A filtering approach based on Gaussian–powerlaw convolutions for local PET verification of proton radiotherapy

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Abstract
Because proton beams activate positron emitters in patients, positron emission tomography (PET) has the potential to play a unique role in the in vivo verification of proton radiotherapy. Unfortunately, the PET image is not directly proportional to the delivered radiation dose distribution. Current treatment verification strategies using PET therefore compare the actual PET image with full-blown Monte Carlo simulations of the PET signal. In this paper, we describe a simpler and more direct way to reconstruct the expected PET signal from the local radiation dose distribution near the distal fall-off region, which is calculated by the treatment planning programme. Under reasonable assumptions, the PET image can be described as a convolution of the dose distribution with a filter function. We develop a formalism to derive the filter function analytically. The main concept is the introduction of 'Q' functions defined as the convolution of a Gaussian with a powerlaw function. Special Q functions are the Gaussian itself and the error function. The convolution of two Q functions is another Q function. By fitting elementary dose distributions and their corresponding PET signals with Q functions, we derive the Q function approximation of the filter. The new filtering method has been validated through comparisons with Monte Carlo calculations and, in one case, with measured data. While the basic concept is developed under idealized conditions assuming that the absorbing medium is homogeneous near the distal fall-off region, a generalization to inhomogeneous situations is also described. As a result, the method can determine the distal fall-off region of the PET signal, and consequently the range of the proton beam, with millimetre accuracy. Quantification of the produced activity is possible. In conclusion, the PET activity resulting from a proton beam treatment can be determined by locally filtering the dose distribution as obtained from the treatment planning system. The filter function can be calculated analytically using convolutions of Gaussians and powerlaw functions.

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