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Evaluating the Precision of Multi-Echo Combination Methods for Susceptibility Mapping by Analysing the Propagation of Single-Echo Phase Noise into Multi-Echo Field and Susceptibility Maps
Emma Biondetti1, Anita Karsa1, David L Thomas2,3, and Karin Shmueli1

1Medical Physics and Biomedical Engineering, University College London, London, United Kingdom, 2Academic Neuroradiological Unit, Department of Brain Repair and Rehabilitation, Institute of Neurology, University College London, London, United Kingdom, 3Leonard Wolfson Experimental Neurology Centre, Institute of Neurology, University College London, London, United Kingdom

Synopsis

In Susceptibility Mapping (SM) using multi-echo acquisitions, noise propagates from the single-echo phase images into the field map in a manner dependent on the method used for multi-echo combination. Field noise then propagates into the susceptibility map, determining the precision of the measured susceptibility. Here, we characterised the propagation of single-echo phase noise into both the combined field and susceptibility maps using three methods for multi-echo combination: fitting, averaging and echo time-weighted averaging. We calculated susceptibility noise maps for both simulated and acquired data, showing that, when choosing a pipeline for multi-echo SM, it is important to consider its precision.

Introduction

Accuracy and precision are two metrics that enable the full characterisation of a processing pipeline for susceptibility mapping (SM). A pipeline’s accuracy depends on the closeness of the measured susceptibility $$$\chi$$$ to $$$\chi$$$’s true value, whereas its precision is determined by how the noise propagates from the signal phase into the measured $$$\chi$$$.

In previous studies of SM of multi-echo gradient-echo (GRE) images1,2, we showed that combining multiple echoes by non-linearly fitting the complex GRE signal over echo times (TEs) (NLFit) gives a more accurate average $$$\chi$$$ than combining multiple echoes by averaging the single-echo phase over TEs, both without weighting (Avg) or with weighting proportional to the TE (wAvg) (Figure 1). Here, we aimed to evaluate how the noise propagates from the single-echo phase images $$$\sigma(\phi(TE))$$$ into the total and local field ($$${\Delta}B_{Tot}$$$ and $$${\Delta}B_{Loc}$$$) and $$$\chi$$$.

We calculated the noise in a $$$\chi$$$ map $$$\sigma(\chi)$$$ as a function of $$$\sigma\left({\Delta}B_{Loc}\right)$$$ for $$$\chi$$$ calculated using Thresholded K-space Division (TKD)3. Previous studies using NLFit4 or Avg5 have calculated the noise in $$${\Delta}B_{Tot}$$$ ($$$\sigma({\Delta}B_{Tot})$$$), but an expression for $$$\sigma({\Delta}B_{Tot})$$$ has not yet been calculated for wAvg6,7 so we derived this here. We compared $$$\sigma({\Delta}B_{Loc})$$$ and $$$\sigma(\chi)$$$ in the three pipelines using data simulated from a realistic head phantom and images of five healthy volunteers (HVs).

Theory

The equations for the calculation of $$$\sigma\left({\Delta}B_{Tot}\right)$$$, including the one we derived for wAvg, are shown at the bottom of Figure 1, with $$$n$$$ the number of TEs and $$$\sigma^2$$$ the variance.

For all pipelines, based on error propagation and the orthogonality8 of $$${\Delta}B_{Loc}$$$ and the background fields ($$${\Delta}B_{Bg}$$$) in the brain:

$$\sigma({\Delta}B_{Loc}(\mathbf{r}))=\sqrt{\sigma^2({\Delta}B_{Tot}(\mathbf{r}))-\sigma^2({\Delta}B_{Bg}(\mathbf{r}))}\qquad[1]$$

with $$$\mathbf{r}$$$ the position of a voxel in image space. Thus,

$$\sigma({\Delta}B_{Loc}(\mathbf{r}))\leq\sigma({\Delta}B_{Tot}(\mathbf{r}))\qquad[2]$$

Based on error propagation, the worst-case noise in $$$\chi$$$ for TKD3 is:

$$\sigma(\chi(\mathbf{r}))=\frac{\sqrt{FT^{-1}(D^{-2}\left(\mathbf{k}\right)FT\left(\sigma^2\left({\Delta}B_{Tot}\left(\mathbf{r}\right)\right)\right)}}{B_0}\qquad[3]$$

with $$$D$$$ the TKD dipole kernel in k-space, $$$\mathbf{k}$$$ the position of a voxel in k-space, $$$FT$$$ the Fourier Transform and $$$B_0$$$ the field strength.

Methods

Multi-echo 3D GRE brain images of five HVs were acquired on a Philips Achieva 3T system (Best, NL) using a 32-channel head coil and a previously described protocol9. Two data sets were acquired in the same session, with $$$M_i,\phi_i$$$ for $$$i=1,2$$$ the magnitude and phase from each data set.

Noisy complex data were simulated at the same TEs, at 3 T, based on a Zubal head phantom10 using realistic ground-truth $$$\chi$$$8,11, proton density ($$$M_0$$$)11 and $$$T_2^*$$$ values12-14 (Figure 2), and a standard deviation (SD) of the magnitude $$$\sigma(M)=0.07$$$ chosen according to the following magnitude signal-to-noise ratio (SNR) criterion for the longest TE image15 in the superior sagittal sinus (SSS):

$$\min_{TE}\left(SNR\right)\geq3{\quad\Leftrightarrow\quad}\sigma(M)\leq\frac{M_{0,SSS}(TE_5)\exp(-TE_5/T_{2,SSS}^*)}{3}=\frac{0.7\exp(-24.6ms/21.2ms)}{3}=0.07\qquad[4]$$

Five 20x20-voxel ROIs16 were drawn on a sagittal slice of HV1’s first-echo $$$M_1$$$ image (Figure 3), then copied to $$$M_1$$$ of HV2-5 after rigid image co-registration with NiftyReg17.

In each HV, ROI-based magnitude ($$$M_{ROI}$$$), magnitude noise ($$$\sigma_{ROI}(M)$$$) and phase noise ($$$\sigma_{ROI}(\phi)$$$) values were calculated as18:

$$M_{ROI}(TE,\mathbf{r})=\frac{1}{2}mean(M_1(TE,\mathbf{r}+M_2(TE,\mathbf{r}))),\quad\mathbf{r}\in\text{ROI}\qquad[5]$$

$$\sigma_{ROI}(M(TE,\mathbf{r}))=\sqrt{R}\frac{1}{\sqrt{2}}SD(M_1(TE,\mathbf{r})-M_2(TE,\mathbf{r})),\quad\mathbf{r}\in\text{ROI}\qquad[6]$$

$$\sigma_{ROI}(\phi(TE,\mathbf{r}))=\sqrt{R}\frac{1}{\sqrt{2}}SD(\phi_1(TE,\mathbf{r})-\phi_2(TE,\mathbf{r})),\quad\mathbf{r}\in\text{ROI}\qquad[7]$$

with $$$R$$$ the 2D-SENSE reduction factor19. Equations [5]-[7] were averaged across ROIs to create summary values of magnitude ($$$\hat{M}_{ROI}$$$), magnitude noise ($$$\hat{\sigma}_{ROI}(M)$$$) and phase noise ($$$\hat{\sigma}_{ROI}(\phi)$$$) at each TE.

For both the simulations and the HVs, a map of $$$\mathbf{\sigma(\phi)}$$$ was calculated at each TE as15:

$$\sigma(\phi(TE,\mathbf{r}))=c(TE)\frac{1}{SNR(TE,\mathbf{r})}=c(TE)\frac{\hat{\sigma}_{ROI}(M(TE))}{M(TE,\mathbf{r})}\qquad[8]$$

with $$$c$$$ a constant of proportionality equal to 1 in the simulations and to

$$c(TE)=\frac{\hat{\sigma}_{ROI}(\phi(TE))}{1/SNR_{ROI}(M)}=\frac{\hat{\sigma}_{ROI}(\phi(TE))\hat{\sigma}_{ROI}(M(TE))}{\hat{M}_{ROI}(TE)}\qquad[9]$$

in the HVs.

For each pipeline, a map of $$$\sigma({\Delta}B_{Tot})$$$ was calculated from the multi-echo $$$\sigma(\phi)$$$ maps using the corresponding equation for $$$\sigma({\Delta}B_{Tot})$$$ (Figure 1, F1-F3). Then, $$$\sigma(\chi)$$$ was calculated using Eq. [3] with $$$\sigma({\Delta}B_{Tot})$$$ for the corresponding pipeline, a threshold=2/3 for $$$D$$$'s inversion, and correction for $$$\chi$$$ underestimation20. To compare $$$\sigma({\Delta}B_{Tot})$$$ and $$$\sigma(\chi)$$$ between pipelines a line profile was traced in the $$$\sigma({\Delta}B_{Tot})$$$ and $$$\sigma(\chi)$$$ images.

Results and Discussion

In both the phantom (Figure 4) and the HVs (Figure 5), wAvg had the lowest $$$\sigma(\chi)$$$, whereas Avg and NLFit had a similar $$$\sigma(\chi)$$$. In both the phantom and the HVs, the difference between NLFit and wAvg/Avg appeared prominent in large-$$$\chi$$$ regions, in line with our previous studies1,2, where $$$\chi$$$ calculated by wAvg always had the smallest SD in all ROIs and $$$\chi$$$ calculated by NLFit always had the largest SD in the veins. $$$\sigma(\chi)$$$ differences between different pipelines, however, were always in the order of a few parts per billion.

Conclusions

We have shown that for multi-echo SM combining the multi-echo phase by fitting vs. averaging over TEs is less precise, although it is more accurate in high-$$$\chi$$$ regions1,2. We suggest that the precision and accuracy of $$$\chi$$$ should be analysed together to assess and compare the relative performance of different SM pipelines for different applications.

Acknowledgements

DLT is supported by the UCL Leonard Wolfson Experimental Neurology Centre (PR/ylr/18575).

References

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Figures

Figure 1 Processing pipelines for multi-echo SM analysed in this study. NLFit (blue) combines multiple TEs by non-linearly fitting the multi-echo complex signal over time11. wAvg (yellow) and Avg (orange) unwrap the single-echo phase using a Laplacian-based method (SHARP20 with a threshold=10-10) then combine multiple TEs by averaging5 (Avg) or averaging6,7 using a TE-based weighting of the phase (wAvg). All the pipelines remove background field variations from $$${\Delta}B_{Tot}$$$ using SHARP20 with a threshold=0.05, then calculate $$$\chi$$$ using TKD3 with correction for $$$\chi$$$ underestimation20.

In the bottom panel, the equations for the calculation of $$${\Delta}B_{Tot}$$$ are shown for each pipeline.


Figure 2 $$$M_0$$$ values in arbitrary units (a. u.), $$$T_2^*$$$ in ms and $$$\chi$$$ in ppm assigned to various ROIs in the phantom (a), and locations of these ROIs in the $$$\chi$$$ (b) and $$$T_2^*$$$ phantoms (c). The superscript number next to a value indicates the reference from which that value was taken. There were two exceptions: $$$T_2^*$$$ in CSF was chosen to be much longer than in the other ROIs; $$$T_2^*$$$ in the rest of the brain was set to an arbitrary value different from $$$T_2^*$$$ in the other ROIs.

Figure 3 The five 20x20-voxel ROIs overlaid on a sagittal slice of the first-echo magnitude image of HV1. These ROIs were drawn including grey and white matter and taking care to exclude/avoid SENSE-, motion- and flow-induced artifacts, then coregistered to the first-echo magnitude image of HV2-5.

Figure 4 $$$\sigma(\chi)$$$ (proportional to $$$\sigma({\Delta}B_{Loc})$$$) in the phantom. The line was drawn including both the superior sagittal sinus (arrows) and the surrounding brain tissue. $$$\sigma({\Delta}B_{Loc})$$$ images are not shown here, as they were proportional to $$$\sigma(\chi)$$$ (see Eq. [3]).

Figure 5 Images of the noise susceptibility maps ($$$\sigma(\chi)$$$) and line profiles are shown for each processing pipeline in each HV. In each HV, the lines were drawn including both high-$$$\chi$$$ regions, e.g. the veins, which appear bright in the images, and the surrounding brain tissue. $$$\sigma({\Delta}B_{Loc})$$$ images are not shown here, as they were proportional to $$$\sigma(\chi)$$$ (see Eq. [3]).

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
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