Comparing dose distributions: The gamma test

In my last post I discussed dose distribution comparison with dose difference and distance-to-agreement (DTA) tests. Another widely used and closely related method for comparing dose distributions is the gamma test.

The gamma test was first introduced by Low et al. in 1998 as a single metric that combined features of both dose difference and DTA, while performing robustly in the regions where those are prone to failure. Conceptually, gamma is very similar to dose difference and DTA, but combines them into an abstract metric resembling a distance (Eq. 1). In this way both dose difference and DTA are taken into account for every point compared (rather than either-or as previously discussed).

Eq. 1

Eq. 2

In the above equations I have used somewhat different notation than Low et al. in an attempt to make things slightly clearer.

If we wish to compare two dose distributions, e.g. a measured versus a calculated distribution, we will have a dose, D_a(r_a), in the first distribution at point r_a, and a dose, D_b(r_b), at the corresponding point r_b in the second distribution. The DTA condition is fulfilled when D_a(r_a) = D_b(r_b+r), where r is an arbitrary point a distance |r| away from r_b. This condition defines an isodose contour in distribution b around point r_b. Away from this contour the DTA, d_DTA, is undefined. DTA is used with a threshold passing value, δ_DTA, e.g. 3mm. A DTA smaller than the threshold is considered passing for a simple DTA test. For gamma, δ_DTA is used to normalize the DTA value, such that a normal “passing” value would then be unity.

Dose difference is simply the difference of the two doses at the corresponding points: |D_a(r_a) = D_b(r_b)|. As with DTA, a pass/fail threshold, δ_DD, is used in the simple dose difference test, but is used to normalize the result in the gamma equation, such that the normal "passing" value would be unity.

We now have two components: normalized DTA and normalized dose difference. By squaring these values, adding, and taking the square root, we have a distance-like metric, Γ, shown in Eq. 1. Because DTA is only defined for values of r, such that D_a(r_a) = D_b(r_b+r), Γ is only defined when that condition is met (geometrically located along the DTA isodose contour).

Finally, the actual gamma index, γ, is determined by finding the minimum value of Γ by varying r. This essentially means traveling along the isodose contour and finding the point at which DTA is smallest.

The convention is for passing γ to be ≤ 1 and failing to be > 1. You will notice that a point yielding normalized DTA = 1 and normalized dose difference = 1 would now fail, since the corresponding γ would be √2.

What γ provides is a single value to evaluate, versus using separate tests and then considering both. As with DTA, γ presents challenges in efficient implementation (clearly Eq.'s 1 and 2 are not hand solvable).

Your comments (especially corrections) are appreciated.

Roy

Further reading:

• D. A. Low, W. B. Harms, S. Mutic, and J. A. Purdy, A technique for the quantitative evaluation of dose distributions, Med. Phys. 25, 656 (1998); http://dx.doi.org/10.1118/1.598248

Comparing dose distributions: DTA and dose-difference

Radiation therapy plan quality assurance often hinges on comparing calculated dose distributions with measured dose distributions. One of the most common techniques to compare dose distributions is the combined use of distance-to-agreement (DTA) and dose difference. In this post I will give an overview of these concepts.

A planar dose distribution from IMRT.

Dose difference is a very straight forward comparison of dose at corresponding points in two distributions. Given a point a_p in the planned distribution and the corresponding point a_m in the measured distribution, the dose difference is simply D(a_m) - D(a_p). A passing criterion is used, e.g. 3% of planned dose, such that if the measured dose difference is <= 3% the measured distribution "passes" at that point. The drawback of the dose difference test is that it is not robust in high gradient regions, as small misalignments can cause large dose differences.

Distance-to-agreement (DTA) is also very straight forward conceptually. Given a point a_p in the planned distribution and the corresponding point a_m in the measured distribution, the distance-to-agreement is the nearest point in the measured distribution from a_m, such that D(a_m + r) = D(a_p). As with dose difference, a passing criterion is chosen, e.g. 3 mm. If the matching dose level is found within a radius of <= 3 mm, the measured distribution "passes" at that point, . This technique is quite robust against misalignments in high gradient regions, as the matching dose level will still be nearby. However, this technique is prone to failure in low gradient regions, where even small misalignments can require a large radius to find the matching dose level. To avoid this to some extent, a dose threshold value can be used, such that dose below of the x% isodose line is not considered, where x% is usually a low dose, penumbra region.

In order to overcome the lack of  robustness in high and low gradient regions for dose difference and DTA respectively, the two tests are often used in conjunction.  This is done by evaluating the tests independently and then defining a point as passing if it passes either test. Thus a distribution might pass 70% for DTA and 70% for dose difference, but 90% for the combined test.

What we've seen is that both of these tests are conceptually quite straight forward, though each has its limitations. The subtleties and challenges arise in the efficient implementation of the algorithms (DTA in particular).

I discuss the gamma test in this post.


Further reading:

• I. J. Yeo and J. O. Kim, A Procedural Guide to Film Dosimetry, Medical Physics Publishing, 2004, ISBN: 9781930524194
• W. B. Harms, Sr., D. A. Low, J. W. Wong, and J. A. Purdy, A software tool for the quantitative evaluation of 3D dose calculation algorithms, Med. Phys. 25, p.c1830 (1998); http://dx.doi.org/10.1118/1.598363
• J. Van Dyk, R. B. Barnett, J. E. Cygler, and P. C. Shragge, "Commissioning and quality assurance of treatment planning computers," Med. Phys. 26, 261–273 (1993). http://dx.doi.org/10.1118/1.598801

Image by MBq and licensed under CC license terms.