Journal of the ICRU Vol 7 No 2 (2007) Report 78 Oxford University Press

doi:10.1093/jicru/ndm042

8 ESTIMATION AND PRESENTATION OF UNCERTAINTY IN THE DELIVERED DOSE 8.1

THE INEVITABILITY OF UNCERTAINTY

† the identification of the tumor and the designation of its histology and staging; † the spatial extent of the tumor and of organs at risk (OARs; images may be incorrectly interpreted, they may be distorted, and so forth); † for external-beam therapy, the immobilization and localization of the patient and of the tumor within the patient, and the effects of physiologic motions on the dose delivered to any point within the patient; † the assessment of the distribution of heterogeneities, the effects of heterogeneities, and imperfections in the techniques to compensate for them; † the algorithms used to estimate dose; † the many parameters involved in the delivery of treatments. Given such uncertainties, one seeks to understand the sources of uncertainty, to reduce them whenever practicable, and to evaluate the magnitude and implications. The mere exercise of identifying the sources and magnitudes of uncertainty can be a valuable aspect of developing and judging a plan. In the practice of radiotherapy, the estimation and reporting of uncertainty has historically been at best implicit. Experienced physicians evaluating a treatment plan undoubtedly make some mental assessment of the magnitude of the known uncertainties and what the consequences may be. However, the current state-of-the-art treatmentdelivery techniques seek greater geometric accuracy in dose delivery, and are more complex than those employed previously. Such methods have

8.2

THE ESTIMATION OF UNCERTAINTY

In analyzing a radiation-treatment plan, there are at least two types of data whose uncertainties need to be estimated. The first involves the estimate of the uncertainty in the dose at selected points in three-dimensions within the patient. The second type involves the estimation of uncertainties in quantities such as D98%, D50%, EUD, TCP, and NTCP or in quantities used for constraints, such as the volume receiving greater than a certain dose D, VD, or the minimum dose that is delivered to a given volume V, DV. In this category is also the quantification of the adequacy of dose coverage of the planning target volume (PTV) and OARs. The terminology of uncertainty analysis has been clarified in ISO (1995). However, little has been reported in the literature in relation to making estimates of uncertainty in radiotherapy. A general

# International Commission on Radiation Units and Measurements 2007

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Radiation therapy is inherently probabilistic. One cannot be certain as to whether a tumor will be controlled, or whether any given normal tissue will be damaged. These outcomes can be stated only in terms of probabilities. In addition, the application of radiation involves very many factors, almost all of which have some level of associated uncertainty. For example, there are uncertainties regarding the following:

many additional potential sources of uncertainty: interplay effects that, with beam scanning, can lead to a degree of dose inhomogeneity within the tumor that are not normally encountered in conventional radiotherapy (see Section 7.6.3), registration uncertainties (e.g., between a physical or virtual compensator and the patient) and, in charged-particle therapy, the effect of inhomogeneities within the patient that can strongly influence the dose distribution in their shadow. In this environment, it can be very difficult to assess the dose implications of the uncertainties through visual inspection, and some form of computational approach is required. Unfortunately, currently available radiotherapy-planning systems have yet to embrace uncertainty analysis. Although, as just emphasized, there are many sources of uncertainty, some of them do not readily lend themselves to computational analysis. The following discussion is limited to the important question of the extent to which the prescribed dose distribution is a true representation of the distribution of dose the patient actually receives.

PRESCRIBING, RECORDING, AND REPORTING PROTON-BEAM THERAPY

8.3

THE PRESENTATION OF UNCERTAINTY

The presentation of the uncertainty in a threedimensional dose distribution presents a challenging problem because of the plethora of data. One approach is described by Goitein (1985) and by Urie et al. (1991), an example of which is shown in Fig. 8.1, taken from Urie et al. (1991). Three dose distributions are juxtaposed: the nominal (most likely) dose distribution, and, separately, the upper and lower bounds on the dose at each point (at the stated probability level). This highlights the scale of potential problems that can arise as a result of a beam juncture from possible treatment uncertainties and, in Fig. 8.1e, how these can be reduced by beam feathering (see Section 6.2.4.5). An alternative approach is described in Lomax (2001). In this method, dose distributions are calculated for a number of translated (or rotated) CT datasets, and, potentially, from datasets with altered CT numbers to simulate density uncertainties. A hybrid dose distribution, which indicates the worst-case dose at any point, is then computed as follows. For points within the PTV and CTV, the dose is set to the lowest dose at that point in any of the calculated dose distributions. For those points

Figure 8.1. Display of the dose distribution in a sagittal section of a patient whose para-aortic nodes are being treated with parallel opposed x-ray beams, using beam junctioning (Urie et al., 1991; reproduced with permission). (a) Nominal dose distribution; (b) absolute dose scale (color from 10 to 80 Gy; color gray ,10 Gy); (c) the upper-bound dose at the 85 percent CL, showing the possibility of a significant region of high dose; (d) the lower-bound dose at the 85 percent CL, showing the possibility of a significant region of low dose; (e) the upper-bound dose when the junction is feathered (21, 0, þ1 cm). A much smaller hot spot is seen in the overlap region [compare (e) with (c)].

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review was presented in Urie et al. (1991). A method has been described (Goitein, 1985) to estimate the uncertainty limits associated with a particular treatment plan (see Section 8.3). Dose uncertainties in proton-beam therapy have also been estimated as suggested in Goitein (1978a; 1982a) and Lomax (2001). At another level, a body of work has appeared in the last few years analyzing patient set-up uncertainties and motions from the point of view of determining the most appropriate safety margin around a tumor volume (van Herk, 2004), and more recently a number of authors have also begun to look into the problem of dealing with uncertainties at the optimization level, mainly from the point of view of organ motion (Beckham et al., 2002; McShan et al., 2002). A confidence level (CL) must be associated with any uncertainty estimate. Without a statement of the CL, an uncertainty estimate is meaningless. It is common in reporting radiotherapy results to indicate the 95 percent (2 SD) confidence intervals. Goitein (1983) argued that, for many purposes in radiation therapy, 1 SD is too low a CL, and 2 SDs are too high, and that an 85 percent confidence interval, corresponding to 1.5 SDs, is a more useful interval for many applications.

UNCERTAINTY IN THE DELIVERED DOSE

outside the PTV (and, hence, within normal tissue), the dose is set to the highest in any of the calculated dose distributions. This then documents potential cold spots within the tumor and, in the same display, potential hot spots within normal tissues. Such an analysis is shown in Fig. 8.2; the potential cool regions in the tumor (colored blue corresponding to a 10–20 percent dose reduction) are because of possible junction problems with the three abutting beams. This form of data presentation has its origins in the display of ‘regions of regret’, which was suggested in Shalev et al. (1988).

Dose –volume histograms (DVHs) (Drzymala et al., 1991; Shipley et al., 1979) are also an important dose-summarizing tool. Techniques for estimating and displaying uncertainty bands for DVHs have been reported (Drzymala et al., 1991; Niemierko and Goitein, 1994; Urie et al., 1991). These techniques lead to the display of a band in dose –volume space, within which a given point of the true DVH lies (at a stated level of confidence). An example of this is shown in Fig. 8.3, which is reproduced from Urie et al. (1991). In this display, as with the uncertainty bounds of Figs 8.1 and 8.2, the uncertainties in the dose at points within the patient are generally highly correlated with one another so that a DVH following one of the uncertainty bounds is generally not physically realizable. Unfortunately, such displays are unable to exhibit this fact. The PTV is defined largely to accommodate alignment and motion uncertainties, and the treatment plan is frequently designed such that the PTV receives a lower dose at its boundaries than in its interior. Because the CTV moment-to-moment or day-to-day is unlikely always to be located near the edge of the PTV, the DVH of the PTV will tend to underestimate the doses in the possible lower-dose regions in the CTV. On the other hand and for the same reasons, the DVH of the CTV will tend to overestimate the doses in the possible lower-dose regions in the CTV. Consequently, the DVHs of the PTV and CTV probably bracket the dose the tumor actually receives and, as such, can be used to estimate an uncertainty band about the ‘true’ (but unknowable) tumor DVH.

Figure 8.3. Dose–volume histogram with upper and lower bound limits (at the 85 percent CL) for the dose distributions shown in Fig. 8.1a, c, and d (Urie et al., 1991; reproduced with permission). The potential hot and cold spots are very evident in the DVH, but their spatial locations, of course, cannot be inferred from the DVH.

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Figure 8.2. (a) The individual beams of a three-beam IMPT plan for a thoracic chordoma, with the nominal combined dose distribution at the bottom. (b) The ‘worst case’ distribution resulting from 5 mm shifts along each major axis of the patient. The worst-case distribution is calculated at each point by taking the minimum dose of these shifted doses within the CTV, and the maximum dose outside. Note the potential cold spots (blue areas) that could occur where beams abut (i.e., along the patch lines of the oblique beams with the posterior beams). (Figure courtesy of A. Lomax, Paul Scherrer Institute, Villigen, Switzerland.)

PRESCRIBING, RECORDING, AND REPORTING PROTON-BEAM THERAPY

8.4 RECOMMENDATIONS FOR THE CONSIDERATION AND REPORTING OF UNCERTAINTY † Those involved in designing radiation treatments should analyze the uncertainties; make an effort to minimize them to the extent practicable; ensure that a quality assurance program is in place to give assurance that the treatment can be given as prescribed; and document their assessment of the remaining uncertainties. † Treatment planning systems should provide tools for the analysis, quantification, and display of uncertainties. † For normal reporting purposes, in uncomplicated cases, the uncertainties in the full threedimensional dose distribution need not be pre-

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sented, but those in summarizing quantities (see Section 5.6.2) should be estimated, together with their corresponding confidence intervals. Such an estimate could be stated as follows: ‘Doses are judged to be accurate to x percent of the prescription dose, or to be within y mm of the true location (at the z percent CL).’ The uncertainty estimate might be based on generic analyses of the particular class of treatment, in which case it should be so-noted. † For cases where unacceptably large uncertainties might exist, and for illustrative purposes in scientific reports: the uncertainties in the dose distribution(s), as well as those in summarizing quantities, should be estimated and presented, together with a statement of the corresponding confidence intervals.

8 estimation and presentation of uncertainty in the delivered dose.

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