Improved Estimation of Myelin Water Fractions with Learned Parameter Distributions

Abstract

Purpose:

To improve estimation of myelin water fraction (MWF) in the brain from multi-echo gradient-echo imaging data.

Methods:

A systematic sensitivity analysis was first conducted to characterize the conventional exponential models used for MWF estimation. A new estimation method was then proposed for improved estimation of MWF from practical gradient-echo imaging data. The proposed method uses an extended signal model that includes a finite impulse response filter to compensate for practical signal variations. This new model also enables the use of pre-learned parameter distributions as well as low-rank signal structures to improve parameter estimation. The resulting parameter estimation problem was solved optimally in the Bayesian sense.

Results:

Our sensitivity analysis results showed that the conventional exponential models were very sensitive to measurement noise and modeling errors. Our simulation and experimental results showed that our proposed method provided a substantial improvement in reliability, reproducibility, and robustness of MWF estimates over the conventional methods. Clinical results obtained from stroke patients indicated that the proposed method, with its improved capability, could reveal the loss of myelin in lesions, demonstrating its translational potentials.

Conclusion:

This paper addressed the problem of robust MWF estimation from gradient-echo imaging data. A new method was proposed to provide improved MWF estimation in the presence of significant noise and modeling errors. The performance of the proposed method has been evaluated using both simulated and experimental data, showing significantly improved robustness over the existing methods. The proposed method may prove useful for quantitative myelin imaging in clinical applications.

1. INTRODUCTION

Myelin water imaging (MWI), or myelin water fraction (MWF) mapping, measures the ratio of myelin water (i.e., water trapped within the myelin sheath) to total water content of brain tissue.¹ Such measurememts have been shown useful for studying demyelination diseases such as multiple sclerosis (MS) and stroke.^2,3 The key to MWI is separation of myelin water signals from other tissue water components. To achieve this, a common approach is to exploit the short T₂ or ${\mathrm{T}}_2^{\ast }$ characteristic of myelin water spins and extract this signal component from either multi-echo spin echo (mSE) or multi-echo gradient echo (mGRE) data using exponential model fitting.^1,4,5 The most widely used models include the multi-exponential model,¹ which expresses the magnitude signal as a linear combination of pre-selected T₂ or ${\mathrm{T}}_2^{\ast }$ basis functions, the magnitude-3-exponential model,⁶ which decomposes the magnitude signal into three real exponentials, and the complex-3-exponential model,^7,8 which decomposes the complex-valued signal into three complex exponentials. However, the exponential model fitting problem is known to be ill-conditioned and the fitting results are often highly sensitive to data noise and modeling errors.^8-16

Several efforts have been made to characterize the effect of measurement noise on MWF estimation. For example, the multi-exponential model was analyzed using numerical simulations under a wide range of noise levels and found to require very high signal-to-noise ratio (SNR) to produce reliable results.^12-14 Similarly, 3-exponential models were also found to be rather sensitive to noise, using simulated data with different SNRs and parameter values.^15,16 To address this issue, several methods have been developed to denoise the measured data before parameter estimation by imposing low-rank constraints^17,18 or sparsity constraints.¹⁸ There are also several methods that apply constraints directly on the model parameters, using constraints such as spectral and local spatial smoothness,^19-23 nonlocal similarity,²⁴ local sparsity,²⁵ and joint sparsity.^26,27 More recently, deep learning-based methods have also been proposed to use prior information embedded in the training data to improve MWF estimation, producing promising results.^28-30

This paper presents a systematic analysis of the MWF estimation problem in the presence of both noise and modeling errors often associated with practical mGRE data. Based on this analysis, a new estimation method has been proposed to improve the robustness of MWF estimation from practical gradient-echo imaging data. The proposed method uses an improved signal model to capture signal deviations from the conventional exponential models due to, for example, field inhomogeneity and drift. This new model also enables effective use of pre-learned parameter distributions and low-rank structures that exist in multi-echo data to reduce estimation uncertainty. The proposed method has been evaluated using both simulated and in vivo experimental data, showing superior performance over several conventional MWF estimation schemes.

2. THEORY

2.1. Conventional Models of Gradient-Echo Water Signals

For MWF mapping, three main models have been used to represent different water components: a) multi-exponential model, b) magnitude 3-exponential model, and c) complex 3-exponential model.

The multi-exponential model represents the magnitude of the acquired signal as a linear combination of M pre-selected ${\mathrm{R}}_2^{\ast }$ basis functions:²⁰

$$s(t_n)=\sum _{m=1}^M{\mathrm{A}}_m{e}^{-{\mathrm{R}}_{2,m}^{\ast }{t}_n},$$ [1]

where A_m is the amplitude of the signal component with relaxation rate ${\mathrm{R}}_{2,m}^{\ast }$, and t_n = TE_n, n = 1, 2, … , N with N being the total number of echoes.

The magnitude-3-exponential (m3e) model exploits the “fact” that human white matter mainly consists of three water compartments, i.e., myelin water (MW), axonal water (AW), and extra-cellular water (EW), and explicitly decomposes the magnitude signal into three components:^4,6

$$s(t_n)=\sum _c{\mathrm{A}}_c{e}^{-{\mathrm{R}}_{2,c}^{\ast }{t}_n},$$ [2]

where c represents tissue component, denoting ‘MW’, ‘AW’, or ‘EW’; {A_c} and {${\mathrm{R}}_{2,c}^{\ast }$} are the corresponding amplitudes and relaxation rates, respectively. Clearly the m3e model is a special case of the multi-exponential model in Eq. [1].

The complex-3-exponential (c3e) model, like the m3e model, also assumes three water components but the model is applied to the complex multi-echo data as:⁸

$$s(t_n)=e^{-i{\varnothing }_0}\sum _c{\mathrm{A}}_c{e}^{-({\mathrm{R}}_{2,c}^{\ast }+i2\pi f_c)t_n}.$$ [3]

The parameters of the multi-exponential model in Eq. [1] are usually solved using a non-negative least squares (NNLS) algorithm,^1,20 while the parameters of both the m3e and c3e models are determined using nonlinear least squares (NLLS) model fitting with preset parameter bounds.^4,6-8

2.2. Proposed Method

The proposed method has three key features: a) an extended exponential model to better handle practical non-exponential signal variations, b) use of learned parameter distributions to stabilize parameter estimation, and c) constrained MWF estimation, which are summarized in Fig. 1. The following sections provides a detailed description of the proposed method, including the signal model and MWF estimation algorithm.

Figure 1.

Illustration of the key features of the proposed method: a) a compensated exponential model to better handle practical non-exponential signal variations, b) use of learned parameter distributions to stabilize parameter estimation, and c) constrained MWF estimation scheme.

2.2.1. Proposed Signal Model

In addition to random noise, in vivo mGRE data usually contain signal distortions due to practical imaging conditions, such as B₀ field inhomogeneity and drift. These signal distortions can lead to significant MWF estimation errors as shown by our analysis results in the Results section. To address this issue, we proposed a new model to represent mGRE data. This model has a built-in capability to absorb signal variations that deviate from the conventional MWF models in Eqs. [1-3]. It also enables incorporation of parameter distributions learned from training data to reduce estimation uncertainty. More specifically, we represent the magnitude signal using the following compensated 3-exponential model that uses a finite impulse response (FIR) filter to compensate for non-exponential decays:

$$\begin{gathered} s(t_n)=\sum _c{\mathrm{A}}_c{e}^{-{\mathrm{R}}_{2,c}^{\ast }{t}_n}\sum _{\ell =-L∕2}^{L∕2}{c}_{\ell }{e}^{i2\pi \ell \Delta f{t}_n} \\ \text{subject to}{c}_{-\ell }=c_{\ell }^{\ast }. \end{gathered}$$ [4]

In Eq. [4], the first term is the conventional m3e model in Eq. [2] and the second term is an (L + 1)-order FIR filter with coefficients {c_ℓ}; Δf denotes the spectral resolution and $c_{\ell }^{\ast }$ the conjugate of c_ℓ. The functional form of the FIR filter was specifically designed to account for non-exponential signal deviations stemmed from intra-voxel field inhomogeneities. A detailed derivation of the FIR filter is provided in the Supporting Information (SI) Text 1. The constraint $c_{-\ell }=c_{\ell }^{\ast }$ was imposed to prevent the FIR filter from introducing undesired signal phases because the model was applied to the magnitude component of the measured GRE data and the FIR filter term should be a real-valued function. This constraint was nicely implemented by invoking the Hermitian symmetry property of Fourier transform, i.e., $c_{-\ell }=c_{\ell }^{\ast }$.

We further imposed statistical distributions on the model parameters to constrain the parameter variations, thus improving estimation robustnes:

$${\mathrm{A}}_c\sim \text{Pr}({\mathrm{A}}_c)\text{and}{\mathrm{R}}_{2,c}^{\ast }\sim \text{Pr}({\mathrm{R}}_{2,c}^{\ast }).$$ [5]

These statistical distributions are assumed to be tissue-dependent and are obtained from training data as described in the subsequent section. For the problem we address, these distributions not only help absorb a priori information but also provide an effective constraint to prevent the estimated values of A_c and ${\mathrm{R}}_{2,c}^{\ast }$ from going outside the “acceptable” ranges due to noise and modeling errors and the ill-conditionedness of the exponential model. Our distribution-based constraint is more informative as compared to the conventionally used hard parameter bounds in NLLS algorithm.^4,6-8 More specifically, the shaped distributions favor those parameter values that occur more frequently in the training samples. In contrast, the conventional parameter bounds essentially assume uniform distributions on the model parameters, and thus are noninformative (other than specifying a large feasible range for the parameters). Additionally, conventional hard parameter bounds are often set empirically; no agreed-upon bounds yet exist in the MWF mapping community.

2.2.2. MWF Estimation

The proposed signal model leads to a constrained MWF estimation problem. The problem was solved by exploiting: a) prior parameter distributions embedded in training data acquired in vivo, b) anatomical priors generated from tissue segmentations of associated anatomical images, and c) low-rank structures inherent in long-${\mathrm{T}}_2^{\ast }$ water components. This was accomplished using the Bayesian statistical estimation framework. Its details are provided as follows.

A. Learning Parameter Distributions from Training Data

In this work, the distributions Pr(A_c) and $\text{Pr}({\mathrm{R}}_{2,c}^{\ast })$ in Eq. [5] were learned from training data. The training data were acquired using mGRE sequence from 10 subjects (see details in the Methods section). To reduce signal distortions and measurement noise, we first exploited the low-rank structure of the water signals and imposed the following subspace constraint:³¹

$$s(t_n)={\sum }_r{b}_r{\varphi }_r(t_n),$$ [6]

where {φ_r(t_n)} are temporal basis functions for the water signal subspace; {φ_r(t_n)} and the rank were determined by principal component analysis (PCA) of all the training data as discussed in SI Text 2. With Eq. [6], training data were filtered by projecting onto the subspace spanned by {φ_r(t_n)} in the least-squares sense. Then, we fit the filtered data using the conventional m3e model via NLLS algorithm:^4,6-8

$$\{{\widehat{\mathrm{A}}}_c\},\{{\widehat{\mathrm{R}}}_{2,c}^{\ast }\}={\text{arg min}}_{{\mathrm{A}}_c,{\mathrm{R}}_{2,c}^{\ast }}\Vert y(t_n)-{\sum }_c{\mathrm{A}}_c{e}^{-{\mathrm{R}}_{2,c}^{\ast }{t}_n}{\Vert }_2^2,$$ [7]

where y denotes the subspace filtered data. If any of the estimated parameters hit the preset parameter bounds, the corresponding voxel was considered to have large signal perturbations and removed from subsequent processing. Some representative maps of the removed voxels are shown in SI Fig. S1. Then, to impose tissue dependence, we classified the estimated parameters {${\widehat{\mathrm{A}}}_c$} and {${\widehat{\mathrm{R}}}_{2,c}^{\ast }$} into two categories: one from myelin-rich tissue (e.g., white matter) and the other from low-myelin tissues (e.g., gray matter and cerebrospinal fluid), based on tissue segmentation results obtained from associated anatomical references. Then, we determined the density functions for these categorized parameters using Gaussian mixture approximation in the maximum likelihood (ML) sense.³² The resulting distributions for myelin-rich and low-myelin tissues are denoted as Pr_M and Pr_LM, respectively.

The final prior distributions used for a newly acquired mGRE data were then determined to further incorporate anatomical priors. Particularly, we calculated Pr(A_c) and $\text{Pr}({\mathrm{R}}_{2,c}^{\ast })$ as:

$$\begin{gathered} \text{Pr}({\mathrm{A}}_c)=\alpha {\text{Pr}}_{\mathrm{M}}({\mathrm{A}}_c)+(1-\alpha ){\text{Pr}}_{\text{LM}}({\mathrm{A}}_c) \\ \text{Pr}({\mathrm{R}}_{2,c}^{\ast })=\alpha {\text{Pr}}_{\mathrm{M}}({\mathrm{R}}_{2,c}^{\ast })+(1-\alpha ){\text{Pr}}_{\text{LM}}({\mathrm{R}}_{2,c}^{\ast }) \end{gathered}$$ [8]

where α is the probability of one voxel being in white matter, derived from segmentation results of the anatomical images associated with this new mGRE data.

B. Determination of the Order of the Compensating FIR Filter

The FIR filter order L in Eq. [4] should be chosen as low as possible with an acceptable level of model fitting error. In our current implementation, L took two possible values, 0 and 4, based on our analysis of in vivo datasets as discussed in SI Text 3. When L = 0, the proposed model becomes the conventional m3e model, which was used for all the voxels if the m3e was judged to be appropriate. For all other voxels, we used L = 4 so that the FIR filter in Eq. [4] was activated to compensate the m3e model.

In this work, the m3e model was evaluated for a given voxel using Cramér-Rao lower bound (CRLB) analysis. More specifically, we first performed m3e model fitting; we then calculated the CRLB of myelin concentration ${\widehat{\mathrm{A}}}_{\text{MW}}$ as follows:

$$\mathbb{E}\left[{\left({\widehat{\mathrm{A}}}_{\text{MW}}-{\mathrm{A}}_{\text{MW}}\right)}^2\right]\ge [{{\mathbf{I}}_{\theta }}^{-1}{]}_{{\mathrm{A}}_{\text{MW}}}={\left[{\left(\frac{({\mathbf{J}}_{\theta }{)}^{\mathrm{T}}({\mathbf{J}}_{\theta })}{{\sigma }^2}\right)}^{-1}\right]}_{{\mathrm{A}}_{\text{MW}}},$$ [9]

where θ denotes the estimated parameters of the m3e model, I_θ and J_θ are the corresponding Fisher information matrix and Jacobian matrix (details on their calculation are provided in SI Text 4), and σ² is the variance of fitting residual. If the CRLB was larger than a preset threshold, then the m3e was insufficient to represent the signal and the FIR filter was activated. Details on selecting the threshold are provided in SI Text 5.

C. Bayesian-Based Parameter Estimation

After Pr(A_c), $\text{Pr}({\mathrm{R}}_{2,c}^{\ast })$ and L were determined, the proposed model was fit to the measured data in the Bayesian sense. More specifically, for the voxels with L = 0, we solved the following optimization problem:

$${\widehat{\mathrm{A}}}_c,{\widehat{\mathrm{R}}}_{2,c}^{\ast }={\text{arg min}}_{{\mathrm{A}}_c,{\mathrm{R}}_{2,c}^{\ast }}\frac{1}{{\sigma }^2}\Vert y(t_n)-{\sum }_c{\mathrm{A}}_c{e}^{-{\mathrm{R}}_{2,c}^{\ast }{t}_n}{\Vert }_2^2-{\sum }_c\{\log (\text{Pr}({\mathrm{A}}_c))+\log (\text{Pr}({\mathrm{R}}_{2,c}^{\ast }))\},$$ [10]

where σ² denotes the noise variance. For the voxels with L ≠ 0, we solved the following problem:

$$\begin{gathered} {\widehat{\mathrm{A}}}_c,{\widehat{\mathrm{R}}}_{2,c}^{\ast },{\widehat{c}}_{\ell }={\text{arg min}}_{{\mathrm{A}}_c,{\mathrm{R}}_{2,c}^{\ast },c_{\ell }}\Vert y(t_n)-{\sum }_c{\mathrm{A}}_c{e}^{-{\mathrm{R}}_{2,c}^{\ast }{t}_n}{\sum }_{\ell =-L∕2}^{L∕2}{c}_{\ell }{e}^{i2\pi \ell \Delta f{t}_n}{\Vert }_2^2+ \\ {\lambda }_{\mathrm{A}}{\sum }_c{\Vert {\mathrm{A}}_c-{\mu }_{{\mathrm{A}}_c}\Vert }_2^2+{\lambda }_{{\mathrm{R}}_2^{\ast }}{\sum }_c{\Vert {\mathrm{R}}_{2,c}^{\ast }-{\mu }_{{\mathrm{R}}_{2,c}^{\ast }}\Vert }_2^2 \\ \text{subject to}{c}_{-\ell }=c_{\ell }^{\ast }, \end{gathered}$$ [11]

where μ_Ac and ${\mu }_{{\mathrm{R}}_{2,c}^{\ast }}$ are the statistical means of the learned distributions; λ_A and ${\lambda }_{{\mathrm{R}}_2^{\ast }}$ are regularization parameters selected based on the variances of noise and parameters. In this work, Eq. [10] was solved using the majorization-minimization algorithm³³ and Eq. [11] was solved using the NLLS algorithm.³⁴ A detailed description of these optimization algorithms is provided in SI Text 6 and 7.

Instead of treating the solutions from Eqs. [10] and [11] as the final estimated values of the model parameters, we further constrained the long-${\mathrm{T}}_2^{\ast }$ components (i.e. the AW and EW pools) by forcing them to belong to a subspace in probabilistic sense, i.e.,

$$s_l(t_n)={\sum }_r{a}_r{\varphi }_r(t_n)\text{with}\{a_r\}\sim \text{Pr}(\{a_r\}).$$ [12]

This additional constraint was inspired by our analysis (see the Results section) which showed that the estimation variation of long-${\mathrm{T}}_2^{\ast }$ components could significantly influence the estimation of the MW component. Therefore, we leveraged the inherent low-rank characteristic of the long-${\mathrm{T}}_2^{\ast }$ signals to decrease their estimation variations. More specifically, basis functions {ϕ_r(t_n)} and coefficient distributions Pr({a_r}) were determined from the initial estimates of s_l(t_n) (obtained based on the solutions from Eqs. [10] and [11]), using PCA and ML-based density estimation, respectively. We then determined the final estimates of long-${\mathrm{T}}_2^{\ast }$ components in the Bayesian sense:

$${\mathbf{min}}_{\{a_r\}}\frac{1}{{\sigma }^2}\Vert s_{\ell }(t_n)-{\sum }_r{a}_r{\varphi }_r(t_n){\Vert }_2^2-\log \text{Pr}(\{a_r\}),$$ [13]

and removed them from the measured signals to determine the MW component. The final MW values were determined from the residual (denoted as s_r) as follows:

$${\tilde{\mathrm{A}}}_{\text{MW}},{\tilde{\mathrm{R}}}_{\text{MW}}^{\ast }=\mathbf{arg}{\mathbf{min}}_{{\mathrm{A}}_{\text{MW}},{\mathrm{R}}_{2,\text{MW}}^{\ast }}\frac{1}{{\sigma }^2}\Vert s_r(t_n)-{\mathrm{A}}_{\text{MW}}{e}^{-{\mathrm{R}}_{2,\text{MW}}^{\ast }{t}_n}{\Vert }_2^2-\log \text{Pr}({\mathrm{A}}_{\text{MW}})\text{Pr}({\mathrm{R}}_{2,\text{MW}}^{\ast }).$$ [14]

Given that the short-${\mathrm{T}}_2^{\ast }$ characteristic of the MW component, we used the first 10 time points for fitting. Eq. [14] was solved using the majorization-minimization algorithm.³³ A detailed description of the optimization algorithm is provided in SI Text 6.

3. METHODS

3.1. Generation of Simulated Data

A numerical phantom was generated with four anatomical regions: white matter (WM), gray matter (GM), cerebrospinal fluid (CSF), and lesions. The ground truth signal for each voxel was synthesized using the conventional c3e model (Eq. [3]). Specific model parameters followed region-dependent Gaussian distributions as detailed in SI Table S1. Imaging-related parameters included: first echo time (TE₁) = 1.93ms, echo spacing (ES) = 0.88ms, and number of echoes (#TE) = 30. To investigate the effect of measurement noise, white Gaussian noise was added at different levels (SNR = 50, 100, 150, and 200). To investigate the effect of field inhomogeneity-related signal perturbations, large spatial and temporal field variations were derived from in vivo experimental data and added to the computational phantom.

3.2. Acquisition of In Vivo Experimental Data

In vivo MR experimental data were collected from twelve healthy volunteers and three patients with ischemic strokes (1 with acute stroke; 2 with subacute strokes). The healthy subjects were scanned on a 3T Siemens Prisma scanner while the patients were scanned from a 3T Siemens Skyra scanner. All studies were approved by local institutional review boards. Written informed consent was obtained from all participants. The scanning protocols included a 3D mGRE scan³⁵ (FOV = 240 × 240 × 72 mm³, matrix = 78 × 110 × 24, flip angle (FA) = 27°, TR = 160 ms, TE₁ = 1.93 ms, ES = 0.88 ms, #TE = 148, bipolar readout gradients), an associated MPRAGE scan for each healthy subject (FOV = 240 × 240 × 192 mm³, matrix size = 256 × 256 × 192, TE = 2.29 ms, TI = 900 ms, TR = 1900 ms) and an associated FLAIR scan for each patient (FOV = 240 × 240 × 164 mm³, matrix size = 480 × 480 × 82, TE = 89 ms, TR = 9000 ms). Data inconsistencies between the odd and even echos acquired using positive and negative gradients, respectively, were corrected using an established method for data preprocessing.³⁶ The anatomical MPRAGE and FLAIR images were co-registered to the corresponding mGRE data and then segmented to provide masks for WM, GM, and CSF as well as three regions-of-interest (ROIs) in WM, i.e., genu, splenium, and internal capsule.

Among all the subjects, seven healthy and three stroke subjects were used to learn the parameter distributions. One of the healthy subjects was scanned twice within one experiment session to test reproducibility.

3.3. Sensitivity Analysis

CRLB analysis was used to characterize the inherent estimation uncertainties associated with different models. A matrix perturbation analysis was performed to quantify the model sensitivity to modeling errors.

A. CRLB Analysis

CRLB is a theoretical lower bound for the variance of an unbiased estimator and commonly used in statistics to characterize the inherent estimation uncertainty.³⁷ In this work, we derived the CRLBs for different signal models under a range of SNR levels (100-1000). Particularly, we quantified the estimation variance of conventional models (Eqs. [1-3]) using the classic unconstrained CRLB:³⁸

$$\mathbb{E}\left[({\widehat{\mathrm{A}}}_{\text{MW}}-{\mathrm{A}}_{\text{MW}}{)}^2\right]\ge [{{\mathbf{I}}_{\theta }}^{-1}{]}_{{\mathrm{A}}_{\text{MW}}}={\left[{\left(\frac{({\mathbf{J}}_{\theta }{)}^{\mathrm{T}}({\mathbf{J}}_{\theta })}{{\sigma }^2}\right)}^{-1}\right]}_{{\mathrm{A}}_{\text{MW}}},$$ [15]

where I_θ and J_θ are the Fisher information matrix and Jacobian matrix associated with Eqs. [1-3] and σ² denotes the noise variance. For the proposed method, we quantified its estimation uncertainty using the Bayesian CRLB considering the imposed parameter distributions:³⁸

$$\mathbb{E}\left[({\widehat{\mathrm{A}}}_{\text{MW}}-{\mathrm{A}}_{\text{MW}}{)}^2\right]\ge [({\mathbf{I}}^{\text{tot}}{)}^{-1}{]}_{{\mathrm{A}}_{\text{MW}}}={\left[{\left(\mathbb{E}\left[\frac{({\mathbf{J}}_{\theta }{)}^{\mathrm{T}}({\mathbf{J}}_{\theta })}{{\sigma }^2}\right]+{{\Sigma }_{\pi }}^{-1}\right)}^{-1}\right]}_{{\mathrm{A}}_{\text{MW}}},$$ [16]

where I^tot is the total Fisher information matrix, calculated from the Jacobian matrix J_θ and the covariance matrix Σ_π of the prior distributions.

In this study, both Eqs. [15] and [16] were computed with the same set of parameters specified in Section 3.1.

B. Matrix Perturbation Analysis

To further study the effect of modeling errors on MWF estimation, we also performed a matrix perturbation analysis.³⁹ Particularly, we represented the signal components and modeling errors using two different subspaces and the measured data a union-of-subspaces:

$$\mathit{y}={\mathbf{M}}_s{\mathit{c}}_s+{\mathbf{M}}_e{\mathit{c}}_e.$$ [17]

In Eq. [17], M_s and M_e are the basis functions for signal components and modeling errors, respectively; c_s and c_e are the corresponding coefficient vectors. Eq. [17] is a generic form and can be adapted to either multi-exponential model by forming M_s with the pre-selected ${\mathrm{R}}_2^{\ast }$ basis functions in Eq. [1], or m3e, c3e and the proposed method (with L = 0) by forming M_s with the 3-exponentials in Eqs. [2-4] (with fixed relaxation rates). M_e, on the other hand, was derived from in vivo fitting residuals.

With Eq. [17], estimation error in the MW component caused by modeling errors has the following analytic form:

$$\Delta \mathit{e}={\mathbf{M}}_{\text{MW}}{\mathbf{M}}_s^{†}{\mathbf{M}}_e{\mathit{c}}_e$$ [18]

for the conventional models and

$$\Delta \mathit{e}={\mathbf{M}}_{\text{MW}}{\left(\begin{array}{c}\hfill {\mathbf{M}}_s\hfill \\ \hfill \mathbf{B}\hfill \end{array}\right)}^{†}\left(\begin{array}{c}\hfill {\mathbf{M}}_e\hfill \\ \hfill 0\hfill \end{array}\right){\mathit{c}}_e={\mathbf{M}}_{\text{MW}}{\tilde{\mathbf{M}}}_s^{†}{\tilde{\mathbf{M}}}_e{\mathit{c}}_e$$ [19]

for the proposed model. In Eqs. [18] and [19], M_MW was formed by basis functions associated with MW; B was the matrix satisfying ${\sigma }^2{\Sigma }_{\pi }^{-1}={\mathbf{B}}^{\mathbf{T}}\mathbf{B}$ with Σ_π being the covariance of the prior distributions.

Since the spectral norms (denoted as ∥·∥) of ${\mathbf{M}}_{\text{MW}}{\mathbf{M}}_s^{†}{\mathbf{M}}_e$ and ${\mathbf{M}}_{\text{MW}}{\tilde{\mathbf{M}}}_s^{†}{\tilde{\mathbf{M}}}_e$ bound the error amplification factors associated with Eqs. [18] and [19], we used them as a measure of model sensitivity to errors.⁴⁰ Similar as the CRLB analysis, they were calculated with the parameters set-up specified in Section 3.1.

3.4. Comparison with Conventional Methods

To evaluate the performance of the proposed method, we compared it with several conventional methods, which included the NNLS algorithm (which fits the multi-exponential model with non-negativity constraints), the rNNLS algorithm (which adds additional regularization term to NNLS), the m3e fitting, and the c3e fitting.^1,4,8,19 For both NNLS and rNNLS, 15 ${\mathrm{T}}_2^{\ast }$ bases were taken from [3ms, 1500ms], among which bases within [3ms, 25 ms] were assigned to MW component.¹⁵ The regularization parameter in rNNLS was chosen manually by examination of the resulting MWF maps. For m3e and c3e fitting, the parameter bounds detailed in SI Table S2 were used. Initial values used for the phantom were the regional mean values while the ones used for the in vivo studies were taken based on literature.¹⁵ All computations were carried out in MATLAB (MathWorks, Natick, MA).

4. RESULTS

Figure 2 summarizes the sensitivity analysis results. The results of CRLB analysis are illustrated in Fig. 2A. As can be seen, all three conventional models are very sensitive to noise perturbations inherently (e.g., CRLB>100% at SNR of 100), although m3e and c3e have lower estimation uncertainty than the multi-exponential model as expected due to their low model complexity. The proposed model with the imposed parameter distributions result in significantly reduced CRLBs, indicating that the parameter priors can effectively reduce the estimation uncertainty. The results of matrix perturbation analysis are illustrated in Fig. 2B. As shown, the error amplification factors for all three conventional models are very large due to their poor numerical conditioning. In contrast, the proprosed model has a much smaller error amplification factor, implying its enhanced robustness in handling practical modeling errors. Interestingly, we also observe that with practical modeling errors, the c3e model becomes a lot less robust than the m3e. Since their main difference is whether to model the signal phases, our result implies that the c3e is very sensitive to practical phase errors, which is consistent to previous studies.^16,41

Figure 2.

Performance analyses. (A) CRLB analysis: model sensitivity to noise at various SNR levels, characterized by the CRLB. As can be seen, all conventional models have large estimation uncertainty while the proposed model with imposed parameter distributions results in significantly reduced CRLBs. (B-C) Matrix perturbation analysis: structured error amplification factors of different models, measured by the spectral norms of the associated error matrices given in Eqs. [18-19]; (B) considers long-${\mathrm{T}}_2^{\ast }$ water in the models while (C) considers only MW. As can be seen, when inter-compartment overlapping exists, all models except the proposed model are highly sensitive to structured perturbations; when only one compartment in presence, all models become more robust.

The sensitivity analysis results in Fig. 2B illustrate the ill-conditionedness of the MWF estimation problem. But the results can not tell if the ill-conditionedness is a result of overlapping between the MW and the other water components (i.e., M_MW and the other bases in M_s in Eqs. [18-19]) or between MW and signal perturbations (i.e., M_MW and M_e in Eqs. [18-19]). To gain a further insight into this, we recalculated the error amplification factors in Fig. 2B without the AW and EW bases in M_s, eliminating the overlapping among the water components. As shown in Fig. 2C, the error amplification factors thus obtained are significantly reduced for all the signal models. These results suggest that signal overlappings among the water components are a key source of model ill-conditionedness; therefore, estimation fluctuations of the long-${\mathrm{T}}_2^{\ast }$ components could have significant effect on the MW estimation. This observation motivated us to impose the additional constraints on the long-${\mathrm{T}}_2^{\ast }$ components, as described in Section 2.2.2C.

Figure 3 summarizes the simulation results using the data described in the Methods section. Different algorithms were first compared under noise perturbations. Figure 3A illustrates the MWF maps obtained at the SNR of 150. As can be seen, the proposed method produced the most accurate MWF map with significantly reduced noise-induced fluctuations. The subtle WM fingers were well-delineated; GM and lesions were also clearly distinguishable from healthy WM. In contrast, NNLS suffered from high estimation uncertainty; rNNLS reduced the signal fluctuations but at the cost of increased bias; while both m3e and c3e fitting were more robust than the NNLS-based methods, the corresponding MWF maps were still rather noisy. To compare the estimation variances of different methods, we also conducted Monte-Carlo simulations (50 trials). As shown in Fig. 3B, the proposed method resulted in the smallest estimation variances at all SNR levels tested here. In addition to noise, the proposed method was also evaluated under field-related structured perturbations. The ΔB₀ map used in this study is illustrated in Fig. 3C. Fig. 3D shows that, in presence of structured perturbations, the proposed fitting scheme offered the most robust MWF estimation, especially in regions with very large B₀ inhomogeneity.

Figure 3.

Simulation results. (A) MWF results obtained in one of the Monte-Carlo trials at SNR = 150. (B) Estimation STD in Monte-Carlo simulations obtained with different SNR levels; (C) Field map used to generate field-related perturbations. (D) MWF results obtained under field-related perturbations at SNR = 150. (E-F) Curve fitting results by c3e under no phase perturbations and practical phase perturbations respectively. In short, the proposed method produced the most accurate and robust MWF estimation under both noise and field-related perturbations.

One point worth mentioning is the high sensitivity of the c3e model to practical phase errors as suggested in Fig. 3D, which is consistent to our analysis shown in Fig. 2B. To further illustrate this issue, we have provided example fitting curves of the c3e without and with practical-level phase perturbations in Figs. 3E and F, respectively. The comparison shows that the unstableness of c3e fitting shown in Fig. 3D indeed came from its large modeling error in the signal phase; significant phase fitting residuals have led to large artifacts in the resulting MWF maps. These results suggest that fitting MWFs from practical mGRE data may be more robust with the magnitude-based models.

The proposed method has also been evaluated under practical conditions using in vivo experimental data. Figure 4 shows a set of representative results obtained from one of the healthy subjects. As can be seen, the proposed method achieved the best MWF estimation among all the algorithms being tested. More specifically, the MWF maps obtained by our method showed reduced spatial fluctuations, better preservations of subtle WM structures, and less image artifacts especially in regions with large susceptibility effect. The over-estimation in CSF was also reduced. To test the reproducibility, the same healthy subject was scanned twice in one experiment session. We have compared the MWF results obtained from these repeated scans using m3e and our method, respectively. As shown in Fig. 5, our method led to significantly reduced inter-scan variability over the conventional method, indicating better reproducibility. The MWF maps obtained from other healthy subjects are summarized in Fig. 6. High-quality MWF maps can be consistently seen across different subjects. Quantitative ROI analysis of the MWF values obtained from all the healthy subjects is illustrated in Table 1 and compared with those values reported in the literature. As can be seen, the mean ROI values were in high agreement with the previous studies. Note that the quality of the MWF maps produced by the conventional methods seems poorer than some of those reported in the literature.²⁸ This may be becasue our measured data have larger field-induced signal perturbations since the data acquisition scheme used in this study was designed mainly for spectroscopic imaging (not optimized for myelin mapping).³⁵

Figure 4.

In vivo results obtained from one of the healthy subjects. As can be seen, the MWF maps obtained by the proposed method have the highest quality as compared to the ones produced by other methods, with more realistic MWF spatial distributions and less image artifacts especially for regions of large susceptibility effect (marked by the blue arrows).

Figure 5.

Reproducibility results from the same subject as in Fig. 4, obtained by (A) m3e and (B) proposed method. As can be seen, the proposed method shows much better inter-scan reproducibility than the conventional fitting scheme.

Figure 6.

In vivo results obtained from more healthy subjects. The MWF maps produced by the proposed method are visually consistent among different subjects.

Table 1.

Comparison between the statistical mean and std of MWF values (%) computed by the proposed method with those in literature. Empty entries denote unspecified values in the respective literature studies.

ROI \ Method	Proposed	Ref. (4)	Ref. (29)	Ref. (11)
Entire WM	9.82 ± 2.52	11.0	10.0 ± 4.7	11.3
Entire GM	2.57 ± 2.00	--	2.8 ± 3.6	3.1
Genu	10.61 ± 2.08	12.0	13.2 ± 6.1	9.9 ± 1.0
Splenium	12.46 ± 1.89	--	16.5 ± 5.7	13.1 ± 1.0
Internal Capsule	10.05 ± 2.12	--	9.6 ± 5.1	--

To demonstrate the robustness and accuracy of the proposed method under clinical conditions, we have also tested it on two patients with subacute strokes and compared to traditional m3e fitting. We excluded these stroke data in paramer distribution learning to avoid potential bias on our evaluation. As shown in Fig. 7, the proposed method had successfully captured the demyelination in lesions for both patients, consistent to the corresponding anatomical references. In addition, the MWF values within normal tissues were consistent to our expectations without significant image artifacts. In contrast, the MWF maps obtained by m3e fitting failed to detect the lesions with large image distortions.

Figure 7.

In vivo results obtained from two stoke patients shown in (A) and (B), respectively, using both m3e model fitting and the proposed method. As can be seen, the proposed method has successfully captured the myelin loss in lesions (marked by blue arrows) in both patients, consistent to the corresponding anatomical references (top panel). In contrast, the MWF maps obtained by m3e fitting has failed to detect the lesions with large image artifacts.

5. DISCUSSION

The proposed method uses learned parameter distributions to improve the robustness of MWF estimation. It is well known that priors could introduce undesired estimation bias in Bayesian data analysis, especially in dealing with limited noisy data. To minimize any potential bias of the priors on MWF estimation, we designed our method with the following considerations. First, for mGRE experiments, we noticed that signal distortions (due to field inhomogeneity and fluctuations) in the mGRE data are getting progressively worse for the echoes with longer TEs. Due to the mathematical ill-conditionedness of the signal model and the presence of these signal distortions, the long-${\mathrm{T}}_2^{\ast }$ components are often poorly estimated without constraints using the conventional methods. So, we used the learned parameter distributions to stabilize the model (or reduce sensitivity to signal distortions and noise) and to eliminate outliers (caused by model ill-conditionedness and signal distortions). Second, for MWF mapping, the long-${\mathrm{T}}_2^{\ast }$ components are just “nuisance” signals and no biological information was derived from those components. We took advantage of this fact and used parameter distributions to capture the effects of B0 inhomogeneity and drift (in a distribution sense) and consequently to improve the estimation of the long-${\mathrm{T}}_2^{\ast }$ components in the presence of model ill-conditionedness and signal distortions. Third, the MW component is less affected by the field inhomogeneity and drift than the long-${\mathrm{T}}_2^{\ast }$ components; so, its parameter distributions reflect the distributions of biological values. The distributions we used in this study covered the entire known range. Finally, after the long-${\mathrm{T}}_2^{\ast }$ components are removed, the model for MW estimation is well conditioned. Given the number of data points (10) we have for fitting a single exponential for the MW, the measured data dominated the parameter inference over the effect of the prior (in contrast to some Bayesian data analysis problems where limited and noisy data are available and as a result, the prior can have dominating effects in inference). We have further investigated the influence of prior distributions on MWF estimation using both simulated and experimental data. A detailed description of the results is provided in SI Text 8.

The proposed method learns the distribution functions for two sets of parameters: 1) the relative concentrations of different water pools, and 2) ${\mathrm{T}}_2^{\ast }$ relaxation rates. The relative water concentration distributions should not be scanner dependent. The ${\mathrm{T}}_2^{\ast }$ effects have some level of scanner-dependence since different types of scanners would have different levels of B₀ field inhomogeneity and field drift induced by the data acquisition. But modern scanners are rather stable; distributions from different scanners should be relatively stable. In our experiments, we had data from 3T Prisma and 3T Skyra; they showed similar distributions (as shown in SI Fig. S2). However, for data acquired using different sequences (say, gradient-echo vs spin-echo) and/or at different field strengths (say, 3T vs 7T), the distributions can vary significantly and need to be re-learned.

It is also worth noting that our current implementation does not separate normal tissues from tissues with lesions in learning the model parameter distributions. Hence, our current parameter distributions are relatively broad. As more high-quality training data are available, we can learn prior distributions for specific lesions; we expect those distributions to be “sharper” than the undifferentiated ones we used in this study. So, although our current parameter distributions have already improved the bound constraints currently used, better and more specific parameter distributions can further improve the performance of our proposed method and enhance its practical utility.

Several methods have been proposed recently to address the robustness issue with MWF estimation. Some of thse methods denoise the measured data before parameter estimation by imposing low-rank constraints^17,18 or sparsity constraints¹⁸, while others apply constraints directly on the model parameters, using constraints such as spectral and local spatial smoothness,^19-23 nonlocal similarity,²⁴ local sparsity,²⁵ and joint sparsity.^26,27 Our proposed method is complementary to these methods; they can be integrated to further improve MWF estimation.

The proposed method has several limitations. First, our current implementation does not specifically compensate for motion-related perturbations. The proposed method can be combined with existing motion correction schemes for mGRE data to reduce the motion artifacts.^42,43 Second, we used a global FIR order to capture the signals that significantly deviate from the m3e model. This global model order may not be optimal to individual voxels. Modern model selection methods, such as Akaike information criterion (AIC),⁴⁴ can be leveraged if an optimal order is desired for some application. Third, the prior parameter distributions used in this study were obtained using conventional fitting methods on the “good” pixels, which are not “optimal”. For practical application of our proposed method, more effort should be made to learn those parameter distributions. For example, the accuracy of prior distribution learning may be improved using more advanced deep learning methods.⁴⁵

6. CONCLUSIONS

This paper addresses the problem of robust MWF estimation from mGRE data in the presence of significant noise and modeling errors. The sensitivity of MWF estimation using the existing exponential signal models was systematically analyzed and a new method was proposed to provide improved estimation of MWF. Our sensitivity analysis revealed that conventional models were highly ill-conditioned and sensitive to signal perturbations. The proposed method used an improved signal model to compensate for data errors and synergistically integrated learned parameter distributions and low-rank structures to reduce MWF estimation uncertainty. The performance of the proposed method has been evaluated using both simulated and experimental data, showing significantly improved robustness over the existing methods. The proposed method may prove useful for quantitative myelin imaging in clinical applications.

Supplementary Material

sm

Supporting Information Table S1. Parameters used to generate simulation phantom. To construct the phantom, random parameters were first drawn from the distributions specified above, then combined in such a way to produce spatially smooth parameter maps and to respect well-established facts on myelin biology such as a higher MWF corresponds to larger ${\mathrm{R}}_2^{\ast }$ values for the different water components.

Supporting Information Table S2. Initial values and bounds used for in-vivo data. y₁ denotes the data at TE₁. For the simulation phantom, the same bounds were used, but the initial values were the regional means as specified in Table S1.

Supporting Information Figure S1. Dropped/accepted voxels in one representative training dataset. (A-C) Estimated ${\mathrm{T}}_2^{\ast }$ maps and identified voxels to be dropped for MW, AM, and EM, respectively. (D) The final maps for dropped and accepted voxels.

Supporting Information Figure S2. Comparison of ${\mathrm{T}}_2^{\ast }$ distributions of data acquired from Prisma and Skyra, respectively. As can be seen, distributions obtained from 3T Prisma and 3T Skyra are similar.

Supporting Information Figure S3. Change of condition number of M_s with the FIR order L.

Supporting Information Figure S4. In vivo data fitting with different FIR model orders. (A) L = 0, (B) L = 2, (C) L = 4, (D) L = 6, and (E) L = 8. The solid lines are averaged signals over different voxels, while the shaded areas indicate the standard deviations.

Supporting Information Figure S5. Regions in which the FIR filter (L = 4) are activated. (A) Numerical phantom. (B-C) In vivo data.

Supporting Information Figure S6. Fitting results obtained from (A) random noise, and (B) in vivo data, including maps of MWF, axonal water fractions (AWF), and extra-cellular water fractions (EWF).

Supporting Information Figure S7. Simulation-based investigation on the influence of prior distributions on lesion detection. (A) MWF maps with a series of myelin loss inside the lesions, ranging between 0% (total myelin loss) and 100% (no myelin loss); the location of the lesion is indicated by the blue arrows. (B) MWF estimation errors within the lesion against different levels of myelin loss.

Supporting Information Figure S8. In vivo investigation on the influence of prior distributions on lesion detection. We compared MWF maps obtained with prior distributions determined from the training data pool with and without stroke data, respectively. The label ‘Previous’ indicates the previously learned distributions from all training data including stroke; the label ‘New’ indicates the new distributions learned without seeing any stroke data.