Unified Bayesian network for uncertainty quantification of physiological parameters in dynamic contrast enhanced (DCE) MRI of the liver

Abstract

Objective. Physiological parameter estimation is affected by intrinsic ambiguity in the data such as noise and model inaccuracies. The aim of this work is to provide a deep learning framework for accurate parameter and uncertainty estimates for DCE-MRI in the liver. Approach. Concentration time curves are simulated to train a Bayesian neural network (BNN). Training of the BNN involves minimization of a loss function that jointly minimizes the aleatoric and epistemic uncertainties. Uncertainty estimation is evaluated for different noise levels and for different out of distribution (OD) cases, i.e. where the data during inference differs strongly to the data during training. The accuracy of parameter estimates are compared to a nonlinear least squares (NLLS) fitting in numerical simulations and in vivo data of a patient suffering from hepatic tumor lesions. Main results. BNN achieved lower root-mean-squared-errors (RMSE) than the NLLS for the simulated data. RMSE of BNN was on overage of all noise levels lower by 33% ± 1.9% for k trans, 22% ± 6% for v e and 89% ± 5% for v p than the NLLS. The aleatoric uncertainties of the parameters increased with increasing noise level, whereas the epistemic uncertainty increased when a BNN was evaluated with OD data. For the in vivo data, more robust parameter estimations were obtained by the BNN than the NLLS fit. In addition, the differences between estimated parameters for healthy and tumor regions-of-interest were significant (p < 0.0001). Significance. The proposed framework allowed for accurate parameter estimates for quantitative DCE-MRI. In addition, the BNN provided uncertainty estimates which highlighted cases of high noise and in which the training data did not match the data during inference. This is important for clinical application because it would indicate cases in which the trained model is inadequate and additional training with an adapted training data set is required.