A study of Bayesian deep network uncertainty and its application to synthetic CT generation for MR‐only radiotherapy treatment planning
Authors: Law Max Wai‐Kong , Tse Mei‐Yan , Ho Leon Chin‐Chak , Lau Ka‐Ki , Wong Oi Lei , Yuan Jing , Cheung Kin Yin , Yu Siu Ki 
Published: September 2023

Background

The use of synthetic computed tomography (CT) for radiotherapy treatment planning has received considerable attention because of the absence of ionizing radiation and close spatial correspondence to source magnetic resonance (MR) images, which have excellent tissue contrast. However, in an MR‐only environment, little effort has been made to examine the quality of synthetic CT images without using the original CT images.

Purpose

To estimate synthetic CT quality without referring to original CT images, this study established the relationship between synthetic CT uncertainty and Bayesian uncertainty, and proposed a new Bayesian deep network for generating synthetic CT images and estimating synthetic CT uncertainty for MR‐only radiotherapy treatment planning.

Methods and Materials

A novel deep Bayesian network was formulated using probabilistic network weights. Two mathematical expressions were proposed to quantify the Bayesian uncertainty of the network and synthetic CT uncertainty, which was closely related to the mean absolute error (MAE) in Hounsfield Unit (HU) of synthetic CT. These uncertainties were examined to demonstrate the accuracy of representing the synthetic CT uncertainty using a Bayesian counterpart. We developed a hybrid Bayesian architecture and a new data normalization scheme, enabling the Bayesian network to generate both accurate synthetic CT and reliable uncertainty information when probabilistic weights were applied. The proposed method was evaluated in 59 patients (13/12/32/2 for training/validation/testing/uncertainty visualization) diagnosed with prostate cancer, who underwent same‐day pelvic CT‐ and MR‐acquisitions. To assess the relationship between Bayesian and synthetic CT uncertainties, linear and non‐linear correlation coefficients were calculated on per‐voxel, per‐tissue, and per‐patient bases. For accessing the accuracy of the CT number and dosimetric accuracy, the proposed method was compared with a commercially available atlas‐based method (MRCAT) and a U‐Net conditional‐generative adversarial network (UcGAN).

Results

The proposed model exhibited 44.33 MAE, outperforming UcGAN 52.51 and MRCAT 54.87. The gamma rate (2%/2 mm dose difference/distance to agreement) of the proposed model was 98.68%, comparable to that of UcGAN (98.60%) and MRCAT (98.56%). The per‐patient and per‐tissue linear correlation coefficients between the Bayesian and synthetic CT uncertainties ranged from 0.53 to 0.83, implying a moderate to strong linear correlation. Per‐voxel correlation coefficients varied from −0.13 to 0.67 depending on the regions‐of‐interest evaluated, indicating tissue‐dependent correlation. The R2 value for estimating MAE solely using Bayesian uncertainty was 0.98, suggesting that the uncertainty of the proposed model was an ideal candidate for predicting synthetic CT error, without referring to the original CT.

Conclusion

This study established a relationship between the Bayesian model uncertainty and synthetic CT uncertainty. A novel Bayesian deep network was proposed to generate a synthetic CT and estimate its uncertainty. Various metrics were used to thoroughly examine the relationship between the uncertainties of the proposed Bayesian model and the generated synthetic CT. Compared with existing approaches, the proposed model showed comparable CT number and dosimetric accuracies. The experiments showed that the proposed Bayesian model was capable of producing accurate synthetic CT, and was an effective indicator of the uncertainty and error associated with synthetic CT in MR‐only workflows.