Introduction

Quantitative susceptibility mapping (QSM) is frequently used in the diagnosis of neurological diseases¹. However, conventional QSM quantification required background field removal and dipole inversion steps^2,3. Single-step QSM quantification methods, such as SS-L2-QSM⁴ and total field inversion (TFI)⁵, were developed to reduce error propagation in the two-step approach. Nevertheless, conventional single-step QSM methods had many parameters to tune, which might limit their applications. Recently, deep learning-based methods solved ill-posed MRI reconstruction and quantification problems, achieving superior performance compared with the conventional regularizations. A single-step deep learning model, autoQSM, generated QSM from the total field map directly using naïve feedforward model⁶. Additionally, the LPCNN⁷ and MoDL-QSM⁸ combined the fidelity term with a proximal network for dipole inversion. However, no deep learning model with data fidelity term was used in single-step QSM quantification. In this study, we proposed a deep-learning-regularized single-step QSM quantification, named SS-POCSnet. We evaluated the model performance on different datasets with varied matrix sizes and resolutions.

Methods

We firstly trained a deep-learning-regularized dipole inversion network, named POCSnet. Then we deployed the deep learning regularization in a neural network called single-step POCSnet (SS-POCSnet). SS-POCSnet regularized both the VSHARP² and dipole inversion terms, mitigating the tissue phase and susceptibility underestimation. In POCSnet, the closed-form solution of susceptibility map was iteratively calculated as follows:
χ =(F^HD^HDF+λ₁I)^-1F^HDF𝝋+ λ₁(F^HD^HDF+λ₁I)^-1χ_net (1)
where 𝝋 is phase map, D the dipole model in Fourier domain, and F the Fourier transform, χ_net the neural network (i.e., V-Net here) output, and ƛ₁ the regularization parameter. (Figure 1a).In SS-POCSnet, the minimization problem for VSHARP² with deep-learning regularization is
𝝋_local =argmin𝝋_local½(M_iF^HC_iF(𝝋_local-𝝋_total))²+λ₂/2(𝝋_local-𝝋_net)² (2)
where M the eroded brain mask, C the SMV kernel in Fourier domain, ƛ₂ the regularization parameter, i index for varied radii SMV kernels in VSHARP method, and tissue phase from the neural network 𝝋_net can be derived as below.
𝝋_net=F^HDFχ_net (3)
Eq.2 is solved using a conjugate gradient algorithm (CG), and the solution is fed into Eq. 1 to update χ_k. The parameters are ƛ₁ = ƛ₂ = 0.1, max iterations = 5, number of CG = 3. SMV kernels of 1, 5, 12 mm were used (Figure 1b).

For training POCSnet, 950 paired susceptibility maps and phase maps in matrix size 64×64×64 and resolution 0.86×0.86×1.00 mm³/pixel were simulated. For model testing, one COSMOS data (matrix size = 160×160×160, resolution = 1.06×1.06×1.06 mm³/pixel) from the 2016 QSM reconstruction challenge⁹, and three patient datasets (QSM acquisition: matrix size = 704 × 704 × 135, nominal resolution = 0.3267 × 0.3267 × 1.0 mm³/pixel; T1w image: matrix size = 800 × 800 × 25, and resolutions = 0.29 × 0.29 × 5.5 mm³/pixel and T2w-FLAIR: matrix size = 512 × 512 × 330, and resolution = 0.49 × 0.49 × 0.56 mm³/pixel) were applied. In addition, SS-L2-QSM⁴ and autoQSM⁶ were configured with reasonable parameters for comparison. And ROIs in deep grey matters of COSMOS were segmented for model analysis.

The NRMSE was used for training POCSnet. Other parameters are ‘ADAM’ optimizer with beta = (0.9, 0.99), number of epochs = 198, and learning rate = 2 × 10^-5. As data augmentation, adjacent batches were randomly scaled and summed during model training. For model inference, the pre-trained POCSnet model was applied to SS-POCSnet directly, without retraining the model. TensorFlow v2.2.0 and Keras v2.4.3 were used on a GPU server equipped with four NVIDIA 3090 Ti GPUs, 16 GB RAM, and 8-core Intel Xeon CPU. It took around 8 hours to train POCSnet on one GPU.

Results

SS-POCSnet had the best quantitative results among methods evaluated (in Table 1). VSHARP+SS-POCSnet and SS-POCSnet were both able to delineate detailed anatomical structures. In Figure 2, the image profile of SS-POCSnet was closer to that of COSMOS compared with the profiles of other methods and the underestimations were reduced in results from SS-POCSnet. Figures 3 compared QSM maps generated by these methods, from patients with CMBs and calcifications. SS-POCSnet provided better image contrasts with generally reduced residual background and underestimations. Moreover, the CMBs were expected to be hyperintensities on QSM and two calcification balls were not visible in results from autoQSM. Finally, MS lesions were indicated by arrows and more detectable on results from SS-POCSnet.

Conclusion

Deep-learning-regularized single-step QSM quantification can mitigate underestimating susceptibility values in deep brain nuclei and generate a QSM volume in ~40 seconds.

Acknowledgements

This work is supported by Hong Kong Health and Medical Research Fund 07182706 and 06172916.