Synopsis

Keywords: Quantitative Imaging, Quantitative Imaging

A dual MDI strategy was developed and demonstrated to achieve T₂* and R₂* mapping that are both highly delineated between noise backgrounds and normal tissues as well as spikes free, thus eliminating the need for routine noise masking or manual segmentation.

Introduction

Noise-mask has always been necessary in T₂* (T₂ as well) mapping to spiking results related to low signal-to-noise ratio(SNR) such as background noise, and to long T₂* species like cerebral spinal fluids (CSF). The need for noise-masking has been essential to model-based curve fitting methods¹, as well as non-model based analytical methods^2,3, as they share two common aspects that are rarely discussed or conveniently ignored. The first one is that their T₂* mapping results are non-specific between long T₂* and low signal-to-noise ratio (SNR) signals, both of which create a ‘flat’ signal-time curve that will always be interpreted with long T2* values, because T2* cannot be 0 in the exponential model. The second is processing with real signals, which shows noise distribution of Rician or Rayleigh⁴ with non-zero means, rather than the zero-mean Gaussian. This noise distribution change will lead to strong T2* variations in the form of spikes for long T₂*, low SNR and noise signals.
Recently, a multi-dimensional integration (MDI) method⁵ was proposed to yield well-delineated T₂* maps, generating high T₂* for CSF but low T₂* for noise background with minimal spikes. However, this only applied to T₂*, not R₂* maps. Nevertheless, the MDI method provided a potential perspective for T₂* mapping via complex signal processing with Gaussian noise.
In this work, we propose a dual MDI strategy, namely dMDI, to achieve independent T₂*/R₂* mapping without the need for noise masking or manual extraction.

Methods

Denote the complex signal of the i_th echo and j_th coil channel as S_i,j. According to the MDI theory⁵, a signal function G_MDI is defined as the signal ratio between consecutive echoes:
$$G_{MDI}(i,j)\equiv \left | \frac{S_{i+1,j}}{S_{i,j}} \right |= e^{-\frac{\Delta TE}{T_{2}^{*}}} [1]$$
And by solving $$$\min_{\Theta }\sum_{i}^{N_{e}-1}\sum_{j}^{N_{c}}\left \| S_{i+1,j}-S_{i,j}\cdot \Theta \right \|_{2}^{2}$$$, the numeric solution of G_MDI can be efficiently calculated as:
$$\Theta _{MDI}=\left | \frac{\sum_{i}^{N_{e}-1}\sum_{j}^{N_{c}}S_{i,j}^{*}\cdot S_{i+1,j}}{\sum_{i}^{N_{e}-1}\sum_{j}^{N_{c}}\left | S_{i,j} \right |^{2}} \right | [2]$$
And the corresponding R₂* and T₂* values can be calculated as:
$$R_{2MDI}^{*}=-\frac{ln(\Theta _{MDI})}{\Delta TE} [3] $$
$$T_{2MDI}^{*}=-\frac{\Delta TE}{ln(\Theta _{MDI})} [4] $$
Eqs.1~4 are described previously⁵ and thus denoted with the subscript ‘MDI’. Similarly, an inversed signal function G_MDIinv can be defined as:
$$G_{MDIinv}(i,j)\equiv \left | \frac{S_{i,j}}{S_{i+1,j}} \right |= e^{\frac{\Delta TE}{T_{2}^{*}}} [5]$$
And similarly:
$$\Theta _{MDIinv}=\left | \frac{\sum_{i}^{N_{e}-1}\sum_{j}^{N_{c}}S_{i+1,j}^{*}\cdot S_{i,j}}{\sum_{i}^{N_{e}-1}\sum_{j}^{N_{c}}\left | S_{i+1,j} \right |^{2}} \right | [6]$$
$$R_{2MDIinv}^{*}=\frac{ln(\Theta _{MDIinv})}{\Delta TE} [7] $$
$$T_{2MDIinv}^{*}=-\frac{\Delta TE}{ln(\Theta _{MDIinv})} [8] $$
Finally, the proposed dMDI strategy for T₂*/R₂* is:
$$R_{2dMDI}^{*}=\frac{-ln(\Theta _{MDI})+ln(\Theta _{MDIinv})}{2\Delta TE} [9] $$
$$T_{2dMDI}^{*}=\left [ \frac{\left | ln(\Theta _{MDI}) \right |+\left | ln(\Theta _{MDIinv}) \right |}{2\Delta TE} \right ]^{-1} [10] $$
Computer simulation and 3T (uMR Omega, United Imaging, Shanghai, China) multi-echo GRE knee data were used for demonstration. Non-linear Levenberg–Marquardt (NLM) curve fitting using the model $$$S(t)=be^{-a\cdot t}$$$, as well as the analytic NumART² and ARLO³ methods, were adopted for comparison.

Results

The simulated Θs and ln(Θ) combinations as functions of SNR are shown in Figure 1. The signal ratios Θ_MDI and Θ_MDIinv approach their respective true values at sufficiently high SNR, but towards zero as SNR decreases (Fig.1a). Among the ln(Θ) combinations, the $$$[-ln(\Theta _{MDI})+ln(\Theta _{MDIinv})]/2$$$(in Fig.1b) and $$$[|ln(\Theta _{MDI})|+|ln(\Theta _{MDIinv})|]/2$$$ (Fig.1c) would also approach zero as SNR decreases. This means the corresponding T₂* and R₂* values of dMDI (Eqs.9&10) will both approach zero at low SNR, while achieving accurate results for decent SNR.
Figure 2 shows T₂*/R₂* maps of the knee from different methods. Spikes and high-value backgrounds can be seen on the R₂* map of MDI and the T₂* maps of NLM, NumART and ARLO, but are totally eliminated in both maps of dMDI.
Figure 3 shows the direct overlaid images of T₂* maps on the PD weighted anatomy magnitude image. The dMDI T₂* of the cartilage and ligaments are well displayed with excellent structure details, while the NLM overlaid results display unreliable visual quality with numerous spikes and inconsistent backgrounds.

Discussion

The original MDI method has shown that, by jointly processing the complex signals from both signal dimensions of echoes and coil channels, the background T₂* values can be suppressed due to the intrinsic cancellation of Gaussian noise. However, its R₂* maps are still calculated as the reciprocal, thus yielding high R₂* values where T₂* is low. Therefore, noise masking is still needed for MDI R₂* maps instead of T₂* maps, merely flipping the situation of previous methods (Fig.2).
On the other hand, by additionally constructing a directionally reversed MDI, i.e. the nominal reciprocal of MDI, opposite trends of SNR response in ln(Θ) can be achieved between MDI and MDI_inv, especially in the low SNR regime. These opposite trends can negate each other when properly combined (Figs.1b&1c, Eqs.9&10), which is the working principle of dMDI to achieve intrinsic SNR dependency and spike elimination in both T₂*/R₂* maps. With virtually zero artifacts, the T₂* maps can be directly overlaid onto the anatomic images without noise masking or manual segmentation, greatly improving the workflow and evaluation reliability (Fig.3).

Conclusion

In conclusion, a dMDI method was proposed and demonstrated. To the best of our knowledge, this is the first demonstration of both T2* and R2* mapping that simultaneously offer exceptional quality and accuracy and as a result, require noise masking for neither.

Acknowledgements

No acknowledgement found.