The probability of correct target dosage: Dose-population histograms for deriving treatment margins in radiotherapy

Purpose: To provide an analytical description of the effect of random and s ystematic geometrical deviations on the target dose in radiotherapy and to derive margin rules. Methods and Materials: The cumulative dose distribution delivered to the cl inical target volume (CTV) is expressed analytically. Geometrical deviation s are separated into treatment execution (random) and treatment preparation (systematic) variations. The analysis relates each possible preparation (s ystematic) error to the dose distribution over the CTV and allows computati on of the probability distribution of, for instance, the minimum dose deliv ered to the CTV, Results: The probability distributions of the cumulative dose over a popula tion of patients are called dose-population histograms in short. Large exec ution (random) variations lead to CTV underdosage for a large number of pat ients, while the same level of preparation (systematic) errors leads to a m uch larger underdosage for some of the patients. A single point on the hist ogram gives a simple "margin recipe." For example, to ensure a minimum dose to the CTV of 95% for 90% of the patients, a margin between CTV and planni ng target volume (PTV) is required of 2.5 times the total standard deviatio n (SD) of preparation (systematic) errors (Sigma) plus 1.64 times the total SD of execution (random) errors (sigma') combined with the penumbra width, minus 1.64 times the SD describing the penumbra width (sigma(p)). For a si gma(p) of 3.2 mm, this recipe can be simplified to 2.5 Sigma + 0.7 sigma', Because this margin excludes rotational errors and shape deviations, it mus t be considered as a lower limit for safe radiotherapy. Conclusion: Dose-population histograms provide insight into the effects of geometrical deviations on a population of patients. Using a dose-probabilit y based approach, simple algorithms for choosing margins were derived. (C) 2000 Elsevier Science Inc.