Gamma index

The gamma index was proposed by Low et al. [1] in 1998. It combines two other simpler methods: Dose Difference (DD) and Distance To Agreement (DTA).

Dose Difference

The DD is based on calculating the dose difference at corresponding points \(\vec{r_e}\) and \(\vec{r_r}\) located on the evaluated image \(D_e\) and the reference image \(D_r\), respectively.

\[\delta(\vec{r_r}, \vec{r_e}) = D_e(\vec{r_e}) - D_r(\vec{r_r})\]

Its acceptance criterion is expressed in percentages [%]. It is relative to global or local value:

• global – a predetermined value, typically the maximum value in the reference image,

• local – value at the current point in the reference image.

If the DD value at \(\vec{r_r}\) point is less than or equal to the value of the acceptance criterion (e.g., 3%, global), then this point passes the test. Otherwise, it fails.

The drawback of this test is that it struggles with high-gradient regions, as small shifts in the spatial distribution of radiation dose can result in large DD values.

Distance To Agreement

The DTA, on the other hand, is determined by calculating the distance between the two closest points with the same dose on two images.

\[r(\vec{r_r}, \vec{r_e}) = | \vec{r_e} - \vec{r_r} |,\]

\[where\ \ D_e(\vec{r_e}) = D_r(\vec{r_r})\]

Its acceptance criterion is expressed in millimeters [mm]. If the DTA value at \(\vec{r_r}\) point is less than or equal to the value of the acceptance criterion (e.g., 3 mm), then this point passes the test. Otherwise, it fails.

The drawback of this method is that it encounters difficulties in low-gradient areas, where minor dose misalignments may require a large search radius, resulting in a large DTA value.

Gamma function and gamma index

To overcome the disadvantages of DD and DTA, the gamma index was developed. First, it is necessary to calculate the gamma function, which combines DD and DTA. In the case of the DTA component, only the distance is calculated without taking into account the condition of equal doses.

\[\Gamma(\vec{r_r}, \vec{r_e}) = \sqrt{\frac{\delta^2(\vec{r_r}, \vec{r_e})}{\Delta D ^2} + \frac{r^2(\vec{r_r}, \vec{r_e})}{\Delta d ^2}}\]

\(\Delta D\) and \(\Delta d\) are acceptance criteria for DD and DTA, respectively. The commonly used criteria values are 3%/3mm with global normalization, abbreviated as 3%G/3mm.

Now, to obtain the gamma index value for a single reference point \(\vec{r_r}\), one must select the minimum value of the gamma function \(\Gamma\) for that point and all evaluated points.

\[\gamma(\vec{r_r}) = \min_{\vec{r_e}}\Gamma(\vec{r_r}, \vec{r_e})\]

It is calculated for each reference point, and in the end, the image containing gamma index values is obtained. It has the same size as the reference image.

The gamma index is not symmetric – swapping the reference and evaluated images can yield a different result.

GIPR

The Gamma Index Passing Rate (GIPR) is a measure indicating how many points passed the test – that is, how many gamma index values are less than or equal to 1.

\[GIPR = \frac{| \{\gamma(\vec{r_r}) \le 1\} |}{| \{\gamma(\vec{r_r})\} |}\]

If this value is sufficiently high (e.g., at least 90%), the evaluated image is acceptable.

Flow diagram

The figure shows the successive stages of the gamma index calculations. GIPR is an optional step that provides a simple single metric, but analyzing the gamma index image can provide a lot more information about the result.

Example

To illustrate how the gamma index works, a simple example is presented here. It is calculated for two-dimensional images, each containing only four elements.

The used parameters are: 3% with global normalization for the DD acceptance criterion and 3 mm for the DTA acceptance criterion.

Below are the calculations of gamma index image values. Note that some of the gamma function calculations are skipped, and only minimum values are shown.

\[\gamma(\vec{r_{r0}}) = \Gamma(\vec{r_{r0}}, \vec{r_{e0}}) = \sqrt{\frac{(0.93 - 0.93)^2}{(0.03*1.00)^2} + \frac{(0 - (-1))^2 + (1 - 0)^2}{3^2}} \approx \sqrt{0 + 0.222} \approx 0.471\]

\[\gamma(\vec{r_{r1}}) = \Gamma(\vec{r_{r1}}, \vec{r_{e1}}) = \sqrt{\frac{(0.96 - 0.95)^2}{(0.03*1.00)^2} + \frac{(2 - 1)^2 + (1 - 0)^2}{3^2}} \approx \sqrt{0.111 + 0.222} \approx 0.577\]

\[\gamma(\vec{r_{r2}}) = \Gamma(\vec{r_{r2}}, \vec{r_{e1}}) = \sqrt{\frac{(0.96 - 0.97)^2}{(0.03*1.00)^2} + \frac{(2 - (-1))^2 + (1 - 2)^2}{3^2}} \approx \sqrt{0.111 + 1.111} \approx 1.106\]

\[\gamma(\vec{r_{r3}}) = \Gamma(\vec{r_{r3}}, \vec{r_{e3}}) = \sqrt{\frac{(1.02 - 1.00)^2}{(0.03*1.00)^2} + \frac{(2 - 1)^2 + (3 - 2)^2}{3^2}} \approx \sqrt{0.444 + 0.222} \approx 0.816\]

\[GIPR = \frac{3}{4} = 75\%\]

The calculation of the gamma index for this example using YAGIT library is presented here.

References

[1]

D. A. Low, W. B. Harms, S. Mutic, and J. A. Purdy, “A technique for the quantitative evaluation of dose distributions,” Medical Physics, vol. 25, no. 5, pp. 656-661, 1998.