Synopsis

Recently, a robust principle component analysis (rPCA) method was implemented to myelin water fraction (MWF) mapping using mGRE data. Based on the intrinsic nature of T2* relaxation, hankelization and non-negative matrix factorization was implemented to enhance low rankness of each rank-1 component. In this study, the noise sensitivity of model-free rPCA was investigated. According to simulation and in vivo analysis, model-free rPCA technique is more robust at noise and other physiological artifacts than model-based fitting technique.

Introduction

Myelin water imaging (MWI) imaging is widely used as a noninvasive imaging technique to diagnose abnormal neuroplasticity such as multiple sclerosis¹. Developments of multi-echo gradient echo (mGRE) based MWI have led to various multi-component T2^* relaxation analysis techniques². However, mGRE-MWI using numerical models (e.g. magnitude three-pool, complex three-pool) suffer from imaging artifacts such as physiological noise^2,3. Recently, as an application of the blind source separation (BSS) technique, a robust principle component analysis (rPCA) has been suggested to separate signal sources of the mGRE data into slow-decaying, fast-decaying and artifact component by enforcing the data-driven properties (e.g. low rankness and sparsity)⁴.

The objective of this study is to investigate the noise sensitivity of model-free rPCA for MWF mapping when compared to model-based fitting techniques. In the simulation, a numerical phantom composed of three-pool of T2^* (myelin/intra/extra water) was investigated in varying SNR situations. In in-vivo, data from two healthy volunteers and two patients were acquired to investigate the effect of various physiological artifacts.

Method

[Model-free rPCA for mGRE data]

Based on the intrinsic characteristics of T2* relaxation (e.g. exponential and non-negativity), the mGRE data was processed with hankelization and non-negative matrix factorization (NMF) to enhance the rank-1 property of each low-rank component (Fig.1). The modified rPCA is as followed:

$$\min_{L_1, L_2, S}\frac{1}{2}\parallel L_1+L_2+S-y\parallel^2_2 + \sum_r \parallel R(H(L_1)) \parallel_*+ \sum_r \parallel R(H(L_2)) \parallel_*+ \parallel \psi(S) \parallel_1$$

where ||·||_* is the nuclear norm using NMF, H(∙) is the hankelization in the echo domain, ψ(∙) is a temporal sparsifying operator (1D-FFT in the echo domain), R(∙) is the extraction of local patches for locally low-rank⁵. The length of hankelization was selected to half the number of echoes⁶. The separated three components represent slow-decaying (L₁), fast-decaying (L₂) and artifact (S) components⁴. MWF was estimated by $$$ \frac{L_2}{L_1+L_2} $$$.

[Data Processing]

Two different fitting models and model-free rPCA were analyzed. The numerical model and parameters of the magnitude three-pool model and complex three-pool model are represented in Table1.

[Simulation]

The three-components of T2^* were assumed to be Gaussian ((μ,σ): T^*_2,1=(10ms, 2ms), T^*_2,2=(70ms, 5ms), T^*_2,3=(50ms, 3ms)) distributed. The MWF map was consist of various range of MWF (=2:2:24%). The histogram of relative error $$$(=\frac{MWF_{estimated} - MWF_{true}}{MWF_{true}})$$$ throughout MWF map was fitted to normalized distribution function to estimate the bias. Standard deviation of MWFestimated and Pearson correlation coefficient $$$(=\frac{cov(MWF_{true}, MWF_{estimated})}{σ_{true} \cdotσ_{estimated} }, r^{2})$$$ between MWFtrue map and MWFestimated map were investigated in various SNR (=50:50:200). Other parameters were as followed: Matrix size = 128x128x5, ΔTE = 2ms, TE1 = 1.7ms, # of echoes = 22.

[Data Acquisition]

Data from two healthy volunteers and two patients with Alzheimer's Disease (AD) were acquired on a clinical 3 Tesla MRI scanner (Tim Trio, Siemens Medical Solution, Erlangen, Germany) using a 12-channel head coil. The 3D mGRE data were acquired as followed: FOV = 256x256x120mm³, spatial resolution = 2x2x3mm³, flip angle = 20°, TR = 60ms, TE1 = 1.6ms, ΔTE = 2.1ms, # of echoes = 15. Additionally, MPRAGE was obtained for anatomy.

Result

Figure 2 shows the effect of noise on MWF map by using model-free rPCA and two different fitting models in simulation. The relative error represented underestimation bias of 23% in model-free rPCA (Fig.2a). The standard deviation of MWFestimated by using model-free rPCA was lower than the model-based fitting (Fig.2b). The correlation coefficient (r²) between MWFtrue map and MWFestimated map was over 0.98 even at SNR = 50 (Fig.2c).

Figure 3 shows the effect of noise on MWF map by using model-free rPCA and complex 3-pool model fitting in healthy volunteer. The correlation coefficient of model-free rPCA was over 0.95 (Fig.3c). Histogram of relative error was similar to simulation results (Fig.3d).

Figure 4 shows the effect of various artifacts in mGRE data on MWF map by using model-free rPCA and complex three-pool model fitting in in-vivo. Motion corrupted artifact (Fig.4a, patient), B0 inhomogeneity (Fig.4b, healthy volunteer) and flow artifact (Fig.4c, patient) in mGRE data deviated from complex three-pool model and represented in MWF map by using model-based fitting.

Conclusion and Discussion

In this study, we investigated the performances of the model-free rPCA compared to the model-based fitting methods with respect to noise and artifacts. In specific, by numerical simulations, model-free rPCA showed more robustness to the noise. Also, visual inspections from representative slices from the in-vivo subjects show that the method is better in three additional circumstances. The benefits of using the method is that the artifacts can be separated as sparse component (S), which is why the method shows better results for the mentioned circumstances. However, limitation of this technique is the slight underestimation for overall MWF values. However, as can be seen in the correlation coefficients, the underestimated level is linearly proportional to the values.

Acknowledgements

References

1. MacKay A, Laule C. Magnetic resonance of myelin water: an in vivo marker for myelin. Brain Plasticity 2016;2(1):71-91.

2. Nam, Y., Lee, J., et al. Improved estimation of myelin water fraction using complex model fitting. Neuroimage. 2015b;116:214-221.

3. Lee, H., Nam, Y., et al Improved three dimensional multi-echo gradient echo based myelin water fraction mapping with phase related artifact correction. Neuroimage. 2018;169:1-10.

4. Shin, J. Lee, H., et al. robust principal analysis in multi-echo gradient-echo imaging. ISMRM 26th Annual Meeting. 2018

5. Trzasko, J.D.,et al. Exploiting local low-rank structure in higher-dimensional MRI applications. SPIE,2013;p.8.6.