Bragi Sveinsson^{1}, Garry Gold^{1}, Brian Hargreaves^{1}, and Daehyun Yoon^{1}

Double-echo in steady-state (DESS) is a 3D sequence which offers both morphological images and quantitative parameter maps (SNR-efficient 3D maps of T2 and apparent diffusion coefficient (ADC)) in various applications, such as breast imaging or knee cartilage imaging. The sequence has less sensitivity to ADC than to T2, sometimes leading to noisy ADC maps. Here, we investigate the effects of using regularized fitting of the signals, with a penalty in ADC variability, to produce less noisy ADC maps. The method is designed to apply less regularization to regions with high SNR. The approach makes use of a recent analytical expression for a ratio between DESS signals.

The DESS sequence produces signals S_{1}
before and S_{2} after a spoiler gradient^{1,2,3}. For
estimating ADC, DESS is run twice, with a large and small spoiler
gradient. This gives high and low diffusion sensitivity for the two scans, labeled
H and L, respectively, producing four signals S_{1H}, S_{2H}, S_{1L},
and S_{2L}. It has been shown that the ratio f=(S_{2H}S_{1L}/S_{1H}S_{2L})
is sensitive to ADC while being insensitive to T_{1} and T_{2}^{4}.
This, combined with a variation of a recently developed expression for f^{5},
allows a simple regularization that penalizes rapid change in ADC and produces
less noisy ADC maps:

$$ \min_{\Delta ADC} \left\lVert \frac{S_{2H}S_{1L}}{S_{1H}S_{2L}} - f(ADC + \Delta ADC)\right\lVert^2 + \lambda \left\lVert D_{xy}(ADC + \Delta ADC) \right\lVert^2 \hspace{5cm}[1]$$

where D_{xy}
is a difference operator between neighboring pixels. By multiplying the
left-hand side with S_{1H}S_{2L}, the penalty term matters less
for pixels with strong signal, thus making the regularization SNR-weighted:

$$ \min_{\Delta ADC} \left\lVert S_{2H}S_{1L} - S_{1H}S_{2L} f(ADC + \Delta ADC)\right\lVert^2 + \lambda \left\lVert D_{xy}(ADC + \Delta ADC) \right\lVert^2 \hspace{5cm}[2]$$

The expression in ref. 5 enables easy linearization of f by doing a first-order Taylor approximation, similar to ref. 6. This turns Eq. 2 into a quadratic equation that is easily solved for a given starting point. This can be iterated until the method converges on a result. This results in fast, regularized estimation.

DESS signals were simulated over 256×64 pixels using
Extended Phase Graphs (EPG)^{7} (scan parameters in Table 1), with
added noise. The tissue parameters were T_{1} = 1.2s, T_{2} =
40ms, and ADC was made to alternate between 1.6μm^{2}/ms and 1.5μm^{2}/ms,
as shown in Fig. 1a. The left half of the image was given twice the signal
strength as the right half. This led to an SNR of 100 in the S_{1L} signal
in the right half and 200 in the left half. Eq. 2 was then applied with λ=0 and
λ=0.1 (the signals were normalized so that S_{1H}S_{2L}=1 on
average over the region of interest, or equivalently λ=0.1S_{1H}S_{2L}).

The method was then tested in phantom scans, running two
DESS scans with differing diffusion weighting (scan parameters in Table 1).
First, a coil giving an SNR of 330 in the S_{1L} signal was used. Next,
the scans were repeated in a coil that gave a lower SNR of 70. ADC was
estimated both with λ=0 and λ=0.1. For reference, a standard DWI scan with
very low resolution was run.

The procedure was then tested in vivo, acquiring a pair of sagittal DESS knee scans in a healthy volunteer and another axial scan pair of another volunteer (parameters in Table 1) and processed with λ=0, 0.02, 0.1.

The simulation results are shown in Figs. 1b-c. Fig. 1b, with no regularization, gives noisy estimates, and the structure in the right half (with lower SNR) is hard to discern. Regularized results (Fig. 1c) show the ADC structure more clearly. Since less regularization is applied on the left (having stronger signal), more of the original structure is retained, while the right half is more smoothed. This is also demonstrated in Fig. 2a, which shows the average of each estimate across y. Fig. 2b shows the Fourier transform of the regularized estimate in Fig. 1c divided by that of the unregularized estimate. The regularization clearly suppresses the higher frequency components more as SNR gets lower.

The results from the phantom scans with high SNR showed
patterns which were similar after regularization. For the case with low
SNR, no particular pattern was discernible without regularization. When
regularization was applied, a similar structure as for the high-SNR case became
apparent.
The estimates are shown in Fig. 3, along with
their means and standard deviations, and were all close to the DWI reference value of 2.0 μm^{2}/ms.

The in vivo scans in Fig. 4 show smoother maps with regularization. The regularized maps indicate higher values in superficial cartilage than in deep cartilage, in agreement with ref. 8, which is difficult to discern from the non-smoothed maps (λ=0).

Table 1: Scan parameters for simulations, phantom scans,
and in vivo scans.

Figure 1: (a) A 256×64 matrix with
T_{1}=1.2s, T_{2}=40ms, ADC = 1.5μm^{2}/ms
and ADC = 1.6μm^{2}/ms. The left half (128 pixels) has SNR=200 while the
right half has SNR=100. (b) Without regularization, this results in noisy ADC
maps, and the true ADC distribution is especially hard to see in the right
half. (c) Using Eq. 2 with λ=0.1
reveals the structure of the ADC distribution.

Figure 2: (a) The average of the estimates In Figs. 1b and
1c across y. Since the regularization is weighted by signal strength, stronger
smoothing is applied to the right half, while the true ADC structure is very
clear in the left half. (b) The ratio of the Fourier transforms of Figs. 1c and
1b demonstrates how much the regularization suppresses higher frequency
components. More high frequency components are retained with higher SNR. The
transforms are averaged across k_{y}, and only half of the frequency
components are shown since the transforms are symmetric.

Figure 3: ADC maps of a phantom from scans with high and
low SNR, with and without regularization. The regularized map with high SNR
retains most of the pattern that is seen without regularization. In the low-SNR
case, no pattern is discernible without regularization, but becomes visible
when regularization is used.

Figure 4:
(a-b) A sagittal and axial knee scan, processed
by applying Eq. 2 with λ = 0. (c-d) The same scan processed with λ =
0.02. (e-f) The same scan processed with λ = 0.1.