Purpose: Proton pencil-beam energy spectrum is an essential parameter for calculations of dose and linear energy transfer(LET). We extract energy spectrum from measured integral depth dose(IDD) by decomposing the measured IDD into IDDs of mono-energetic protons. The decomposition is unique if the mono-energetic IDDs are linearly independent, but this condition may not be satisfied especially at low energy when the proton range < the step size that is used in generating the mono-energetic IDDs. In this work, we examined the condition of linear independence and improved the uniqueness.
Methods: Spectra with single and double peaked Gaussian distribution were used to simulate multi-energetic proton beams. The IDD of the beam was synthesized from mono-energetic IDDs with its energy spectrum. The energy interval of mono-energetic IDDs is 0.05 MeV so that the calculation error of LET will not be larger than 0.5 keV/μm. Optimal decomposition algorithms were performed on the synthesized IDD. The uniqueness of the extracted energy spectrum was examined by the difference of the extracted energy spectrum from the original. To improve the uniqueness and meanwhile save the memory and computation, we used multiple resolution mono-energetic IDDs. The step sizes of the IDDs of mono-energetic pencil beams are 1 and 0.1 mm.
Results: The difference of the extracted and original Gaussian energy spectrum peaked at 75 and 80 MeV was <1%. The difference increased as the energy deceased. With 1 mm step size of IDDs, we obtained well extracted Gaussian spectrum peaked at and larger than 30 MeV, and with 0.1 mm step size of IDDs, peaked at and larger than 20 MeV.
Conclusion: High resolution mono-energetic IDDs are necessary to extract low energy spectrum from measured IDDs. Use of energy interval larger than 0.05 MeV in the mono-energetic IDDs will be investigated to optimize the calculated dose and LET.