Quantitative susceptibility mapping (QSM) (de Rochefort et al., 2010) is an image contrast in magnetic resonance imaging (MRI) to measure the underlying tissue apparent magnetic susceptibility, which enables quantification of specific biomarkers such as iron, calcium and gadolinium (Wang and Liu, 2015). The forward model of QSM in three dimensional image space is:

where $\chi$ is the tissue susceptibility, $b$ is the measured local magnetic field, $d$ is the spatial dipole convolution kernel, and $n$ is the measurement noise. Dipole convolution can also be defined in k-space (Fourier space) as follows:

where $F$ is the Fourier transform operator and $D$ is the point-wise multiplication operator with the dipole kernel in k-space. The k-space formulation is more computationally efficient because of the fast Fourier transform, so Eq. 2 is often implemented in practice. The standard deviation (SD) of the Gaussion noise $n$ is obtained by computing the variance of the least squares fit of the magnetic field $b$ from the acquired multi-echo data (Liu et al., 2013). The problem is to recover $\chi$ from $b$ due to the ill-posedness of the dipole kernel in QSM (Wang and Liu, 2015; Kee et al., 2017).

Two representative methods have been proposed to solve the QSM inverse problem. The first one is called COSMOS (calculation of susceptibility through multiple orientation sampling) (Liu et al., 2009). COSMOS relies on multiple orientation scans to calculate the susceptibility map with high accuracy. As a result, it has been used as the gold standard reference when developing new QSM algorithms. However, the drawback of COSMOS is that it requires at least three orientation scans, which is infeasible for clinical use. Another method called MEDI (morphology enabled dipole inversion) (Liu et al., 2011b) was proposed to solve the QSM problem with a single orientation scan. MEDI uses a morphology-related regularization term and solves the following optimization problem:

where $W$ is derived from the observation noise covariance matrix, $\lambda$ is the tunable parameter of weighted total variation (TV) regularization (Rudin et al., 1992) with binary weighting matrix $M$ of susceptibility’s spatial gradients which only penalizes regions outside the brain tissue edges in order to suppress image-space artifacts introduced by dipole inversion (Liu et al., 2011b). Numerical optimization algorithms for solving Eq. 3 are reviewed by Kee et al. (2017). With efficient numerical solvers, MEDI generates reasonable susceptibility maps compared to COSMOS as a reference (Liu et al., 2011b) and requires only single orientation scan. As a result, MEDI has been used for clinical applications in the last decade (Wang et al., 2017; Chen et al., 2014).

From the Bayesian point of view, Eq. 3 belongs to the maximum a posteriori probability (MAP) estimation problem with the likelihood distribution defined as a multivariate Gaussian:

where $n\sim\mathcal{N}(0,\Sigma_{b|\chi})$ with $\Sigma_{b|\chi}$ diagonal, and the prior distribution defined as the Laplace of the spatial gradient:

Based on Bayes’s rule, the full posterior distribution $p(\chi|b)$ given the measured local field $b$ can also be estimated in principle, which will quantify the uncertainty in the solutions delivered and may have some clinical implications. In this paper, motivated by the posterior distribution estimation problem in QSM and advances in deep learning based density estimation techniques, we introduce a set of neural network parameterized distributions to learn an approximate posterior distribution of susceptibility $\chi$ for any given $b$ with an adaptive training strategy. We validate our method on both healthy subjects and patients and show good performance of the proposed method. This paper is extended from previously published work (Zhang et al., 2020b) at MIDL 2020. The additions include a detailed methodology section, comparisons to PDI-VI0 as another baseline in Figures 2-4 and Table 1, an experiment on multiple sclerosis patients in Figure 3, amortized versus subject-specific variational inference in Figure 5 and 6, uncertainty estimation evaluation in Figure 7, and the discussion section.

In recent years, posterior distribution estimation in imaging inverse problems has become a new topic in medical imaging (Repetti et al., 2019; Chappell et al., 2009), in which variance of a random variable is provided from posterior distribution to measure the uncertainty of the solution. However, posterior distribution estimation requires a complicated or even intractable integral from Bayes formula. Therefore, approximate inference methods are used to reduce the computational cost and intractability of the problem. Markov chain Monte Carlo (MCMC) (Andrieu et al., 2003) and variational inference (VI) (Bishop, 2006) are two classes of approximate inference approaches to the Bayesian estimation problem. In MCMC, Markov chain based sampling methods are used to generate random samples from the true posterior distribution in order to represent an empirical distribution which resembles the true distribution. MCMC is general in that it is able to achieve the exact inference given infinite computational time. However, in imaging inverse problems, the computational cost of MCMC for Bayesian estimation is often several magnitudes higher than that of the optimization method of MAP estimation, because of the curse of dimensionality (Pereyra, 2017). In addition, convergence of the Markov chain is hard to diagnose, raising concerns on the quality of the samples.

An alternative approach is to use VI, in which an approximate distribution is proposed with tractable function form and unknown parameters, and an optimization algorithm is used (for example, expectation-maximization (EM) algorithm (Blei et al., 2017)) to learn these parameters by minimizing the divergence between the true and approximate posterior distributions. After convergence, the approximate posterior distribution is expected to represent the true posterior distribution. Compared to MCMC, VI is fairly efficient as the inference problem is reduced to the optimization problem with respect to the distribution parameters. However, VI may make the model less expressive and thus lead to suboptimal performance. Although more complicated approximate function has a better representation ability in some cases, it introduces higher computational cost. Such accuracy-computation trade-off cannot be achieved easily as the inference performance depends on the tricky design of the approximate distribution form.

Due to advances in deep learning in the past few years, using deep neural network as the approximate function has become a new trend in VI. This is especially true for generative models such as variational auto-encoder (VAE) (Rezende et al., 2014; Kingma and Welling, 2013), in which an encoder network is built to approximate the latent variable distribution conditioned on the observed data and a decoder network is built to represent the observed data distribution conditioned on the latent variable. In addition, because of the generalization ability of a deep neural network with millions of trainable weights, amortized formulation with regularization is applied on the training dataset to learn the network weights for faster inference on the test dataset than classic VI per data, but at the expense of lower precision (Cremer et al., 2018). As a result, this leads to a new trade-off between inference speed and amortization accuracy.

Another topic related to posterior distribution estimation with deep learning are the deep generative models trained with maximum likelihood, such as autoregressive (Oord et al., 2016) and flow models (Dinh et al., 2016). In these models, neural network parameterized models are built to deploy tractable maximum likelihood training and generate new samples after training. If the parameterized model family is highly expressive with enough training samples, maximum likelihood training is expected to learn parameters which fit to the true data density well and generate new data with high fidelity. Autoregressive and flow models differ from VAE in that exact likelihood is evaluated in the former while approximate evaluation is applied for the latter. Such tractable inference makes training simpler but models less expressive, except for flow models which provide a combination of tractability and high expressiveness.

In this work, we propose to solve the posterior distribution estimation problem in QSM using a neural network parameterized distribution family by combining posterior density estimation from samples with posterior distribution approximation via VI for domain adaptation. Assuming multivariate Gaussian represented by a CNN as the posterior distribution of susceptibility given the input local field, a COSMOS (Liu et al., 2009) dataset of field susceptibility pairs were used as samples from the true posterior distribution to train such CNN with an MAP loss function. Applying the likelihood in Eq. 4 and prior in Eq. 5, the pre-trained CNN was adapted using VI posterior distribution approximation on different patient datasets which only contained input measured fields. Compared to MAP estimation MEDI (Liu et al., 2011b) in Eq. 3 and other deep learning QSM methods, QSMnet (Yoon et al., 2018) and FINE (Zhang et al., 2020a), the proposed method estimated the full posterior distribution of susceptibility with uncertainty quantification, while achieving domain adaptations on various datasets.

Based on the assumption that the pattern from field $b$ to $p(\chi|b)$ is recoverable, a general distribution $p_{data}(\chi|b)$ for any given $b$ can be approximated with a learning-based approach. To accomplish that, a set of parameterized distributions $q_{\psi}(\chi|b)$ using a neural network with parameters $\psi$ are introduced and learned on a cohort of datasets including healthy subjects and patients. In this work, we assume a multivariate Gaussian distribution with diagonal covariance matrix as the approximate posterior distribution, i.e., $q_{\psi}(\chi|b)=\mathcal{N}(\mu_{\chi|b},\Sigma_{\chi|b})$, and use a dual-decoder network architecture (Figure 1) extended from 3D U-Net (Ronneberger et al., 2015; Çiçek et al., 2016) to represent $q_{\psi}(\chi|b)$, with dual decoders’ outputs representing mean $\mu_{\chi|b}$ and variance $\Sigma_{\chi|b}$ maps.

The adaptive learning strategy proposed in this paper tackles the domain adaptation challenge in medical imaging with deep learning from a probabilistic distribution refinement point of view. Since the high quality COSMOS samples are acquired only from healthy subjects, posterior density estimation with COSMOS samples may not generalize well to the patients with pathology not covered by the COSMOS dataset. As a result, even though the COSMOS pre-trained PDI performs well on COSMOS test dataset from the same distribution (Figure 2), inferior mapping happens evidenced by lower susceptibility values for lesions when applying PDI to the patients directly (Figures 3 and 4). Based on the distribution approximation principle, the pre-trained density estimation network PDI needs fine-tuning in order to fit to the patient data distribution as well. VI with KL divergence as a measure of similarity between two distributions is used for approximate distribution refinement, which helps reduce the generalization error of PDI (Figure 3 and 4). However, in terms of other domain generalizations such as different imaging resolutions, PDI-VI with KL divergence loss function for weight adjustment has not been tested and may suffer in accuracy, which will be explored in the future work. Another domain adaptation method FINE works better than PDI-VI (Figure 4) to reduce generalization error of the pre-trained network, since FINE fits to every test case by minimizing the fidelity loss, which has the major drawback of significantly increased computational time (Table 1).

The relationship between PDI-VI (Eq. 9) and VAE (Kingma and Welling, 2013) is described in the methods section. The key point is that the generative network (the decoder) from latent variable to data in VAE is replaced by a physics-based likelihood model (Eq. 4) in PDI-VI. This implies a general way of learning the posterior distribution of image data conditioned on the measured signal for any imaging modality, where a specific forward imaging model is used to form the ”decoder” and only the ”encoder” is learned with input measured signals in an unsupervised fashion like VAE. The training strategy of PDI-VI utilizes both widely available measured signals in clinic and well-defined imaging physical models to improve the reconstruction fidelity of the trained model. When gold standard reconstructions are available for training, as in the COSMOS dataset, combining direct conditional density estimation using high quality images with VI domain adaptation on measured input signals could improve the performance of VI trained on the measured signals alone (Figure 3).

PDI defines a set of parameterized distributions using a neural network and learns these parameters from samples to approximate the true distribution, where the expressiveness of the distribution family affects their approximation ability. The network architecture (Figure 1) is inspired by 3D U-Net (Ronneberger et al., 2015; Çiçek et al., 2016), which was originally proposed for medical image segmentation tasks and has also been successfully used in deep QSM reconstructions (Yoon et al., 2018; Zhang et al., 2020a; Bollmann et al., 2019), therefore such architecture should be expressive enough for field-to-susceptibility mapping. The COSMOS experiment indicates satisfactory image-to-image mapping ability of the proposed architecture (Figure 2 and Table 1). The simulation experiment verifies correlation between the predicted SD map and the error map, indicating reasonable uncertainty quantification of PDI and PDI-VI. Despite such merits, the choice of variational posterior form in this work is simply a Gaussian distribution with diagonal covariance matrix, which is known as the mean field approximation for modeling and calculation simplicity in classic VI. This factorized Gaussian does not consider correlation between voxels in the reconstructed susceptibility map, but in view of the forward convolution operation (Eq. 1) which aggregates the global susceptibility into the measured field at each location, taking into account the dependency between local voxels in the susceptibility map may make the variational posterior more expressive. Possible options could be improving the Gaussian posterior with a non-diagonal covariance matrix and using an autoregressive (Oord et al., 2016) or flow-based (Dinh et al., 2016) model to capture the dependency.

The prior distribution of susceptibility (Eq. 5) used in PDI-VI comes from MEDI (Liu et al., 2011b), where weighted TV regularization was used to suppress streaking artifacts appeared on QSM dipole inversion (Kee et al., 2017). In general, the prior distribution $p(x)$ captures the density of data $x$ from a prior knowledge, where higher quality data $x$ has a higher density value. In this sense, estimating the density from sufficient data may build a more comprehensive prior distribution and therefore become more efficient to regularize the inverse problem solution. In fact, learning the prior density for MAP estimation of the imaging inverse problem has been explored by Tezcan et al. (2018) and Luo et al. (2019), where VAE and PixelCNN++ (Salimans et al., 2017) were deployed to learn the explicit prior distribution of MR images. Lønning et al. (2019) proposed the recurrent inference machine by implicitly encoding the prior distribution for MAP estimation. These deep prior approaches inspire us to extend our work in the future by learning and evaluating a prior density from data and inserting them into Eqs. 9 and 10 for VI.

The inference gap (Eq. 11) summarizes two types of errors when applying the amortized inference strategy. Amortized VI has the advantage of fast inference during test time. However, it has slightly worse visual quality inside and surrounding the hemorrhage than subject-specific VI (Figure 5a). Even though an almost zero amortization gap was achieved (Figure 5b), the regularization term of KL divergence (Eq. 10) was still better imposed in subject-specific VI, which may contribute to its better reconstruction of the hemorrhage. However, such advantage comes at a cost of extra inference time. To accelerate the inference speed of subject-specific VI, optimizing the initialization of variational parameters is useful to reduce the number of VI optimization steps. Meta-learning (Naik and Mammone, 1992; Hochreiter et al., 2001) may be applied to optimize the optimization process of VI per data, where a learner can be designed during pre-training to learn an inference algorithm that generalizes well to the data of interest.

In conclusion, we demonstrate a neural network parametrized distribution which yields an approximate posterior distribution of susceptibility given an input local field map for the Bayesian QSM inverse problem. The network was pre-trained on a COSMOS dataset by fitting to the empirical distribution and adapted to different domains using amortized VI. The proposed method computes adaptive reconstructions of susceptibility together with an uncertainty estimation. Future work will include building a more expressive posterior distribution family, learning a deep prior density for regularization and optimizing the subject-specific VI algorithm using meta-learning.

In this appendix we show the steps of simulation in section 4.6:

• •

  Synthesize local magnetic field data $b_{syn}$ from COSMOS gold standard susceptibility $\chi_{COSMOS}$ using dipole convolution model:

  $$b_{syn}=\chi_{COSMOS}*d$$

• •

  Synthesize multi-echo MR images $S_{j}$ from $b_{syn}$ (above), $R2^{*}$ ($T2^{*}$ decay rate) and $M_{0}$ (water) using forward physical model:

  $$S_{j}=M_{0}e^{-R2^{*}\cdot t_{j}}e^{i(\phi_{0}+b_{syn}\cdot t_{j})}+n_{j},$$

  where $t_{j}$ is the echo time of the $j$-th echo, $n_{j}$ is the i.i.d. Gaussian noise on real and imag parts for all voxels and initial phase $\phi_{0}=0$ is assumed for all voxels.

• •

  Deploy nonlinear field fitting and spatial phase unwrapping to estimate the noisy local field data.