1 Introduction
Quantitative susceptibility mapping (QSM) is a recent phasebased quantitative magnetic resonance imaging (MRI) technique that enables in vivo quantification of magnetic susceptibility, a tissue parameter that is altered in variety of neurological disorders[1, 2]. QSM has been shown to quantify changes in vascular injury such as the formation of cerebral microbleeds over time, hemorrhage, and stoke [3, 4]. Iron deposition in the deep gray matter due to aging or disease can also be investigated using QSM[5]. In neurodegenerative diseases such as Parkinson’s disease[6], Alzheimer’s disease[7] and Huntington’s disease[8], QSM can quantify the paramagnetic iron deposition related to the progression of these diseases and potentially becomes a biomarker for diagnosis and managing neurodegenerative disease patients.
Although QSM has been demonstrated to have great potential in both research studies and clinical practice, accurate and reproducible quantification of tissue susceptibility requires multiple steps of careful data processing including phase reconstruction, coil combination (for multichannel coils)[9], multiecho phase combination (for multiecho sequences)[10], background phase removal[11, 12, 13] and phasesusceptibility dipole inversion[14, 12, 15]. Among them, the dipole inversion step is considered the most difficult because it is intrinsically an illposed inverse problem[16]. This is due to the representation of the relationship between magnetic field perturbation and susceptibility distribution as a convolution, which can be more efficiently calculated as a pointwise multiplication in frequency space except along the conical surface where zero values result in missing data or noise amplification when solving for the inverse .
To overcome this issue, we can sample the missing data by acquiring at least three scans with different relative orientations of the volumeofinterest in the main magnetic field and apply the algorithm of calculation of susceptibility through a multipleorientation sampling (COSMOS)[15]. This requires a subject to change the head orientation between the repeated scans, which has several disadvantages that has significantly limited its application in practice: 1) the scan time is prolonged since multiple repeated scans are required, increasing both the cost of QSM and the risk of motion artifact; 2) coregistration of the different orientation images are required, which both increases processing time and potentially introduces errors due to misalignment; and 3) modern highfield head coils are usually configured to be very close to the subject’s head for higher sensitivity, limiting the ability to rotate one’s head and degrading the quality of the QSM calculation. As a result, COSMOS is usually considered impractical for patient studies despite its superior potential to alleviate the illposed dipole inversion.
In the last decade, many algorithms have been proposed to get around this inverse problem. Thresholded Kspace Division (TKD) simply thresholds the dipole kernel to a predetermined nonzero value to avoid dividingbyzero problem[17]. Morphology Enabled Dipole Inversion (MEDI) regularized the illposed inversion problem by imposing edge preservation according to the information from the magnitude images[12]. Compressed Sensing Compensated inversion (CSC) adopts the observation that the missing kspace satisfies the compressed sensing requirement and regularizes the problem using sparse L1 norm[18]. QSIP approaches the problem by inversion of a perturbation model and makes use of tissue/air susceptibility atlas[19]
. These traditional methods have three major limitations: 1) they either suffer from significant streaking artifacts or require careful hyperparameter tuning process; 2) they are iterative and therefore can take hours to compute, greatly reducing their practical implementation; 3) they give vastly different susceptibility quantifications and reproducibility, making it difficult to compare studies that use different algorithms.
Recently, Deep Convolutional Neural Networks (DCNNs) have showed great potential in computer vision tasks such as image classification
[20], semantic segmentation[21] and object detection[22]. Among various deep neural network architecture, UNet[23] becomes the most popular backbone for many medical imagerelated problems[24, 25, 26] due to its effectiveness and universality. [27] and [28] adopted the UNet structure and extended it to 3D to solve the dipole inversion problem of QSM by training the network to learn the inversion using patches of various sizes as the input. Since its inception in 2014, Generative Adversarial Networks (GANs)[29] have been incorporated into CNNs to further improve performance of segmentation, classification, and especially contrast generation tasks[30, 31, 32, 33, 34, 35] by combining a generator that is trained to generate more realistic and accurate images with a discriminator who is trained to distinguish real from the generated images. In the method described in this study (QSMGAN) , we 1) modified the structure of the 3D UNet proposed by [27] and [28] to incorporate the physical indication into the model and 2) utilized the power of GAN to regularize the model training process to further improve the accuracy of QSM dipole inversion.2 Materials and Methods
2.1 Theory of QSM dipole inversion and generative adversarial networks
Assuming that the susceptibilityinduced magnetization is regarded as a magnetic dipole and the orientation of the main magnetic field is defined as the zaxis in the imaging Cartesian coordinate, the magnetic field perturbation and susceptibility distribution is related by a convolution, which can be efficiently calculated by a pointwise multiplication in frequency space[1].
(1) 
Where is the local field perturbation, is the main magnetic field, represents the tissue susceptibility,
is the frequency space vector and
is the zcomponent. In practice, we measure by phase variation and solve the inverse problem for the susceptibility distribution . However, notice that when , the bracket term on the righthand side becomes close to zero, which causes missing measurement or noise amplification when solving the inverse problem, making it illposed.Assume is the acquired tissue phase of the subject and is the susceptibility map of the subject we want to solve in the illposed phasesusceptibility dipole inversion problem, and function f represents the relationship between them, then we can simplify equation (1) with:
To solve the dipole inversion problem, we are finding a function that gives:
Where
is an estimate of the true susceptibility map
. The idea of GANs is to define a game between two competing components (networks): the discriminator (D) and the generator (G). G takes an input and generates a sample that D receives and tries to distinguish from a real sample. The goal of G is to “fool” D by generating more realistic samples. In this case, we use G as the function h:The adversarial game between G and D is a minimax objective:
where is the distribution of true susceptibility maps and is the distribution of tissue phases. To stabilize the training process, we adopt the method of Wasserstein GAN (WGAN)[30], and the value function for WGAN is:
(2) 
where is the set of 1Lipschitz functions, which can be enforced by adding a gradient penalty term to the value function[36]:
where is a parameter that controls the weight of the gradient penalty. Since the goal for G in this task is to recover/reconstruct QSM from a certain input tissue phase, we also included an L1 loss as content loss in the objective function of G:
where is the adversarial loss indicated in equation (2).
2.2 QSMGAN framework
We designed a 3D UNet like architecture as the generator part of the QSMGAN framework as shown in Figure 1
. In each UNet block, there are two 3x3x3 Conv3dBatchNormLeakyReLU (negative slope of 0.2) layers. 3D average pooling is used to downsample the image patch, while 3D transpose convolution is applied to restore the resolution in the upsampling path. At the end of the generator, we applied a cropping layer to focus the training on only the center part of the patch. For the discriminator part of the QSMGAN, we designed a 3D patchbased convolutional neural network where each block of the network is composed of a 3D convolution (4x4x4 kernel size and stride 2) and a LeakyReLU (negative slope of 0.2). The four blocks in the network lowers the input patch to 1/16 of the original size and the 3D convolution layer at the end converts the resulting patch to a binary output corresponding to the prediction of true and fake QSM patches.
2.3 Subjects and data acquisition
Eight healthy volunteers (average age 28, M/F=3/5) were recruited in this study as the training and validation dataset for QSMGAN. All volunteers were scanned with a 3D multiecho gradientrecalled sequence (4 Echoes, TE= 6/9.5/13/16.5ms, TR=50ms, FA=20, bandwidth=50kHz, 0.8mm isotropic resolution, FOV=24x24x15cm) using a 32channel phasearray coil on a 7T MRI scanner (GE Healthcare Technologies, Milwaukee, WI, USA). The sequence was repeated for three times on each volunteer with different head orientation (normal position, tilted forward and tilted left) to acquire data for COSMOS reconstruction. GRAPPAbased parallel imaging[37] with an acceleration factor of 3 and 16 autocalibration lines were also adopted to reduce the scan time of each orientation to about 17 minutes.
2.4 QSM data processing and dataset preparation
The raw kspace data were retrieved from the scanner and processed on a Linux workstation using inhouse software developed with Matlab 2015b (Mathworks Inc., Natick, MA, USA). The following processing steps (summarized in Figure 2
) were performed to obtain the tissue phase maps required for input to the QSMGAN and calculation for the gold standard COSMOSQSM which was used as the learning target of QSMGAN: 1) GRAPPA reconstruction was applied to interpolate the missing kspace lines due to parallel imaging acceleration and channelwise inverse Fourier transform was applied to obtain the coil magnitude and phase images; 2) coil images were combined to obtain robust echo magnitude and phase images using the MCPC3DS method
[10]; 3) raw phase was unwrapped using a Laplacianbased algorithm[38]; 4) FSL BET[39] was applied on magnitude images from all echoes to obtain a composite brain mask from the intersection of each individual echo mask; 5) VSHARP[18] was used to remove the background field phase to get the tissue phase map; 6) images from different orientations were coregistered using magnitude images with FSL FLIRT[39]; 7) the dipole field inversion was solved using the COSMOS algorithm[15]. In addition, TKD[17], MEDI[12] and iLSQR[14] QSM maps were also reconstructed from single orientation data for evaluation and comparison. A threshold of 0.15 was selected for TKD algorithm, and =2000 was used in MEDI.2.5 Training and validation
The 8 subjects were divided into 5 for training, 1 for validation, and 2 for testing. Scans with three orientations were all included in the dataset so the total number of scans in training/validation/test set are 15/3/6. To build the training set, tissue phase and susceptibility patches were sampled by center coordinates with a gap of 8 voxels in all three spatial dimensions. Since background occupies most of the image volume, we sampled 90% patches from inside the brain and only 10% from the background to increase the efficiency of the training. For validation and testing, the input tissue phase volume is divided into nonoverlapping patches according to the output patch size and the susceptibility map is reconstructed patchwise by feeding the input tissue phase patch to the trained network. Figure 3 demonstrates the relationship between receptive field and input/output patch size.
To assist the neural network training, we multiplied the input phase by a scale factor of 100 and the transform the output x by a scaled hyperbolic tangent operation to get the surrogate target :
This transform not only converts the range of the target susceptibility map to [1, 1], which aids in the network training, but also results in a more Gaussian distributed histogram, helping the network learn values in different ranges (Figure
4).As the baseline network, we first trained the UNet based generator separately with the pairs of input and output patch sizes listed in Table 1
. To train the generator, an Adam optimizer with a learning rate of 1e4 was used and betas were set to (0.5, 0.999). The network was trained for 40,000 iterations with a batch size of 16 that was lowered to 8 for larger input patch sizes. L1 loss was used as the loss function for the baseline network.
3D UNet Patch Size (inputoutput)  L1 error (1e3)  PSNR  NMSE 

3232  1.4900.184  42.251.01  0.3020.056 
4832  1.4030.204  43.071.22  0.2520.063 
6432  1.3160.230  43.391.37  0.2370.072 
9632  1.3190.216  43.381.32  0.2370.068 
4848  1.4240.195  42.581.13  0.2810.061 
6448  1.3090.210  43.531.31  0.2290.065 
9648  1.3100.212  43.371.28  0.2370.067 
12848  1.3110.215  43.401.31  0.2360.068 
6464  1.3890.211  42.871.21  0.2640.063 
9664  1.3160.207  43.461.28  0.2330.066 
12864  1.3220.211  43.321.27  0.2400.067 
To train the QSMGAN, we again started with the baseline network and then: 1) fixed the generator G and trained D for 20,000 iterations to ensure that D was well trained, as suggested by [36]; and 2) trained G and D together for 40,000 iterations. During each iteration, D (the critic) was updated 5 times with the gradient penalty . Adam optimizers were used for both G and D and the learning rate was lowered to 1e5. To balance the content loss and adversarial loss, was set to 1 and to 0.01.
2.6 Evaluation metrics
To evaluate the quality of the predicted QSM map reconstructed by the network (), we calculated and compared the following metrics: 1) L1 error = ; 2) Peak SignaltoNoise Ratio (PSNR) = , where computes the voxel value range of the input image and computes the mean squared error between the reconstructed image and target image; and 3) Normalized Mean Squared Error (NMSE) = .
3 Results
3.1 Baseline 3D UNet
We experimented with combinations of three different input patch sizes (323, 483, 643) and 5 output patch sizes (323, 483, 643, 963, 1283, with input > output) for the baseline 3D UNet. Figure 5 demonstrates example axial slices of the effects of different inputoutput size pairs while Table 1 lists the comparison of the quantitative metrics (L1, PSNR, NMSE) that evaluate the quality of the resulting QSM map. When input patch size was the same as the output patch size, inversion error increased towards the edge of the patch, resulting in visible discontinuities in a gridlike pattern in the reconstructed QSM map. The higher L1 error, lower PSNR and NMSE also supports this phenomenon quantitatively. When we increased the input patch size and applied center cropping at the end of the UNet as shown in Figure 3, the patch edge artifact reduced and the metrics improved. Among the different combinations of patch sizes, the input patch size of 643 and output patch size of 483 (6448) provided the best balance between sufficient accuracy of the UNet dipole inversion and low computation burden/efficiency. Therefore, for the QSMGAN evaluation we used the 6448 3D UNet as a basic building block.
3.2 Effectiveness of QSMGAN
Using the 6448 3D UNet as the generator, the added benefit of using QSMGAN over the 3D UNet is shown by the quantitative metrics listed in Table 2. Figure 6 demonstrates the visual comparison of reconstructed QSM of 3D UNet and QSMGAN, where the adversarial training further improved the quality of the reconstructed QSM map by reducing both residual blurring and the remaining edge discontinuity artifacts from the relatively smaller input patch size, providing a more accurate and detailed mapping of susceptibility compared to the 3D UNet baseline.
Methods  L1 error (1e3)  PSNR  NMSE 

TKD  2.8260.178  38.821.69  0.4960.076 
MEDI  2.9090.194  41.241.71  0.5390.059 
iLSQR  2.1930.227  42.031.45  0.4100.088 
3D UNet 6448  1.3090.210  43.531.31  0.2290.065 
QSMGAN 6448  1.2620.248  43.721.55  0.2210.078 
3.3 Comparison with nonlearningbased methods
Compared to 3 common ‘nonlearningbased’ QSM dipole inversion algorithms (TKD, MEDI and iLSQR), our QSMGAN approach had 4259% reductions in NMSE and L1 error in the test datasets while increasing PSNR by 413% as shown in Table 2. Figure 7 shows example QSM slices from the two test subjects generated from QSMGAN and the nonlearningbased algorithms. Although TKD had the lowest computational complexity, it also resulted in the most streaking artifacts. Despite its smoothe appearance, MEDI was the least uniform with relatively high L1 error and inaccurate contrast of some fine structures such as vessels. It also required the longest computation time of all of the methods (about 2 hours on a regular desktop workstation). Although iLSQR QSM had lower L1 error than TKD and MEDI, it was visually noisier than all other methods. QSMGAN not only resulted in the best L1 error, PSNR, and NMSE but achieved the most similar QSM map to COSMOS in only 2 srconds of reconstruction time per scan, the same order of time complexity as with TKD method.
4 Discussion
Although in theory, the phasesusceptibility relationship in QSM is global, meaning the tissue phase is determined by the susceptibility of all locations in the imaging volume, we still adopted a patchbased deep learning approach similar to
[28] for several reasons. Since the network is 3D, the patchbased method can significantly reduce the computation complexity and memory requirement compared to wholevolume based methods like that described in [27], especially when conducting highresolution QSM. For example, if we needed to generate a full QSM volume with 256x256x150 matrix size using the entire volume as an input to the 3D UNet architecture, even the most advanced GPU with 32GB graphics memory would not be able to hold the whole training sample. The patchbased method also converts one single scan into hundreds of input images, even before data augmentation. Since COSMOS takes a relatively long scan time and is cumbersome to conduct, training a more generalizable deep convolutional network is beneficial when only a limited amount of data is available. Because the phase is mostly determined by nearby susceptibilities due to the properties of the susceptibilityphase convolutional kernel, the patchbased approach a good approximate of the dipole inversion.As Table 1 demonstrates, increasing the input patch size and applying center cropping at the end of the 3D UNet significantly improved the quality of the reconstructed QSM maps. This can be intuitively described by Figure 3, where when the input patch size applied equaled the output patch, an output voxel near the center of the patch (Figure 3a) could receive information from the entire patch. However, a voxel near the edge of the output patch (Figure 3b) would only receive information from the orange region and a large portion of the phase information from the gray region would be missing, reducing the ability of the network to accurately solve for the susceptibility. When we increase the input patch size (Figure 3c) and crop the output patch such that only the center of the patch is considered a valid QSM prediction, voxels near the edge of the patch regain phase input information thereby increasing the accuracy of the quantified susceptibility values.
Another observation from Table 1 is that the medium output patch size (483) achieved the best QSM reconstruction performance. The smaller patch size (323) performed worse because the output voxels received less information, introducing more error to the patch approximation of global convolution. Unexpectedly, the larger patch size (643) didn’t provide any extra benefit to the dipole inversion either. This might be due to the fact that it introduces more variables into the computation process and increased the difficulty of training a good network for QSM reconstruction. In addition, for each output patch size, using excessively large input patches (such as 96 to 32) did not further reduce the error but slightly downgraded the QSM quality. This might be due to the increased information far from the output patch that could interfere with the dipole inversion.
A disadvantage of using an excessively large input patch size is the dramatically increased computational complexity and GPU memory requirement. Note that the network is threedimensional, the computational complexity and memory requirement of training the networks roughly increases with the input patch size by O(). The center cropping we applied to ensure a large enough receptive field, only exacerbated this problem, greatly reducing the efficiency of the prediction process. For example, if we increased the input patch size from 323 to 643, the training/prediction time and memory becomes 8x as long and only 1/8 of the computed patches are utilizied. Based on the observation that excessively large input patch sizes greatly increased the computation burden without improving the quality of the resulting QSM map, we selected the 4832 3D UNet as the base network to integrate with the GAN.
The rationale for the GAN training, which included adding a discriminator or “critic”, was to guide the generator (or the 3D UNet) to further refine its result so that it cannot be distinguished from a real COSMOS QSM patch. Although it took a long time (48 hours) to train the QSMGAN, once the training was finished, the discriminator was no longer needed. As a result, reconstruction or prediction of the QSM map for a new scan/subject from tissue phase only required one forward pass through the 3D UNet for each input patch, thereby resulting in a computation complexity that is identical to the 3D UNet baseline.
5 Conslusions
In this study, we implemented a 3D UNet deep convolutional neural network approach to improve the dipole inversion problem in quantitative susceptibility mapping reconstruction. To better approximate the global convolution property in the phasesusceptibility relationship though patchbased neural networks, we enlarged the input patch size and introduced center cropping to ensure an increased receptive field for all neural network outputs. This cropping technique provided significantly lower edge discontinuity artifacts and higher accuracy. Including a generative adversarial network based on the WGANGP technique further improved the stability of training process, the image quality, and accuracy of the susceptibility quantification. Compared the other traditional nonlearning dipole inversion algorithms such as TKD, MEDI and iLSQR, our proposed method could efficiently generate more accurate, COSMOSlike QSM maps from single orientation, backgroundfieldremoved, tissue phase images. Future directions will investigate the network’s ability to generalize to other scan parameters such as TE, TR, and image resolution as well as test the performance of the QSMGAN on patients with different pathologies.
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