evaluation of radiation dose distribution is essential in radiation therapy . due to the existence of a high - dose gradient region with variations of dose up to 30% per centimeter in the two - dimensional 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 two - dimensional dose distributions .
the algorithm was originally developed as a binary  composite  map and was then improved as positive continuous values .
two criteria were defined as the dose difference ( d ) and the distance to agreement ( dta ) , d , to evaluate low - and high - dose 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 d = 3% and d = 3 mm.[246 ] the concept of the gamma function algorithm has been discussed in detail by low et al .
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 two - dimensional film dose distribution measurement , while the evaluated distribution may be a full three - dimensional dose distribution calculation . 
but there is no report of the development of the gamma function for a three - dimensional practical comparison .
in addition , as current treatment planning systems are able to provide a three - dimensional dose distribution , their output can be used as a reference dose distribution while the evaluated dose distribution remains a two - dimensional dose map .
several attempts were reported to extend the gamma function as a three - dimensional dose distribution evaluation tool . based on the gamma function definition , a new function ,  ,
the  function produces similar results , however , it does not involve a search algorithm .
in addition , wendling et al . reported a three - dimensional gamma function applied to evaluate a three - dimensional dose distribution grid calculated using a back projection technique .
the results were , however , segmented two - dimensional gamma maps arranged for each distance . the gamma function was used to evaluate two - dimensional dose distributions in numerous reports to quantify the agreement between a reference and an evaluated dose distribution map.[4571015 ] however , there are several issues , which need to be taken into consideration .
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 . moreover , if there is a search for a consistent dose value in neighboring points in a two - dimensional area , this can not 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 under - dose areas for regions of disagreement .
several digital images were acquired using a scanning liquid ionization chamber electronic portal imaging device ( slic - epid ) for 10  10 cm , 20  20 cm open , 16  21 cm   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.[1719 ] 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 in - house 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 rdmi , j and edmi , j are corresponding pixel values in reference and evaluated dose maps , respectively .
the modified gamma function is shown in equation 1 . where (rref ) is the conventional gamma function , developed by low et al . and
based on a quantitative evaluation method to compare the measured and calculated dose distribution values : where   and   are points of interest in evaluated and reference dose maps respectively .
the dose difference criterion can be expressed as : where   and   are evaluated dose and reference dose dr at positions   and , respectively .
output of the gamma function can be categorized as the pass - fail criteria : in order to extend the dta to a three dimensional criterion , the conventional dta was extended as follows : where ( zeva - zref ) was added into the conventional formula . in order to create a three - dimensional map for analysis , two two - dimensional fluence maps located above and below the reference two - dimensional fluence map were extracted from a dose grid calculated using the pinnacle treatment - planning system . the three - dimensional map was then compared with that obtained from an epid image ( the reference dose map ) using the modified three - dimensional 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 slic - epid 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 treatment planning system ( adac inc .
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 10-cm 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 ,
conventional gamma maps ( a series ) , relative dose difference maps ( b series ) and modified gamma maps for over- and under - dose area enhancement ( c series ) for a 2020 cm ( 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 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 over - dose / under - dose areas .
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 the sign term added to the conventional gamma function is able to produce positive and negative values corresponding to the over- and under - dose 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 over - dose region can be a helpful tool in optimizing image alignment using gamma function results .
the results of the gamma function modified for three - dimensional 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 sign - gamma functions is also shown in figure 2 as differences between two- and three - dimensional 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 .
two - dimensional ( a series ) , three - dimensional modified gamma maps ( b series ) and the difference between two- and three - dimensional gamma maps ( c series ) for a 2020 cm ( first and second rows ) , and a mlc field ( third and fourth rows ) for a 5 pixel transition and for a 5 rotation ( first and third rows , respectively ) with 3%/2.5 mm criteria percentage of agreement and disagreement achieved using modified two- and three - dimensional gamma functions and the contribution of over- and under - dose regions in disagreed regions for a range of radiation fields with a 5 pixel translation and 5 rotation of reference dose maps for a two - dimensional dose distribution evaluation , in order to control unavoidable misalignments between reference and evaluated dose distributions , a two - dimensional 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 off - plane area .
this kind of mismatching can come from vertical displacement of treatment couch , geometrical patient positioning during set - up procedure , undesired organ motion , which cause more or less attenuation of the beam on - axis and perpendicular to the two - dimensional dose map . as a result
the important thing is that the inconsistency can be detected by a both two- and three - dimensional gamma functions . however , if the dose variation is within the tolerance with other points in a three dimensional comparison , the three - dimensional gamma function classifies the point as agreed dose area while the two - dimensional classifies it as a disagreed area . as results in tables 1 and 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 two - dimensional 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 two - dimensional gamma function mostly concentrates on low gradient and high gradient regions . in this case , a three - dimensional 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 three - dimensional dose distribution evaluation gamma functions
the results of three - dimensional and signed three - dimensional gamma functions are shown in figure 3 .
the results show that like the non - clinical cases shown previously , the three - dimensional gamma function improves the agreement between reference and evaluated dose maps .
in addition , a signed gamma map for both approaches ( two- and three - dimensional ) is able to provide more information about the location of under- and over - dose 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 slic - epid and an image resizing tool is used to make them the same size compared to measured dose maps .
two - dimensional relative dose map ( a series ) measured using a slic - epid ( 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 percentage of agreements and disagreements achieved using conventional , modified two- and three - dimensional gamma functions and the contribution of over- and under - dose areas in disagreed regions for typical breast and lung cases although the reference dose map is a two - dimensional dose distribution , each point of a two - dimensional dose maps can be compared with other peripheral points .
the significance of the difference is that the two - dimensional case underestimates the agreement between two two - dimensional dose maps .
in contrast , a three - dimensional gamma map is more practical and results are closer to the real situation .
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 10-cm 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 ,
conventional gamma maps ( a series ) , relative dose difference maps ( b series ) and modified gamma maps for over- and under - dose area enhancement ( c series ) for a 2020 cm ( 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 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 over - dose / under - dose areas .
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 the sign term added to the conventional gamma function is able to produce positive and negative values corresponding to the over- and under - dose 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 over - dose region can be a helpful tool in optimizing image alignment using gamma function results .
the results of the gamma function modified for three - dimensional 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 sign - gamma functions is also shown in figure 2 as differences between two- and three - dimensional 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 .
two - dimensional ( a series ) , three - dimensional modified gamma maps ( b series ) and the difference between two- and three - dimensional gamma maps ( c series ) for a 2020 cm ( first and second rows ) , and a mlc field ( third and fourth rows ) for a 5 pixel transition and for a 5 rotation ( first and third rows , respectively ) with 3%/2.5 mm criteria percentage of agreement and disagreement achieved using modified two- and three - dimensional gamma functions and the contribution of over- and under - dose regions in disagreed regions for a range of radiation fields with a 5 pixel translation and 5 rotation of reference dose maps for a two - dimensional dose distribution evaluation , in order to control unavoidable misalignments between reference and evaluated dose distributions , a two - dimensional 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 off - plane area .
this kind of mismatching can come from vertical displacement of treatment couch , geometrical patient positioning during set - up procedure , undesired organ motion , which cause more or less attenuation of the beam on - axis and perpendicular to the two - dimensional dose map . as a result
the important thing is that the inconsistency can be detected by a both two- and three - dimensional gamma functions .
however , if the dose variation is within the tolerance with other points in a three dimensional comparison , the three - dimensional gamma function classifies the point as agreed dose area while the two - dimensional classifies it as a disagreed area . as results in tables 1 and 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 two - dimensional 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 two - dimensional gamma function mostly concentrates on low gradient and high gradient regions . in this case , a three - dimensional 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 three - dimensional dose distribution evaluation gamma functions were applied for two lung and breast cases .
the results of three - dimensional and signed three - dimensional gamma functions are shown in figure 3 .
the results show that like the non - clinical cases shown previously , the three - dimensional gamma function improves the agreement between reference and evaluated dose maps .
in addition , a signed gamma map for both approaches ( two- and three - dimensional ) is able to provide more information about the location of under- and over - dose 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 slic - epid and an image resizing tool is used to make them the same size compared to measured dose maps .
two - dimensional relative dose map ( a series ) measured using a slic - epid ( 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 percentage of agreements and disagreements achieved using conventional , modified two- and three - dimensional gamma functions and the contribution of over- and under - dose areas in disagreed regions for typical breast and lung cases although the reference dose map is a two - dimensional dose distribution , each point of a two - dimensional dose maps can be compared with other peripheral points .
the significance of the difference is that the two - dimensional case underestimates the agreement between two two - dimensional dose maps .
in contrast , a three - dimensional gamma map is more practical and results are closer to the real situation .
although the conventional gamma function is able to recognize the agreed and disagreed regions , it is not able to illustrate the contribution of over - and under - dose 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 under - dose regions in the disagreed areas .
the desired dose map for a three - dimensional dose distribution evaluated in the current study is an array of 3 two - dimensional 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 two - and three - dimensional dosimetry .
 in order to evaluate two - dimensional 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 three - dimensional dose distribution evaluation tool , and includes the recognition of over - dose / under - dose areas . using a sign term 
 , the conventional gamma function separates the disagreed areas into two parts : over - dose and under - dose areas . 
 the new gamma function was modified using an extension of the dose difference criterion , d , from two dimensions into three dimensions . in order to provide two - dimensional 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 two - dimensional 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 under - dose regions . 
 in addition , a slight increase of the agreement percentage for reference and evaluated dose maps were observed by the extension of d to three dimensions . it is concluded that the modified method is more realistic and applicable for the evaluation of both two - dimensional and three - dimensional dose distributions . 
Introduction
Materials and Methods
Results and Discussion
Modification of the gamma function to enhance over- and under-dose regions
Modification of the gamma function as a three dimensional dose distribution assessment tool
Conclusion