On the possibility of determining an effective energy spectrum of clinical electron beams from percentage depth dose (PDD) data of broad beams

Abstract

Experiments have already shown that obvious differences exist between the dose distribution of electron beams of a clinical accelerator in a water phantom and the dose distribution of monoenergetic electrons of nominal energy of the clinical accelerator in water, because the electron beams which reach the water surface travelling through the collimation system of the accelerator are no longer monoenergetic. It is evident that, while calculating precisely the dose distribution of any incident electron beams, the energy spectrum of the incident electron beam must be taken into consideration. In this note we shall present a method for determining an effective energy spectrum of clinical electron beams from PDD data (percentage depth dose data). It is well known that there is an integral equation of the first kind which links the energy spectrum of an incident electron beam with PDD through the dose distribution of monoenergetic electrons in the medium, as a kernel function in the integral equation. In this note, the integral equation of the first kind will be solved by using the regularization method. The bipartition model of electron transport will be used to calculate the kernel function, namely the energy deposition due to monoenergetic electron beams in the medium.

Tikhonov regularization modified as improved method for determining electron energy spectra from percentage depth dose curves data of broad beams

“…It is well known that Tikhonov regularization method can solve this type of problems. Its basic purpose is put additional information about the desired solution in order to stabilize the problem and to single out a useful and stable solution 1,9,10 .…”

Angular correction in reconstruction of electron spectra from depth dose distributions

“…These procedures are based on the depth-dose data that assumes the electrons were at normal incidence to the surface of the absorbing medium. [2][3][4][5] This assumption is not a good one for clinical electron beams and so the electron spectra produced by a reconstruction procedure invoking this assumption will inherently contain a systematic error. Depthdose distributions from clinical electron beams are most affected in the shallow depth region by incident electrons having a large angular distribution.…”

Reconstruction of electron spectra using singular component decomposition

“…Therefore, many investigators have tried inverse reconstruction to derive electron beam parameters from the measured data such as the central axis depth dose curves which are already included into the standard commissioning procedure for medical accelerators. [3][4][5][6][7][8] There are several advantages to using inverse reconstruction.…”

“…In this article, we refer to this method as a spectral weight technique. The spectral weight technique with different methods for solving the integral Fredholm equation of the first kind was used by Zhengming and Jette, 3 Faddegon and Blevis, 4 and Deng et al 5 The spectral weight technique is universal because it is suitable for simulation of any complicated form of energy spectra. However, it inevitably will have problems with the steep gradients and sharp peaks in an energy spectrum.…”

“…The importance of the regularization techniques to correct for the ill-conditioning property was pointed out by Zhengming and Jette. 3 An alternative approach to the reconstruction of electron spectra was used by Kawrakow et al 7 In this approach, parameters of the analytic trial functions are sought instead of the multiple spectral weights. The form of the trial functions can be easily selected because the general form of electron spectra of medical accelerators is well known.…”