Two datasets of points of known spatial positions and an associated absorbed dose value are often compared for quality assurance purposes in External Beam Radiation Therapy (EBRT). Some problems usually arise regarding the pass fail criterion to accept both datasets as close enough for practical purposes. Instances of this kind of comparisons are fluence or dose checks for intensity modulated radiation therapy, modelling of a treatment unit in a treatment planning system, and so forth. The gamma index is a figure of merit that can be obtained from both datasets; it is widely used, as well as other indices, as part of a comparison procedure. However, it is recognized that false negatives may take place (there are acceptable cases where a certain number of points do not pass the test) due in part to computation and experimental uncertainty. This work utilizes mathematical methods to analyse comparisons, so that uncertainty can be taken into account. Therefore, false rejections due to uncertainty do not take place and there is no need to expand tolerances to take uncertainty into account. The methods provided are based on the rules of uncertainty propagation and help obtain rigorous pass/fail criteria, based on experimental information.
Modern radiation therapy aims at a high level of accuracy and, as a consequence, becomes more demanding regarding quality assurance checks (even patient-specific checks) and measurement and computation performance. The use of comparisons of two datasets consisting of a sample of measured or computed absorbed dose points covering the treatment field or a patient tomographic slice is frequently performed on a routine basis. Therefore, the acceptance method should be both straightforward and reliable.
Traditionally, treatment goals in radiation therapy were achieved by choosing several directions around the patient so that the dose from all the beams was conformed to the target volume, sparing healthy tissues. Nowadays, it is possible to improve the homogeneity of absorbed dose in the planning target volume (PTV) and reduce the absorbed dose to healthy organs using several fields of non-uniform intensity (IMRT) designed to combine in an optimised dose distribution inside the patient [
The process is more complex than the one involved in conventional radiation therapy. The way the different beam orientations are combined could lead to practical problems, due to several issues: small and elongated beams are used, there are high dose gradients inside the fields, some features of the linear accelerator could have a noticeable effect, and treatment planning computation could not be accurate enough. These issues can make a particular plan unsuitable for treatment, and this is the reason why a comprehensive quality control of the technique and checks for each plan are often recommended [
Two main types of patient specific checks have been recommended in the literature [
(1) The first one consists on recomputing the plan substituting the representation of a suitable phantom for the patient representation and obtaining the 2D dose distribution on several planes inside the phantom. Radiographic or radiochromic film is inserted in the phantom in the same positions where the 2D doses where computed, and it is irradiated with the whole treatment. These films are scanned with an appropriate device and compared with the computed dose planes; this is a way to check the combined dose distribution. Some 3D measurement devices are also available [
(2) Irradiations are carried out for each beam, with the film or 2D detector placed perpendicular to the beam direction. The dose distributions have been previously computed with the treatment planning system, and a corresponding set of 2D computed dose distributions (or fluence maps) has to be compared with the measured ones. A check of each fluence map is obtained with this technique.
In either case, a comparison of two datasets with a great number of points has to be performed. Similar situations arise when commissioning a treatment planning system, since computation results have to be checked against measurement results. This leads to the following problem.
Given two arrays of values (absorbed dose), maybe with different spacing, find a convenient criterion to decide whether or not they can be considered as coincident for practical purposes. The dose distributions to be expected in radiation therapy can have sharp gradients in the field boundary, and possibly also inside the field, where the dose is not homogeneous. Wherever a sharp gradient is present, the result could be affected by geometrical errors (i.e., error in the position of a collimator leaf, error in the computation of the dose on the edge of the collimator leaf) and the check method should be able to cope with this. A small geometrical error is considered acceptable, but a direct comparison of the measured and reference dose in this area could result in a value out of dose tolerance. This is the reason why acceptance criteria based on distance to agreement (DTA) were developed [
A solution was proposed by Low et al. [
A point passes the check if this index is less or equal than to 1.
The gamma index can be easily generalized to a 3D comparison, if DTA is computed with a 3D search [
It is widely acknowledged that in few occasions measured and computed datasets pass this gamma test for every measurement point, and it is customary to allow for some percentage of points failing the test [
Palta et al. [
In other kinds of comparisons (like comparisons between computed datasets), statistical information is not available and a decision about the percentage of failing points that can be tolerated has to be based on other considerations.
In this work, a novel method is presented that modifies the gamma index check, introducing uncertainty features into its computation. This method has some interesting properties: first, it is a direct propagation of experimental uncertainty, allowing for uncertainty analysis. Second, tolerance levels are not modified because of uncertainty; using this method, tolerance levels can be set to values close to the accuracy actually sought for. And third, experimental devices, computations, and their uncertainties are characterized by simple and physically meaningful parameters. Therefore, the study of the check procedure is reduced to the
Each of the datasets has been represented as several arrays of random variables. Their mean values are the values in the dataset, labelled as small-case letters with subscripts for their position and superscripts for the dataset:
A pass/fail test has to be performed for each possible pair of points, one from each dataset: point
Compute the following parameters:
Compute
For each point in the reference dataset, the value
A global modified pass rate can be reported with the results of this test for every point in the reference dataset. A value of
In a similar fashion, the test can be carried out for 3D datasets.
Compute the following parameters:
Compute
For each point in the reference dataset, the value
As in the 2D case, a global modified pass rate can be reported with the results of this test for every point in the reference dataset. A value of
A probabilistic method to check test datasets for coincidence with a reference dataset, taking uncertainty into account, was tested with an example. It has to be remarked that for the new test to be passed, every point has to pass the test, that is, the probability test has to be passed for each pair of points drawn from the reference and test datasets. Common practice when using classic gamma test is to allow a limited percentage of points to fail the test. For the application of the present method, the probability comparison will only be passed if all points pass the test.
A practical example with 5 segments was set up. Figure
Reference dataset. (a) Composite irradiation. (b) Segment 1. (c) Segment 2. (d) Segment 3. (e) Segment 4. (f) Segment 5.
A 1 mm shift along
Same modifications, but the increment of dose in the second segment is 3% and the third is shifted 4 mm.
The first segment has 2% less dose than the reference and is shifted 4 mm along
All segments but the smallest one were shifted 4 mm along the
All segments but the smallest one were shifted 4 mm along the
Therefore, each of the cases corresponds to a set of shifts and changes of intensities for every segment as exemplified in Figure
Example of setup of a test dataset by shifting and modifying segment intensity.
The modified planar distributions (test datasets) were compared with the original one (reference dataset) with the following uncertainty parameters: 0.2% dose and 0.5 mm, 0.5% dose and 0.5 mm, and 0.2% dose and 1 mm. Dose uncertainty is relative, and this fact has been taken into account in the computation of the indices. Tests were performed for tolerances 2% dose and 2 mm and 3% dose and 3 mm.
A function in
Results for the gamma test are shown in Table
Gamma results for Cases
Case | Case | Case | Case | Case | ||
---|---|---|---|---|---|---|
Tolerance 2%—2 mm | Dose Unc. 0.2%. Dist. Unc. 0.2 mm | 0.9929 | 0.9926 | 0.9696 | 0.9920 | |
Dose Unc. 0.2%. Dist. Unc. 0.5 mm | 0.9931 | 0.9930 | 0.9718 | 0.9927 | ||
Dose Unc. 0.5%. Dist. Unc. 0.5 mm | 0.9932 | 0.9932 | 0.9726 | 0.9938 | ||
Dose Unc. 0.2%. Dist. Unc. 1.0 mm | 0.9966 | 0.9966 | 0.9876 | 0.9972 | ||
Classic test | 0.9615 | |||||
Tolerance 3%—3 mm | Dose Unc. 0.2%. Dist. Unc. 0.2 mm | 0.9957 | 0.9957 | 0.9847 | 0.9965 | |
Dose Unc. 0.2%. Dist. Unc. 0.5 mm | 0.9967 | 0.9968 | 0.9898 | 0.9979 | ||
Dose Unc. 0.5%. Dist. Unc. 0.5 mm | 0.9968 | 0.9968 | 0.9909 | 0.9979 | ||
Dose Unc. 0.2%. Dist. Unc. 1.0 mm | ||||||
Classic test | 0.9794 |
Figure
Image of pass probability for Case
Figure
Image of pass probability for the new test for Case
Case
On the other hand, Figure
The method presented in this work is potentially applicable to a broad set of comparisons: computer versus measured dose distributions for planning system commissioning, IMRT commissioning and patient checks, commissioning of measurement devices, and so forth. For any real experimental case, care should be taken to characterize its uncertainty. Furthermore, this method could be used to evaluate whether experimental uncertainties could deteriorate the sensitivity of a test. Accuracy requirements in IMRT patient plan checks are very high, and it is useful to know if the checking device uncertainty could induce the checker to accept plans too easily.
Some alternative methods have been described in the literature in order to refine the standard gamma index test; but the result is a consensus about tolerances and pass rate criteria. It is interesting to look at some conclusions in the ESTRO Booklet no. 9 [
In the survey performed by Nelms and Simon [
This work shows a practical application of several results about the probability distribution of quadratic forms of normal random variables. Since no
A classical test (with
When the new method is used, it becomes feasible to ensure whether or not points failing a classic test are a consequence of measurement limitations. If the new test does not yield a 100% pass rate it is possible to assert that the failing points cannot have been caused solely by the measurement procedure but there is also a problem with the irradiation. Therefore, no failing points are allowed.
As pointed out previously, this novel method relates experimental features (uncertainty) with test results. A well-defined answer in terms of probability, whether or not the probability of failing a gamma test at the point
Gamma is often described as a distance in a
Two sets of 2D dose distributions are defined: the reference one and the test one. Both are regular arrays but their spacing could be different. The reference points are labelled with subscripts (
In the general case of a quadratic form, the noncentrality parameters are linear combinations of the means. Thus, a quadratic form of central normal variables results in a linear combination of central chi-squared variables. The normal variables involved in Γ are noncentral ones; their means are the differences between doses or between spatial coordinates in the test and the reference datasets.
Different expansions of the distribution function of a weighted sum of noncentral chi-squared variables can be found in the literature, and they could be used for this problem. Shah and Khatri [
According to Imhoff, if
It is possible to modify the original gamma test for the
The squared gamma random variable is now
No conflict of interests exists for any of the authors.
The authors are indebted to C. Weatherill from Boston University for some very valuable suggestions. They are also indebted to E. Overton and A. Baker for their careful revision of this paper and valuable suggestions.