantial variability in motility between fractions and patients with maximum aberrations from isocenter ranging from 0.5 mm to over 10 mm in one patient alone. In conclusion, evidence strongly suggests that intra-fraction motion of the prostate is a random walk and neither static (like inter-fraction setup errors) nor stationary (like a cyclic motion such as breathing, for example). The prostate tends to drift away from the isocenter during a fraction, and this variance increases with time, such that shorter fractions are beneficial to the problem of intra-fraction motion. As a consequence, fixed safety margins (which would over-compensate at the beginning and under-compensate at the end of a fraction) cannot optimally account for intra-fraction motion. Instead, online tracking and position correction on-the-fly should be considered as the preferred approach to counter intra-fraction motion.

We present a calculation of critical phenomena directly in continuous dimension d employing an exact renormalization group equation for the effective average action. For an Ising-type scalar field theory we calculate the critical exponents ν(d) and η(d) both from a lowest-order and a complete first-order derivative expansion of the effective average action. In particular, this can be used to study critical behavior as a function of dimensionality at fixed temperature.  2004 Published by Elsevier B.V.

The results indicate that 3D-CRT plans in DIBH pose the lowest risk for both major coronary events and secondary lung cancer.

BackgroundThe accuracy of the Elekta Clarity™ three-dimensional ultrasound system (3DUS) was assessed for prostate positioning and compared to seed- and bone-based positioning in kilo-voltage cone-beam computed tomography (CBCT) during a definitive radiotherapy.MethodsThe prostate positioning of 6 patients, with fiducial markers implanted into the prostate, was controlled by 3DUS and CBCT. In total, 78 ultrasound scans were performed trans-abdominally and compared to bone-matches and seed-matches in CBCT scans. Setup errors detected by the different modalities were compared. Systematic and random errors were analysed, and optimal setup margins were calculated.ResultsThe discrepancy between 3DUS and seed-match in CBCT was −0.2 ± 2.7 mm laterally, −1.9 ± 2.3 mm longitudinally and 0.0 ± 3.0 mm vertically and significant only in longitudinal direction. Using seed-match as reference, systematic errors of 3DUS were 1.3 mm laterally, 0.8 mm longitudinally and 1.4 mm vertically, and random errors were 2.5 mm laterally, 2.3 mm longitudinally, and 2.7 mm vertically. No significant difference could be detected for 3DUS in comparison to bone-match in CBCT.Conclusions3DUS is feasible for image guidance for patients with prostate cancer and appears comparable to CBCT based image guidance in the retrospective study. While 3DUS offers some distinct advantages such as no need of invasive fiducial implantation and avoidance of extra radiation, its disadvantages include the operator dependence of the technique and dependence on sufficient bladder filling. Further study of 3DUS for image guidance in a large patient cohort is warranted.