Nathan Tibbitts Roberts^{1,2}, Timothy J Colgan^{1}, Kang Wang^{3}, and Diego Hernando^{1,4}

Many coil combination methods have been developed, including methods that require pre-calibrated coil sensitivity maps, as well as methods that do not require additional sensitivity maps. Several of these methods have been adapted to CSE-MRI, where accurate signal combination is particularly critical as it needs to preserve consistent phase and magnitude information across echoes; however, their relative performance remains unknown. Therefore, the purpose of this work is to compare theoretically, in simulation, and experimentally the bias and noise performance of quantitative parameter maps resulting from five commonly used coil multi-echo coil combination techniques.

Coil Combination

Our work focuses on five coil combination methods, abbreviated as follows:

1. Roemer: The optimal method for pre-calibrated coil combination (1,2).

2. Walsh: A calibrationless technique described by Walsh et al (6).

3. MIP-TE (Most In-Phase Echo): A common calibrationless multi-echo coil combination method that calculates the coil sensitivity maps by applying low-pass filtering and normalization to the most in-phase echo.

4. ESPIRiT: A calibrationless technique described by Uecker et al (7).

5. Joint Fit: Direct (and calibrationless) estimation of unknown parameters from multi-coil data (i.e. no coil combination)(8).

CSE-MRI

All parameter estimations/CSE-MRI reconstructions were performed using nonlinear least squares and the following signal model for each coil combination method:

$$A_{c}e^{i\phi_{c}}M_{0}(1-\eta+\eta\sum_{p=1}^{6}{a_{p}e^{i2\pi*f_{p}*TE}})e^{i\phi_{m}}e^{-R2^{*}TE}e^{i2\pi*\psi*TE}~~~~~~~~~~~~~~~~~~~~~\text{[1]}\\\text{Where:}~~~~~~A_{c}e^{i\phi_{c}}~\text{=}~\text{coil}~\text{sensitivities}\\~~~~~~~~~~~~~~~M_{0}e^{i\phi_{m}}~\text{=}~\text{complex}~\text{proton}~\text{density}\\~~~~~~~~~~~~~~\eta~\text{=}~\text{PDFF,}~~~\psi~\text{=}~\text{fieldmap}\\~~~~~~~\sum_{p=1}^{6}{a_{p}e^{i2\pi*f_{p}TE}}~\text{=}~\text{fat}~\text{spectrum}$$

Experiment 1. CRLB Compared to Simulated Data

Two Cramer-Rao lower bounds (CRLB) (9)
of Eq. 1 were calculated: 1) known coil sensitivities (providing a bound for
the Roemer method), 2) unknown coil sensitivities (providing a bound for the
remaining methods). CRLBs for 8 channels, PDFF=5%, R2*=30, TE_{1}=2.6ms,
and ΔTE=1.8ms were compared to Monte-Carlo simulations (Figure 1) over a range of SNR
values [6-60] for each coil combination method. Statistics of estimated
parameters were computed pixel-wise across 1000 repetitions per SNR.

Experiment 2. Performance Over Ranges of Simulated PDFF, R2* and SNR

Monte-Carlo simulations (Figure 1) were performed for a range of PDFF
[0%-30%], R2* [20s^{-1}-200s^{-1}], and SNR [5-35] values. Average
absolute error of PDFF (bias corrected (10))
and R2* was computed in an ensemble of pixels and across 25 repetitions per
PDFF/R2*/SNR combination.

Experiment 3. Phantom Acquisitions

A multi-echo 2D spoiled gradient recalled echo (SGRE) acquisition
was repeated 2000 times (4 acquisitions x 500 repetitions/acquisition) in a
fat/water agar gel phantom on a 3.0T MR system (GE Healthcare Discovery MR750,
Waukesha, WI) using the following parameters: 2.19x2.19x1.6mm^{3}
voxels size, 128x128 matrix, 6 echoes (single shot), flip angle 3^{o}, BW=±90.91kHz.
The first 25 repetitions of each acquisition were discarded and a third order
phase correction was applied on a pixel-by-pixel basis over repetitions to
correct for phase drift over the scan duration. Subsets of N repetitions were then
averaged to modulate SNR (N=1,2,4,8,16,19). After coil combining the resulting
datasets (100 repetitions per SNR), PDFF, R2* and B0 were estimated and pixel-wise
statistics were computed across repetitions.

Experiment 1. CRLB Compared to Simulated Data

CRLB analysis, coupled with Monte-Carlo simulations, revealed the standard deviation of parameter estimates of all five methods match and are theoretically bounded by the same lower limit (Figure 2).

Experiment 2. Performance Over Ranges of Simulated PDFF, R2* and SNR

Monte-Carlo simulations over a range of PDFF, R2*, and SNR revealed all methods behave similarly at low and high SNR, except MIP-TE which exhibited bias in both PDFF and R2* in low SNR regimes (Figure 3).

Experiment 3. Phantom Acquisitions

Phantom experiments confirmed the Monte-Carlo simulation results, except bias in MIP-TE PDFF was reduced (Figure 4).

**Discussion**

This work compared the bias and noise performance in quantitative parameter maps resulting from multi-echo coil combination techniques. We examined five commonly used coil combination methods and found the variability of all five methods is theoretically bounded by the same lower limit. Importantly, simulations and phantom experiments confirmed this result over a range of PDFF, R2*, and SNR.

Our results suggest that separate calibration of coil sensitivities may be unnecessary for coil-combination in CSE-MRI. Of the remaining four methods, MIP-TE was biased at low SNR and Joint Fit was computationally costly (results not included). In contrast, either Walsh or ESPIRiT provide unbiased estimation with moderate computation. This work had several limitations: specifically, it did not include parallel imaging acceleration, for simplicity of analysis.

In summary, theoretical analysis, simulations, and phantom results demonstrated nearly equivalent performance of a wide array of coil combination methods, regardless of the use of pre-calibrated sensitivity maps.

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8. Hernando D, Haldar JP, Sutton BP, Ma J, Kellman P, Liang ZP. Joint estimation of water/fat images and field inhomogeneity map. Magn Reson Med 2008;59(3):571-580.

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10. Roberts NT, Hernando D, Holmes JH, Wiens CN, Reeder SB. Noise properties of proton density fat fraction estimated using chemical shift-encoded MRI. Magn Reson Med 2018;80(2):685-695.

Figure 1. Multi-channel
CSE-MRI data were generated as depicted above. Synthetic parameter maps and
acquisition parameters (1) are given as input in a T1/R2* weighted fat/water
CSE-MRI SGRE signal model (2). Resulting single-channel, multi-echo images (3)
are multiplied by simulated coil sensitivity maps (4, magnitude and phase) to
produce the final multi-echo, multi-channel dataset (5). Subsequent simulations
added zero-mean complex Gaussian noise on a channel-by-channel basis to the multi-echo,
multi-channel dataset and estimated input parameters using nonlinear least
squares and Eq. 1.

Figure 2. CRLB
analysis, coupled with matching Monte-Carlo simulations, revealed the standard
deviation of parameter estimates of the five coil combination methods in
question are theoretically bounded by the same lower limit. The top row plots
the CRLB of both known and unknown coil sensitivities overlaid by the standard
deviation of PDFF, R2*, and fieldmap, respectively, estimated by simulated
CSE-MRI. The bottom row plots the mean of PDFF, R2*, and fieldmap,
respectively, of the same simulated estimates. SNR is determined in first echo
of the coil combined Roemer data as average signal over standard deviation of
the noise across repetitions.

Figure 3. Repeated Monte-Carlo
simulations over a range of PDFF, R2*, and SNR revealed all methods behave
similarly at low and high SNR, except MIP-TE which exhibited bias in both PDFF
and R2* estimates in low SNR regimes. Average absolute error of estimated PDFF
(row 1), R2* (row 2) and fieldmap (row 3) are plotted over a range of PDFF
(columns 1-2) and R2* (columns 3-4). SNR is determined in first echo of the
coil combined Roemer data as average signal over standard deviation of the
noise across repetitions (Low SNR = 5, high SNR = 35).

Figure 4. Phantom
experiments in a fat/water/R2* gel agar phantom demonstrate similar performance
of all coil combination methods, except MIP-TE which becomes more biased as SNR
decreases and R2* increases. Subsets of N repetitions of 1900 repeated
multi-echo 2D SGRE acquisitions were averaged to modulate SNR
(N=1,2,4,8,16,19). Averages (across 100 repetitions) of PDFF and R2* estimates
from CSE-MRI performed after each respective coil combination method are
plotted above. SNR is determined in the first echo of the coil combined Roemer
data as average signal over standard deviation of the noise across repetitions.