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.
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).
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.
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.
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.
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.