Fitting of Radial Dose Function for 192ir Brachytherapy Sources

This article presents an improved configuration for fitting the radial dose functions of sealed 192Ir brachytherapy sources. The proposed functions allow for precise fitting of radial dose function in the range of 0.5 20 cm. The maximum deviations between fitting data and Monte Carlo modeling data do not exceed 0.6%, while average deviations are in the order of 0.1%. The fitting coefficients for the proposed functions are specified for 11 sources from different manufacturers. The proposed functions could be used in treatment planning systems for more accurate calculation of the dose distributions around the sources using a wide range of data according to TG-43 dosimetry protocol.


Introduction
In order to calculate the absorbed dose distribution near the brachytherapy source many commercial treatment planning systems use functional dependencies obtained by fitting a set of values of the radial dose function. This dataset could be determined by experimental dosimetry studies or by computer simulation. The American Association of Physicists in Medicine (AAPM) Task Group 43 (TG-43) dosimetry protocol recommends using third to fifth order polynomials in the following form [1]. ( ) . g r a a r a r a r a r a r = + + + + + The fit coefficients are available in scientific publications for a large number of currently used sources. Although the accuracy of the fifth order polynomials is sufficient for fitting and interpolation of the radial dose function within the range 0.5 20 cm r cm ≤ ≤ he fitting quality decreases sharply beyond this range. Also, the behavior of the function is 'non-physical' outside this range. This is manifested by the large number of minima and maxima. Various authors have proposed many alternatives to polynomials, including a double exponential fit Meigooni [4] Even though all of the above functions allow for a more accurate fitting of data, yet they don't always accurately describe the behavior of the radial dose function at small distances from the source, which is especially crucial for sources with 125 I and 103 Pd radionuclides. Expression (2) has some physical sense if we assume that the spectrum of the photons emitted by the radionuclide, consists of two main lines, which is true for 60 Co. In order to better reflect the rapid change of the radial dose function at small distances from the source, expression (4) could be modified as follows:

r a r a a r a r a r e P
or in the form [5]: All expressions are compiled in such a way that the number of fitting parameters a 0 , a 1 ,.., would not exceed 7. In this research, we calculated the radial dose function for source BEBIGIr2.A95-2,

Materials and Methods
Two types of brachytherapy sources containing 192 Ir as an active substance are commercially available. The first type is characterized by a gradually increasing radial dose function which reaches its maximum at the distance of 1 cm from the source; whereas the second type is characterized by a dose decrease with distance at small values of r. Among the type one sources are BEBIG GI192M11, Flexisource, Gammamed 12i, Gammamed plus, Microselectron V1 and V2, SPEC M19, Vari Source classic and Vary Source VS200; the type two sources are BEBIG Ir2.A95-2 and Buchler G0814. Values of the radial dose functions for source BEBIG Ir2.A95-2 were obtained by the authors [6], and taken from publication [7]; data for all other sources is taken from the database of Carleton University [8]. In all cases, the nonlinear least-squares fitting was performed using OriginPro software. The fitting quality was assessed by two criteria: by the error of fit coefficients and by the average and maximum deviations of the fitting results from the tabulated data, according to the following expressions:

Results and Discussion
The best fitting results for the radial dose functions of the type one sources is provided by formulas (5) through (7). For the type two sources, the best quality fitting is obtained by formulas (6) and (7)( Table 1). Overall, fitting by formula (7) provides the most satisfactory results for all sources studied, because at equal quantity of parameters provides a smaller average derivations fitting data and a smaller error in parameter definitions. Fitting parameters for formulas (5) -up and (7) -down, are presented in Table 1. The maximum deviations between fitting data and Monte Carlo modeling data do not exceed 0.4% for all source, whereas average deviations are in the order of 0.1sssssss%. The proposed functions could be used in treatment planning systems for more accurate calculation of the dose distributions around the sources using a wide range of data according to TG-43 dosimetry protocol.

Conclusion
In this article, we propose fitting functions for the radial dose function of sealed high-dose 192 Ir brachytherapy sources. The proposed functions were used for fitting the data for 11 sources obtained by the authors using the Monte Carlo method as well as data available from the database of Carleton University. The proposed functions result in higher quality fitting compared to fifth order polynomial fits recommended by AAPM TG-43. These functions can be used at distances 0.5-20 cm from the source. The maximum deviations between fitting data and Monte Carlo modeling data do not exceed 0.6%, whereas average deviations are in the order of 0.1%. We believe that the proposed functions are excellent candidates for use in calculating dose distributions in treatment planning systems.