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) images^1,2, 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 wAvg^6,7 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 $T_2^*$ values^12-14 (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 studies^1,2, 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

Acknowledgements

References

1. Biondetti E, Thomas D L and Shmueli K. Application of Laplacian-based Methods to Multi-Echo Phase Data for Accurate Susceptibility Mapping. Proceedings of the International Society for Magnetic Resonance in Medicine. 2016; 24:1547.
2. Biondetti E, Karsa A, Thomas D L, et al. The Effect of Averaging the Laplacian-Processed Phase over Echo Times on the Accuracy of Local Field and Susceptibility Maps. Graz, Austria: 4th International Workshop on MRI Phase Contrast & Quantitative Susceptibility Mapping; 2016.
3. Shmueli K, de Zwart JA, van Gelderen P, et al. Magnetic susceptibility mapping of brain tissue in vivo using MRI phase data. Magnetic Resonance in Medicine. 2009;62(6):1510–1522.
4. Kressler B, De Rochefort L, Liu T, et al. Nonlinear regularization for per voxel estimation of magnetic susceptibility distributions from MRI field maps. IEEE Transactions on Medical Imaging. 2010;29(2):273–281.
5. Wu B, Li W, Avram AV, et al. Fast and tissue-optimized mapping of magnetic susceptibility and T2* with multi-echo and multi-shot spirals. NeuroImage. 2012;59(1):297–305.
6. Li W, Wang N, Yu F, et al. A method for estimating and removing streaking artifacts in quantitative susceptibility mapping. NeuroImage. 2015;108:111–122.
7. Wei H, Dibb R, Zhou Y, et al. Streaking artifact reduction for quantitative susceptibility mapping of sources with large dynamic range. NMR in Biomedicine. 2015;28(10):1294–1303.
8. Liu T, Khalidov I, de Rochefort L, et al. A novel background field removal method for MRI using projection onto dipole fields (PDF). NMR in Biomedicine. 2011;24(9):1129–1136.
9. Biondetti E, Karsa A, Thomas DL, et al. Evaluating The Accuracy of Susceptibility Maps Calculated from Single-Echo versus Multi- Echo Gradient-Echo Acquisitions. Proceedings of the International Society of Magnetic Resonance in Medicine. 2017;25:1955.
10. Zubal IG, Harrell CR, Smith EO, et al. Two dedicated software, voxel-based, anthropomorphic (torso and head) phantoms. Proceeding of the International Conference at the National Radiological Protection Board. 1995:105–111.
11. Liu T, Wisnieff C, Lou M, et al. Nonlinear formulation of the magnetic field to source relationship for robust quantitative susceptibility mapping. Magnetic Resonance in Medicine. 2013;69(2):467–476.
12. Peters AM, Brookes MJ, Hoogenraad FG, et al. T2* measurements in human brain at 1.5, 3 and 7 T. Proceedings of the International Society for Magnetic Resonance in Medicine 2006;14:926.
13. Zhao JM, Clingman CS, Närväinen MJ, et al. Oxygenation and hematocrit dependence of transverse relaxation rates of blood at 3T. Magnetic Resonance in Medicine. 2007;58(3):592–597.
14. Yao B, Li T-Q, van Gelderen P, et al. Susceptibility contrast in high field MRI of human brain as a function of tissue iron content. NeuroImage. 2009;44(4):1259–1266.
15. Gudbjartsson H and Patz S. The Rician distribution of noisy MRI data. Magnetic Resonance in Medicine. 1995;34(6):910–914.
16. Lerski RA, de Wilde J, Boyce D, et al. Quality Control in Magnetic Resonance Imaging. IPEM Report 80. York, UK: The Institute of Physics and Engineering in Medicine, 1998:14-16.
17. Ourselin S, Roche A, Subsol G, et al. Reconstructing a 3D structure from serial histological sections. Image and Vision Computing. 2001;19(1–2):25–31.
18. Dietrich O, Raya JG, Reeder SB, et al. Measurement of signal-to-noise ratios in MR images: Influence of multichannel coils, parallel imaging, and reconstruction filters. Journal of Magnetic Resonance Imaging. 2007;26(2):375–385.
19. Weiger M, Pruessmann KP and Boesiger P. 2D SENSE for faster 3D MRI. Magnetic Resonance Materials in Physics, Biology and Medicine. 2002;14(1):10–19.
20. Schweser F, Deistung A, Sommer K, et al. Toward online reconstruction of quantitative susceptibility maps: Superfast dipole inversion. Magnetic Resonance in Medicine. 2013;69(6):1582–1594.