Victor N. D. Carvalho^{1,2}, Olivier M. Girard^{1}, Andreea Hertanu^{1}, Samira Mchinda^{1}, Lucas Soustelle^{1}, Axelle Grélard^{3}, Antoine Loquet^{3}, Erick J. Dufourc^{3}, Gopal Varma^{4}, David C. Alsop^{4}, Pierre Thureau^{2}, and Guillaume Duhamel^{1}

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_{1D}s 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. T_{1} and T_{2}) 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 spins^{3}, which are much weaker than the external magnetic fields. Thus, as compared to T_{1}, 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 study^{6}, since assessing myelin fluidity may yield further understanding of demyelinating diseases, such as multiple sclerosis^{7}.

IhMT, inhomogeneous magnetization transfer^{8}, is a T_{1D}-weighted MRI technique, whose measurement sequence can be tailored to enable T_{1D} measurements^{9} (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_{1D}^{10}. At lower power, only the long T_{1D}s are revealed and contribute to ihMTR signal^{11}. 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_{1D}s.

To include a second dipolar reservoir in the ihMT model, we have applied the matrix formalism to the ihMT theory^{12} and solved the differential equations at the steady-state by using the matrix exponential solution^{13}. 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 theory^{11}.

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.

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 pulse^{13}:

\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*}

[1] Dufourc EJ, Mayer C, Stohrer J, Althoff G, Kothe G. Dynamics of phosphate head groups in biomembranes. Comprehensive analysis using phosphorus-31 nuclear magnetic resonance lineshape and relaxation time measurements. Biophys J. 1992;61(1):42-57.

[2] Trivikram R. Molugu, Soohyun Lee, and Michael F. Brown. Concepts and Methods of Solid-State NMR Spectroscopy Applied to Biomembranes. Chemical Reviews 2017 117 (19), 12087-12132. doi: 10.1021/acs.chemrev.6b00619

[3] Slichter, Charles P. Principles of Magnetic Resonance. Harper & Row Publishers, New York, 1963.

[4] R. Gaspar, E.R. Andrew, D.J. Bryant, and E.M. Cashell. Dipolar relaxation and slow molecular motions in solid proteins. Chemical Physics Letters, 86(4):327 – 330, 1982.

[5] O. Mensio, R.C. Zamar, F. Casanova, D.J. Pusiol, R.Y. Dong. Intramolecular character of the intrapair dipolar order relaxation in the methyl deuterated nematic para-azoxyanisole. Chemical Physics Letters 356 (2002) 457–461.

[6] Cornelia Laule, Irene M. Vavasour, Shannon H. Kolind, David K.B. Li, Tony L. Traboulsee, G.R. Wayne Moore, and Alex L. MacKay. Magnetic resonance imaging of myelin. Neurotherapeutics, 4(3):460 – 484, 2007. Advances in Neuroimaging/Neuroethics.

[7] Benjamin Ohler, Karlheinz Graf, Richard Bragg, Travis Lemons, Robert Coe, Claude Genain, Jacob Israelachvili, Cynthia Husted. Role of lipid interactions in autoimmune demyelination. Biochimica et Biophysica Acta 1688 (2004) 10–17

[8] G. Varma, O.M. Girard, V.H. Prevost, A.K. Grant, G. Duhamel, and D.C. Alsop. Interpretation of magnetization transfer from inhomogeneously broadened lines (ihMT) in tissues as a dipolar order effect within motion restricted molecules. Journal of Magnetic Resonance 260 (2015) 67–76.

[9] Gopal Varma, Olivier M. Girard, Valentin H. Prevost, Aaron K. Grant, Guillaume Duhamel, and David C. Alsop. In vivo measurement of a new source of contrast, the dipolar relaxation time, T1D, using a modified inhomogeneous magnetization transfer (ihMT) sequence. Magnetic Resonance in Medicine, 2017 Oct; 78(4): 1362-1372.

[10] Swanson, S. D., Malyarenko, D. I., Fabiilli, M. L., Welsh, R. C., Nielsen, J. and Srinivasan, A. (2017), Molecular, dynamic, and structural origin of inhomogeneous magnetization transfer in lipid membranes. Magn. Reson. Med., 77: 1318-1328. doi:10.1002/mrm.26210

[11] Alan P. Manning, Kimberley L. Chang, Alex L. MacKay, Carl A. Michal. The physical mechanism of ‘‘inhomogeneous” magnetization transfer MRI. Journal of Magnetic Resonance 274 (2017) 125–136.

[12] G. Varma, O.M. Girard, V.H. Prevost, A.K. Grant, G. Duhamel, and D.C. Alsop. Interpretation of magnetization transfer from inhomogeneously broadened lines (ihMT) in tissues as a dipolar order effect within motion restricted molecules. Journal of Magnetic Resonance 260 (2015) 67–76.

[13] Portnoy, S. and Stanisz, G. J. (2007), Modeling pulsed magnetization transfer. Magn. Reson. Med., 58: 144-155. doi:10.1002/mrm.21244

Figure 1.

A) Comparing T_{1} with T_{1D}: T_{1} is the relaxation time of the Zeeman order
and is associated with fast molecular motions in membranes. On the other hand,
T_{1D} is the relaxation time of the dipolar order and is also associated with
slower and collective motions. Probing collective motions through T_{1D} can be
used to assess membrane integrity.

B)Modified
ihMT preparation to assess T_{1D}. The modification proposed in [9] allows the
switching time, Δt, to be incremented, while keeping all other parameters constant.

C) ihMTR
images of a rat spinal cord (ex-vivo) acquired with incremental Δt. The ihMTR
attenuation is a function of T_{1D}.

Figure 2.

A) Single-component T_{1D}
model of ihMT, with only one dipolar reservoir.

B) White
matter (WM) and gray matter (GM) T_{1D} fits of a rat spinal cord. Note that the
apparent T_{1D} decreases as B_{1RMS} increases. This is likely explained because membranes exhibit multiple T_{1D }components. As the irradiated RF power is increased, the short T_{1D}
components contribute more to ihMT. Since the single-component T_{1D} model cannot account for this
effect, it leads to a decrease of the apparent T_{1D}.

C) T_{1D} and A
maps acquired with different B_{1RMS}. A is the scaling factor^{9} of the
function ihMTR(Δt).

Figure 3.

A)Bi-component T_{1D} model of ihMT, with two dipolar reservoirs. The fraction f_{D}
represents the fraction of the semisolid pool associated with T_{1D1}, while 1-f_{D}
represents the fraction associated with T_{1D2}.

B)Left:
1000 realizations of noisy ihMTR were generated with the proposed bi-component T_{1D} model (T_{1D1}=10 ms, T_{1D2}=0.4 ms, f_{D}= 0.4) for different B_{1RMS}s.
Right: Histograms of the apparent T_{1D} distribution estimated by using the single T_{1D} model
(Fig. 2A) for B_{1RMS}=3.5 and 9.0 μT.

C)Single-T_{1D} fits (model of Fig. 2A) of bi-T_{1D} simulated ihMTR data (model
of Fig.3A) for different B_{1RMS}s. The apparent T_{1D} decreases with B_{1RMS}, as observed in the
experimental results.

Figure 4.

A) Bi-component T_{1D}
fits of a rat spinal cord. T_{1D1} is in the order of 10 ms, and T_{1D2} is
in the order of 400 μs. The fraction of long T_{1D}, f_{D}, is around 0.35-0.45 in WM and stable around 0.25 in GM. One can observe that the fraction of short T_{1D}, 1-f_{D}, increases in WM, suggesting that short T_{1D} components are revealed.

B) T_{1D1}, fraction f_{D}, T_{1D2} and fraction 1-f_{D} maps
acquired with different B_{1RMS}. Observe that the T_{1D2} map, the short T_{1D}
component, is revealed as B_{1RMS} is increased.

C) Example of curve fit with the proposed bi-component T_{1D} model.

Table 1. Experimental parameters. The MT model parameters was based on [9].