A Comprehensive Framework for Uncertainty Quantification of Voxel‐Wise Supervised Deep Learning Models in IVIM MRI

Accurate estimation of intravoxel incoherent motion (IVIM) parameters from diffusion-weighted MRI remains challenging due to the ill-posed nature of the inverse problem and high sensitivity to noise, particularly in the perfusion compartment. In this work, we propose a probabilistic deep learning framework based on deep ensembles (DEs) of mixture density networks (MDNs), enabling estimation of total predictive uncertainty and decomposition into aleatoric (AU) and epistemic (EU) components. The method was benchmarked against nonprobabilistic neural networks, a Bayesian fitting approach, and a probabilistic network with single Gaussian parametrization. Supervised training was performed on synthetic data, and evaluation was conducted on both simulated and in vivo brain mouse dataset. The reliability of the quantified uncertainties was assessed using calibration curves, output distribution sharpness, and the continuous ranked probability score (CRPS). MDNs produced more calibrated and sharper predictive distributions for the diffusion coefficient ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>D</mml:mi></mml:mrow> <mml:annotation>$$ D $$</mml:annotation></mml:semantics> </mml:math> ) and the perfusion fraction ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>f</mml:mi></mml:mrow> <mml:annotation>$$ f $$</mml:annotation></mml:semantics> </mml:math> ) parameters, although slight overconfidence was observed in the pseudodiffusion coefficient ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow> <mml:mrow><mml:mo>∗</mml:mo></mml:mrow> </mml:msup> </mml:mrow> <mml:annotation>$$ {D}^{\ast } $$</mml:annotation></mml:semantics> </mml:math> ). The robust coefficient of variation (RCV) indicated smoother in vivo estimates for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msup><mml:mrow><mml:mi>D</mml:mi></mml:mrow> <mml:mrow><mml:mo>∗</mml:mo></mml:mrow> </mml:msup> </mml:mrow> <mml:annotation>$$ {D}^{\ast } $$</mml:annotation></mml:semantics> </mml:math> with MDNs compared with Gaussian model. Despite the training data covering the expected physiological range, elevated EU in vivo suggests a mismatch with real acquisition conditions, highlighting the importance of incorporating EU, which was allowed by DE. Overall, we present a comprehensive framework for IVIM fitting with uncertainty quantification, which enables the identification and interpretation of unreliable estimates. The proposed approach can also be adopted for fitting other physical models through appropriate architectural and simulation adjustments.