Riccardo Metere^{1}, Samuel A. Hurley^{2}, André Pampel^{1}, Karla Loreen Miller^{2}, and Harald E. Möller^{1}

Quantitative magnetization transfer experiments require extensive sampling of the off-resonance spectrum to obtain information of the relaxation properties of non-water protons. To reduce the acquisition times, the off-resonance sampling has been optimized in previous works based on the stability of approximate biophysical model fits. Here, we use a singular value decomposition approach for the analysis of the principal components. In particular we propose a data-driven optimization method for the acquisition scheme and we discuss the potential impact of applying this analysis for parameter estimations, including potential extensions of the classical biophysical models.

We acquired images of a post-mortem full human brain using a Siemens MAGNETOM 7T MRI scanner, a circularly polarized Tx / 32-Channel Rx head coil, and modified FLASH sequence ($$$T_R=70\;\mathrm{ms}$$$, $$$T_E=7.88\;\mathrm{ms}$$$, $$$\alpha=30\mathrm{°}$$$, resolution: $$$1\;\mathrm{mm}$$$ isotropic) with a saturation pulse before the acquisition of each k-space line. The saturation pulse was tuned logarithmically in the frequency offset range $$$100\mathrm{-}100000\mathrm{Hz}$$$ with varying power level corresponding to nominal on-resonance flip angles of $$$200°,\,300°,\,400°,\,500°,\,800°,\,1100°$$$, with an additional acquisition without saturation pulse.

The aforementioned simulations were performed using a binary spin bath model and matrix-algebra-based solutions of the Bloch equations^{5}, and typical values for post-mortem white matter.

To investigate a data-driven optimal sequence for magnetization transfer mapping we iteratively found the best (i.e. introducing the largest variance) z-spectrum point to include to a given MT dataset.

Specifically, we proceeded to:

- construct a matrix M using as columns the data reshaped (due to memory limits only a single central slice was used) corresponding to different z-spectrum points;

- start with only one column containing data without the saturation pulse;

- look for the optimal marginally larger z-spectrum sub-sampling by adding a single z-spectrum point to the matrix and perform an SVD to obtain the principal linear components; the data point adding the largest eigenvalue (normalized) is the one with more variance with the original dataset;

- once the marginally optimal acquisition is included in the matrix, repeat the procedure until all data is included.

To obtain the desired reduced dataset, a threshold value on the marginal variance can be chosen, for example, from signal-to-noise ratio considerations.

The list in Fig.[2],[3],[4] can be used directly to progressively measure (depending on acquisition time) the most relevant z-spectrum points.

The SVD technique performs well for principal component analysis under the assumption of linearity, which may not be satisfied for all components. Other analyses, for example those based on different statistical metrics, may yield different - and possibly more accurate - results. Additionally, the choice of the most relevant z-spectrum points may prove valuable for obtaining MT fits using biophysical models with comparable quality as when more data points are acquired. This was postponed in this work due to the relatively long computation times of these fits. The SVD analysis can also be used to make predictions on the number of components that can be extracted using biophysical models. For example, it is possible to use Otsu's method^{6} to define a variance-optimum threshold for signal/noise separation, and use that for thresholding the largest eigenvalues.

Finally, in the presence of a linear formulation of the biophysical model, i.e. $$$S = YX$$$, where $$$S$$$ is the measured data, $$$Y$$$ is the matrix of unknown parameters and $$$X$$$ is the biophysical model matrix, the SVD approach can reduce the fitting procedure to the determination of a rotation matrix $$$T$$$ mapping the (reduced) right-singular matrix $$$\tilde{V}^\dagger$$$ into the model $$$X$$$:$$S=U\Sigma{}V^\dagger\approx{}\tilde{U}\tilde{\Sigma}\tilde{V}^\dagger=\tilde{U}\tilde{\Sigma}T^{-1}T\tilde{V}^\dagger$$with $$$Y=\tilde{U}\tilde{\Sigma}T^{-1}$$$ and $$$X=T\tilde{V}^\dagger$$$ and the $$$\tilde{\phantom{x}}$$$ symbol indicate the truncated SVD matrices (largest eigenvalues and corresponding eigenvectors) matching the biophysical model considered.

1. Cercignani, M., Alexander, D.C., 2006. Optimal acquisition schemes for in vivo quantitative magnetization transfer MRI. Magn. Reson. Med. 56, 803–810. doi:10.1002/mrm.21003

2. Henkelman, R.M., Huang, X., Xiang, Q.-S., Stanisz, G.J., Swanson, S.D., Bronskill, M.J., 1993. Quantitative interpretation of magnetization transfer. Magn. Reson. Med. 29, 759–766. doi:10.1002/mrm.1910290607

3. Sled, J.G., Pike, G.B., 2000. Quantitative Interpretation of Magnetization Transfer in Spoiled Gradient Echo MRI Sequences. Journal of Magnetic Resonance 145, 24–36. doi:10.1006/jmre.2000.2059

4. Barta, R., Kalantari, S., Laule, C., Vavasour, I.M., MacKay, A.L., Michal, C.A., 2015. Modeling T1 and T2 relaxation in bovine white matter. Journal of Magnetic Resonance 259, 56–67. doi:10.1016/j.jmr.2015.08.001

5. Müller, D.K., Pampel, A., Möller, H.E., 2013. Matrix-algebra-based calculations of the time evolution of the binary spin-bath model for magnetization transfer. Journal of Magnetic Resonance 230, 88–97. doi:10.1016/j.jmr.2013.01.013

6. Otsu, N., 1979. A Threshold Selection Method from Gray-Level Histograms. IEEE Transactions on Systems, Man, and Cybernetics 9, 62–66. doi:10.1109/TSMC.1979.4310076

Fig.1: The simulation for a typical white matter MT signal. The simulations were obtained using a binary spin bath model with the following parameters:$$$M_{0,A}=0.8681$$$, $$$M_{0,B}=0.1319$$$, $$$R_{1,A}=1.8\;\mathrm{Hz}$$$, $$$R_{1,B}=1\;\mathrm{Hz}$$$, $$$R_{2,A}=32.2581\;\mathrm{Hz}$$$, $$$R_{2,B}=84746\;\mathrm{Hz}$$$, $$$k_{A,B}=0.3456\;\mathrm{Hz}$$$ where: $$$M_0$$$ indicate the equilibrium magnetization, $$$R_1,R_2$$$ the longitudinal and transverse relaxation times, $$$k$$$ the exchange constant, and $$$A$$$ refers to the liquid pool and $$$B$$$ to the semi-solid pool. A superlorentzian line-shape was used for describing $$$R_{2,B}$$$ effects.

Fig.2: The exact values used for off-resonance excitation and the marginal variance added to the existing measurement subset starting from an acquisition without off-resonance excitation. The saturation pulse was tuned logarithmically in the frequency offset range $$$100\mathrm{-}100000\mathrm{Hz}$$$ with varying power level corresponding to on-resonance flip angle of $$$200°,\,300°,\,400°,\,500°,\,800°,\,1100°$$$. Some combination of low power and high frequency offsets were skipped because simulations indicated little contributions to magnetization transfer.

Fig.3: Graphical plot of the marginal variance. The threshold level indicated is obtained through the approximate Elbow estimation, in analogy to the traditional approach of principal component analysis with SVD. Only 10 measurements are above the indicated threshold. Different threshold can be used, possibly from the perspective of determining the number of relevant components through other means (for example analyzing the SVD results on the full dataset) and retrospectively determining the threshold.

Fig.4: Position of the acquired off-resonance data points in the simulated MT signal in the corresponding portion of the z-spectrum. The simulated MT signal is shown in gray levels (using the same parameters as in Fig.[1]). The acquired off-resonance data points are displayed in levels of orange, where we display the marginal variance as obtained in this work (using a log10 scale for improved visualization).