Convolution method and CTV-to-PTV margins for finite fractions and small systematic errors.

The van Herk margin formula (VHMF) relies on the accuracy of the convolution method (CM) to determine clinical target volume (CTV) to planning target volume (PTV) margins. This work (1) evaluates the accuracy of the CM and VHMF as a function of the number of fractions N and other parameters, and (2) proposes an alternative margin algorithm which ensures target coverage for a wider range of parameter values. Dose coverage was evaluated for a spherical target with uniform margin, using the same simplified dose model and CTV coverage criterion as were used in development of the VHMF. Systematic and random setup errors were assumed to be normally distributed with standard deviations Sigma and sigma. For clinically relevant combinations of sigma, Sigma and N, margins were determined by requiring that 90% of treatment course simulations have a CTV minimum dose greater than or equal to the static PTV minimum dose. Simulation results were compared with the VHMF and the alternative margin algorithm. The CM and VHMF were found to be accurate for parameter values satisfying the approximate criterion: sigma[1 - gammaN/25] < 0.2, where gamma = Sigma/sigma. They were found to be inaccurate for sigma[1 - gammaN/25] > 0.2, because they failed to account for the non-negligible dose variability associated with random setup errors. These criteria are applicable when sigma greater than or approximately egual sigma(P), where sigma(P) = 0.32 cm is the standard deviation of the normal dose penumbra. (Qualitative behaviour of the CM and VHMF will remain the same, though the criteria might vary if sigma(P) takes values other than 0.32 cm.) When sigma << sigma(P), dose variability due to random setup errors becomes negligible, and the CM and VHMF are valid regardless of the values of Sigma and N. When sigma greater than or approximately egual sigma(P), consistent with the above criteria, it was found that the VHMF can underestimate margins for large sigma, small Sigma and small N. A potential consequence of this underestimate is that the CTV minimum dose can fall below its planned value in more than the prescribed 10% of treatments. The proposed alternative margin algorithm provides better margin estimates and CTV coverage over the parameter ranges examined here. This algorithm is not amenable to expression as a simple formula (e.g., as a linear combination of Sigma and sigma). However, it can be easily calculated. For 0.1 cm < or = sigma < or = 0.75 cm, 0 < or = gamma < or = 1 and 5 < or = N < or = 30, the VHMF underestimates margins by as much as 33%. With the alternative margin algorithm, the maximum underestimate is 7%. These results suggest that the VHMF should be used with caution for hypofractionated treatment and in adaptive therapy.