Introduction

QSM is a valuable MRI post-processing technique, which extracts magnetic susceptibility distributions from GRE phase images. However, QSM reconstruction has long been challenging due to the requisite ill-posed dipole inversion. Deep learning frameworks are emerging as alternative methods for solving ill-conditioned inverse problems in medical imaging, including QSM^1,2. Two methods, DeepQSM¹ and QSMnet², have successfully trained the U-net³ for QSM using synthetic phantoms and in vivo COSMOS maps. However, susceptibility contrast loss was observed in iron-rich deep grey matter regions. In this study, we propose an enhanced U-net framework, named xQSM, via replacing traditional convolution with the recently proposed Octave Convolution (OctConv)⁴ to recover the susceptibility contrast loss in the original U-net based methods.

Methods

xQSM deep neural network
The xQSM framework is shown in Figure 1. As illustrated in the middle row, OctConv⁴ explicitly factorises the input feature maps into high and low resolution groups {X_H , X_L} and introduces an “X”-shaped communication between the feature maps. We also modified the original OctConv⁴ by replacing the nearest-neighbour interpolation with a transposed convolution to allow more learnable parameters. The modified OctConv operation is formulated as:

Ƒ_HH = Conv_HH(X_H) , Ƒ_HL = AvgPool(Conv_HL(X_H))
Ƒ_LH =ConvT(Conv_LH(X_L)) , Ƒ_LL = Conv_LL(X_L)

where Conv(·) represents convolution operations with subscripts representing different kernels; ConvT(·) represents transposed convolution of kernel size 2, which doubles the resolution of feature maps; AvgPool(·) is the average pooling operation of stride 2, which halves the resolution of feature maps.
The xQSM is built on a backbone U-net and comprises 2 OctConv layers, 2 max pooling layers, 2 transposed convolution layers, 1 final convolutional layers, 12 batch normalization layers, and 2 concatenations, as shown in Figure 1 bottom row. A total of 15,000 cropped small patches (size: 48³) were generated from 96 in vivo QSM subjects to train both xQSM and the backbone U-net. All network parameters were initialised with normally distributed random numbers of zero mean and 0.01 standard deviation. Adaptive moment estimation was used to optimise the network. It took 6-8 hours (i.e., 40 epochs) to complete the network training on two Tesla V100 GPUs using MATLAB, with the mini-batch size of 32 and mean squared error as the loss function.

Validation with simulated and in vivo datasets
Simulated magnetic field maps were generated, by a forward calculation, from a 3D Shepp-Logan phantom (susceptibilities of 0, 0.2, 0.3, and 1 ppm for various ellipsoids) and a COSMOS acquisition (1 mm isotropic) at 7 T. In vivo local field maps post-processed from a healthy subject at 7 T (0.6 mm isotropic) and another 3 healthy subjects at 3 T (1 mm isotropic) were also obtained. The proposed xQSM was compared with the original U-net and conventional dipole inversion methods (iLSQR⁵ and MEDI⁶). Local field maps from a patient with multiple sclerosis at 3 T (1 mm isotropic) and a mouse brain at 9.4 T (0.1 mm isotropic) were also tested for the generalization of the xQSM network.

Results

Figure 2 compares xQSM and U-net results on the 3D Shepp-Logan phantom. Although both networks were trained with in vivo brain images, they were able to recover QSM distribution of the simplified geometrical shapes. This suggests that the deep networks have learned the underpinning dipole inversion process instead of merely memorising the features of training sets. Both the error maps and the ROI measurements in the bar chart showed that our xQSM incorporating OctConv surpassed the backbone U-net with conventional convolution.

Simulation results from COSMOS are shown in Figure 3. The proposed xQSM method achieved similar accuracies (PSNR and SSIM) as conventional iLSQR and MEDI methods, while the original U-net substantially underestimated susceptibilities of deep grey matter (orange arrows), especially Globus Pallidus. The improved susceptibility contrast from U-net to xQSM is also confirmed by the ROI measurements in the bar graph.

QSM results from an in vivo local field map (0.6 mm isotropic) acquired at 7 T are shown in Figure 4 top two rows. The proposed xQSM successfully recovered the susceptibility contrast loss as appeared in the original U-net. Susceptibility measurements of deep grey matter from 4 healthy subjects are reported in the bar graph. The xQSM measurements are consistent with conventional iLSQR and MEDI methods, while U-net significantly underestimated globus pallidus (P = 0.018) and caudate (P = 0.034) susceptibilities.

Figure 5 demonstrates that xQSM can detect the white matter lesions in a multiple sclerosis patient (red arrows), and can also reconstruct mouse brain QSM with a much higher spatial resolution (100 microns), which suggests the general capability and robustness of the method.

Discussion and conclusion

The suppressed susceptibility contrast in deep grey matter from the original U-net may be due to the truncation of dipole field in the cropped training patches. The state-of-the-art OctConv is specifically designed for such multi-resolution tasks via focusing on multi-resolution/scale learning ability of deep networks. In this work, we proposed an xQSM network implementing OctConv to reduce the redundancy of the network parameters and enhance the multi-scale learning ability. The xQSM led to high-quality QSM reconstructions with improved deep grey matter susceptibility contrast than the original U-net in both simulation and in vivo experiments.

Acknowledgements

No acknowledgement found.