Validation of Provotorov theory of RF saturation to describe inhomogeneous magnetization transfer (ihMT)

Scott D. Swanson^{1}

Magnetization transfer (MT) provides information about the immobile components of tissue. Recent studies have used double-sided RF saturation to create an enhanced MT (eMT), particularly in white matter, and have identified dipolar order in lipid systems as a possible explanation of difference between the MT and eMT; termed inhomogeneous magnetization transfer (ihMT) (1). This study measures RF saturation of semisolid and liquid components in model systems with a 2D matrix of frequencies and fits results to Provotorov's theory of RF saturation (2-4).

A 2D array of saturation frequencies, $$$(\Delta_1 , \Delta_2)$$$ (Fig. 1), was used and the solid-state NMR spectrum of adamantane and 5% Prolipid 161 (PL161) in deuterium oxide were recorded. Adamantane is a well-studied plastic crystal with partially averaged dipolar couplings and a long proton $$$T_{1D}$$$ time (5), ideally suited for testing Provotorov theory. PL161 has been identified as a lipid system that shows a large ihMT effect. NMR spectra of adamantane (Fig. 2a) were integrated, normalized to 100%, and the results plotted as a function of $$$\Delta_1$$$ and $$$ \Delta_2$$$ (Fig. 2b). Similar data were collected and processed in PL161 for both the semisolid lipid component (Fig. 3b) and the water component (Fig. 3c).

$$$T_{1D}$$$ times were measured by decay of Jeener-Broekaert echo and found to be 670 ± 10 ms in adamantane and 40 ± 2 ms in 5% PL161. The RF field used in these saturation studies was approximately 100 Hz.

A Fermi function was used for adamantane line shape $$$g_i(\Delta)$$$ and data in Fig. 2b were fitted to (2) ...

$$M_z=\frac{\omega_{loc}^2+T_{1D}(\Delta_1^2 R^{RF}_1+\Delta_2^2 R^{RF}_2)}{\omega_{loc}^2+\omega_{loc}^2(R^{RF}_1+R^{RF}_2) T_1+T_{1D}(\Delta_1^2 R^{RF}_1+\Delta_2^2 R^{RF}_2)+(\Delta_1-\Delta_2)^2R^{RF}_1 R^{RF}_2 T_1 T_{1D}},$$

where $$$R^{RF}_i$$$ is the transition rate given by,

$$R^{RF}_i(\omega_1,\Delta) = \frac{\pi}{2}\omega_1^2 g_i(\Delta)$$.

The local field, $$$\omega_{loc}$$$ and line shape function were used as adjustable parameters in the fit.

The long $$$T_{1D}$$$ times in adamantane create inhomogeneous broadening that is asymmetrically saturated by single-sided RF (Fig. 2a). Coupling to the dipolar pool is removed by dual-sided RF saturation (Fig 2a and 2b). Data in Fig. 2b are fitted to the above equation (Fig. 2c) by varying the width and slope of the Fermi function and the local magnetic field $$$\omega_{loc}$$$. The local field was found to be $$$\omega_{loc} = 13,801\pm80$$$ rad/sec (R-square = 0.9968).

Similar asymmetric RF saturation is observed for PL161 spectra (Fig. 3a). Figs. 3b and 3c show nearly identical saturation patterns for the lipid and water components, with the saturation of water diminished by cross-relaxation and the amount of PL161. Fitting of results to theory using a Gaussian line shape reveals $$$\omega_{loc} = 42,420\pm680$$$ rad/sec and $$$T_{2b} = 14.18\pm0.08 \mu s$$$ (R-square = 0.9828). For a pure Gaussian lineshape, $$$ \omega_{loc}^2 = \frac{1}{3 T_{2b}^2}$$$. Fitting these as independent variables we find $$$ \omega_{loc}^2 = 1.799x10^9 $$$ and $$$\frac{1}{3 T_{2b}^2} = 1.657x10^9$$$, showing reasonable agreement.

The diagonal of Fig. 3c is the conventional MT profile with $$$\Delta_1 = \Delta_2$$$, whereas the anti-diagonal is the eMT profile with $$$\Delta_1 = - \Delta_2$$$. ihMT is the difference between MT and eMT, shown in Fig 3d.

The data and fitting of data to theory unequivocally demonstrate that the Provotorv model of RF saturation is an appropriate description of ihMT. Results of 2D saturation profiles in adamantane and PL161 show very good agreement with Provotorov theory.

A detailed, microscopic description of RF saturation of lipids requires an inhomogeneously broadened line generated by, e.g., motionally averaged Pake patterns. This microscopic view is visualized in the asymmetric RF saturation of both adamantane (Fig. 2a) and PL161 (Fig. 3a). However, this microscopic description not needed to describe MT. MT and eMT are determined the fraction of the semisolid resonance that is saturated by RF and not by the detailed shape of the saturated line. The shape of the unsaturated semisolid line is important, but not the shape of the saturated line. This effect is demonstrated in Figs. 3b and 3c, where the amount MT (Fig 3c) is shown to be proportional to the amount of saturation of the semisolid component (Fig. 3b).

Understanding the origins of ihMT allows interpretation of imaging results with respect to molecular structure and dynamics of semisolid components in tissue. Combination of Provotorov and MT theory (1,2,4) shows that ihMT signal increases with increasing proton $$$T_{1D}$$$ times. $$$T_{1D}$$$ is sensitive to slow molecular motions, a dynamic region not measured by conventional T1 and T2 sequences. ihMT images should provide sensitivity to slow lipid motions that may appear as a preclinical indicators in diseases such as multiple sclerosis.

1. Varma G, Girard OM, Prevost VH, Grant AK, Duhamel G, Alsop DC. Interpretation of magnetization transfer from inhomogeneously broadened lines (ihMT) in tissues as a dipolar order effect within motion restricted molecules. Journal of magnetic resonance 2015;260:67-76.

2. Lee J-S, Khitrin AK, Regatte RR, Jerschow A. Uniform saturation of a strongly coupled spin system by two-frequency irradiation. Journal of Chemical Physics 2011;134(23).

3. Provotorov BN. Magnetic Resonance Saturation in Crystals. Soviet Physics Jetp-Ussr 1962;14(5):1126-1131.

4. Yeung HN, Adler RS, Swanson SD. Transient Decay of Longitudinal Magnetization in Heterogeneous Spin Systems under Selective Saturation: Reformulation of the Spin-Bath-Model Equations by the Redfield-Provotorov Theory. Journal of Magnetic Resonance Series A 1994;106(1):37-45.

5. Resing HA. NMR Relaxation in Adamantane and Hexamethylenetetramine - Diffusion and Rotation. Molecular Crystals and Liquid Crystals 1969;9:101-132.

Fig. 1 NMR sequence to map dual frequency RF saturation.
Frequencies $$$\Delta_1$$$ and $$$\Delta_2$$$ are independently varied to
create an array of off-resonance conditions using 5ms Gaussian pulses; n = 200. In adamantane the range is -20 to
20 kHz and in PL161 the range is -50 to 50 kHz.

Fig. 2. NMR spectra of adamantane (a) (black), RF saturation
at +4 kHz (blue), -4 kHz (red), and +/- 4 kHz (yellow). Summing 1681spectra and
plotting results as a function of the saturation frequencies $$$\Delta_1$$$ and
$$$\Delta_2$$$ from -20 to +20 kHz (b). Fit of Provotorov model to data (c).

Fig. 3. NMR spectra (a) of PL161 (black), at +10 kHz (blue), -10 kHz (red) and +/- 10 kHz (yellow). Summing lipid component of PL161 spectra (b) as above. Similarly for water (c). The diagonal of (c) is the MT profile and the anti-diagonal is the eMT profile (d).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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