Spinet-QSM: model-based deep learning with schatten p-norm regularization for improved quantitative susceptibility mapping

Quantitative susceptibility mapping (QSM) provides an estimate of the magnetic susceptibility of tissue using magnetic resonance (MR) phase measurements. The tissue magnetic susceptibility (source) from the measured magnetic field distribution/local tissue field (effect) inherent in the MR phase images is estimated by numerically solving the inverse source-effect problem. This study aims to develop an effective model-based deep-learning framework to solve the inverse problem of QSM.

This work proposes a Schatten \(\textit{p}\) $\mathit{p}$ -norm-driven model-based deep learning framework for QSM with a learnable norm parameter \(\textit{p}\) $\mathit{p}$ to adapt to the data. In contrast to other model-based architectures that enforce the l \(_{\text {2}}\) ${}_{\text{2}}$ -norm or l \(_{\text {1}}\) ${}_{\text{1}}$ -norm for the denoiser, the proposed approach can enforce any \(\textit{p}\) $\mathit{p}$ -norm ( \(\text {0}<\textit{p}\le \text {2}\) $\text{0}<\mathit{p}\le \text{2}$ ) on a trainable regulariser.

The proposed method was compared with deep learning-based approaches, such as QSMnet, and model-based deep learning approaches, such as learned proximal convolutional neural network (LPCNN). Reconstructions performed using 77 imaging volumes with different acquisition protocols and clinical conditions, such as hemorrhage and multiple sclerosis, showed that the proposed approach outperformed existing state-of-the-art methods by a significant margin in terms of quantitative merits.

The proposed SpiNet-QSM showed a consistent improvement of at least 5% in terms of the high-frequency error norm (HFEN) and normalized root mean squared error (NRMSE) over other QSM reconstruction methods with limited training data.