The purpose of this note is to evaluate the relationship between the stochastic errors in CT numbers and the standard deviation of the computed proton beam range in radiotherapy planning. The stochastic voxel-to-voxel variation in CT numbers called 'noise,' may be due to signal registration, processing and numerical image reconstruction technique. Noise in CT images may cause a deviation in the computed proton range from the physical proton range, even assuming that the error due to CT number-stopping power calibration is removed. To obtain the probability density function (PDF) of the computed proton range, we have used the continuing slowing down approximation (CSDA) and the uncorrelated white Gaussian noise along the proton path. The model of white noise was accepted because for the slice-based fan-beam CT scanner; the power-spectrum properties apply only to the axial (x, y) domain and the noise is uncorrelated in the z domain. However, the possible influence of the noise power spectrum on the standard deviation of the range should be investigated in the future. A random number generator was utilized for noise simulation and this procedure was iteratively repeated to obtain convergence of range PDF, which approached a Gaussian distribution. We showed that the standard deviation of the range, sigma, increases linearly with the initial proton energy, computational grid size and standard deviation of the voxel values. The 95% confidence interval width of the range PDF, which is defined as 4sigma, may reach 0.6 cm for the initial proton energy of 200 MeV, computational grid 0.25 cm and 5% standard deviation of CT voxel values. Our results show that the range uncertainty due to random errors in CT numbers may be significant and comparable to the uncertainties due to calibration of CT numbers.

Purpose: To evaluate the uncertainty of computed proton range in radiotherapy treatment planning which is attributed to random component in CT numbers. Method and Materials: We utilize a random number generator to simulate a white Gaussian noise in CT numbers along the proton pathlength. The proton range is computed using continuous slowing down approximation which is valid for most of proton range. To simulate the statistical straggling of computed proton range, this procedure is iteratively repeated to obtain convergence of proton range PDF which is approaching a Gaussian. The FWHM (full‐width at half maximum) of the range PDF is used as a measure of uncertainty. Results: We investigate parameters which affect the proton range uncertainty in the presence of CT image noise. These parameters may include 1) initial proton energy, 2) noise period and 3) noise amplitude. The FWHM of range PDF increases linearly with the noise period. These results indicate that low frequency fluctuations in CT image noise can significantly increase the range uncertainty. We have also computed the range PDF as a function of initial proton energy. The FWHM of range PDF increases linearly with the initial proton energy. For the maximum proton energy of 250 MeV, the FWHM of proton range PDF can achieve a value of 5 mm in the presence of CT image noise. We note that the ratio FWHM/range increases as the proton range decreases; therefore, the relative range uncertainty is larger for smaller ranges. Conclusions: Range uncertainties due to CT image noise can be significant and comparable to the uncertainties attributed to the calibration of CT numbers. The relative range uncertainty increases as the range decreases. Noise reduction in CT images using smoothing and denoising algorithms can be recommended to reduce the standard deviation of range PDF.