Error Bound Analysis of Physics-Informed Neural Networks-Driven T2 Quantification in Cardiac Magnetic Resonance Imaging

Physics-Informed Neural Networks (PINN) are emerging as a promising approach for quantitative parameter estimation of Magnetic Resonance Imaging (MRI). While existing deep learning methods can provide an accurate quantitative estimation of the T2 parameter, they still require large amounts of training data and lack theoretical support and a recognized gold standard. Thus, given the absence of PINN-based approaches for T2 estimation, we propose embedding the fundamental physics of MRI, the Bloch equation, in the loss of PINN, which is solely based on target scan data and does not require a pre-defined training database. Furthermore, by deriving rigorous upper bounds for both the T2 estimation error and the generalization error of the Bloch equation solution, we establish a theoretical foundation for evaluating the PINN's quantitative accuracy. Even without access to the ground truth or a gold standard, this theory enables us to estimate the error with respect to the real quantitative parameter T2. The accuracy of T2 mapping and the validity of the theoretical analysis are demonstrated on a numerical cardiac model and a water phantom, where our method exhibits excellent quantitative precision in the myocardial T2 range. Clinical applicability is confirmed in 94 acute myocardial infarction (AMI) patients, achieving low-error quantitative T2 estimation under the theoretical error bound, highlighting the robustness and potential of PINN.

Convex Dual Theory Analysis of Two-Layer Convolutional Neural Networks with Soft-Thresholding

Soft-thresholding has been widely used in neural networks. Its basic network structure is a two-layer convolution neural network with soft-thresholding. Due to the network's nature of nonlinearity and nonconvexity, the training process heavily depends on an appropriate initialization of network parameters, resulting in the difficulty of obtaining a globally optimal solution. To address this issue, a convex dual network is designed here. We theoretically analyze the network convexity and numerically confirm that the strong duality holds. This conclusion is further verified in the linear fitting and denoising experiments. This work provides a new way to convexify soft-thresholding neural networks.