

ORIGINAL ARTICLE 



Year : 2012  Volume
: 37
 Issue : 4  Page : 200206 

Modification of the gamma function for the recognition of over and underdose regions in three dimensions
Mohammad Mohammadi^{1}, Nima Rostampour^{2}, Thomas P Rutten^{3}
^{1} Department of Medical Physics, Royal Adelaide Hospital; School of Chemistry and Physics, University of Adelaide, Adelaide, SA 5000, Australia and Department of Medical Physics, School of Medicine, Hamadan University of Medical Sciences, Hamadan, Iran ^{2} Department of Medical Physics, School of Medicine, Hamadan University of Medical Sciences, Hamadan, Iran ^{3} Department of Medical Physics, Royal Adelaide Hospital, Adelaide, SA 5000, Australia
Date of Submission  07Dec2011 
Date of Decision  19Jul2012 
Date of Acceptance  10Oct2012 
Date of Web Publication  20Nov2012 
Correspondence Address: Mohammad Mohammadi Department of Medical Physics, Royal Adelaide Hospital, Adelaide, SA5000, Australia
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09716203.103605
Abstract   
In order to evaluate twodimensional radiation dose distributions, an algorithm called the Gamma function has recently been modified. The current study concentrates on modification of the gamma function as a threedimensional dose distribution evaluation tool, and includes the recognition of overdose/underdose areas. Using a sign term, the conventional gamma function separates the disagreed areas into two parts: overdose and underdose areas. The new gamma function was modified using an extension of the dose difference criterion, ΔD, from two dimensions into three dimensions. In order to provide twodimensional dose maps for analysis, several images were acquired for a range of regular and irregular radiation fields using a Scanning Liquid Ionization Chamber Electronic Portal Imaging Device. The raw images were then converted into twodimensional transmitted dose maps using an empirical method. They were utilized as reference dose maps. Translational and rotational manipulations were performed on the reference dose distribution maps to provide evaluated dose maps. The reference and evaluated dose maps were then compared using conventional and modified gamma tools. The results indicated that the modified algorithm is able to enhance the over and underdose regions. In addition, a slight increase of the agreement percentage for reference and evaluated dose maps were observed by the extension of ΔD to three dimensions. It is concluded that the modified method is more realistic and applicable for the evaluation of both twodimensional and threedimensional dose distributions.
Keywords: Dose distribution, gamma function, twodimensional dose distribution, twodimensional dosimetry
How to cite this article: Mohammadi M, Rostampour N, Rutten TP. Modification of the gamma function for the recognition of over and underdose regions in three dimensions. J Med Phys 2012;37:2006 
How to cite this URL: Mohammadi M, Rostampour N, Rutten TP. Modification of the gamma function for the recognition of over and underdose regions in three dimensions. J Med Phys [serial online] 2012 [cited 2019 Jun 24];37:2006. Available from: http://www.jmp.org.in/text.asp?2012/37/4/200/103605 
Introduction   
Evaluation of radiation dose distribution is essential in radiation therapy. Due to the existence of a highdose gradient region with variations of dose up to 30% per centimeter in the twodimensional dose distribution, the evaluation of dose delivery is more complicated. In order to cope with this problem, the gamma function is the most popular algorithm reported so far to evaluate twodimensional dose distributions. The algorithm was originally developed as a binary "composite" map ^{[1]} and was then improved as positive continuous values. ^{[2],[3]} Two criteria were defined as the dose difference (ΔD) and the distance to agreement (DTA), Δd, to evaluate lowand highdose gradient regions, respectively. The gamma response, which is called the gamma index, is a positive value which is ≤1 for agreed areas and >1 for disagreed areas. The typical gamma criteria utilized for gamma analysis for clinical purposes are ΔD = 3% and Δd = 3 mm. ^{[2],[4],[5],[6]}
The concept of the gamma function algorithm has been discussed in detail by Low et al.^{[3]} It has been mentioned that "the evaluated distribution will have at least as high a dimensionality as the reference distribution. For example, the reference distribution may be a twodimensional film dose distribution measurement, while the evaluated distribution may be a full threedimensional dose distribution calculation." But there is no report of the development of the gamma function for a threedimensional practical comparison. In addition, as current treatment planning systems are able to provide a threedimensional dose distribution, their output can be used as a reference dose distribution while the evaluated dose distribution remains a twodimensional dose map.
Several attempts were reported to extend the gamma function as a threedimensional dose distribution evaluation tool. Based on the gamma function definition, a new function, χ, was proposed by Bakai et al. and the results were then compared with those achieved using the gamma function. The χ function produces similar results, however, it does not involve a search algorithm. ^{[7]} In addition, Wendling et al. reported a threedimensional gamma function applied to evaluate a threedimensional dose distribution grid calculated using a back projection technique. The results were, however, segmented twodimensional gamma maps arranged for each distance. ^{[8],[9]}
The gamma function was used to evaluate twodimensional dose distributions in numerous reports to quantify the agreement between a reference and an evaluated dose distribution map. ^{[4],[5],[7],[10],[11],[12],[13],[14],[15]} However, there are several issues, which need to be taken into consideration. Firstly, the dose distribution is a threedimensional concept. Therefore, the relevant uncertainties should be addressed in a threedimensional volume. In addition, although the gamma function is a powerful tool to recognize the agreed and disagreed areas, it is not able to separate the over and underdose regions. ^{[16]} Moreover, if there is a search for a consistent dose value in neighboring points in a twodimensional area, this cannot be extended into a volume. The hypothesis of this study is that "with extension into the third dimension, consideration of the "less or more" fluctuations in the z direction, caused by patient positioning, organ motion, couch positioning, and isocenter calibration, for the point dose displacement in the z dimension can be achieved".
The current work concentrates on the evaluation of a typical measured dose distribution as an evaluated dose map and its deviation from a simulated reference dose distribution and vice versa. The gamma function, as one of the popular dose distribution evaluation tools, is used in the current study and modified to enhance the over and underdose areas for regions of disagreement. In addition, the gamma function is modified for a threedimensional assessment.
Materials and Methods   
Several digital images were acquired using a Scanning Liquid Ionization Chamber Electronic Portal Imaging Device (SLICEPID) for 10 × 10 cm^{2}, 20 × 20 cm^{2} open, 16 × 21 cm ^{2} wedged and multileaf collimated fields. In order to create high dose gradient regions on image central area, several images were also acquired in the presence of a 10 cm attenuator layer, covering half of the radiation field. The raw EPID images were then converted into two dimensional dose maps using an appropriate method reported elsewhere. ^{[17],[18],[19]} The acquired fluence maps were translated 1 to 5 pixels and rotated 1 to 5° then compared with the original fluence maps. All of the work procedures were performed using inhouse codes written in MATLAB 7 (MathWorks Inc, Natick, MA).
In order to recognize and enhance the over and underdose areas in a gamma map, a sign matrix was added to the gamma map as follows:
Where RDM_{i,j} and EDM_{i,j} are corresponding pixel values in reference and evaluated dose maps, respectively. The sign matrix is able to create a binary map of 1 and 1. The modified gamma function is shown in Equation 1.
where γ(r _{ref} ) is the conventional gamma function, developed by Low et al. ^{[2]} and based on a quantitative evaluation method to compare the measured and calculated dose distribution values: ^{[20]}
where
Where and are points of interest in evaluated and reference dose maps respectively. Δd and ΔD are DTA and dose difference criteria at the position.
The dose difference criterion can be expressed as:
where and are evaluated dose and reference dose at positions and , respectively. ^{[3]} Output of the gamma function can be categorized as the passfail criteria:
In order to extend the DTA to a three dimensional criterion, the conventional DTA was extended as follows:
where (z _{eva}  z _{ref} ) ^{2} was added into the conventional formula. In order to create a threedimensional map for analysis, two twodimensional fluence maps located above and below the reference twodimensional fluence map were extracted from a dose grid calculated using the Pinnacle ^{[3]} treatmentplanning system. The threedimensional map was then compared with that obtained from an EPID image (the reference dose map) using the modified threedimensional gamma algorithm.
Since different approaches were used to obtain reference and evaluated dose maps, the pixel/voxel size of primary images may alter the accuracy of the current study. In order to prevent any interpolation in image resizing and due to the SLICEPID pixel size of 1.27 mm × 1.27 mm, the dose difference and DTA criteria was selected as 3% and 2.54 mm, respectively.
In order to verify the results of the modified gamma for clinical cases, the dose distribution evaluation tools were also applied for two lung and breast cases. To do this, several portal images from two anonymous patients were collected, and after conversion, the raw portal images were converted to dose maps using a developed empirical method and the results were compared with the corresponding predicted portal dose maps calculated using the Pinnacle ^{[3]} treatment planning system (ADAC Inc., PHILIPS Medical System, Milpitas, CA, USA). The details of portal dose calculation and measurement were explained extensively in previous reports. ^{[19],[21],[22]}
Results and Discussion   
Modification of the gamma function to enhance over and underdose regions
The outcome of gamma evaluation was investigated for a range of rotated and translated open, wedged and irregular fields. Irregular fields were created using MLCs and also open fields with a 10cm attenuator layer covering half of the field. Typical results of conventional gamma, relative dose difference maps and the gamma modified with the sign term are shown in [Figure 1] for regular fields and for an irregular field with 5 pixel translation and 5 degree rotation around the image centre. The values for conventional and modified gamma distributions in agreed areas are shown in grey and in contrast, the disagreed regions for conventional gamma are shown in color. For modified gamma maps, over and underdose regions are shown in red and blue respectively.  Figure 1: Conventional gamma maps (a series), relative dose difference maps (b series) and modified gamma maps for over and underdose area enhancement (c series) for a 20×20 cm^{2} (first and second rows), and a MLC field (third and fourth rows) for 5 pixels transition and for 5° rotation (first and third rows, respectively) with 3%/2.5 mm criteria
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Quantitatively, percentages of agreement and disagreement for the conventional and modified gamma in the above mentioned cases are shown in [Table 1] for a 5 pixel translation and 5° image rotation. As [Table 1] shows no significant difference was observed in the percentage of agreed areas using the modified gamma algorithm for a 5 pixel translation. The modified gamma method is able to enhance the overdose/underdose areas.  Table 1: Percentage of agreement and disagreement achieved and the contribution of over and underdose regions in the disagreed regions for a range of radiation fields for a 5 pixels translation and 5 rotation of reference dose maps
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The sign term added to the conventional gamma function is able to produce positive and negative values corresponding to the over and underdose areas. This could be a helpful tool to find an appropriate approach including possible translational and rotational misalignments to find the position of the evaluated dose map compared to the reference one. In addition, the dose distribution evaluation, indicating the under and overdose region can be a helpful tool in optimizing image alignment using gamma function results.
Modification of the gamma function as a three dimensional dose distribution assessment tool
The results of the gamma function modified for threedimensional DTA and three dimensional dose difference is shown in [Figure 2] for a 5 pixels translational and for a 5° rotational manipulation. The comparison of two and three dimensional signgamma functions is also shown in [Figure 2] as differences between two and threedimensional gamma assessment (C series). Results show that in all cases studied in the current work, applying the three dimensional gamma function slightly increases the percentage of agreement compared to the two dimensional gamma function. Tabulated results in [Table 2] indicate that the extension of gamma function to three dimensions, which basically produces a three dimensional DTA, increases the percentage of agreement. Although the differences in the percentage of agreement for open fields is low, but comparisons of high dose gradient regions in the dose distribution maps, the difference increases significantly. This can be observed for wedged field and an open field where an attenuator is positioned in half of the radiation field.  Figure 2: Twodimensional (a series), threedimensional modified gamma maps (b series) and the difference between two and threedimensional gamma maps (c series) for a 20×20 cmj (first and second rows), and a MLC field (third and fourth rows) for a 5 pixel transition and for a 5º rotation (first and third rows, respectively) with 3%/2.5 mm criteria
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 Table 2: Percentage of agreement and disagreement achieved using modified two and threedimensional gamma functions and the contribution of over and underdose regions in disagreed regions for a range of radiation fields with a 5 pixel translation and 5 rotation of reference dose maps
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For a twodimensional dose distribution evaluation, in order to control unavoidable misalignments between reference and evaluated dose distributions, a twodimensional DTA criterion is defined. However, several misalignments, and consequently misreading of dose values, due to more or less attenuations and different contribution of scattered radiation reaching to the point of interest can possibly occur in the offplane area. This kind of mismatching can come from vertical displacement of treatment couch, geometrical patient positioning during setup procedure, undesired organ motion, which cause more or less attenuation of the beam onaxis and perpendicular to the twodimensional dose map. As a result, variations of dose values in an evaluated dose distribution map can occur. The important thing is that the inconsistency can be detected by a both two and threedimensional gamma functions. However, if the dose variation is within the tolerance with other points in a three dimensional comparison, the threedimensional gamma function classifies the point as agreed dose area while the twodimensional classifies it as a disagreed area.
As results in [Table 1] and [Table 2] shows the wedged condition, which can be categorized as a medium gradient region, has significant differences between conventional and modified methods. This has led us to conclude that the twodimensional gamma in the medium gradient region is not able to show the real agreement between two reference and evaluated dose distributions. It should be noted that the twodimensional gamma function mostly concentrates on low gradient and high gradient regions. In this case, a threedimensional gamma may improve the results of comparison at medium gradient regions. Regarding the abovementioned issues, the real dose values can be found in upper or lower dose maps in the third dimension.
In order to evaluate the results with clinical situations, the modified sign and threedimensional dose distribution evaluation gamma functions were applied for two lung and breast cases. The results of threedimensional and signed threedimensional gamma functions are shown in [Figure 3]. The numerical values of the investigation are also provided in [Table 3]. The results show that like the nonclinical cases shown previously, the threedimensional gamma function improves the agreement between reference and evaluated dose maps. In addition, a signed gamma map for both approaches (two and threedimensional) is able to provide more information about the location of under and overdose regions. In the clinical cases, the pixel size in the predicted dose image matrix is not identical with the corresponding transmitted dose maps measured using the SLICEPID and an image resizing tool is used to make them the same size compared to measured dose maps. This perhaps decreases the slope of dose gradient in high gradient regions.  Figure 3: Twodimensional relative dose map (a series) measured using a SLICEPID (b series), calculated using a treatment planning system (c series), three dimensional gamma map (d series) and three dimensional signed gamma map for a lung (first row) and breast (second row) cases with 3%/2.54 mm criteria
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 Table 3: Percentage of agreements and disagreements achieved using conventional, modified two and threedimensional gamma functions and the contribution of over and underdose areas in disagreed regions for typical breast and lung cases
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Although the reference dose map is a twodimensional dose distribution, each point of a twodimensional dose maps can be compared with other peripheral points. This leads to a threedimensional dose comparison. The significance of the difference is that the twodimensional case underestimates the agreement between two twodimensional dose maps. In contrast, a threedimensional gamma map is more practical and results are closer to the real situation.
Conclusion   
Although the conventional gamma function is able to recognize the agreed and disagreed regions, it is not able to illustrate the contribution of overand underdose regions. In the current work, it has been shown that by combining the dose difference map with a sign term it is possible to highlight the over and underdose regions in the disagreed areas. The enhancement can also be extended for agreed areas if required.
The desired dose map for a threedimensional dose distribution evaluated in the current study is an array of 3 twodimensional dose maps and it can be extended into 5 or 7 dose maps pending DTA criterion and the size of dose grid voxels. This extension could be a helpful effort to approach more realistic dose distribution comparisons for twoand threedimensional dosimetry.
Acknowledgments   
The authors express their gratitude to the referees for fruitful comments and advice. The work was supported by the research deputy of Hamadan University of Medical Sciences.
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[Figure 1], [Figure 2], [Figure 3]
[Table 1], [Table 2], [Table 3]
