Gerhard Drenthen^{1,2}, Walter Backes^{1,2}, Albert Aldenkamp^{3}, and Jacobus Jansen^{1,2}

Myelin-water quantification
relies on modeling of multi-exponential T2-relaxation time decay. For this, we
explore the greedy Orthogonal Matching Pursuit (OMP) method and compare it to the
most commonly applied non-negative least squares (NNLS) method. The two methods
are evaluated by means of simulations, phantom measurements and *in vivo* image data.

The greedy OMP
algorithm builds a sparse representation of the signal by iteratively selecting
that particular element from a dictionary which correlates most with the
current residual^{2}. Therefore, similarly to NNLS, no starting model
is needed. Non-negativity is guaranteed for each iteration by solving a
non-negative least squares problem for the current selection of elements^{2}.
To increase the stability of the algorithm, it is called 20 times per
voxel (the OMP is roughly 20 times faster than NNLS).

To estimate the
theoretical errors of the NNLS and OMP algorithms, relaxation data were synthesized
using the extended phase graph (EPG)^{3} algorithm (flip angle 150⁰).
The relaxation times were chosen to match those of the *in vivo* content
(T1=860ms, T2_{long}=88ms, and T2_{myelin}=30ms). For
simulations, noise was estimated from the *in
vivo* image acquisition. To assess the effects of myelin content on the
analysis, 1000 Gaussian noise realizations were performed for a varying myelin-water
content (range 0-30%).

To estimate errors in MWF estimation due to the MSE sequence, two mono-exponential manganese(II)-chloride vials were constructed with T2 values of 30 and 110ms. Multi-exponential decays were synthesized by summing two randomly chosen voxels from each phantom and weighing them such that a varying MWF (range 0-30%) was obtained. This process was repeated 1000 times.

To visualize the
difference of the two methods *in vivo*,
three healthy male volunteers were scanned (aged 28-30).
The vials and volunteers
were scanned on a 3.0T unit (Philips Achieva) using a single-slice MSE sequence^{4}
(TR=3000ms, TE=12ms, 32 echoes, voxel
size 1.5x1.5x4 mm, 2 signal averages). The EPG algorithm was used to correct
for B1-inhomogeneities.

Lastly, a dictionary with 120 and 1000 logarithmically spaced T2 values between 10 and 3500ms was used for the NNLS and OMP, respectively.

In Figure 1 an example solution is shown to visualize the difference between the NNLS and OMP output spectra. In this example a noisy signal comprising of two components (at 30 and 88ms) and a MWF of 15% is solved by both methods.

In Figure 2 the results of the simulations and phantom measurements are shown. For a specific MWF of 15% the analysis of the simulated data results in an underestimated mean MWF of 13.8% (OMP) and 11.8% (NNLS), for the phantom measurements the estimated MWF is 14% (OMP) and 12.2% (NNLS). Therefore, the OMP is approximately 3.5 times ($$$\left|\frac{15\%-\mu_{NNLS}}{15\%-\mu_{OMP}}\right|$$$) closer to the true mean compared to NNLS for the simulated data, and approximately 3 times for the phantom measurements. Both methods show a similar precision (variation on the mean).

The MWF map of a healthy volunteer is shown in Figure 3. As region of interests, the splenium
and the genu of the corpus callosum were delineated in the white matter. In
each region the MWFs obtained with the OMP were, on average, higher in
comparison to the NNLS (splenium: 21.1%±4.4% vs
17.7%±4.9% and genu:
11.3%±5.1% vs
8.0%±4.3%). These MWF values agree well to those reported in
previous studies^{5}.

OMP is an approximately 3 times more accurate measure for MWF quantification compared to NNLS. Most likely this is due to the increased resolution of the T2 relaxation time spectrum (e.g. larger dictionary). The simulations and phantom measurements show that both the NNLS and OMP tend to underestimate the true MWF. Possibly the underestimation occurs when part of the amplitude peak has a too short T2 value, which is compensated by a higher amplitude. This effect is stronger for higher T2 values where the resolution is lower due to the logarithmic spacing. Therefore, the OMP, having a higher resolution dictionary, is less prone than NNLS to the MWF underestimation and therefore more accurate for myelin-water quantification.

1. Whittall K. Quantitative interpretation of NMR relaxation data. J Magn Reson. 1989;84:134-152.

2. Yaghoobi M, Wu D, Davies M. Fast Non-negative orthogonal matching pursuit. IEEE Signal Processing Letters. 2015;22(9):1229-1233.

3. Hennig J. Multiecho imaging sequences with low refocusing flip angles. J Magn Reson. 1988;78:397–407.

4. Kolind S, Madler B, Fisher S, et al. Myelin water imaging: implementation and development at 3.0T and comparison to 1.5T measurements. Magn Reson Med. 2009;62:106-115.

5. Prasloski T, Madler B, Ziang Q, et al. Applications of stimulated echo correction to multicomponent T2 analysis. Magn Reson Med. 2012;67:1803-1814.

Figure 1: Graphical representation of the input
(MWF, 15%), and output of both the NNLS (MWF, 14.3%) and OMP (MWF, 14.8%) method.

Figure 2: The mean and standard deviation of
the MWFs estimated by the OMP (solid green curve) and NNLS (dotted red curve)
are shown in reference to the true myelin-water content for both the simulated
data (A) and the phantom measurements (B). Additionally, the deviation from the
true myelin-water content is shown at the top of the graphs.

Figure 3: The MWF map obtained using the NNLS
(A) and OMP (B) as well as the two regions of interest (C) are shown. The
differences between the methods is shown in the scatter plot (D), the MWFs for
the specific regions, as well as those for all white matter voxels, are shown.