Improved Determination of the Myelin Water Fraction in Human Brain using Magnetic Resonance Imaging through Bayesian Analysis of mcDESPOT

Abstract

Myelin water fraction (MWF) mapping with magnetic resonance imaging has led to the ability to directly observe myelination and demyelination in both the developing brain and in disease. Multicomponent driven equilibrium single pulse observation of T₁ and T₂ (mcDESPOT) has been proposed as a rapid approach for multicomponent relaxometry and has been applied to map MWF in human brain. However, even for the simplest two-pool signal model consisting of MWF and non-myelin-associated water, the dimensionality of the parameter space for obtaining MWF estimates remains high. This renders parameter estimation difficult, especially at low-to-moderate signal-to-noise ratios (SNR), due to the presence of local minima and the flatness of the fit residual energy surface used for parameter determination using conventional nonlinear least squares (NLLS)-based algorithms. In this study, we introduce three Bayesian approaches for analysis of the mcDESPOT signal model to determine MWF. Given the high dimensional nature of mcDESPOT signal model, and, thereby, the high dimensional marginalizations over nuisance parameters needed to derive the posterior probability distribution of MWF parameter, the introduced Bayesian analyses use different approaches to reduce the dimensionality of the parameter space. The first approach uses normalization by average signal amplitude, and assumes that noise can be accurately estimated from signal-free regions of the image. The second approach likewise uses average amplitude normalization, but incorporates a full treatment of noise as an unknown variable through marginalization. The third approach does not use amplitude normalization and incorporates marginalization over both noise and signal amplitude. Through extensive Monte Carlo numerical simulations and analysis of in-vivo human brain datasets exhibiting a range of SNR and spatial resolution, we demonstrated the markedly improved accuracy and precision in the estimation of MWF using these Bayesian methods as compared to the stochastic region contraction (SRC) implementation of NLLS.

1. INTRODUCTION

Myelination changes are an important feature of brain development (1) and of the pathophysiology of a number of central nervous system diseases including multiple sclerosis (2-3), phenylketonuria (4-5), schizophrenia (6), epilepsy (7) and psychotic disorders (8), and has been recently associated with Alzheimer's disease (9). Magnetic resonance imaging (MRI) methods for assessing myelination have been developed around the concept of the MRI-visible myelin water fraction (MWF) (10), conventionally measured through multicomponent T₂ analysis of transverse signal decay data acquired using multiple spin echo-based sequences, such as the classic Carr-Purcell-Meiboom-Gill (11-16) or the gradient- and spin-echo (17) sequences.

Multicomponent driven equilibrium single pulse observation of T₁ and T₂ (mcDESPOT) has been proposed as an alternative approach to multicomponent relaxometry with greatly reduced acquisition time and making use of conventional MR acquisition sequences (18), namely, spoiled gradient recalled echo (SPGR) and fully balanced steady state free precession (bSSFP), with, for both sequences, imaging data acquired over multiple flip angles (FA) with a very short repetition time (TR). mcDESPOT has been applied to map MWF in human brain (19-23) with data acquisition sufficiently rapid to permit whole brain coverage. However, Bouhrara et al. (24) showed that, even for a simple two-pool model, quantitative parameter estimation from mcDESPOT is problematic, especially at low-to-moderate SNR. This parameter instability was found to persist in spite of the use of stochastic region contraction (SRC) in combination with nonlinear least squares (SRC-NLLS) algorithm, which has been proposed as an effective procedure for extracting parameter values from mcDESPOT data (25-26). It has been found (24) that given the flatness of NLLS parameter fit energy surfaces, SRC-NLLS remains highly sensitive to local minima and requires, therefore, high SNR data for reliable estimation of MWF. This limits the utility of mcDESPOT for the high resolution whole brain mapping studies required to detect spatial changes in MWF and to reduce partial volume effects.

Several previous studies have established the potential benefits of the Bayesian probability approach to parameter estimation (27-31). Indeed, unlike NLLS-based algorithms, Bayesian analysis does not require initial estimates, so that derivation of the posterior probability distribution function (PDF) for a given parameter is less sensitive to the local minimum structure of the global fit function. In addition, Bayesian analysis permits the incorporation of prior probability distributions. In favorable cases, the product of a prior and the likelihood function can be integrated analytically, serving to decrease the dimensionality of the integrations required to derive the posterior PDF of a given parameter (30).

In this study, we investigated the accuracy and precision of MWF determination from a two-component mcDESPOT model using three Bayesian-based methods. As is conventional, the MWF was defined as the amplitude of the component with the smaller value for T₂, reflecting a motionally-restricted water pool. We compared these results with those derived using SRC-NLLS. We first review the theory of the two-pool mcDESPOT signal model. We then describe in detail our Bayesian approaches for MWF mapping using mcDESPOT. Next, we present extensive Monte Carlo (MC) simulations that give the relative bias and dispersion in the estimation of MWF for several underlying input parameters as a function of SNR, using the SRC-NLLS- and the Bayesian-based approaches. Finally, we present MWF maps obtained from human brain datasets that exhibit a range of SNR and spatial resolution.

2. THEORY

2.1. The Two-pool mcDESPOT Signal Model

mcDESPOT is based on the use of SPGR and bSSFP datasets, each acquired over a range of FAs, with very short TR (18, 32). Neglecting exchange between components, the two-component SPGR signal is conventionally given by (33)

$$M_{SPGR}^k=M_{SPGR}^0\text{sin}\left(B_1{\alpha }_k\right)\left(f_s\frac{1-E_{1,s}}{1-E_{1,s}\text{cos}\left(B_1{\alpha }_k\right)}+(1-f_s)\frac{1-E_{1,l}}{1-E_{1,l}\text{cos}\left(B_1{\alpha }_k\right)}\right)$$ [1]

where the dimensionless quantity B₁ denotes an RF transmit scaling factor, s and l respectively denote the short- and long- T₂ components, f_s is the fraction of the short T₂ component, $M_0^{SPGR}$ represents the signal amplitude at echo time TE = 0 and incorporates proton density and machine factors, and α_k is the k^th excitation FA out of a total of K FAs. We also define E_1,j = exp(−TR_SPGR/T_1,j), where T_1,j is the spin-lattice relaxation time of the j^th component.

Similarly, the two-component bSSFP signal in the absence of exchange between components is conventionally given by (34)

$$M_n^{bSSFP}\mid M_{bSSFP}^0\left(f_s(M_{a,s}^n+i{M}_{b,s}^n)+(1-f_s)(M_{a,l}^n+i{M}_{b,l}^n)\right)\mid ,$$ [2]

where

$$M_{a,j}^n=\frac{E_{2,j}(1-E_{1,j})\text{sin}\left(B_1{\beta }_n\right)\text{sin}\phi }{(1-E_{1,j}\text{cos}\left(B_1{\beta }_n\right))(1-E_{2,j}\text{cos}\phi )-E_{2,j}(E_{1,j}-\text{cos}\left(B_1{\beta }_n\right))(E_{2,j}-\text{cos}\phi )}$$

and

$$M_{a,j}^n=\frac{E_{2,j}(1-E_{1,j})(\text{cos}\phi -E_{2,j})\text{sin}\left(B_1{\beta }_n\right)}{(1-E_{1,j}\text{cos}\left(B_1{\beta }_n\right))(1-E_{2,j}\text{cos}\phi )-E_{2,j}(E_{1,j}-\text{cos}\left(B_1{\beta }_n\right))(E_{2,j}-\text{cos}\phi )}.$$

Here, j indicates the short (s) or long (l) T₂ species, $M_{bSSFP}^0$ represents the bSSFP signal amplitude at echo time TE = 0 and incorporates proton density and machine factors, β_n is the n^th excitation FA out of a total of N FAs, and φ = 2π · Δω · TR_bSSFP + ϑ, with Δω the off-resonance frequency of proton pools with the assumption that both proton pools exhibit the same chemical shift, and ϑ the phase increment of the applied radio-frequency (RF) pulse. E_1,j = exp(−TR_bSSFP/T_1,j) and E_2,j = exp(−TR_bSSFP/T_2,j), where T_2,j is the spin-spin relaxation time of the j^th component.

In our simulation analyses, the FAs were assumed to be well-calibrated (i.e. B₁ = 1) and the two proton pools were assumed to be on-resonance (i.e. Δω = 0 Hz), while in the experimental analyses, B₁ and Δω were calculated using the saturated double angle method (S-DAM) (35) and the DESPOT₂-FM method (36), respectively, and used as known parameters for the calculation of the f_s maps, reflecting MWF. Further, in all cases, two different bSSFP datasets were obtained with phase increments ϑ equal to 0 or π (bSSFP₀ and bSSFP_π) respectively, corresponding to the usual implementation of mcDESPOT (18-24, 26, 32).

2.2. Bayesian Analysis of mcDESPOT

Bayesian analysis can be particularly robust for parameter estimation in models with several unknown parameters, especially in the presence of noise (27-31). Unlike NLLS, it does not require initial estimates and so is much less susceptible to problems with local minima. These characteristics arise fundamentally from the process of marginalization over nuisance parameters, defined as those parameters besides the one under current consideration, and are independent of the additional fact that Bayesian inference also provides a natural framework for the incorporation of prior knowledge when available.

Marginalization results in a considerable computational burden, but provides a posterior PDF for each individual parameter. In the case of mcDESPOT, the posterior PDF of f_s follows form:

$$\text{P}(f_s\mid \mathbf{S})=\int \frac{\text{P}\left({\Lambda }^{\ast }\right)L(\mathbf{S}\mid \Lambda )}{\text{P}\left(\mathbf{S}\right)}\text{d}{\Lambda }^{\ast },$$ [3]

where $\Lambda \left(f_s{T}_{2,s}{T}_{2,l}{T}_{1,s}{T}_{1,l}{M}_{SPGR}^0{M}_{bSSF{P}_0}^0{M}_{bSSF{P}_{\pi }}^0{\sigma }_{SPGER}{\sigma }_{bSSF{P}_0}{\sigma }_{bSSF{P}_{\pi }}\right)$ is the vector of unknown parameters, $\sigma =\left({\sigma }_{SPGR}{\sigma }_{bSSF{P}_0}{\sigma }_{bSSF{P}_{\pi }}\right)$ is the vector of standard deviations, SDs, of the noise, Λ* is equivalent to Λ but without f_s and represents the vector of nuisance parameters in the determination of f_s, P(Λ*) is the product of prior parameter distributions for the elements of Λ*, P(S) = ∫ P(Λ*) L(S | Λ) dΛ* is a normalization constant, $\mathbf{S}=\left({\mathbf{S}}_{SPGR}{\mathbf{S}}_{bSSF{P}_0}{\mathbf{S}}_{bSSF{P}_{\pi }}\right)$ is the vector of measured signals, each of which is itself a vector describing the signal amplitude for each value of FA, and L(S | Λ) is the joint likelihood function of S given Λ. Given the independent and identically Gaussian distributed elements of S, the joint likelihood function for mcDESPOT is

$$\begin{array}{cc}\hfill & L(\mathbf{S}\mid \Lambda )=L({\mathbf{S}}_{SPGR}\mid \Lambda )L({\mathbf{S}}_{bSSF{P}_0}\mid \lambda )L({\mathbf{S}}_{bSSF{P}_{\pi }}\mid \lambda )\hfill \\ \hfill & =\prod _{k=1}^{K}P(S_{SPGR}^k\mid \lambda )\prod _{n=1}^{N}P(S_{bSSF{P}_0}^n\mid \lambda )\prod _{n=1}^{N}P(S_{bSSF{P}_{\pi }}^{}\mid \lambda )\hfill \\ \hfill & ={\left({\sigma }_{SPGR}\sqrt{2\pi }\right)}^{-K}{\left({\sigma }_{bSSF{P}_0}\sqrt{2\pi }\right)}^{-2}{\left({\sigma }_{bSSF{P}_{\pi }}\right)}^{-N}exp(-\frac{({\mathbf{S}}_{SPGR}-{\mathbf{M}}_{SPGR}){({\mathbf{S}}_{SPGR}-{\mathbf{M}}_{SPGR})}^T}{2{\sigma }_{SPGR}^2}\hfill \\ \hfill & -\frac{({\mathbf{S}}_{bSSF{P}_0}-{\mathbf{M}}_{bSSF{P}_0}){({\mathbf{S}}_{bSSF{P}_0}-{\mathbf{M}}_{bSSF{P}_0})}^T}{2{\sigma }_{bSSF{P}_0}^2}-\frac{({\mathbf{S}}_{bSSF{P}_{\pi }}-{\mathbf{M}}_{bSSF{P}_{\pi }}){({\mathbf{S}}_{bSSF{P}_{\pi }}-{\mathbf{M}}_{bSSF{P}_{\pi }})}^T}{2{\sigma }_{bSSF{P}_{\pi }}^2})\hfill \end{array}$$

where P(S | Λ) is the PDF of S given Λ with obvious subscripting as indicated, the terms of the form (S − M) are difference of row vectors S and M across FAs, the superscript T indicates transpose operation, and M_SPGR and M_bSSFP are the row vectors of ideal measurement values at different FAs given by Equations 1 and 2, respectively.

Finally, an estimate, f̂_s, of the short fraction, f_s, can be derived as the first moment of its posterior PDF (30) through

$${\widehat{f}}_s\sqrt{}{f}_s\text{P}(f_s\mid \mathbf{S})\text{d}{f}_s,$$ [4]

As can be seen from Equations 3 and 4, the estimation of f̂_s requires a high dimensional integration of its posterior PDF in order to perform a complete marginalization over nuisance parameters. One useful technique for reducing the dimensionality of required marginalizations is to use a so-called conjugate prior, by which we mean a prior that can be coupled with a likelihood function to permit analytic closed form integrals over certain nuisance parameters (37). In the following, we describe three different approaches to reduce the dimensionality of parameter space for the estimation of f_s. The first approach is based on data normalization, the second approach on the use of noninformative priors and data normalization, and the final approach on the exclusive use of priors that are noninformative.

2.2.1. Approach 1: Normalized data and known standard deviation of noise

If sufficiently large background regions are available in the SPGR and bSSFP images, σ_SPGR, ${\sigma }_{bSSF{P}_0}\text{and}{\sigma }_{bSSF{P}_{\pi }}$ can be estimated with high accuracy and precision (38), avoiding the need to marginalize over these variables. Further, we can avoid marginalization over $M_{SPGR}^0,M_{bSSF{P}_0}^0\text{and}{M}_{bSSF{P}_{\pi }}^0$ by normalizing both the experimental and theoretical signals by their respective mean values calculated over FAs (18, 24, 26, 32): ${\tilde{S}}_{SPGR}^k=K\frac{S_{SPGR}^k}{{\sum }_{k=1}^K{S}_{SPGR}^k},{\tilde{S}}_{bSSF{P}_0}^n=N\frac{S_{bSSF{P}_0}^n}{{\sum }_{k=1}^N{S}_{bSSF{P}_0}^n}\text{and}{\tilde{S}}_{bSSF{P}_{\pi }}^n=N\frac{S_{bSSF{P}_{\pi }}^n}{{\sum }_{k=1}^N{S}_{bSSF{P}_{\pi }}^n},\text{and}{\tilde{M}}_{SPGR}^k=K\frac{M_{SPGR}^k}{{\sum }_{k=1}^K{M}_{SPGR}^k},{\tilde{M}}_{bSSF{P}_0}^n=N\frac{M_{bSSF{P}_0}^n}{{\sum }_{k=1}^N{M}_{bSSF{P}_0}^n}\text{and}{\tilde{M}}_{bSSF{P}_{\pi }}^n=N\frac{M_{bSSF{P}_{\pi }}^n}{{\sum }_{k=1}^N{M}_{bSSF{P}_{\pi }}^n}$. In effect, this further reduces the dimensionality of parameter space from 10 to 4. Equation 3 can be rewritten as

Equation 3 can be rewritten as

$$\begin{array}{cc}\hfill \text{P}(f_s\mid \tilde{S},\tilde{\sigma })& ={{\tilde{\sigma }}_{SPGR}\sqrt{2\pi })}^{-K}{{\tilde{\sigma }}_{bSSF{P}_0}\sqrt{2\pi })}^{-N}{\left({\tilde{\sigma }}_{bSSF{P}_{\pi }}\sqrt{2\pi }\right)}^{-N}\hfill \\ \hfill & \frac{1}{\text{P}\left(\tilde{\mathbf{S}}\right)}\sqrt{}\frac{1}{{\lambda }^{\ast }}exp(-\frac{({\tilde{\mathbf{S}}}_{SPGR}-{\tilde{\mathbf{M}}}_{SPGR}){({\tilde{\mathbf{S}}}_{SPGR}-{\tilde{\mathbf{M}}}_{SPGR})}^T}{2\tilde{\sigma }{\sigma }_{SPGR}^2}\hfill \\ \hfill & -\frac{({\tilde{\mathbf{S}}}_{bSSF{P}_0}-{\tilde{\mathbf{M}}}_{bSSF{P}_0}){({\tilde{\mathbf{S}}}_{bSSF{P}_0}-{\tilde{\mathbf{M}}}_{bSSF{P}_0})}^T}{2\tilde{\sigma }{\sigma }_{bSSF{P}_0}^2}\hfill \\ \hfill & -\frac{({\tilde{\mathbf{S}}}_{bSSF{P}_{\pi }}-{\tilde{\mathbf{M}}}_{bSSF{P}_{\pi }}){({\tilde{\mathbf{S}}}_{bSSF{P}_{\pi }}-{\tilde{\mathbf{M}}}_{bSSF{P}_{\pi }})}^T}{2\tilde{\sigma }{\sigma }_{bSSF{P}_{\pi }}^2})\text{d}{\lambda }^{\ast },\hfill \end{array}$$ [5]

where λ* = ( T_2,s T_2,l T_1,s T_1,l ) is the vector of the remaining parameters, $\frac{1}{\lambda }=\frac{1}{T_{2,s}}\frac{1}{T_{2,l}}\frac{1}{T_{1,s}}\frac{1}{T_{1,l}},\text{d}{\lambda }^{\ast }=\text{d}{T}_{2,l}\text{d}{T}_{1,s}\text{d}{T}_{1,l},\text{and}{\tilde{\sigma }}_{SPGR}=K\frac{{\sigma }_{SPGR}}{{\sum }_{k=1}^K{S}_{SPGR}^k},{\tilde{\sigma }}_{bSSF{P}_0}=N\frac{{\sigma }_{bSSF{P}_0}}{{\sum }_{n=1}^N{S}_{bSSFP}^0}\text{and}{\tilde{\sigma }}_{bSSF{P}_{\pi }}=N\frac{{\sigma }_{bSSF{P}_{\pi }}}{{\sum }_{n=1}^N{S}_{bSSFP}^{\pi }}$ represent the normalized SDs of the noise. In the present study, the number of measurements s much greater than the number of derived parameters, so that results are largely independent of reasonable selections of prior probabilities (37). We assumed a form of noninformative Jeffreys priors, that is, P(λ*) = 1/λ* with obvious notation for multiplication of parameter reciprocals (27-28, 39).

2.2.2. Approach 2: Normalized data and unknown standard deviation of noise: Jeffreys priors σ

We now describe the analysis for the case in which there is insufficient background in the SPGR and bSSFP images to ensure accurate estimation of the noise SD. We again assume Jeffreys priors on σ, so that $\text{P}\left(\sigma \right)=\frac{1}{{\sigma }_{SPGR}}\frac{1}{{\sigma }_{bSSF{P}_0}}\frac{1}{{\sigma }_{bSSF{P}_{\pi }}}$ with the same assumption on λ* as above. Then, using ${\int }_0^{\infty }{\text{y}}^{-\text{a}∕2}exp(-\frac{b}{2y})\text{dy}=\Gamma \left(\frac{a-2}{2}\right){\left(\frac{2}{b}\right)}^{(a-2)∕2}$, where Γ is the gamma function, Equation 3 for the posterior PDF of f_s becomes (see Appendix)

$$\begin{array}{cc}\hfill \text{P}(f_s\mid \tilde{\mathbf{S}})& =\frac{\Gamma K∕2)\Gamma (N∕2)\Gamma (N∕2)}{8\sqrt{{\pi }^K}\sqrt{{\pi }^N}\sqrt{{\pi }^N}}\int \frac{1}{\text{P}\left(\tilde{\mathbf{S}}\right)}-\frac{1}{{\Lambda }^{\ast }}{\left(({\tilde{\mathbf{S}}}_{SPGR}-{\tilde{\mathbf{M}}}_{SPGR}){({\tilde{\mathbf{S}}}_{SPGR}-{\tilde{\mathbf{M}}}_{SPGR})}^T\right)}^{\frac{K}{2}}(({\tilde{\mathbf{S}}}_{bSSF{P}_0}\hfill \\ \hfill & -{{\tilde{\mathbf{M}}}_{bSSF{P}_0}){({\tilde{\mathbf{S}}}_{bSSF{P}_0}-{\tilde{\mathbf{M}}}_{bSSF{P}_0})}^T)}^{-\frac{-N}{2}}(({\tilde{\mathbf{S}}}_{bSSF{P}_{\pi }}\hfill \\ \hfill & -{{\tilde{\mathbf{M}}}_{bSSF{P}_{\pi }}){({\tilde{\mathbf{S}}}_{bSSF{P}_{\pi }}-{\tilde{\mathbf{M}}}_{bSSF{P}_{\pi }})}^T)}^{-\frac{N}{2}}\text{d}{\lambda }^{\ast },\hfill \end{array}$$ [6]

where again, λ* = ( T_2,s T_2,l T_1,s T_1,l ). As can be seen from Equation 6, the combination of the Jeffreys priors and the Gaussian likelihood functions permitted us to obtain a posterior PDF of f_s, P(f_s | S̃), that is independent of σ_SPGR, ${\sigma }_{bSSF{P}_0}\text{and}{\sigma }_{bSSF{P}_{\pi }}$, eliminating the need to marginalize over these variables. As in the above, the experimental and the theoretical signals were normalized in order to avoid marginalization over $M_{SPGR}^0,M_{bSSF{P}_0}^0\text{and}{M}_{bSSF{P}_{\pi }}^0$. Therefore, the dimensionality of the parameter space is again reduced from 10 to 4.

2.2.3. Approach 3: Non-normalized data and unknown standard deviations of the noise: NIG priors on M₀ and σ

Normalization with respect to the mean of the measured data as described above may introduce a bias due to the potential propagation of uncertainties in these mean values due to noise, especially in the case of a limited number of measurements. To overcome this potential limitation, we also address the case of non-normalized data with unknown noise SD using conjugate priors that permit closed-form solutions. For this, conjugate normal-inverse-Gamma distributions, NIG(M₀, M²), on M₀ and σ, given by

$$\text{NIG}({\mathbf{M}}_0,{\sigma }^2)=\text{NIG}(M_{SPGR}^0,{\sigma }_{SPGR}^2)\text{NIG}(M_{bSSF{P}_0}^0,{\sigma }_{bSSF{P}_0}^2)\text{NIG}(M_{bSSF{P}_{\pi }}^0,{\sigma }_{bSSF{P}_{\pi }}^2).$$

were taken as priors (40). Thus, our prior distributions were given by

$$\text{NIG}(M_0,{\sigma }^2\mid V,\alpha ,\beta )=\text{N}(0\mid V{\sigma }^2)\text{IG}({\sigma }^2\mid \alpha \beta )=\sqrt{\frac{V}{2\pi }}\frac{{\beta }^{\alpha }}{\Sigma \left(\alpha \right)}{\left(\frac{1}{{\sigma }^2}\right)}^{2\alpha +3}exp\left(-\frac{2\beta +V{M}_0^2}{2{\sigma }^2}\right),$$

where N(0 | Vσ²) is a normal distribution over M₀ with zero mean and variance Vσ² , with V = δ/(g g^T), where g represents the vector of theoretical signal measurements and δ is a scale parameter, with large δ indicating a minimally-informative prior. IG(σ² | α, β) is an inverse Gamma prior on σ² with shape parameter α and scale parameter β, with small values of α and β indicating a minimally-informative prior. A detailed description of the NIG prior can be found in (30, 40-41). The combination of the NIG prior distributions with their corresponding Gaussian likelihood functions allows one to obtain a closed form of marginalized joint likelihood function, L(S | λ), which is explicitly independent of M₀ and σ through

$$\begin{array}{cc}\hfill L(\mathbf{S}\mid \lambda )=& \int \text{NIG}({\mathbf{M}}_0,{\sigma }^2)L(\mathbf{S}\mid \Lambda )\text{d}{\sigma }^2\hfill \\ \hfill & C{\left(\left({\mathbf{S}}_{SPGR}{{\mathbf{S}}_{SPGR}}^T\right)-\frac{{\left({\mathbf{g}}_{SPGR}{{\mathbf{S}}_{SPGR}}^T\right)}^2}{\left({\mathbf{g}}_{SPGR}{{\mathbf{g}}_{SPGR}}^T\right)}\right)}^{-\frac{K}{2}}(\left({\mathbf{S}}_{bSSF{P}_0}{{\mathbf{S}}_{bSSF{P}_0}}^T\right)\hfill \\ \hfill & {\frac{{\left({\mathbf{g}}_{bSSF{P}_0}{{\mathbf{S}}_{bSSF{P}_0}}^T\right)}^2}{\left({\mathbf{g}}_{bSSF{P}_0}{{\mathbf{g}}_{bSSF{P}_0}}^T\right)})}^{-\frac{N}{2}}(\left({\mathbf{S}}_{bSSF{P}_{\pi }}{{\mathbf{S}}_{bSSF{P}_{\pi }}}^T\right){\frac{{\left({\mathbf{g}}_{bSSF{P}_{\pi }}{{\mathbf{S}}_{bSSF{P}_{\pi }}}^T\right)}^2}{\left({\mathbf{g}}_{bSSF{P}_{\pi }}{{\mathbf{g}}_{bSSF{P}_{\pi }}}^T\right)})}^{-\frac{N}{2}}\hfill \end{array}$$

where λ = (f₂ T_2,s T_2,l T_1,s T_1,l ), $\text{d}{\mathbf{M}}_0=\text{d}{M}_0^{SPGR}\text{d}{M}_0^{bSSF{P}_0}\text{d}{M}_0^{bSSF{P}_{\pi }}\text{d}{\sigma }^2=\text{d}{\sigma }_{SPGR}^2\text{d}{\sigma }_{bSSF{P}_0}^2\text{d}{\sigma }_{bSSF{P}_{\pi }}^2$ C is a proportionality constant that depends only on α, β, δ, K and N and is independent of $\lambda ,{\mathbf{g}}_{SPGR}=\text{sin}\alpha \left(f_s\frac{1-E_{1,s}}{1-E_{1,s}\text{cos}\alpha }+(1-f_s)\frac{1-E_{1,l}}{1-E_{1,l}\text{cos}\alpha }\right)\text{and}{\mathbf{g}}_{bSSFP}=\mid (f_s({\mathbf{M}}_{a,s}+i{\mathbf{M}}_{b,s})+(1-f_s)({\mathbf{M}}_{a,l}+i{\mathbf{M}}_{b,l}))\mid$, where M_a and M_b were defined in Equation 2. Therefore, the posterior PDF of f_s given by Equation 3 can be rewritten as

$$\text{P}(f_s\mid \mathbf{S})=\frac{1}{\text{P}\left(\mathbf{S}\right)}\int \frac{1}{{\lambda }^{\ast }}L(\mathbf{S}\mid \lambda )\text{d}{\lambda }^{\ast },$$ [7]

where, again, λ* = ( T_2,s T_2,l T_1,s T_1,l ). As can be seen from Equation 7, this third approach also permits reduction of the dimensionality of the parameter space from 10 to 4.

2.3. Bayesian Monte Carlo Analysis of mcDESPOT

Although the dimensionality of the parameter space is considerably reduced using the Bayesian analyses described above, the estimation of f_s still requires a four dimensional integration of the posterior PDF defined in Equation 4, as indicated in Equations 5-7. This remains an intractable high dimensional integration requiring extensive computational time. To overcome this difficulty, rather than standard fully sampled integrations, we explored the use of standard Monte Carlo (MC) integration (42). In brief, MC integration is performed by averaging the values of the integrand, the posterior PDF defined in Equation 5, over a large random set, M, of parameter combinations. By assuming random sampling from a uniform PDF on a grid defined over the integration parameter ranges, the posterior PDF of f_s can be approximated as

$$\text{P}(f_s\mid \mathbf{S})\simeq \frac{V}{M}\sum _{m=1}^M\frac{\text{P}\left({\lambda }_m^{\ast }\right)L(\mathbf{S}\mid {\lambda }_m)}{\text{P}\left(\mathbf{S}\right)},$$ [8]

where V is a volume constant. In the following, we will refer to this method as Bayesian Monte Carlo (BMC).

3. MATERIALS and METHODS

3.1. Simulation Analysis

3.1.1. Input Parameters, Parameter Bounds, and SNR Definition

The following input parameters were used for all simulations: TR_SPGR = TR_bSSFP = 6.5 ms, α_SPGR = [2 4 6 8 10 12 14 16 18 20]°, β_bSSFP = [2 6 14 22 30 38 46 54 62 70]°, with two bSSFP datasets generated with RF phase increments of 0 or π. The limits of the parameter search space for the SRC-NLLS algorithm were: 0 ≤ f_s ≤ 1, 2 ms ≤ T_2,s ≤ 45 ms, 45 ms ≤ T_2,l ≤ 200 ms, 100 ms ≤ T_1,s ≤ 700 ms and 700 ms ≤ T_1,l ≤ 3000 ms. These values were also used as limits of the integrals needed in the fully sampled Bayesian marginalization (Eqs. 5-7) and as ranges for the random sampling needed in the BMC integration (Eq. 8). Details of the SRC-NLLS algorithm can be found in (25-26). Briefly, for each iteration, 20,000 random samples were generated within the specified parameter bounds. The 50 solutions with the lowest least-squares residuals (LSR) were used to construct the bounds of the next iteration. Iterations were terminated after convergence was achieved, defined as a relative difference between the minimum and maximum values of all parameters of less than 1%, or after 30 iterations.

Zero-mean Gaussian white noise of SD σ was added to simulated signals in order to investigate results across a range of SNR, defined as SNR = M₀ / σ, where M₀ represents the signal amplitude at TE = 0 ms. All numerical calculations were performed with MATLAB (MathWorks, Natick, MA, USA).

3.1.2. Comparisons of the BMC and Fully Sampled Integrations

Input parameter values of T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms were used in the following analyses, and were selected based on values obtained in human brain studies (10-12, 14-16, 18-21).

Determination of the optimal total number of samples for the BMC integration

In order to define a value for the number of random samples, M, used in the BMC integration (Eq. 8) that maximizes integration accuracy in the determination of f_s while maintaining reduced computational time, estimations of f_s were performed for different values of M between 10 and 100,000. Analyses were performed for an input f_s value of 0.15 and two SNR values of 500 and 2000. For each combination of SNR and M, results are presented as the mean-squared-error, MSE, that incorporates information about both bias and dispersion (38). MSE was calculated as the difference between the true value, f_s, used in generating the MR signal, and estimated values, ${\widehat{f}}_s^{BMC}$, obtained using BMC integration, according to

$$\text{MSE}=R^{-1}\sum _{r=1}^R{(f_s-{\widehat{f}}_{s,r}^{BMC})}^2,$$ [9]

where R, the total number of noise realizations, was fixed at R = 1001. Analyses were performed for all three Bayesian-based approaches described above.

Comparison of the BMC and fully sampled integrations

The BMC (Eq. 8) and fully sampled (Eq. 5) integration techniques for the estimation of f_s were compared for f_s values of 0.05, 0.15 and 0.3 at SNR values of 500, 1000 and 2000. For each SNR and f_s combination, BMC integration was performed with M = 20,000, which was determined to be an optimal value from the analysis described above. Results are presented as the mean, SD, and mean relative error, MRE, in estimated f_s values over 1001 noise realizations. The MRE were calculated as the difference between estimated values obtained using fully sampled, ${\widehat{f}}_{FS}^s,\text{and}\text{BMC},{\widehat{f}}_s^{BMC}$, integrations through

$$\text{MRE}(\%)=\frac{100}{R}\sum _{r=1}^R\frac{\mid {\widehat{f}}_{s,r}^{FS}-{\widehat{f}}_{s,r}^{BMC}}{{\widehat{f}}_{s,r}^{FS}},$$ [10]

where R is the total number of noise realizations. MRE was used instead of MSE, as defined above, in order to assess relative error for different values of f_s. Extensive simulations demonstrated that the three Bayesian approaches showed very similar performance, so that results for the comparison of BMC and fully sampled integrations are shown for the first approach only (Eq. 5).

3.1.3. Comparison of the SRC Algorithm and the BMC Approaches

Comparison of the SRC- and BMC-mcDESPOT methods for the estimation of f_s were evaluated using numerical simulations. The first evaluation was performed for different f_s values of 0.05, 0.1, 0.15, 0.2, 0.3 and 0.5 at SNR values of 500, 1000 and 2000, where the other underlying input parameters were fixed at T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms. The second evaluation was performed by fixing the f_s and SNR values to 0.15 and 500, respectively, and exploring several values of the remaining parameters, with T_2,s = (10, 15, 20, 25) ms, T_2,l = (70, 100, 110, 150) ms, T_1,s = (300, 400, 450, 550) ms and T_1,l = (1400, 1800, 2000, 2500) ms.

For each parameter combination and SNR, the performance of each method was evaluated by calculating the relative bias, relative dispersion and relative root-mean-squared-error, RMSE, in the estimation of f_s (43). The relative bias, a measure of accuracy, was defined as

$$\text{Bias}(\%)=100\ast \frac{\mid f_s-{\overline{\widehat{f}}}_s\mid }{f_s},$$ [11]

where ${\overline{\widehat{f}}}_s=(1∕R){\sum }_{r=1}^R{\widehat{f}}_s^r$ and R is the total number of noise realizations, R = 1001. The relative dispersion, a measure of precision, was defined as

$$\text{Dispersion}(\%)=100\ast \frac{\sqrt{(1∕(R-1))\sum _{r=1}^R({\widehat{f}}_s^r-{\widehat{f}}_s)}}{f_s},$$ [12]

and finally the RMSE, a measure including both precision and accuracy (38), was defined as the root-square of the sum of the squared relative bias of the estimator and its relative variance by the expression

$$\text{RMSE}(\%)=\sqrt{{\text{Bias}}^2+{\text{Dispersion}}^2.}$$ [13]

3.1.4. Energy Surface and Posterior PDF Analyses

For NLLS, including SRC-NLLS, the least-squares energy surface structure defines in part the reliability of parameter fits. In contrast, for Bayesian analysis, reliability of estimates are defined by the derived posterior PDFs. Therefore, to characterize the SRC-NLLS analysis, minimum LSR (MLSR) was calculated as a function of f_s by fixing f_s to a given value and varying all other parameters. The LSR was calculated over the entire combined multidimensional range of these other parameters for each fixed value of f_s, with the minimum calculated LSR designated as the MLSR. This process was repeated for different fixed values of f_s (24). For the BMC analyses, the posterior PDF of f_s was calculated by fixing f_s to a given value and marginalizing over all remaining parameters as described above. This process was again repeated for different fixed values of f_s. In both cases, the signal was generated with underlying input parameters of T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms, with noise added as described above. All analyses were performed for f_s values of 0.15 and 0.3 and with SNR of 500.

3.2. Experimental Analysis

3.2.1. Data Acquisition

All MRI experiments were performed on a 3T whole body MRI Philips system (Achieva, Best, The Netherlands) using the internal quadrature body coil for transmission and an eight-channel phased-array head coil for reception. Data were collected from the brain of a 23 year-old healthy female, from whom written informed consent was obtained prior to participation. All examinations were performed with approval of the local institutional review board.

Three separate sets of acquisition and reconstruction parameters were used to generate mcDESPOT data with a range of SNR and spatial resolution (SR). We selected oblique imaging slices that contained well-recognized and well-represented white and grey matter, as well as clear delineation of ventricles. The oblique slice package was centered on a plane roughly 4 cm superior to the external auditory meatus. In the SNR_High protocol, that with the highest SNR, all SPGR and bSSFP datasets were acquired with 4 signal averages (SA) and with SR = 1 × 1 × 3 mm³, and then reconstructed to SR = 0.45 × 0.45 × 3 mm³. In the SNR_Intermediate protocol, with intermediate SNR datasets, SPGR and bSSFP datasets were acquired with 1 SA and SR = 1 × 1 × 3 mm³, reconstructed to SR = 0.45 × 0.45 × 3 mm³. In the SNR_Low protocol, with the lowest SNR, SPGR and bSSFP datasets were acquired with 1 SA and SR = 1 × 1 × 1 mm³ and reconstructed to SR = 0.45 × 0.45 × 1 mm³. For each of these three protocols, the 3D SPGR images were acquired with FAs α_SPGR = [2 4 6 8 10 12 14 16 18 20]°, with TE_SPGR = 1.6 ms and TR_SPGR = 6.6 ms, while the 3D bSSFP images were acquired with FAs β_bSSFP = [2 6 14 22 30 38 46 54 62 70]°, with TE_bSSFP = 3.3 ms and TR_bSSFP = 6.6 ms. For all datasets, the bSSFP images were acquired twice, once with Δω = 0 Hz (i.e. bSSFP_π) and once with Δω = 1/(2 * TR) = 76 Hz (i.e. bSSFP₀), with, in both cases, ϑ = π. All images were obtained with field of view (FOV) = 200 × 200 × 9 mm³. Further, SPGR and bSSFP images with a null FA were acquired for each protocol and used to estimate the noise SD needed for BMC1.

In this in-vivo study, the SNR was defined as (44-45)

$$\text{SNR}=\frac{\sqrt{{\stackrel{‒}{S}}_{SPGR}{({\alpha }_{SPGR}=8^o)}^2-2L{\widehat{\sigma }}^2({\alpha }_{SPGR}=0^o)}}{\widehat{\sigma }({\alpha }_{SPGR}=0^o},$$

where L = 8 is the total number of channels of the phased-array head coil and S̄_SPGR(α_SPGR = 8°)² was calculated as the square of the mean signal from all regions in the SPGR image obtained with α_SPGR = 8° corresponding to the maximum signal image in the regions over which the MWF maps were calculated (See Data Analysis section). In addition, $\widehat{\sigma }({\alpha }_{SPGR}=0^o)$ is the noise SD calculated through $\widehat{\sigma }({\alpha }_{SPGR}=0^o)=\sqrt{{\stackrel{‒}{S}}_{SPGR}^2({\alpha }_{SPGR}=0^o)∕2L,}\text{where}{\stackrel{‒}{S}}_{SPGR}^2({\alpha }_{SPGR}=0^o)$ is the mean of the squared pixel intensity values calculated from a large region in the SPGR image obtained with α_SPGR = 0° (44). The SNR_High , SNR_Intermediate and SNR_Low values calculated for each protocol described above were ~90, ~40 and ~25, respectively.

For the B₁ calibration needed for mcDESPOT (32), the S-DAM method (35) was used. Two 3D images with excitation FAs of 70° and 140° were acquired, with SR = 2 × 2 × 3 mm³, and then reconstructed to SR = 0.45 × 0.45 × 3 mm³ for both SNR_High and SNR_Intermediate protocols, or to SR = 0.45 × 0.45 × 1 mm³ for SNR_Low protocol. The other acquisition parameters were: FOV = 200 × 200 × 9 mm³, TE_S-DAM = 1.73 ms and TR_S-DAM = 1538 ms.

3.2.2 Data Analysis

Images from each of the three mcDESPOT datasets with different SNR and SR, as described above, were analyzed on a voxel-wise basis. First, the S-DAM datasets were used to generate B₁ maps. These were then used to generate T₁ maps through a fit of the SPGR data as a function of FAs to a functional form incorporating only a single underlying component. Similarly, T₂ and Δω * TR maps were then generated according to the DESPOT₂-FM method (36) by fitting the bSSFP datasets as a function of FA to a single component form of the signal, using the voxel-wise B₁ and T₁ values obtained as outlined above. Finally, the Δω * TR and B₁ maps, combined with the SPGR and bSSFP datasets, were used to generate f_s maps reflecting MWF (18-23).

For each mcDESPOT dataset, MWF maps were generated using the SRC-NLLS algorithm and the BMC approaches as described above. To correspond to actual practice (24, 26, 46), the boundary conditions for the SRC-NLLS algorithm were set to 0 ≤ f_s ≤ 0.6, 2 ms ≤ T_2,s ≤ 45 ms, 75 ms ≤ T_2,l ≤ 200 ms, 100 ms ≤ T_1,s ≤ 700 ms and 700 ms ≤ T_1,l ≤ 3000 ms. The same boundary conditions were used as sampling ranges for BMC. Further, manual segmentation was performed to eliminate ventricles and non-brain regions of the images, with statistical analyses and MWF maps generated for the remaining regions of interest. The statistical analyses, consisting of calculations of the mean, SD, and coefficient of variation, CoV, for the pixel-wise estimate of MWF, were performed on three different image regions as indicated in Figure 9. The first region encompasses the frontal lobes, the second region encloses the temporal lobes, and the third region encloses the occipital lobes. For each region, all the statistical analyses were calculated from pixels with MWF > 1% defined from the MWF maps generated using BMC3 approach. Finally, MLSR and posterior PDF analyses were performed on a single voxel from data obtained with the SNR_Low dataset as described in the Simulation Analysis section above.

Figure 9.

Left column: means and standard deviations of estimates of f_s, reflecting MWF, calculated in the three brain regions indicated in the above illustration from the SNR_High (≃ 90), SNR_Intermediate (≃ 40),and SNR_Low (≃ 25) protocols. In each case, results obtained from SRC-NLLS and the three BMC approaches described in the text are presented. Right column: the corresponding coefficients of variation (CoV).

4. RESULTS

4.1. Simulation Analysis Results

The effect of varying the number of random samples used in BMC integration

Figure 1 shows the MSE, as defined in Equation 9, for estimation of the short fraction, f_s, as a function of the total number of samples, M, used in the BMC integration (Eq. 8). Results are presented for SNR values of 500 and 2000 using the three different BMC approaches described in the Materials and Methods section. For each of the three and for both values of SNR, the MSE decreased with increasing M, as expected. However, minimal further decrease in MSE as a function of M was seen for M ≃ 6000 for SNR = 500, and for M ≃ 10000 for SNR = 2000, indicating that, as expected, higher SNR permits further refinement of the accuracy of the integration through use of additional sampled points.

Figure 1.

MSE (Eq. 9) in the estimation of the short fraction, f_s, as a function of the total number of random samples, M, used in the BMC integration. Results are presented for two different SNR values, 500 and 2000, using the three different BMC approaches described in the text (cf. "Bayesian Analysis of mcDESPOT" section). The other underlying input parameters were fixed to values of T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms.

Accuracy of f_s determination as a function of SNR for BMC compared to fully sampled integration

Figure 2 shows the mean, SD and MRE, as defined in Equation 10, for the estimation of f_s calculated over 1001 noise realizations using fully sampled (Eq. 5) or BMC (Eq. 8) integration for three different f_s and SNR combinations. As is evident, BMC integration provides results that are virtually indistinguishable from those obtained with the fully sampled integration. In spite of the large number of randomly sampled points used for BMC integration, processing time was reduced to a few minutes from the several hours required for the fully sampled integration.

Figure 2.

Mean and standard deviation (top row), and MRE (bottom row) of f_s estimated over 1001 noise realizations obtained using fully sampled (Eq. 5) and BMC (Eq. 8) integrations for different SNR and f_s values. The other underlying input parameters were fixed to values of T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms.

Comparison of the accuracy of SRC-NLLS and BMC

Figure 3 shows the relative bias, dispersion and RMSE in the estimation of f_s for several combinations of SNR and true input f_s values. As expected, relative bias, dispersion and RMSE decreased with increasing SNR or f_s values in all cases. Comparing the methods, the BMC approaches demonstrated a substantial reduction of relative bias, dispersion and RMSE compared to the SRC-NLLS algorithm, especially at low-to-moderate SNR (i.e. SNR < 2000). However, at high SNR, the SRC-NLLS was able to perform, overall, almost as well as the BMC, especially for f_s ≥ 0.15, that is, with more equal component fractions. In addition, at low SNR (i.e. SNR < 1000), BMC3 showed a slight overall decrease in relative bias, dispersion and RMSE compared to BMC1 and BMC2. Using average values over all parameter combinations as a benchmark for performance, the averaged values for relative bias were 18/4/4/4%, for relative dispersion 44/24/25/24% and for RMSE 48/24/25/24% for SRC-NLLS, BMC1, BMC2 and BMC3, respectively, indicating the superior performance of BMC analysis. For SNR = 500, the mean computation times over several noise realizations for the calculation of f_s using a 2.4 GHz computer for SRC-NLLS and for each of the BMC were ~0.6 ms and ~1.7 ms, respectively, with the computation time for the three BMC analyses being essentially identical.

Figure 3.

Relative bias, dispersion and RMSE in the estimation of f_s using the SRC-NLLS algorithm and the three BMC approaches. Results are presented for several combinations of SNR and f_s values. The other underlying input parameters were fixed to values of T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms.

Figure 4 shows the relative bias, dispersion and RMSE in the estimation of f_s using SRC-NLLS and BMC analyses. Results were obtained by fixing the f_s and SNR values to 0.15 and 500, respectively, and varying the values of the remaining, nuisance, parameters. In all cases, the relative dispersion and RMSE of estimated f_s values varied minimally with respect to the values of the nuisance parameters. Although the variation in relative bias was greater, it nevertheless remained relatively small for all BMC methods, between 2% and 13%, as compared to the SRC-NLLS, for which bias ranged between 12% and 44%. These results showed that the undesired dependence on the derived value of f_s on nuisance parameters is considerably attenuated through use of Bayesian marginalization. For all parameter combinations investigated, the Bayesian approaches demonstrated substantially reduced relative bias, dispersion and RMSE as compared to the SRC-NLLS algorithm. We found values for relative bias of 32/5/5/4%, for relative dispersion 69/30/31/28%, and for RMSE 76/30/32/28% for the SRC-NLLS, BMC1, BMC2 and BMC3, respectively, again indicating the substantial improvement in the quality of estimates of f_s through use of BMC integration.

Figure 4.

Relative bias, dispersion and RMSE in the estimation of f_s using the SRC-NLLS algorithm and the three BMC approaches. Results were obtained by fixing the f_s and SNR values to 0.15 and 500, respectively, and varying the values of the remaining parameters. Values used were T_2,s = (10, 15, 20, 25) ms, T_2,l = (70, 100, 110, 150) ms, T_1,s = (350, 400, 450, 550) ms and T_1,l = (1400, 1800, 2000, 2500) ms.

Reliability measures of estimates of f_s for SRC-NLLS and BMC

Figure 5 shows the histogram of f_s estimates obtained from 1001 noise realizations using SRC-NLLS and BMC analyses (top row), and the MLSR and posterior PDF for f_s obtained for a single noise realization as a function of f_s (middle and bottom rows, respectively). Results were obtained from signals generated with SNR = 500. It is clear from the histograms that the bias and dispersion in f_s estimates were much greater for SRC-NLLS than for the BMC. The MLSR and posterior PDF illustrations indicate the underlying basis for the superior performance of the Bayesian methods as compared to SRC-NLLS. As is readily seen, the MLSR curves for f_s exhibit an extensive flat region extending on either side of the true input f_s values. In contrast, the posterior PDF of f_s is sharply peaked around the true input f_s value.

Figure 5.

Top row: The histogram of f_s estimates obtained from 1001 noise realizations using the SRC-NLLS algorithm and BMC. Middle row: Minimum least squares residual (MLSR) of f_s as a function of f_s for a single noise realization. Bottom row: The normalized posterior probability density function (PDF) of f_s as a function of f_s obtained using BMC for a single noise realization. Normalization was performed with respect to the maximum value of each corresponding posterior PDF. In each case, the vertical red line indicates the true input value of f_s. Results are presented for data generated with SNR of 500 and for two true input f_s values of 0.15 (left column) and 0.3 (right column). The other underlying input parameters were fixed to T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms.

4.2. Experimental Analysis Results

B₁ and off-resonance frequency, Δω, mapping from noisy images

Figure 6 shows representative bSSFP and SPGR images and corresponding calculated B₁ and Δω * TR maps for each of the three SNR protocol acquisitions. The impact of SNR is readily seen, with small anatomic structures, such as blood vessels and white-grey matter boundaries, increasingly obscured for decreasing SNR. Nevertheless, for each of the three levels of SNR, high quality B₁ and Δω * TR parameter maps were derived through a mono-component fit. Further, the calculated B₁ and Δω * TR maps clearly demonstrate the spatial extent and the degree of RF inhomogeneity and off-resonance effects, respectively.

Figure 6.

Representative examples of axial bSSFP and SPGR weighted-images and the corresponding B₁ and Δω * TR maps calculated using S-DAM and DESPOT2-FM, respectively. Results are presented for each of the three protocols described in the text (see "Materials and Methods"). SNR_High (≃ 90) represents the highest SNR datasets, SNR_Intermediate (≃ 40) represents the intermediate SNR datasets, and SNR_Low (≃ 25) represents the lowest SNR datasets.

Qualitative and quantitative comparison of MWF maps derived using SRC-NLLS and BMC as a function of SNR

Figure 7 shows f_s parameter maps, reflecting MWF, derived from the in-vivo brain images obtained with the SNR_High, SNR_Intermediate and SNR_Low protocols using SRC-NLLS, and BMC1, BMC2, and BMC3 analyses. At SNR_High, all methods performed well, but with superior performance demonstrated with BMC2 and BMC3 as compared to BMC1 and SRC-NLLS. This is evidenced by the much more homogenous MWF maps for BMC2 and BMC3. For the lower SNR datasets (Figs. 7-8), both SRC-NLLS and BMC1 results show substantial random variation in estimated MWF in several image regions, while BMC2 and BMC3 performed similarly and produced much higher quality MWF maps. This is clearly seen from Figure 8 showing MWF calculated from different slices of the SNR_Low dataset. To examine this quantitatively, the mean, SD and CoV of the estimates of MWF were calculated for three different brain regions corresponding to the frontal, temporal and occipital lobes, as shown in Fig 9. In each case, although the mean values varied little with decreasing SNR, both SD and CoV increased with decreasing SNR. The results clearly indicate the superior overall performance of BMC2 and BMC3 as compared to BMC1 and SRC-NLLS.

Figure 7.

Representative f_s maps, reflecting the spatial distribution of MWF, derived from in-vivo brain images using the SRC-NLLS algorithm and the three BMC approaches described in the text. MWF maps were derived from the SNR_High (≃ 90), SNR_Intermediate (≃ 40) and SNR_Low (≃ 25) datasets as described in the text.

Figure 8.

MWF maps derived from in-vivo brain images using the SRC-NLLS algorithm and the three BMC approaches described in the text. MWF maps were derived for different slices from the SNR_Low (≃ 25) datasets.

Reliability measures of estimates of MWF for SRC-NLLS and BMC

Figure 10 shows the MLSR obtained through SRC-NLLS and the Bayesian posterior PDF of MWF as a function of MWF, that is, of f_s, for the SNR_Low brain data shown in Figure 7. As is readily seen, the MLSR curve for MWF exhibits an extensive flat region. In contrast, the posterior PDF of MWF is peaked around a unique MWF value. These results correspond to the simulation results shown in Figure 5.

Figure 10.

Minimum least squares residual (MLSR) of f_s, representing MWF, as a function of f_s, and the normalized posterior probability density function (PDF) of f_s as a function of f_s obtained using the three BMC methods. Results were obtained from a single voxel from datasets obtained with SNR_Low (≃ 25). Normalization was performed with respect to the maximum value for each posterior PDF.

5. DISCUSSION

5.1. BMC vs. SRC-NLLS

We investigated three different implementations of Bayesian probability analyses for the estimation of the short-T₂ fraction, f_s, representing MWF, from the two-component mcDESPOT signal model. BMC1 uses normalization by average signal amplitude, and assumes that noise can be accurately estimated from signal-free regions of the image. BMC2 likewise uses average amplitude normalization, but incorporates a full treatment of noise as an unknown variable through marginalization. BMC3, the most general approach, does not use amplitude normalization and incorporates marginalization over both noise and signal amplitude.

We evaluated the performance of these implementations numerically over wide ranges of SNR and underlying input parameter values, and experimentally from MR imaging data of human brain obtained through use of three different protocols reflecting different noise levels and spatial resolutions. The accuracy and precision in the estimation of f_s (or MWF) obtained with each of the three BMC analyses were systematically compared to those obtained using the SRC-NLLS algorithm, conventionally used for parameter estimation from mcDESPOT. Overall, we found greatly improved accuracy and precision in the estimation of f_s, that is MWF, using the BMC as compared to the SRC-NLLS, especially at lower-to-moderate SNRs (Figs. 3-4 and 7-9). This may be attributable to several facts. First, in contrast to SRC-NLLS, noise SD was either explicitly (BMC1) or implicitly (BMC2 and BMC3) incorporated into the BMC approaches. In addition, in the Bayesian analyses, the parameter of interest, f_s (or MWF), is estimated independently through marginalization over all values of the remaining nuisance parameters. This is in contrast to the SRC-NLLS algorithm in which all unknown parameters are estimated simultaneously. This leads to potential difficulties with local minima, especially in higher-dimensional problems (24). Fundamentally, this arises from the multiple compensating degrees of freedom available through the various parameters, so that different parameter combinations can result in nearly-equivalent LSR. This is reflected in the flattening of the energy surface of MLSR (Figs. 5 and 10) seen with increasing model complexity (24).

Our results are in agreement with the previous work of Neil and Bretthorst (27) and Bouhrara et al. (31) which showed improved accuracy and precision in the estimation of the diffusion and transverse relaxation parameters, respectively, from two-component signal models through use of Bayesian-based approaches compared to NLLS-based methods. In addition, the Bayesian analysis that we have outlined is widely applicable and has great potential to improve the quality of derived MWF from other MRI techniques, such as gradient recalled echo- or other steady state-based analyses (47-49, 50).

In the present work, we based our analysis on the conventional SPGR and bSSFP signal models (Eqs. 1-2) incorporated into mcDESPOT. These models ignore the transverse signal decay that occurs during the time between the preceding pulse and the echo formation in SPGR, and underestimates the bSSFP signal by neglecting the fact that echo formation occurs at a time TR_bSSFP/2, rather than TR_bSSFP, after the pulse that precedes acquisition. This neglect of finite TE effects can introduces bias in parameter estimates, especially for data obtained with relatively long TEs (24, 51). However, the focus of the present work was to analyze the mcDESPOT method as usually implemented.

5.2. BMC Analysis

Our simulations showed that the performance of the three BMC methods investigated here were very similar with, in general, a slightly improved accuracy and precision found using BMC3 (Figs. 3-4). We attribute this to the fact that, unlike BMC3, BMC1 and BMC2 use normalization with respect to the mean amplitude of noisy data. Errors in the calculation of this mean due to noise may introduce a bias in the derivation of the remaining signal parameters, including f_s (or MWF), especially in the case of a limited number of measurements. In our study, the mean amplitude was calculated over a large number of data points, so that the improved accuracy and precision using BMC3 was marginal.

In this work, all priors were taken as noninformative in order to avoid introducing biases into the derived posterior PDF. In addition, the likelihood functions were the dominant terms in the posterior PDFs, so that use of conjugate priors also resulted in negligible bias, as seen from our results. These results motivate investigation of other forms of prior probabilities, such as spatial priors (30), which may further reduce the dispersion seen in the derived MWF maps at low SNR (Figs. 7-8).

Comparison of the three BMC methods was performed under the assumption that the correct values of σ_SPGR, σ_bSSFP0 and σ_bSSFPπ, needed for BMC1, are known precisely in the numerical simulations. However, the quality of the estimate of f_s, that is, MWF, will depend upon the quality of the estimates ${\widehat{\sigma }}_{SPGR},{\widehat{\sigma }}_{bSSF{P}_0}\text{and}{\widehat{\sigma }}_{bSSF{P}_{\pi }}$. Simulation analysis (Table 1) indicates that, as expected, moderate errors (≤ 20%) in the estimated values of noise parameters, such as would be the case for estimation from large background regions of the images (38, 52), have a negligible impact on parameter estimates. Further, the in-vivo analyses highlighted the limited performance of BMC1 due to the lack of spatial invariance of the noise (Fig. 11), likely due to the correlation between the channels of the phased-array coil. This spatial variation of noise may explain the inferiority of parameter estimates from BMC1 as compared to BMC2 and BMC3 for the in-vivo analyses (Fig. 7-9). This problem may be addressed through use of techniques that consider the non-uniformity of noise (53).

Table 1.

Effect of error in σ on the degree of bias, calculated as ΔBias = |Bias (exact σ) - Bias (estimated σ)| and on the degree of variation, calculated as ΔDispersion = |Dispersion (exact σ) - Dispersion (estimated σ)| in estimated f_s using the BMC 1 approach. Results were obtained from 1001 noise realizations for three representative SNRs and for assumed errors in the estimation of σ of 10% and 20% and 50%. The underlying input parameters were fixed at f_s = 0.15, T_2,s = 15 ms, T_2,l = 100 ms, T_1,s = 450 ms and T_1,l = 1800 ms. As indicated, ΔBias and ΔDispersion were negligible for moderate errors (≤ 20%) in σ.

Error (%)	10			20			50
SNR	500	1000	2000	500	1000	2000	500	1000	2000
ΔBias (%)	4	0	0	5	1	1	7	4	3
ΔDispersion (%)	3	2	1	6	4	2	10	8	5

Figure 11.

Examples of magnitude noise images obtained using the SPGR and bSSFP sequences with null FA. As can clearly be seen, noise exhibited substantial spatial variation.

In this work, we assumed a Gaussian noise distribution. This allowed us to obtain closed form expressions for marginalized likelihood functions, thereby reducing the required integrations. However, it is well-known that the noise in magnitude MR images can be described by a Gaussian distribution only for SNR >~ 3 (43, 54), which is therefore a required approximation for the applicability of our specific results. However, the extension of BMC1 to the case of Rician or, more generally, non-central-chi distributed noise, as is appropriate for multiple receive coils (44-45), is straightforward through appropriate modification of the likelihood function given by Equation 4. In contrast, application of BMC2 and BMC3 as implemented here for non-Gaussian noise would require construction of appropriate priors to permit derivation of closed form expressions for marginalized likelihood functions. Lacking this, the required integrations could, in principle, be performed numerically.

BMC integration allows a considerable reduction in processing time as compared to fully sampled integration. In our work, we used random sampling over a uniform grid to compute the posterior PDF of f_s , that is, MWF. Careful selection of M, the number of sampled points used in the BMC integration, was required, given the expected slow rate of convergence of only $~1\sqrt{M}$ (41-42); we found that M ≃ 20,000 samples led to an extremely accurate results (Fig. 2). Other strategies may be preferred to the BMC integrations performed here, in accordance with the comment of Jaynes: "Whenever there is a randomized way of doing something, there is a non-randomized way which yields better results, but requires more thinking" (55). Indeed, other sampling strategies such as stratified sampling or Markov Chain MC may prove to be more efficient for calculating the posterior PDF of f_s (or MWF) when high-dimensional marginalization is required, as is the case here.

5.3. SRC-NLLS Analysis

The SRC algorithm as applied here and in the conventional implementation of mcDESPOT is a particular implementation of NLLS. Our results indicate that SRC-NLLS provides accurate and precise estimates of f_s, that is, MWF, from high SNR datasets (Figs. 3-4 and 7-9), in agreement with previous studies (24, 26). Further, unlike BMC, in which all parameters except the one under consideration are subject to marginalization, SRC-NLLS provides a rapid estimate of all unknown parameters simultaneously with modest computational requirements. Using SRC-NLLS, the mean ± SD of estimated values of T_2,s, T_2,l, T_1,s and T_1,l obtained from SNR_High protocol data were 17 ± 10 ms, 81 ± 13 ms, 641 ± 53 ms, and 2127 ± 229 ms, respectively. Those values are in good agreement with values reported in the literature (46, 56). However, as has been previously pointed out (46), some of those parameters may exhibit unreasonable values in certain circumstances, such as upon the incorporation of chemical exchange into the formalism. This further highlights the potential for BMC-mcDESPOT to improve upon existing higher-dimensional analyses.

Furthermore, as with any least squares-based algorithm, the accuracy of the SRC-NNLS algorithm is limited by the presence of local minima and flat LSR energy surfaces (24); this greatly reduces the quality of parameter estimates at low-to-moderate SNR (Figs. 3-5 and 7-9). These considerations may represent an important limitation in the application of SRC-NLLS to mcDESPOT parameter estimates in the setting of modest SNR, including high-resolution whole-brain studies and studies limited by patient compliance.

5.4. mcDESPOT Analysis

We analyzed the mcDESPOT signal model without exchange for two reasons. First and foremost, the most basic form of mcDESPOT still requires a fit of five unknown parameters using SRC-NLLS or, with Bayesian analysis, marginalization over four nuisance parameters. Further, a recent study showed that the incorporation of two-site exchange has a minimal effect on estimated MWF values (46), which is the main parameter of interest. However, the extension of the BMC to incorporate exchange is straightforward. In addition, to correspond to the conventional analysis of mcDESPOT, we neglected the effects of MT between free water protons and macromolecules, inherent to all steady-state measurements on biological systems (57). However, in a recent study, Liu et al. (50) developed a new method termed mcRISE to account for MT effects through incorporation of a macromolecular proton pool in exchange with mcDESPOT water pools. It has been shown that MT effects have notable impact on derived short fraction estimate in human knee cartilage corresponding, presumably, to proteoglycan water fraction. On the other hand, Zhang et al. (56) compared mcDESPOT measurements obtained using long or short RF pulses, which result in differing amounts of MT. The results indicated that the extracted MWF values are relatively insensitive to changes in MT. Further work, including the application of the mcRISE approach to brain analysis, is required to further understand the impact of MT on MWF.

Of note is the fact that a recent in-vivo study indicated that MWF values derived from mcDESPOT were substantially greater than those derived from a CPMG analysis (46). This was attributed to fact that mcDESPOT and CPMG are influenced to substantially different degrees by the experimental and physiological effects that cause received signals to deviate from ideal models (24, 46). These effects include diffusion, exchange, off-resonance effects, magnetization transfer, J-coupling, spin locking, internal gradients, magnetization spoiling and noise. The importance of these effects in a particular experiment will depend upon both the specifics of the sample or subject under investigation and the details of the pulse sequence, including the selection of parameters such as TE, TR, FAs, and gradient durations and amplitudes. Given the substantial differences between conventional SRC-NLLS and BMC analysis of mcDESPOT, and what we believe to be the substantially improved accuracy of the latter, it is not unexpected that our results for MWF would differ from published values. Indeed, the derived MWF estimates using BMC2 and BMC3 were somewhat slightly lower than those derived using SRC-NLLS. Nevertheless, our in-vivo results were, overall, consistent with values reported in the literature (18-21, 32, 46, 56).

6. CONCLUSIONS

Estimation of MWF in the human brain from mcDESPOT was markedly improved through use of Bayesian analysis, especially at low-to-moderate SNR. We found that Bayesian approaches that incorporated marginalization over noise, designated BMC2 and BMC3, demonstrated superior accuracy and precision in parameter estimation as compared to SRC-NLLS or a Bayesian method, BMC1, that incorporates naive estimation of noise assuming spatially-invariant noise. These results will be of significance for high resolution whole-brain MWF mapping. Further, BMC-mcDESPOT can readily be extended to incorporate exchange between pools and to other tissues, including, for example, the estimation of the proteoglycan-associated water fraction in cartilage (58-59).

Highlights.

• We propose a significantly improved technique for estimating myelin water fraction in human brain using magnetic resonance imaging

• The proposed procedure is based on Bayesian analysis of the mcDESPOT signal model

• Detailed simulations and in-vivo studies indicate the superiority of this Bayesian approach as compared to conventional approach based on least-squares estimates

• These results will substantially improve the quality of high-resolution whole-brain mapping of myelination in human brain

7. APPENDIX.

Given an i.i.d sequence of measurements {S} of normal random variables having mean μ and standard deviation σ, the Gaussian likelihood function, L(S | μ, σ), of {S} is given by

$$L\left(\mathbf{S}\mid \mu ,\sigma \right)={\left(\frac{1}{\sigma \sqrt{2\pi }}\right)}^{R}exp\left(-\frac{\sum _{r=1}^R{(S^r-\mu )}^2}{2{\sigma }^2}\right)$$

where R is the total length of the sequence of measurements. Assuming a Jeffreys prior on σ, that is, P(σ) ∝ 1/σ, the marginalized likelihood function over σ is given by

$$\begin{array}{cc}\hfill L(\mathbf{S}\mid \mu )& ={\int }_0^{\infty }\text{P}\left(\sigma \right)l(\mathbf{S}\mid \mu ,\sigma )\text{d}\sigma ={\int }_0^{\infty }\frac{1}{\sigma }{\left(\frac{1}{\sigma \sqrt{2\pi }}\right)}^{R}exp\left(-\frac{\sum _{r=1}^R{(S^r-\mu )}^2}{2{\sigma }^2}\right)\text{d}\sigma \hfill \\ \hfill & =\left(2\pi \right)\frac{R}{2}{\int }_0^{\infty }{\sigma }^{-(R+1)}exp\left(-\frac{\sum _{r=1}^R{(S^r-\mu )}^2}{2{\sigma }^2}\right)\text{d}\sigma \hfill \end{array}$$ [A.1]

Using the following change of variables: $b_{r=1}^R{(S^r-\mu )}^2$ and y = σ² , Equation A.1 becomes

$$\begin{array}{cc}\hfill L(\mathbf{S}\mid \mu )& ={\left(2\pi \right)}^{\frac{R}{2}}{\int }_0^{\infty }{y}^{-\frac{(R+1)}{2}}exp(-\frac{b}{2y})\frac{1}{2}{y}^{-\frac{1}{2}}\text{dy}\hfill \\ \hfill & =\frac{1}{2}{\left(2\pi \right)}^{\frac{R}{2}}{\int }_0^{\infty }{y}^{-\frac{a}{2}}exp(-\frac{b}{2y})\text{dy}\hfill \end{array}$$ [A.2]

where a (R + 2).

Finally, using ${\int }_0^{\infty }{\text{y}}^{-a∕2}exp(-\frac{b}{2y})\text{dy}=\Gamma \left(\frac{a-2}{2}\right){\left(\frac{2}{b}\right)}^{(a-2)∕2}$, where and Γ is the gamma function, the marginalized likelihood function given by Equation A.2 becomes

$$L(\mathbf{S}\mid \mu )=\frac{1}{2}{\left(2\pi \right)}^{-\frac{R}{2}}\Gamma \left(\frac{a-2}{2}\right){\left(\frac{2}{b}\right)}^{(a-2)∕2}=\frac{\Gamma (R∕2)}{2\sqrt{{\pi }^R}}{\left(\sum _{r=1}^R{(S^r-\mu )}^2\right)}^{-\frac{R}{2}}$$ [A.3]

which is independent of σ.