Johannes Breitling^{1,2,3}, Anagha Deshmane^{4}, Steffen Goerke^{1}, Kai Herz^{4}, Mark E. Ladd^{1,2,5}, Klaus Scheffler^{4,6}, Peter Bachert^{1,2}, and Moritz Zaiss^{4}

Chemical exchange saturation transfer (CEST) MRI allows for the indirect detection of low-concentration biomolecules by their saturation transfer to the abundant water pool. However, reliable quantification of CEST effects remains challenging and requires a high image signal-to-noise ratio. In this study, we show that principle component analysis can provide a denoising capability which is comparable or better than 6-fold averaging. Principle component analysis allows identifying similarities across all noisy Z-spectra, and thus, extracting the relevant information. The resulting denoised Z-spectra provide a more stable basis for quantification of selective CEST effects, without requiring additional measurements.

The proposed denoising algorithm (Fig. 1) is applied
after motion correction using a rigid-registration-algorithm
in MITK^{3}, normalization, correction of B_{0}-inhomogeneities,
and segmentation of brain tissues (cerebrospinal fluid, gray matter, and white
matter). CEST data, consisting of images of size
$$${u}\times{v}\times{w}$$$ for the $$$n$$$ saturation frequency offsets, is
reshaped into a Casorati matrix
$$$\textbf{C}$$$ of size
$$$ {m}\times{n}$$$
, with $$${m}\leq{u}\cdot{v}\cdot{w}$$$
being the
number of brain voxels. Each row of
$$$\textbf{C}$$$ represents the
Z-spectrum of one voxel and each column represents a complete segmented image
for one saturation frequency offset.
PCA is performed by eigendecomposition of the
covariance matrix

$$\mathrm{cov}(\textbf{C})=\frac{1}{n-1}\widetilde{\textbf{C}}^\textbf{T}\widetilde{\textbf{C}}$$

$$\quad=\bf\Phi^\textbf{T}\Lambda\Phi$$

with
$$$\widetilde{\textbf{C}}$$$ being the
column-wise mean-centered Casorati matrix,
$$${\bf\Phi}=({\bf\varphi}_1{\bf\varphi}_2\ldots{\bf\varphi}_n)$$$ being the
$$${n}\times{n}$$$ orthonormal
eigenvector matrix and $$${\bf\Lambda}=\mathrm{diag}(\lambda_1,\lambda_2,\ldots,\lambda_n)$$$ being the
associated diagonal eigenvalue matrix with
$$$\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n$$$
. The variance i.e. information content of a signal
will concentrate in the first few PCs, whereas the noise is spread evenly over
the dataset. Therefore preserving the first few PCs will remove noise from the
data set. The optimal number of components
can be determined
by an empirical indicator function applied to the eigenvalues.^{4}

$$k=\underset{i}{\mathrm{argmin}}\left[\frac{\sum_{l=i+1}^{l=n}\lambda_l}{m(n-i)^5}\right]^\frac{1}{2}$$

Projection of $$$\widetilde{\textbf{C}}$$$ onto the reduced set of the first $$$k$$$ eigenvectors $$${\bf\Phi}_{(k)}$$$

$$\widetilde{\textbf{C}}_{(k)}=\widetilde{\textbf{C}}{\bf\Phi}_{(k)}{{\bf\Phi}_{(k)}}^\textbf{T}$$

and addition of the mean Z-spectrum results in the denoised Casorati matrix. Denoised Z-spectra are reformatted into a final denoised image series with dimensions $$${u}\times{v}\times{w}\times{n}$$$.

In vivo 3D-CEST-MRI
(1.7×1.7×3 mm^{3}, 12 slices) was performed on a 7T whole-body scanner
(Siemens Healthineers, Germany) using the snapshot-CEST approach^{5}.
Pre-saturation by 140 Gaussian-shaped pulses (t_{p} = 15 ms, duty cycle
= 60%, t_{sat} = 3.5 s) was applied at 54 unevenly distributed offsets for
two different mean B_{1} = 0.6 and 0.9 µT. Each Z-spectrum was acquired
six times to enable comparison with high-SNR data obtained by averaging. Conventional,
averaged and denoised Z-spectra were fitted pixel-wise with a Lorentzian 5-pool
fit model. Lorentzian difference images were calculated according to $$$\mathrm{MTR_{LD}}=Z_{ref}-Z$$$ and corrected for B_{1}-inhomogeneities^{6}.

- Hotelling, H. Analysis of a Complex of Statistical Variables into Principal Components. Journal of Educational Psychology 1933;24:417-441,498-520.
- Döpfert J, Witte C, Kunth M, and Schröder L. Sensitivity enhancement of (Hyper-)CEST image series by exploiting redundancies in the spectral domain. Contrast Media Mol. Imaging 2014;9:100-107.
- Nolden M, Zelzer S, Seitel A, et al. The Medical Imaging Interaction Toolkit: challenges and advances. Int J CARS 2013; 8(4):607-620.
- Malinowski ER. Determination of the number of factors and the experimental error in a data matrix. Anal. Chem. 1977;49:612-617.
- Zaiss M, Ehses P, and Scheffler K. Snapshot-CEST: Optimizing spiral-centric-reordered gradient echo acquisition for fast and robust 3D CEST MRI at 9.4 T. NMR Biomed 2018;31:e3879.
- Windschuh J, Zaiss M, Meissner JE, et al. Correction of B1-inhomogeneities for relaxation-compensated CEST imaging at 7 T. NMR Biomed 2015;28:529-537.