Synopsis

T_1D, the relaxation time of the dipolar order, is a probe for membrane dynamics and organization that could be used to assess myelin integrity. A single-component T_1D model associated with a modified ihMT sequence had been proposed for in vivo evaluation of T_1D with MRI. However, experiments and simulations revealed that myelinated tissues exhibit multiple T_1D components. A bi-component T_1D model is proposed and validated. Fits in a rat spinal cord yield two T_1Ds of about 10 ms and 400 μs. The results suggest that myelin has a dynamically heterogeneous organization.

Motivation: T_1D as a probe of membrane molecular dynamics.

Nuclear spin relaxation is characterized by relaxation times (e.g. T₁ and T₂) which are sensitive to different ranges of motional frequencies. So nuclear spin relaxation can in principle deliver quantitative information about membrane molecular dynamics and organization^1-2, such as correlation times and activation energy to describe motions.

T_1D, the relaxation time of the dipolar order, can provide new information about membrane dynamics. The dipolar order is built due to the local fields of dipolar-coupled spins³, which are much weaker than the external magnetic fields. Thus, as compared to T₁, T_1D is more sensitive to slower motional processes, such as membrane collective motions^4-5 (Fig.1A).

Collective motions in a membrane are hydrodynamical deformations, such as bending, stretching and compression. They could be probed to quantify membrane fluidity and elasticity. This is interesting for myelin, a membrane that is particularly important to study⁶, since assessing myelin fluidity may yield further understanding of demyelinating diseases, such as multiple sclerosis⁷.

IhMT, inhomogeneous magnetization transfer⁸, is a T_1D-weighted MRI technique, whose measurement sequence can be tailored to enable T_1D measurements⁹ (Fig.1B).

Background: limitations of the single-component T_1D model.

The ihMT model presented in [8] considers only one dipolar reservoir (Fig.2A), and therefore a single T_1D value.

This single-component T_1D model was used to fit T_1D of a rat spinal cord ex-vivo for different irradiation RF powers (B_1RMS). The experimental parameters are listed in Table 1. The results (Fig.2B-C) show that the apparent T_1D values in both white (WM) and gray matter (GM) decrease as B_1RMS is increased.

This dependence on power could be explained by the existence of multiple T_1D components (long and short T_1D) in spinal cord. If we assume that, in lipids, there are several components that contribute to ihMT, each one with different molecular dynamics, each component would be associated with a different T_1D¹⁰. At lower power, only the long T_1Ds are revealed and contribute to ihMTR signal¹¹. However, since the single-component T_1D model only reports one apparent T_1D, which is expected to represent a weighted average of all T_1D components, it tends to reduce T_1D values when power is increased as short T_1D components are revealed.

This phenomenon was further confirmed by simulations. By using a bi-component T_1D model, with two dipolar reservoirs (Fig.3A), noisy synthetic ihMTR data (Fig.3B) were generated with the values: T_1D1=4 ms, T_1D2=0.4 ms and f_D=0.4, where f_D represents the fraction of the semisolid pool associated with T_1D1. The fraction associated with T_1D2 is (1-f_D).

Rician noise, with standard deviation set to 0.0001, was added 1000 times and the synthetic data were subsequently fitted with the single-T_1D model (Fig. 2A). These steps were repeated for different B_1RMS.

The same behavior as for the experimental data was observed: the apparent T_1D decreases with B_1RMS (Fig.3C). This suggests that the bi-component T_1D model should be more appropriate to fit experimental data.

Proposed model: fitting two T_1Ds.

To include a second dipolar reservoir in the ihMT model, we have applied the matrix formalism to the ihMT theory¹² and solved the differential equations at the steady-state by using the matrix exponential solution¹³. The mathematical formalism of this model is detailed in the Appendix.

The same spinal cord experimental data were fitted again with the proposed bi-component T_1D model (Fig.4). Long T_1D (T_1D1) in the order of 10 ms and short T_1D (T_1D2) in the order of 400 μs were found. In WM, the fraction of short T_1D, (1-f_D), increases with B_1RMS, suggesting that short T_1D components are revealed. Likewise, the short T_1D component (T_1D2) maps are revealed as B_1RMS is increased, which is consistent with the theory¹¹.

The increasing trend of T_1D1 and T_1D2 with power for B_1RMS>5.8uT will require further investigations. At this stage, care must be taken when interpreting these results biophysically.

Conclusions: myelinated tissues contain multiple T_1D components.

The existence of multiple T_1D components in rat spinal cord is suggested by both experimental and simulated data. The proposed bi-component T_1D model allowed measuring, in spinal cord, long and short T_1D (in the order of 10 ms and 400 μs, respectively), the latter being revealed under strong RF power irradiation. Overall, the hypothesis of multi-component T_1D in myelin is reasonable because myelin has a complex and dynamically heterogeneous organization.

Although a theory that would link the actual T_1D values to membrane properties, such as fluidity and elasticity, is still missing, this work is an important step for assessing myelin integrity in vivo.

Appendix: the mathematical formalism of the proposed model.

Proposed model:

\begin{equation*}\dot{M}=AM+B\end{equation*}

where M is the magnetization of each reservoir (Fig.3A):

\begin{gather*}M=\begin{bmatrix}M_{ZA}\\M_{ZB1}\\\beta_{1}\\M_{ZB2}\\\beta_{2}\end{bmatrix}\end{gather*}

and A and B are:

\begin{gather*}A=\begin{bmatrix}-\left(R_{1A}+RM^{B}_{0}+R_{RFA}\right)&RM^A_0&0&RM^A_0&0\\Rf_DM^B_0&-\left(R_{1B}+RM^A_0+R_{RFB}\right)&R_{RFB}2\pi\Delta&0&0\\0&R_{RFB}\frac{2\pi\Delta}{D^2}&-\left[\frac{1}{T_{1D1}}+R_{RFB}\left(\frac{2\pi\Delta}{D}\right)^2\right]&0&0\\R(1-f_D)M^B_0&0&0&-\left(R_{1B}+RM^A_0+R_{RFB}\right)&R_{RFB}2\pi\Delta\\0&0&0&R_{RFB}\frac{2\pi\Delta}{D^2}&-\left[\frac{1}{T_{1D2}}+R_{RFB}\left(\frac{2\pi\Delta}{D}\right)^2\right]\\\end{bmatrix}\end{gather*}

and

\begin{gather*}B=\begin{bmatrix}R_{1A}M^A_0\\R_{1B}f_DM^B_0\\0\\R_{1B}(1-f_D)M^B_0\\0\end{bmatrix}.\end{gather*}

We want to know the values of M after a rectangular pulse¹³:

\begin{equation*}M_{t _1}= e^{At_1}M_{t=0}+A^{-1}\left(e^{At_1}-I\right)B.\end{equation*}

Then, we can calculate $$$ihMTR(Δt)$$$ at steady-state corresponding to all sequence variants shown in Fig.1B as:

\begin{equation*}ihMTR(Δt)=2\frac{M^+-M^{+-}(Δt)}{M_0}.\end{equation*}

Acknowledgements

References

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