Calibrated uncertainty estimation for interpretable proton computed tomography image correction using Bayesian deep learning

Integrated-type proton computed tomography (pCT) measures proton stopping power ratio (SPR) images for proton therapy treatment planning, but its image quality is degraded due to noise and scatter. Although several correction methods have been proposed, techniques that include estimation of uncertainty are limited. This study proposes a novel uncertainty-aware pCT image correction method using a Bayesian convolutional neural network (BCNN). A DenseNet-based BCNN was constructed to predict both a corrected SPR image and its uncertainty from a noisy SPR image. A total 432 noisy SPR images of 6 non-anthropomorphic and 3 head phantoms were collected with Monte Carlo simulations, while true noise-free images were calculated with known geometric and chemical components. Heteroscedastic loss and deep ensemble techniques were performed to estimate aleatoric and epistemic uncertainties by training 25 unique BCNN models. 200-epoch end-to-end training was performed for each model independently. Feasibility of the predicted uncertainty was demonstrated after applying two post-hoc calibrations and calculating spot-specific path length uncertainty distribution. For evaluation, accuracy of head SPR images and water-equivalent thickness (WET) corrected by the trained BCNN models was compared with a conventional method and non-Bayesian CNN model. BCNN-corrected SPR images represent noise-free images with high accuracy. Mean absolute error in test data was improved from 0.263 for uncorrected images to 0.0538 for BCNN-corrected images. Moreover, the calibrated uncertainty represents accurate confidence levels, and the BCNN-corrected calibrated WET was more accurate than non-Bayesian CNN with high statistical significance. Computation time for calculating one image and its uncertainties with 25 BCNN models is 0.7 s with a consumer grade GPU. Our model is able to predict accurate pCT images as well as two types of uncertainty. These uncertainties will be useful to identify potential cause of SPR errors and develop a spot-specific range margin criterion, toward elaboration of uncertainty-guided proton therapy.