SPIRiT-Diffusion: Self-Consistency Driven Diffusion Model for Accelerated MRI

TL;DR

This work introduces SPIRiT-Diffusion, a diffusion model for $k$-space interpolation that is driven by the SPIRiT self-consistency prior. By deriving the forward SDE from SPIRiT and incorporating coil redundancy into the diffusion term, the method achieves robust multi-coil MRI reconstructions without heavy reliance on precise coil sensitivity maps. The paper demonstrates superior performance over image-domain diffusion and several DL-based baselines on a 3D intracranial and carotid vessel wall dataset, including retrospective and prospective acquisitions with high acceleration factors, and discusses model-driven diffusion as a broader paradigm for physics-guided diffusion modeling. Limitations include substantial memory demands and slower sampling, with future work aimed at accelerating inference. Overall, SPIRiT-Diffusion offers a robust, physics-informed diffusion framework for accelerated multi-coil MRI with strong potential for clinical adoption.

Abstract

Diffusion models have emerged as a leading methodology for image generation and have proven successful in the realm of magnetic resonance imaging (MRI) reconstruction. However, existing reconstruction methods based on diffusion models are primarily formulated in the image domain, making the reconstruction quality susceptible to inaccuracies in coil sensitivity maps (CSMs). k-space interpolation methods can effectively address this issue but conventional diffusion models are not readily applicable in k-space interpolation. To overcome this challenge, we introduce a novel approach called SPIRiT-Diffusion, which is a diffusion model for k-space interpolation inspired by the iterative self-consistent SPIRiT method. Specifically, we utilize the iterative solver of the self-consistent term (i.e., k-space physical prior) in SPIRiT to formulate a novel stochastic differential equation (SDE) governing the diffusion process. Subsequently, k-space data can be interpolated by executing the diffusion process. This innovative approach highlights the optimization model's role in designing the SDE in diffusion models, enabling the diffusion process to align closely with the physics inherent in the optimization model, a concept referred to as model-driven diffusion. We evaluated the proposed SPIRiT-Diffusion method using a 3D joint intracranial and carotid vessel wall imaging dataset. The results convincingly demonstrate its superiority over image-domain reconstruction methods, achieving high reconstruction quality even at a substantial acceleration rate of 10.

Figures (10)

• Figure 1: The SPIRiT-Diffusion framework. (a) Forward process: Self-consistent noise at different scales is gradually injected into the multi-coil images. (b) Reverse process: From time $t$ to $t-1$, first, transform $\mathbf{x}(t)$ to the $k$-space domain, execute one iteration about the self-consistency term and the learned probability density prior term (i.e., score function), and finally transform the $k$-space data back to the image domain. By iteratively executing the reverse process, missing data in $k$-space are gradually filled in. It is important to note that the image related to self-interpolation of $k$-space data is referenced from lustig2010spirit.
• Figure 2: Training Flowchart: Generate $\mathbf{x}(t)$ by adding self-consistent noise $\mathbf{z}$ to $\mathbf{x}(0)$ through a forward SDE, where $\mathbf{w}_t$ represents random noise at time $t$ and $\mathbf{S}$ denotes coil sensitivity. Feed $\mathbf{x}(t)$ into the network $\mathbf{s}_{\boldsymbol{\theta}}$ with $\mathbf{z}$ as the label, and use the network's output in conjunction with $\mathbf{z}$ to compute the training loss Eq. \ref{['estimate score']}.
• Figure 3: Reconstruction results of VWI data at absolutely R = 7.6 The top row shows the ground truth and the reconstructions obtained using different methods. The second row shows an enlarged view of the ROI, and the third row displays the error map of the reconstructions.
• Figure 4: Reconstruction results of VWI data at R = 10. The top row shows the ground truth and the reconstructions obtained using different methods. The second row shows an enlarged view of the ROI, and the third row displays the error map of the reconstructions.
• Figure 5: Phase reconstruction results of VWI data at R= 7.6. The top row shows the ground truth and the reconstructions obtained using different methods. The second row shows an enlarged view of the ROI, and the third row displays the error map of the reconstructions.