Rapid Simultaneous High-resolution Mapping of Myelin Water Fraction and Relaxation Times in Human Brain using BMC-mcDESPOT

Abstract

A number of central nervous system (CNS) diseases exhibit changes in myelin content and magnetic resonance longitudinal, T₁, and transverse, T₂, relaxation times, which therefore represent important biomarkers of CNS pathology. Among the methods applied for measurement of myelin water fraction (MWF) and relaxation times, the multicomponent driven equilibrium single pulse observation of T₁ and T₂ (mcDESPOT) approach is of particular interest. mcDESPOT permits whole brain mapping of multicomponent T₁ and T₂, with data acquisition accomplished within a clinically realistic acquisition time. Unfortunately, previous studies have indicated the limited performance of mcDESPOT in the setting of the modest signal-to-noise range of high-resolution mapping, required for the depiction of small structures and to reduce partial volume effects. Recently, we showed that a new Bayesian Monte Carlo (BMC) analysis substantially improved determination of MWF from mcDESPOT imaging data. However, our previous study was limited in that it did not discuss determination of relaxation times. Here, we extend the BMC analysis to the simultaneous determination of whole-brain MWF and relaxation times using the two-component mcDESPOT signal model. Simulation analyses and in-vivo human brain studies indicate the overall greater performance of this approach compared to the stochastic region contraction (SRC) algorithm, conventionally used to derive parameter estimates from mcDESPOT data. SRC estimates of the transverse relaxation time of the long T₂ fraction, T_2,l, and the longitudinal relaxation time of the short T₁ fraction, T_1,s, clustered towards the lower and upper parameter search space limits, respectively, indicating failure of the fitting procedure. We demonstrate that this effect is absent in the BMC analysis. Our results also showed improved parameter estimation for BMC as compared to SRC for high-resolution mapping. Overall we find that the combination of BMC analysis and mcDESPOT, BMC-mcDESPOT, shows excellent performance for accurate high-resolution whole-brain mapping of MWF and bi-component transverse and longitudinal relaxation times within a clinically realistic acquisition time.

1. INTRODUCTION

Alterations in myelin content and in magnetic resonance relaxation times T₁ and T₂ have been shown to be sensitive biomarkers for a number of central nervous system diseases. These include multiple sclerosis (1–5), brain atrophy (6, 7), epilepsy (8–11), Parkinson’s disease (7), Alzheimer's disease (12, 13), phenylketonuria (14, 15), psychotic disorders (16), and schizophrenia (17–20). While it is most often assumed that relaxation within each imaging voxel may be described by a single T₁ or T₂ value, this assumption does not capture the structural and compositional complexity of brain tissue. In fact, previous studies have demonstrated the presence of multi-component T₁ and T₂ relaxation processes in brain as an indicator of compartmentation (5, 21–33). Multicomponent relaxometry (MCR) analysis has characterized two main water pools, with distinct relaxation times and fractions. The pool exhibiting the more rapid transverse relaxation and smaller fraction size has been attributed to myelin-bound water, while the more slowly-relaxing pool has been assigned to relatively unbound intra- and extracellular water (21, 23, 28, 34).

A variety of approaches to MCR have been applied to in-vivo clinical studies (28, 29, 32, 35–43), and have been reviewed recently by Alonso-Ortiz et al. (23). Among these methods, the multicomponent driven equilibrium single pulse observation of T₁ and T₂ (mcDESPOT) method, based on steady-state magnetic resonance imaging (MRI) sequences, is of particular interest (28, 44). mcDESPOT allows simultaneous mapping of multicomponent T₁ and T₂ relaxation times and myelin water fraction (MWF), providing improved sensitivity and specificity to tissue changes associated with development or pathology. Moreover, mcDESPOT permits the use of a relatively short echo time, TE, thereby allowing improved detection of the short-T₂ component of the signal, representing the MWF. Finally, mcDESPOT makes use of conventional MR acquisition sequences, namely, fully balanced steady state free precession (bSSFP) and spoiled gradient recalled echo (SPGR), widely available on clinical MRI systems. Overall, then, mcDESPOT is particularly attractive for clinical investigations (26, 30, 45).

In mcDESPOT analysis, SPGR and bSSFP datasets are acquired over a range of flip angles (FAs), with very short repetition times, TRs. Two different bSSFP datasets are acquired respectively with radio-frequency (RF) phase increments equal to 0 or π (bSSFP₀ and bSSFP_π) to correct for off-resonance effects (44). Although this formalism has been extended to include a third pool to account for either partial volume effects (27) or magnetization transfer (46), mcDESPOT modeling is generally restricted to a two-pool model. Even with this restriction, quantitative parameter estimation from mcDESPOT is problematic, especially at the low-to-moderate signal-to-noise ratios (SNRs) typical of high-resolution imaging (25, 47, 48). In recent studies, Zhang et al. (33, 49), Lankford and Does (50) and Bouhrara et al. (47) showed that for a two-component implementation of mcDESPOT using stochastic region contraction (SRC) with nonlinear least squares (NLLS), determination of relaxation times was problematic due to the flatness of the parameter least-squares energy surfaces. It was found, for example, that estimates of T₁ for the more rapidly relaxing component, and of T₂ for the more slowly relaxing component, showed a tendency to cluster respectively at the upper and the lower limits of the specified parameter spaces (33, 49).

In a recent study, Bouhrara and Spencer showed that the quality of MWF estimates from the two-component mcDESPOT signal model was greatly enhanced through use of a new Bayesian Monte Carlo (BMC) analysis (25). However, that work was limited to MWF analysis and did not address the important issue of relaxation time estimation. In this work we therefore extend BMC-mcDESPOT analysis to the simultaneous estimation of MWF and relaxation times over the whole brain with a voxel volume of 1 mm³, with an acquisition time of under 15 minutes. In addition, we directly demonstrate the superior performance of BMC compared to the conventional approach using SRC.

2. MATERIALS & METHODS

2.1. Experimental Analysis

2.1.1. Data Acquisition

All experiments were performed on a 3T whole body Philips MRI 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 at low-resolution (LR) or high-resolution (HR) from four volunteers, from whom written informed consent was obtained prior to participation. All examinations were performed with approval of the local Institutional Review Board. Table 1 summarizes the experimental acquisition and reconstruction parameters of mcDESPOT imaging data for each volunteer. LR images were obtained with an acquisition voxel size of 2 mm x 2 mm x 2 mm and reconstructed to this same voxel volume of 8 mm³ while all HR images were acquired with an acquisition voxel size of 1.5 mm x 1.5 mm x 1.5 mm and reconstructed to a voxel volume of 1 mm³ using zero-filling to improve visual quality.

Table 1.

In-vivo acquisition parameters

Volunteers	Acquisition parameters
	Low-resolution (LR) protocol	High-resolution (HR) protocol
Volunteer #1 Healthy 23 year-old male	3D SPGR: αSPGR = [2 4 6 8 10 12 14 16 18 20]o, TESPGR = 1.2 ms, TRSPGR = 6.5 ms. 3D bSSFP: βbSSFP = [2 6 14 22 30 38 46 54 62 70]o, TEbSSFP = 3.2 ms and TRbSSFP = 6.5 ms.	3D SPGR: αSPGR = [2 4 6 8 10 12 14 16 18 20]o, TESPGR = 1.2 ms, TRSPGR = 6.5 ms. 3D bSSFP: βbSSFP = [2 6 14 22 30 38 46 54 62 70]o, TEbSSFP = 3.2 ms and TRbSSFP = 6.5 ms.
Volunteer #2 Healthy 22 year-old male	FoV = 230 x 180 x 130 mm3, matrix size = 116 x 90 x 65, acquisition voxel size ≃ 2 mm x 2 mm x 2 mm, total acquisition time ≃ 10 min.	FoV = 230 x 180 x 130 mm3, matrix size = 153 x 120 x 86, acquisition voxel size ≃ 1.5 mm x 1.5 mm x 1.5 mm, total acquisition time ≃ 17 min.
Volunteer #3 Healthy 29 year-old male	-	3D SPGR: αSPGR = [2 4 6 8 10 12 14 16 18]o, TESPGR = 0.83 ms, TRSPGR = 6 ms. 3D bSSFP: βbSSFP = [2 4 7 11 16 24 32 40 50]o, TEbSSFP = 2.8 ms and TRbSSFP = 5.8 ms. FoV = 230 x 190 x 140 mm3, matrix size = 154 x 127 x 94, acquisition voxel size ≃ 1.5 mm x 1.5 mm x 1.5 mm, total acquisition time ≃ 15 min.
Volunteer #4 Healthy 22 year-old male	3D SPGR: αSPGR = [2 4 6 8 10 12 14 16 18 20]o, TESPGR = 0.83 ms, TRSPGR = 6 ms. 3D bSSFP: βbSSFP = [2 6 14 22 30 38 46 54 62 70]o, TEbSSFP = 2.8 ms and TRbSSFP = 5.8 ms. FoV = 200 x 216 x 140 mm3, matrix size = 100 x 108 x 70, acquisition voxel size ≃ 2 mm x 2 mm x 2 mm, total acquisition time ≃ 11 min.	-

^*FoV: Field of view. The bSSFP images were acquired twice, once with RF phase increment, θ_RF, of 0 (bSSFP₀) and once with θ_RF of π (bSSFP_π). All images were acquired with SENSE factor = 2. LR images were reconstructed to voxel volume = 8 mm³ while HR images were reconstructed to voxel volume = 1 mm³.

In our in-vivo studies, SNR was estimated as the mean signal value within a large region in SPGR images obtained over all non-zero FAs divided by the mean signal value for FA = 0^o. These SNR values were ~30 and ~50 for the HR and LR protocols (Table 1), respectively.

2.1.2 Data Analysis

We assumed a two-component non-exchanging system consisting of a short, s, and long, l, T₁ and T₂ components (25, 28, 51). Analysis was performed explicitly accounting for nonzero TE as incorporated into the TE-corrected-mcDESPOT (TEC-mcDESPOT) signal model (52). Images from each mcDESPOT dataset were analyzed on a voxel-wise basis. First, monoexponential T₁ maps were generated 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 off-resonance maps were then generated according to the DESPOT₂-FM method (53) by fitting the bSSFP datasets as a function of FAs to a single component form of the signal, using the voxel-wise T₁ value obtained as outlined above. Finally, the off-resonance map was combined with the SPGR and bSSFP datasets to simultaneously generate f_s, T_2,s, T_2,l, T_1,s, and T_1,l maps, with f_s defining the MWF. Manual segmentation was performed to eliminate ventricles and non-brain regions within the images, with parameter maps generated for the remaining regions of interest.

The first analysis consisted of investigating the performance of the BMC approach for generating high-resolution MWF, T_2,s, T_2,l, T_1,s, and T_1,l maps. Analysis was performed for all LR and HR datasets described in Table 1. Details regarding the mathematical aspects of BMC-mcDESPOT analysis for simultaneous estimation of all system parameters are given in the Appendix.

In recent studies, Zhang et al. (33, 49) showed that determination of relaxation times from mcDESPOT using SRC (54) can be problematic. Motivated by their observations, we performed a second analysis in which we generated MWF, T_2,s, T_2,l, T_1,s, and T_1,l maps using SRC. This analysis was performed on the LR and HR datasets acquired from Volunteer #1 (Table 1). Wide limits for the initial parameter bounds were used to avoid an over-constrained parameter space (33, 47, 51). These limits were 1 ms ≤ T_2,s ≤ 60 ms, 60 ms ≤ T_2,l ≤ 200 ms, 10 ms ≤ T_1,s ≤ 650 ms, 10 ms ≤ T_1,l ≤ 3500 ms and 0 ≤ f_s ≤ 0.45. These same limits were used in all BMC analyses as well. Histograms of parameter values were calculated from white matter (WM) voxels in several slices, with the WM regions defined using manual segmentation of T₂ maps generated using the DESPOT₂-FM method described above.

Finally, the mean and SD for each estimated parameter were calculated over three regions of interest (ROIs) in several slices. The first ROI (ROI #1) lay within the frontal lobes, the second ROI (ROI #2) lay within the regions surrounded by the insular cortex, internal capsule, thalamus and putamen, and the third ROI (ROI #3) enclosed all WM regions. This analysis was performed on the parameter maps obtained from HR datasets using BMC or SRC.

2.2. Simulation Analysis

Numerical simulations were used to further assess the performance of parameter determination using BMC-mcDESPOT and for comparison with those computed using SRC. The following input parameters describing acquisition were used: TR_SPGR = TR_bSSFP = 6.5 ms, TE_SPGR = 1 ms, α_SPGR = [2 4 6 8 10 12 14 16 18 20]^o, and β_bSSFP = [2 6 14 22 30 38 46 54 62 70]^o, with two bSSFP datasets generated with RF phase increments of 0 or π respectively. Input parameters defining the system were: f_s (i.e. MWF) = 0.2, T_2,s = 15 ms, T_2,l = 75 ms, T_1,s = 500 ms, and T_1,l = 1500 ms, corresponding to WM values obtained from the in-vivo analysis described above. Results were obtained over 1001 noise realizations and are presented as histograms. All analyses were performed at steady-state SNR of 500, defined as SNR = M₀ / σ, where M₀ represents the signal amplitude at TE = 0 ms. This steady state SNR value corresponds to an in-vivo SNR of ~20–30 as defined above. More extensive numerical analyses are presented in the Supplementary Material.

3. RESULTS

Figure 1 shows examples of MWF, T_2,s, T_2,l, T_1,s and T_1,l maps derived from brain images using BMC-mcDESPOT. HR datasets and LR datasets were acquired with isotropic voxel volumes of 1 mm³ and 8 mm³ respectively. Parameter maps derived from the HR and LR datasets were similar overall, especially in the WM regions. However, as expected, the HR maps better define anatomic details and regional patterns. The LR maps showed, overall, greater dispersion in derived parameter estimates especially at interfaces between gray and white matter, likely due to partial volume effects. Moreover, in contrast to the WM regions, all T_2,s maps showed high dispersion in the CSF and gray matter. This is a direct consequence of the very low values of MWF in these regions, which renders it impossible to assign meaningful values of T_2,s.

Figure 1.

Figure 1a. Representative MWF, T_2,s, T_2,l, T_1,s, and T_1,l maps calculated for Volunteer #1 (Table 1) using BMC-mcDESPOT. Results are presented for both high-resolution (HR) and low-resolution (LR) datasets. The HR maps allows greater depiction of anatomical detail and regional patterns as compared to the LR maps. Note that the displayed color scales are different from the boundary limits of the parameter space used in the fitting process for improved visualization. 1b. Representative MWF, T_2,s, T_2,l, T_1,s, and T_1,l maps calculated for Volunteer #3 (HR, left panels) or Volunteer #4 (LR, right panels) (Table 1) using BMC-mcDESPOT. The HR maps allows greater depiction of anatomic detail and regional patterns as compared to the LR maps. Note that the displayed color scales are different from the boundary limits of the parameter space used in the fitting process. 1c. High-resolution MWF, T_2,s, T_2,l, T_1,s, and T_1,l maps calculated from brain images of Volunteer #3 (Table 1) using BMC-mcDESPOT. Results are presented for four different slices. Note that the displayed color scales are different from the boundary limits of the parameter space used in the fitting process.

Figure 2 shows examples of MWF, T_2,s, T_2,l, T_1,s, and T_1,l parameter maps derived from the HR and LR human brain images using SRC. Parameter values derived from HR or LR were again similar, especially in WM regions. However, visual inspection (Figs. 1–2) as well as histogram analysis (Fig. 3) demonstrate that the derived parameter estimates using SRC (Fig. 2–3) differed substantially from those obtained with BMC (Figs. 1 and 3), particularly for T_2,s, T_2,l and T_1,s. In addition, unlike BMC, SRC resulted in substantial random variation in the estimation of T_2,s, and clustering of T_2,l and T_1,s estimates towards the lower and upper search space limits, respectively. T_1,l and MWF maps derived from SRC and BMC showed similar regional patterns, although lower dispersion was evident in the MWF estimates derived using BMC. These results indicate the superior overall performance of BMC as compared to SRC. We also observed that the clustering of T_1,s estimates towards the upper search space limits using SRC was reduced to some extent in the LR parameter maps as compared to the HR maps. This is likely due to the relatively higher SNR of the former.

Figure 2.

Representative MWF, T_2,s, T_2,l, T_1,s, and T_1,l maps calculated for Volunteer #1 using SRC. Results are presented for both HR and LR datasets. Note that the displayed color scales are different from the boundary limits of the parameter space used in the fitting process.

Figure 3.

Representative histograms of MWF, T_2,s, T_2,l, T_1,s and T_1,l obtained using BMC or SRC analysis from white matter regions within the HR parameter maps calculated for Volunteer #1. The mean and standard deviation values are as indicated, with BMC generally outperforming SRC. Unlike BMC, SRC resulted in a clustering of T_2,l and T_1,s estimates towards the lower and upper search space limits, respectively. These results indicate the superior overall performance of BMC as compared to SRC.

Figure 4 shows parameter histograms of f_s (i.e. MWF), T_2,s, T_2,l, T_1,s, and T_1,l obtained using Monte Carlo simulations over 1001 noise realizations. Results were obtained at an SNR value of 500 using SRC or BMC. Using SRC, T_2,l estimates were clustered towards the lower search space limit while T_1,s estimates were clustered towards the upper search space limit; this behavior is in good agreement with our in-vivo observations (Figs. 2–3). In contrast, BMC parameter estimates were preferentially distributed around the true input values and avoided search space boundaries. Further, parameter estimates using SRC showed larger bias and dispersion compared to BMC. Extensive numerical analysis further demonstrating the overall superior performance of BMC is presented in the Supplementary Material (Fig. Sup.1).

Figure 4.

Histograms of MWF, T_2,s, T_2,l, T_1,s and T_1,l obtained using BMC or SRC analysis over 1001 noise realizations using numerical simulations. Vertical red dashed lines indicate true input parameter values: f_s = 0.2, T_2,s = 15 ms, T_2,l = 75 ms, T_1,s = 500 ms and T_1,l = 1500 ms. The mean and standard deviations are as indicated. Note that using SRC, T_2,l estimates were clustered towards the lower search space limit while T_1,s estimates were clustered towards the upper search space limit. In contrast, BMC parameter estimates were distributed around the true input values and avoided search space boundaries.

Figure 5 shows parameter means and SDs of estimates of MWF, T_2,s, T_2,l, T_1,s, and T_1,l calculated from the three HR datasets (Table 1). For each HR dataset, results were obtained over three ROIs using BMC and SRC analyses. The results indicate good reproducibility for both analysis methods between the three subjects for all estimated parameters. However, BMC analysis was able to detect regional variation in all system parameters, while SRC showed substantial regional changes in MWF, T_2,s and T_1,l only. In fact, T_2,l and T_1,s calculated using SRC showed artifactually decreased regional variation; the averaged values calculated over the three subjects were T_2,l = 64 ± 3.9 ms and T_1,s = 626 ± 7.6 ms, which are very close to the lower and upper search space limits of 60 ms and 650 ms, respectively. In contrast, the averaged values of these parameters calculated using BMC, 77 ± 2.1 ms and 517 ± 13.2 ms, were well-within the search space limits. This is consistent with Figs. 2–4 and the comments above. Finally, parameter estimates of MWF, T_2,s and T_1,l derived using SRC showed, overall, higher SDs as compared to those derived using BMC. In fact, the average SD values over the three volunteers in the estimation of MWF, T_2,s and T_1,l, respectively, were 0.1, 8.2 ms and 425 ms using SRC, and 0.06, 5.4 ms and 360 ms using BMC. This is in good agreement with Fig. 2, which shows substantial spatial irregularities, especially in derived MWF and T_2,s values, as compared to the BMC results of Fig. 1.

Figure 5.

Means and standard deviations of estimates of MWF, T_2,s, T_2,l, T_1,s and T_1,l calculated from the HR datasets acquired from the brains of three volunteers (Vol) (Table 1). For each HR dataset, results were obtained in three different regions of interest (ROIs) using BMC or SRC analysis. ROI #1 lay within the frontal lobes, ROI #2 lay within the regions surrounded by the insular lobes, internal capsules, thalamus and putamen, and ROI #3 incorporated all WM regions.

4. DISCUSSION

In 1994, Mackay et al. demonstrated in-vivo MCR in the human brain for myelin mapping using MRI (21). Since then, MCR has become an active area of research, with several methods having been developed to either accelerate data acquisition (23, 28, 29, 32, 36, 37, 39, 40, 55) or to improve parameter determination (23, 25, 36, 44, 56–65). Among these, mcDESPOT permits rapid whole-brain coverage, and so may have particular potential for clinical investigations, especially those involving patients with limited ability to remain stationary for the lengthy duration of MRI studies. However, it has been shown (25, 33, 47, 48) that the high dimensionality of the mcDESPOT signal models renders parameter estimation problematic when the acquired images exhibit substantial noise, such as in the case of high-resolution mapping. However, we showed that the quality of derived MWF estimates is greatly enhanced using BMC analysis (25). In the present work, we generalized BMC-mcDESPOT to the simultaneous estimation of MWF and relaxation times for a two-component model; simulation analysis demonstrated that BMC allows high quality determination of all system parameters. Further, our in-vivo analysis showed that high-quality, high-resolution MWF and relaxation time maps may be obtained with an isotropic voxel volume of 1 mm³ within a clinically realistic acquisition time, permitting accurate depiction of anatomical detail and regional patterns compared to LR mapping (Fig. 1).

In this study we compared the performance of the BMC-mcDESPOT analysis with that of the SRC algorithm, conventionally used for parameter estimation with mcDESPOT, for determination of MWF and relaxation times. Overall, we found greatly reduced bias and dispersion in the estimation of all system parameters using BMC (Figs. 1–5, Fig. Sup.1). We attribute this to several factors, including the implicit incorporation of noise SD and the marginalization over nuisance parameters in the BMC analysis (25). In fact, a fundamental difference between Bayesian methods and the several variants of NLLS algorithms, such as SRC, is that the posterior probabilities resulting from the former are obtained through incorporation of the entire specified ranges of nuisance parameters through marginalizations. In contrast, the evaluation of single combinations of estimated parameters required to optimize the objective function in NLLS leads to potential difficulties with local minima, especially in higher-dimensional problems such as mcDESPOT (47). A drawback to Bayesian analysis is the requirement for high dimensional integration in order to perform the marginalizations. To overcome this, we have implemented Monte Carlo sampling to perform the required integrations, although more advanced sampling strategies such as Markov chain Monte Carlo may further improve performance (66).

Our numerical and in-vivo analyses showed that using SRC, T_2,l estimates were preferentially clustered close to the lower search space limit, while T_1,s estimates were preferentially close to the upper search space limit (Fig. 2–4, Fig. Sup.1). This is in good agreement with the observations of Zhang et al. (33, 49). This problem was markedly reduced using BMC, with parameter estimates distributed around mean values that were clearly separated from the search space limits. However, visual inspection of the parameter maps (Figs. 1–2) as well as more quantitative analysis (Fig. 3–5) showed higher dispersion in derived parameter estimates as compared to the literature (26, 28, 33, 44, 49, 67), especially using SRC. We attribute this to our use of relatively large parameter space bounds to avoid bias (33, 47, 51); as has previously been noted, this comes at the expense of increased dispersion (47, 51). A priori restriction of the allowable parameter space therefore represents a tradeoff between bias and dispersion.

In agreement with recent studies (33, 49), our results showed that MWF and relaxation parameters values derived from mcDESPOT were substantially different from those derived from a CPMG analysis. This can be attributed to the fact that MRI pulse sequences, including mcDESPOT and CPMG, are influenced to substantially different degrees by experimental and physiological effects. Therefore, received signals will deviate from their ideal models due to diffusion (68–71), exchange (33, 72, 73), off-resonance effects (74–76), magnetization transfer (49, 77–79), J-coupling (80–82), spin locking (83, 84), internal gradients (85, 86), and magnetization spoiling (87, 88). The importance of these effects will depend both upon the specifics of the sample or subject under investigation and on the details of the pulse sequence, including the selection of parameters such as TE, TR, FAs, RF pulse shape, and gradient durations and amplitudes (89–93).

Our HR analyses were performed on three healthy young volunteers. While BMC showed good reproducibility between these subjects for all estimated parameters (Fig. 5), larger groups of subjects would be required for statistical comparisons between groups varying by, for example, age or disease status.

In the present work, we based our analysis on SPGR and bSSFP signal models that account for nonzero TE, that is, we used the TEC-mcDESPOT signal model (52). The neglect of the transverse signal decay that occurs during TE has been recently shown to potentially introduce large biases in parameter estimates from mcDESPOT (52). Nevertheless, limitations remain in the signal modeling of mcDESPOT, which represent limitations of the present work as well as in certain literature results. First, the mcDESPOT signal models analyzed by ourselves and others neglect finite RF pulse length (94–98). Similarly, we assumed perfect spoiling of transverse magnetization before each excitation pulse in the SPGR sequence; it has been shown that there may be a degree of preservation of transverse coherence in fast SPGR sequences with concomitant deviation of the signal behavior from the idealized model (87, 88, 99). Further, we assumed a two-component signal model; it has been shown that a three-component model improves the performance of mcDESPOT analysis in regions exhibiting partial volume effects, such as boundaries between brain tissue and ventricles and meninges (27). In addition, we assumed no exchange between the two proton pools. Extension of the BMC analysis to incorporate exchange is straightforward in principle and would permit evaluation of the impact of exchange on relaxation times, as well as on the estimate of MWF. Our analysis also assumes that the effect of magnetization transfer between free water protons and macromolecules is small (49). Moreover, we assumed a Gaussian noise distribution for our Bayesian analysis to obtain closed form expressions for marginalized joint likelihood functions (25). However, it is well-known that the noise in magnitude MR images follows a noncentral-χ distribution (100), which can be approximated by a Gaussian distribution only for relatively high SNR. Finally, as with any MRI sequence, the spatial encoding gradients of steady-state sequences modulate the degree of signal attenuation due to water diffusion (101, 102). However, water diffusion in different compartments, such as myelin sheets and intra- and extra-cellular spaces, is not incorporated into the mcDESPOT formalism. While extensive further investigation would be required to analyze all of the above effects, the present work stands to demonstrate the feasibility of high-resolution mapping of MWF and relaxation times under clinical MRI acquisition conditions.

5. CONCLUSIONS

We have demonstrated the feasibility of simultaneous HR mapping of MWF and relaxation times in white matter regions of human brain using BMC-mcDESPOT. High-resolution whole-brain maps of MWF and transverse and longitudinal bi-component relaxation times can be obtained in less than 15 min.

Highlights.

• We extend Bayesian Monte Carlo (BMC) analysis of the two-component mcDESPOT signal model to simultaneous estimation of myelin water fraction (MWF) and relaxation times.

• Simulations and in-vivo studies indicate the superiority of BMC-mcDESPOT as compared to the conventional approach based on nonlinear least squares analysis.

• We demonstrate that use of BMC-mcDESPOT permits accurate high-resolution mapping of MWF and relaxation times within a clinically realistic acquisition time.

6. Appendix

6.1. BMC-mcDESPOT Analysis for the estimation of all system parameters

The estimate, λ̂_i, of a given parameter, λ_i, belonging to the set of unknown parameters λ is given by (25):

$${\widehat{\lambda }}_i=\int {\lambda }_i\mathrm{P}({\lambda }_i\mid \mathbf{S})\mathrm{d}{\lambda }_i=\int \cdots \int {\lambda }_i\frac{P({\mathbf{\lambda }}^{\ast })L(\mathbf{S}\mid \mathbf{\lambda })}{P(\mathbf{S})}\mathrm{d}\mathbf{\lambda }\simeq \sum _{m=1}^M{\lambda }_{i,m}\frac{P({\mathbf{\lambda }}_m^{\ast })L(\mathbf{S}\mid {\mathbf{\lambda }}_m)}{P(\mathbf{S})}.$$ [A.1]

Here, P(λ_i|S)is the posterior distribution of λ_i given the vector S = (S_SPGR S_bSSFP0 S_bSSFPπ) of measured signals, each of which is itself a vector describing the signal amplitude for each value of FA, m denotes one of a total of M random sets of parameter combinations sampled from a grid defining the range of our Monte Carlo integration, λ* is equivalent to λ but excludes the parameter λ_i being estimated and so represents the vector of nuisance parameters in the determination of λ_i, and P(λ*) = 1/λ* is the product of prior distributions for the elements of λ* taken as noninformative Jeffreys priors (25, 103). For example, the prior distribution in the estimation of T_1,s is given by: P(λ*) = 1/λ* = (1/f_s) * (1/T_2,s) * (1/T_2,l) * (1/T_1,l). This is a widely-used though strictly improper prior. P(S) = ∫P(λ*) L(S|λ) dλ is a normalization constant. We note that, in our previous paper (25), dλ in the definition of P(S) was erroneously written as dλ*, although the numerical calculations were performed with the correct expression. Finally, L(S|λ) is the marginalized joint likelihood function of S given λ (25):

$$\begin{array}{l}L(\mathbf{S}\mid \mathbf{\lambda })\propto {\left(({\stackrel{\sim }{\mathbf{S}}}_{\mathit{SPGR}}-{\stackrel{\sim }{\mathbf{M}}}_{\mathit{SPGR}}){({\stackrel{\sim }{\mathbf{S}}}_{\mathit{SPGR}}-{\stackrel{\sim }{\mathbf{M}}}_{\mathit{SPGR}})}^T\right)}^{-\frac{K}{2}}(({\stackrel{\sim }{\mathbf{S}}}_{{\mathit{bSSFP}}_0}\\ {-{\stackrel{\sim }{\mathbf{M}}}_{{\mathit{bSSFP}}_0}){({\stackrel{\sim }{\mathbf{S}}}_{{\mathit{bSSFP}}_0}-{\stackrel{\sim }{\mathbf{M}}}_{{\mathit{bSSFP}}_0})}^T)}^{-\frac{N}{2}}(({\stackrel{\sim }{\mathbf{S}}}_{{\mathit{bSSFP}}_{\pi }}\\ {-{\stackrel{\sim }{\mathbf{M}}}_{{\mathit{bSSFP}}_{\pi }}){({\stackrel{\sim }{\mathbf{S}}}_{{\mathit{bSSFP}}_{\pi }}-{\stackrel{\sim }{\mathbf{M}}}_{{\mathit{bSSFP}}_{\pi }})}^T)}^{-\frac{N}{2}},\end{array}$$ [A.2]

where S̃ and M̃ are respectively the experimental and theoretical signals normalized by their respective mean values calculated over K SPGR or N bSSFP FAs (25). The theoretical SPGR and bSSFP signals are given in the following section. For computational efficiency, the likelihood function given in Eq. A. 2 is first calculated for all random parameter combinations. Then, for each parameter, the likelihood function is combined with the corresponding joint prior to create the joint posterior distribution for each of those random parameter combinations. Finally, the parameter estimate is obtained based on the first moment of the joint posterior distribution after marginalization over nuisance parameters.

6.2. SPGR and bSSFP signal models

Neglecting exchange between components, the two-component SPGR signal is given by (52)

$$M_{\mathit{SPGR}}^k=M_{\mathit{SPGR}}^{0}sin({\alpha }_k)\left(f_s\frac{E_{2,s}^{†}\left(1-E_{1,s}\right)}{1-E_{1,s}cos({\alpha }_k)}+(1-f_s)\frac{E_{2,l}^{†}\left(1-E_{1,l}\right)}{1-E_{1,l}cos({\alpha }_k)}\right),$$ [1]

where s and l respectively denote the short- and long- T₂ components, f_s is the fraction of the short T₂ component, $M_{\mathit{SPGR}}^0$ represents the signal amplitude at echo time TE = 0 and incorporates proton density, T₂′ and machine factors (47, 52), and α_k is the k^th excitation FA out of a total of K FAs. We also define $E_{2,j}^{†}=\mathit{exp}\left(-{\text{TE}}_{\mathit{SPGR}}/T_{2,j}\right)$ and E_1,j = exp(−TR_SPGR/T_1,j) , where T_1,j is the spin-lattice and T_2,j is the spin-spin relaxation time of the j^th component.

Similarly, the two-component bSSFP signal in the absence of exchange between components is given by (52)

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

where

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

and

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

where $M_{\mathit{bSSFP}}^0$ 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 RF pulse. Finally, E_1,j = exp(−TR_bSSFP/T_1,j) and E_2,j = exp(−TR_bSSFP/T_2,j).