IHMT: Is it misnamed? A simple theoretical description of "inhomogeneous" MT.

Alan P Manning^{1}, Kimberley L Chang^{2}, Alex MacKay^{1,3}, and Carl A Michal^{1}

Inhomogeneous Magnetization Transfer (IHMT) appears to be myelin-selective^{[1,2,3]}. In IHMT, images are acquired with four different soft prepulses: no prepulse (*S _{0}*), prepulse at offset +Δ (

$$\text{IHMTR}=\frac{S_+ + S_- - 2S_{\text{both}}}{2S_0} \le 1$$

IHMTR≠0 in systems with lipid lamellar structures (e.g. myelin, hair conditioner)^{[1,2]}, but ≈0 in many other systems (e.g. gelatin, agarose)^{[4]}.

Lipid hydrocarbon chain spin systems are thermally-averaged, resulting in spectral behaviour dominated by intra-methylene dipolar couplings. These methylenes resemble an ensemble of spin-1 systems^{[5,6]}, and others have suggested this underlies the IHMT mechanism^{[1,7]}. Recent work has also highlighted IHMT's sensitivity to *T _{1D}*, the dipolar order relaxation time

Here, we present experimental results on *homogeneously*-broadened systems with IHMTR≠0, contradicting the inhomogeneous explanation. The following simple spin-1 model shows how this is possible.

We consider the non-aqueous protons only, since their behaviour under the prepulses determines if IHMT will occur. The Hamiltonian of a dipolar-coupled methylene pair is $$${\cal H} = -\omega_0 I_z -\omega_D I_z^2$$$ (ω_{0}=Larmor frequency, ω_{D}=dipolar interaction strength, ω_{0}»ω_{D}). The equilibrium density matrix is

$$\rho = M_0 I_z = M_0\, \text{diag}(1,0,-1).$$

Then, assuming the prepulses are calibrated for inversion, *S*_{+} exchanges *ρ*_{11}/*ρ*_{22}, *S*_{-} exchanges *ρ*_{22}/*ρ*_{33}, and *S*_{both} exchanges *ρ*_{11}/*ρ*_{33} (using 2× the power to invert both transitions). Following the prepulse, $$$\langle I_z \rangle$$$ is the magnetization, and $$$\langle I_z^2 \rangle$$$ the dipolar order (nonzero only in *S*_{+}/*S*_{-}).

We form the "non-aqueous IHMTR":

$$\begin{align}\text{non-aqueous IHMTR} &=\frac{(\langle I_z \rangle_+ -\langle I_z \rangle_0)+(\langle I_z \rangle_- -\langle I_z \rangle_0) - (\langle I_z \rangle_{\text{both}} - \langle I_z \rangle_0)}{2\langle I_z \rangle_0} \\&= \frac{ (-\frac{1}{2}) + (-\frac{1}{2}) - (-2) }{2} =\frac{1}{2}.\end{align}$$

We don't multiply $$$(\langle I_z \rangle_0 - \langle I_z \rangle_{\text{both}})$$$ by 2 because we assumed 2× the power was used to invert both transitions.

Following a hard observe pulse of flip angle α, the spectral line amplitudes *A*_{±} at ω_{0}±ω_{D} are

$$A_\pm = [\frac{1}{2}(\rho_{11}-\rho_{33})]\sin\alpha\pm [\frac{1}{2}(\rho_{11}+\rho_{33}) -\rho_{22}] \cos\alpha \sin\alpha. ~~~~~\text{(1)}$$

Evidently, irradiating one transition affects the other's amplitude, and both amplitudes are equal when α=π/2. The lines cannot be independently saturated, as would be the case in an inhomogeneously-broadened system, *yet it has a nonzero IHMTR*. This equation's second term produces spectral asymmetry from dipolar order, and can only be observed when α≠*n*π/2.

The PL161 and fir non-aqueous proton spectra following IHMT prepulses (Figure 2) showed no evidence of hole-burning. However, with α=33^{o}, the characteristic spectral asymmetry from dipolar order in *S*_{+}/*S*_{-} is obvious.

As a function of α, the ω>0/ω<0 spectral integrals of PL161 were fit to (1) (Figure 3). Inhomogeneously-broadened spectra would only have a sin(α) dependence, but we clearly see more complex behaviour consistent with dipolar order.

Our samples' non-aqueous and aqueous IHMTRs vs. Δ and their *T _{1D}*s are given in Figures 4 and 5, respectively. Fir and hair have IHMTR≠0,

Many tissues and phantoms in which IHMTR≈0 are highly dynamic systems, so dipolar couplings are transient. For IHMT to occur, we believe the coupling lifetime must be long enough for dipolar order to develop: τ_{D}$$$\gtrsim$$$1/(4ν_{1}) (τ_{D} = dipolar coupling correlation time, ν_{1} = prepulse nutation frequency). ν_{1} must be small enough to only excite one side of the spectrum, setting a lower limit for τ_{D}. In a system with transient dipolar couplings, *T _{1D}* is constrained to be no greater than τ

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Pulse sequences. **A/B**: Single soft Gaussian/rectangular prepulses at +Δ or -Δ, (giving *S*_{+} or *S*_{-}). **C**: Via sine modulation, the prepulse is applied simultaneously at ±Δ (giving *S*_{both}). **D**: ADRF/ARRF to measure *T*_{1D}. (*B*_{1}=pulse amplitude, τ=pulse time, δ=delay time, α=hard observe pulse flip angle.)

Non-aqueous PL161 and Douglas Fir proton spectra following IHMT prepulses. *S*_{+ }and *S*_{-} spectra show dipolar order (manifesting as spectral asymmetry), but not *S*_{0} or *S*_{both}. No evidence of isolated "hole burning" characteristic of inhomogeneous broadening is seen. Parameters: τ=20 ms, |Δ|=10 kHz, B_{1}=1 kHz, α=33^{o}.

PL161 spectrum vs. observe pulse flip angle α, compared to the spin-1 model. *S*_{+} used sequence A (Δ=+2.5 kHz, *B*_{1} = 2.5 kHz, α=0.5 ms). *I*_{< }/ *I*_{>} are non-aqueous intensities of (*S*_{+} - *S*_{0}), integrated between (-80,-3)/(3,80) kHz. An exp(-*C*α) multiplicative term (*C*=parameter) was included to account for B_{1} inhomogeneities.

Aqueous and non-aqueous IHMTRs vs. prepulse offset. Samples known to have homogeneously broadened spectra (wood and hair) still have IHMTR≠0. Sequences A and B were used (τ=500 ms, *B*_{1}=415 Hz, δ=0.5 ms, α=33^{o}). The non-aqueous and aqueous intensities were integrated between ((-80,3) and (3,80) kHz) and (-1,1) kHz, respectively.

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

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