Markus Zimmermann^{1}, Ana-Maria Oros-Peusquens^{1}, Zaheer Abbas^{1,2}, Elene Iordanishvili^{1}, Seonyeong Shin^{1}, Seong Dae Yun^{1}, and N. Jon Shah^{1,2,3,4}

A new parameter estimation algorithm, MERLIN, is presented for
accurate and robust multi-exponential relaxometry using MRI.
Multi-exponential relaxometry is fundamentally ill-conditioned, and
as such, is extremely sensitive to noise. MERLIN is a fully
automated, multi-voxel approach that incorporates $$$\ell_1$$$-regularization
to enforce sparsity and spatial consistency of the estimated
distributions. The proposed method is compared to the conventional
$$$\ell_2$$$-regularized NNLS (rNNLS) in simulations
and *in vivo* experiments, using a multi-echo gradient-echo
(MEGE) sequence at 3T. The estimated water fraction maps from MERLIN
are spatially more consistent, more precise, and more accurate,
reducing the root-mean-squared-error by up to 90 percent in
simulations.

Let $$$\mathbf{x}$$$ be the distributions and $$$\mathbf{d}$$$ the data acquired at
all voxel positions in vectorized form. Let $$$\bar{\mathbf{A}}=\mathrm{diag}(\mathbf{A},\dots,\mathbf{A})$$$
be a block-diagonal matrix, where $$$\mathrm{A}_{m,n}=\exp(-\mathrm{TE}_m/T_{2,n}^*)$$$,
with the acquired echo times, $$$\mathrm{TE}_m$$$, and the estimated
relaxation times, $$$T_{2,n}^*$$$. MERLIN
minimizes $$\min_{\mathbf{x}\geq0}\lambda_\mathrm{S}\lVert \mathbf{x}\rVert_1+\lambda_\mathrm{TV}\lVert\mathbf{x}\rVert_\mathrm{TV} \\
\mathrm{s.t.}\lVert\mathbf{\bar{A}x}-\mathbf{d}\rVert_2^2\leq\alpha\eta^2$$ with the regularization parameters $$$\lambda_\mathrm{S}=0.01$$$
and $$$\lambda_\mathrm{TV}=0.1$$$,
and the isotropic
total variation norm, $$$\lVert\cdot\rVert_\mathrm{TV}$$$,
applied along the spatial dimensions. MERLIN ensures that
the data residual is not greater than $$$\alpha=1.025$$$ times the minimal data residual of the non-regularized problem,
$$$\eta^2 = \min_{\mathbf{x}\geq0}\lVert\bar{\mathbf{A}}\mathbf{x}-\mathbf{d}\rVert_2^2$$$.
The split Bregman method^{4}
is used to solve this problem.

Numerical validations were performed using a phantom with 16
regions-of-interest (ROI), containing two water pools, whereas the
background (BG) contained only a single pool. The $$$T_2^*$$$
of the slow-relaxing compartment was set to 50 ms for all ROIs and
the background. This represents the intra-/extracellular water pool
(IE) of white matter at 3T. The fast-relaxing compartment of each ROI
had a water fraction of 5%, 10%, 15%, and 20% and a $$$T_2^*$$$
of 3, 6, 9, and 12 ms, being a proxy for the myelin water pool (M).
Each pool was modeled by a Gaussian distribution centered at position
$$$T_2^*$$$, with a standard deviation of 10%^{3}.
To quantify the accuracy of the estimated
parameter maps, the root mean squared error (RMSE) is computed for
the MWF and the geometric mean of $$$T_2^*$$$ (gmT2*)
of all ROIs and, separately, for the background. Finally, the median
and quartile deviation of the MWF and gmT2* (M)
of each ROI are individually compared against the ground truth.
*In vivo* experiments were conducted using a
commercial 3T scanner (MAGNETOM Prisma, Siemens Healthineers,
Erlangen, Germany). A 2D MEGE sequence was used with TE_{1}=3ms,
ΔTE=1ms, 80 echoes, TR=2000ms, FA=85°, matrix size 130 x 130,
resolution 1.6mm x 1.6mm x 2mm, 21 slices, BW=1282 Hz/px. The total
scan time was 4min 40s. Following prior written informed consent, MR
data was acquired from a representative healthy volunteer. The data
was corrected for static magnetic field inhomogeneity^{7}.
ROI analysis was performed on the MWF maps and the results were
compared to the literature^{8–11}.

For the simulation results, the estimated MWF and gmT2*
for the fast-relaxing pool (M) are shown in Fig 1. The parameter maps estimated by MERLIN appear spatial more consistent. The RMSE of the
estimated maps is given in Table I, showing a reduced RMSE of up to 90% compared to rNNLS. The estimated distributions as
well as the median estimated MWF and gmT2*
(M) versus the ground truth are displayed in Fig. 2. The estimated distributions are more narrow and the estimated parameters more precise compared to rNNLS. For the *in vivo *results, the MWF and gmT2*
(M) maps of three representative slices are shown in Fig. 3, together
with the contrast images of the first echo. Again, the parameter maps estimated by MERLIN appear spatially more consistent. Finally, the results of
the ROI analysis are given in Table II. The results are largely in agreement with the literature for rNNLS and MERLIN and the standard deviation of the estimated MWF is consistently lower compared to rNNLS.

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Fig. 1. Simulation results showing (**a**) the ground truth, (**b**)
the estimated myelin water fraction (MWF) maps, (**c**) the ground
truth and (**d**) the estimated gmT2* maps
for the myelin pool (M) computed with rNNLS and MERLIN for different
SNR levels. For visual clarity, the background was masked out in the
gmT2* (M) maps. The maps estimated by MERLIN appear less noisy and spatially more consistent.

Table I: Root-mean-squared-error of the estimated myelin water
fraction (MWF) and gmT2* maps for the myelin
and intra- and extra-cellular pool (M/IE) for the ROIs and the
background, respectively. The error is consistently reduced for MERLIN with a relative gain between 65% and 90%.

Fig. 2. Simulation results showing (**a**) the estimated
distributions and (**b**) the median estimated versus the true
myelin water fraction (MWF) and gmT2* (M) for
the myelin pool (M) for different SNR levels, with the error bars
indicating the quartile deviation. The black line in (**a**)
indicates the true gmT2* and the dashed lines
the distance of 2 standard deviations. The distributions estimated by MERLIN are more narrow and the parameter maps estimated by MERLIN show a consistently lower quartile deviation. In particular for gmT2*, the results of MERLIN are less biased.

Fig. 4. *In vivo* results showing (**a**) the T2*-weighted
images at the first echo time, (**b**) the myelin water fraction
(MWF) and (**c**) the gmT2* for the
myelin pool (M) for rNNLS and MERLIN for 3 different slices.
The maps estimated by MERLIN appear less noisy and spatially more consistent.

Table II: Region-of-interest analysis of the estimated myelin water
fraction. The results are largely in agreement with the literature. The standard deviation of the estimated MWF is consistently lower compared to rNNLS.