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

Part of this material was presented at the 1998 ASTRO meeting in Phoenix, AZ.
https://doi.org/10.1016/S0360-3016(00)00518-6Get rights and content

Abstract

Purpose: To provide an analytical description of the effect of random and systematic geometrical deviations on the target dose in radiotherapy and to derive margin rules.

Methods and Materials: The cumulative dose distribution delivered to the clinical target volume (CTV) is expressed analytically. Geometrical deviations are separated into treatment execution (random) and treatment preparation (systematic) variations. The analysis relates each possible preparation (systematic) error to the dose distribution over the CTV and allows computation of the probability distribution of, for instance, the minimum dose delivered to the CTV.

Results: The probability distributions of the cumulative dose over a population of patients are called dose-population histograms in short. Large execution (random) variations lead to CTV underdosage for a large number of patients, while the same level of preparation (systematic) errors leads to a much larger underdosage for some of the patients. A single point on the histogram 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 planning target volume (PTV) is required of 2.5 times the total standard deviation (SD) of preparation (systematic) errors (Σ) plus 1.64 times the total SD of execution (random) errors (σ′) combined with the penumbra width, minus 1.64 times the SD describing the penumbra width (σp). For a σp of 3.2 mm, this recipe can be simplified to 2.5 Σ + 0.7 σ′. Because this margin excludes rotational errors and shape deviations, it must 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-probability based approach, simple algorithms for choosing margins were derived.

Introduction

Knowledge of geometrical uncertainties in radiotherapy is steadily increasing. Numerous publications (of which only a few recent ones are referenced as example) have presented data on the accuracy of target volume delineation 1, 2, 3, organ motion 4, 5, and setup accuracy 6, 7, 8. In principle, the full characterization of all geometrical uncertainties should lead to objective choices for treatment margins (9). However, no unambiguous guidelines have been presented thus far for this purpose, indicating that the statistical methods required to choose treatment margins are still unclear. Many studies and reports describing margin choice do not fully address the difference between (treatment) execution variations (often called random or day-to-day variations) and (treatment) preparation variations 10, 11. The latter errors are often called systematic, because they are systematic for a single radiotherapy course of a single patient, but they are stochastic over a group of patients. In those studies that do separate these two types of variations, the authors are unsure how to combine them (12). The most advanced approaches that have been presented to date concern Monte Carlo-like simulations based on real patient data to determine the impact of geometrical variations on the dose delivery 13, 14. These studies are highly specific to a certain technique and their results are sometimes difficult to extrapolate to other situations. Another numerical approach is to use coverage probabilities to derive margins 15, 16.

In this paper, we will use the terms clinical target volume (CTV) and planning target volume (PTV) as defined by the ICRU (17). The definition of CTV is: “The CTV is a tissue volume that contains a gross tumor volume (GTV) which is the gross palpable or visible/demonstrable extent and location of the malignant growth, and/or subclinical microscopic malignant disease, which has to be eliminated. This volume has to be treated adequately in order to reach the aim of therapy: cure or palliation.” The definition of PTV is: “The PTV is a geometrical concept, and it is defined to select appropriate beam sizes and beam arrangements, taking into consideration the net effect of all the possible geometrical variations and inaccuracies in order to ensure that the prescribed dose is actually absorbed in the CTV.”

The aim of this study is to provide an analytical description of the influence of geometrical deviations occurring during both the preparation and execution stage of treatment on the dose delivered to the CTV. Next, this model will be used to derive the margin that is required between the CTV and the PTV. The computations will be based on the probability distribution of the dose delivered to the CTV. Solutions will be given for a simplified situation. Finally, we will discuss the validity of the current PTV definition and propose some refinements.

Section snippets

Theory

In this study, the impact of geometric deviations will be quantified in terms of the dose delivered to the CTV. The method is based on the probability distribution of geometrical deviations. First, the cumulative dose distribution (including geometric deviations) is computed. Next, the probability that the CTV actually receives a given dose is computed. The derivations in this paper are classical in the sense that no biological response parameters are used but that the effects are solely

Dose-population histograms

Table 1 lists some recent data on the accuracy of prostate irradiation from studies performed by our group. We will use these data to derive more or less realistic minimum dose-population histograms for various margins and an ideal homogeneous dose distribution. However, one should consider that these computations exclude rotations and shape variations (thereby underestimating some of the error sources) and assume perfect conformity of the dose distribution (thereby overestimating the impact of

Discussion

In our analysis we have derived margins based on the probability of correct target dosage. In our opinion, the definition of PTV should therefore be modified and be: “The PTV is a geometrical concept, and it is defined to select appropriate beam sizes and beam arrangements, taking into consideration the net effect of all the possible geometrical variations and inaccuracies inorder to obtain a clinically acceptable and specified probability that the prescribed dose is actually absorbed in the CTV

Conclusion

Probability histograms of the cumulative dose in a population of patients (dose-population histograms) are a simple and effective means for describing the effect of a specific choice of a margin on the population of patients and allow the design of simple margin recipes. The impact of treatment preparation (systematic) errors is much larger than the impact of treatment execution (random) variations. Large treatment execution (random) variations lead to a moderate underdosage for a large number

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