In my last post I discussed dose distribution comparison with
dose difference and distancetoagreement (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 eitheror 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 distancelike 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