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On the dipolar order underlying broad macromolecular lines
Olivier M. Girard1, Victor N.D. Carvalho1, Ludovic de Rochefort1, Andreea Hertanu1, Pierre Thureau2, Gopal Varma3, David Alsop3, and Guillaume Duhamel1

1Aix Marseille Univ, CNRS, CRMBM, Marseille, France, 2Aix Marseille Univ, CNRS, ICR, Marseille, France, 3Division of MR Research, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA, United States

Synopsis

Dipolar order has recently regained attention in MRI to analyze dipolar broadened lines in CEST and inhomogeneous Magnetization Transfer (ihMT), leading to new frequency irradiation patterns for enhanced saturation and access to an unexplored degree of freedom. A better understanding of dipolar order is of great interest to guide intuition and may lead to fundamental optimization of the ihMT technique, which is a promising tool providing new tissue contrasts. In this contribution we propose to review this concept, considering a simplified model of isolated proton pairs and the general Provotorov theory of RF saturation which applies to an ensemble of coupled spin.

Introduction

Dipolar order has recently regained attention in MRI to analyze dipolar broadened lines in CEST (1) and inhomogeneous Magnetization Transfer (ihMT) (2,3), leading to new frequency irradiation patterns for enhanced saturation and access to new contrast mechanisms sensitive to underlying molecular structures. A better understanding of dipolar order is of great interest to guide intuition and may lead to fundamental optimization of the ihMT MRI technique, which is a promising tool providing access to an unexplored degree of freedom to generate new tissue contrasts. However, dipolar order and the associated spin temperature concept (4) are somewhat difficult to comprehend under a classical picture. In this contribution we propose to provide insight into these concepts, considering a simplified model of isolated proton pairs and the general Provotorov theory of RF saturation which applies to an ensemble of coupled spin.

The isolated ½ spin pair model

We consider a pair of magnetically equivalent 1H nuclei (as a model for methylene groups). The dipolar interaction between the two nuclei generates local magnetic field perturbations leading to a dipolar splitting of the resonance line into a doublet. Under the quantum mechanical description, the dipolar splitting is induced by the secular component of the dipolar Hamiltonian. The remaining time varying non-secular term is associated with relaxation processes. The system is described in the singlet-triplet eigenbasis (5) (Fig.1) with the triplet state being equivalent to a spin-1 model (6). The two transitions between the triplet states correspond to the two peaks of the so-called dipolar doublet. Within such a system the Zeeman and dipolar temperatures (Fig.2) are the two constants of motion (7) and are related to the energy level population differences. At thermal equilibrium the Zeeman order is driven by the lattice temperature (leading to the classical M0), and the dipolar order is negligible.

Biophysical lineshape modeling and observation of the dipolar order

To deal with ensembles of dipolar coupled spins and describe the RF absorption lineshape of biological tissues or lipid membranes, biophysical models have been proposed (8–11). Within motion restricted macromolecules, the dominant source of line broadening is the non-zero averaging of the secular dipolar interaction generating a Residual Dipolar Coupling (RDC). From the simple model of an oriented dipolar interaction modeled as a Gaussian-broadened singlet or doublet (9) with a width that depends on orientation (8), one needs to consider temporal and spatial averaging processes. Temporal averaging of molecular motions reduces the instantaneous dipolar coupling to the RDC, expressed as a function of the molecular order parameter. Spatial averaging of the lineshape is then performed over all molecular orientations. Considering an isotropic averaging of all RDC orientations and strong broadening conditions leads to the well-known Super-Lorentzian (SL) lineshape model (Fig.3) typically observed for biological tissues and lipid systems. Whereas strong broadening conditions indicates that the spin pairs are not isolated from neighboring spins for such lineshapes, the ½ spin pair model has the conceptual advantage to consider the dipolar splitting providing insight into the underlying dipolar order that may be revealed by a spectral asymmetry (6,12,13) following off-resonance irradiation. To illustrate this, we report our results of a Jeener-Broekaert NMR acquisition allowing observation of dipolar order relaxation on Adamantane (Fig.4).

Thermodynamical modeling of dipolar order

Whereas the isolated ½ spin pair model in which the dipolar order is explicit do not apply to complex spin systems, the spin temperature formalism (4) allows generalizing the concept of dipolar order to an ensemble of coupled spins characterized by an arbitrary lineshape. The dipolar order is consistently modeled as an additional constant of motion and represented as an additional reservoir (Fig.5). The Provotorov theory (15), initially developed for solids, describes the coupling between Zeeman and dipolar orders under weak RF irradiation applied close to resonance. This theory has been successfully applied to model the ihMT signal dependency with various saturation parameters (power, frequency offset, duty cycle) (3,14), underlying the fundamental role of dipolar order for ihMT.

Conclusion

The investigation of dipolar order theory reveals an unexplored degree of freedom within dipolar-broadened macromolecular lineshapes, which cannot be described solely by the usual Zeeman order (classical magnetization). Dipolar order is intrinsically related to the dipolar splitting underpinning broad macromolecular lines. It can be described by an additional constant of motion coupled with the Zeeman order. Whereas it can be intuitively understood considering a simple spin-pair model (Fig.2), the spin temperature formalism provides a framework to deal with more complex spin systems. Dipolar order imaging such as ihMT offers a new endogenous contrast mechanism and opens a new perceptive to study broad macromolecular lines, such as found in myelin.

Acknowledgements

This work was supported by the French National Research Agency ANR (n° ANR‐17‐ CE18‐0030).

This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement n°713750. Also, it has been carried out with the financial support of the Regional Council of Provence-Alpes-Côte d’Azur and with the financial support of the A*MIDEX (n° ANR- 11-IDEX-0001-02), funded by the Investissements d'Avenir project funded by the French Government, managed by the French National Research Agency (ANR).

References

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2. Varma G, Duhamel G, de Bazelaire C, Alsop DC. Magnetization Transfer from Inhomogeneously Broadened Lines: A Potential Marker for Myelin. Magn. Reson. Med. 2015;73:614.

3. 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. J. Magn. Reson. 2015;260:67.

4. Goldman M. Spin Temperature and Nuclear Magnetic Resonance in Solids. Oxford: University Press. 1970.

5. Levitt MH. Spin Dynamics: Basics of Nuclear Magnetic Resonance, 2nd Edition. Wiley. 2008.

6. Manning AP, Chang KL, MacKay AL, Michal CA. The physical mechanism of “inhomogeneous” magnetization transfer MRI. J. Magn. Reson. 2017;274:125.

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11. Wilhelm MJ, Ong HH, Wehrli SL, et al. Direct magnetic resonance detection of myelin and prospects for quantitative imaging of myelin density. Proc. Natl. Acad. Sci. 2012;109:9605.

12. Korb J-P, Maruani J. Pulse saturation behavior of inhomogeneously broadened lines. I. Slow spectral diffusion case. Phys. Rev. B 1981;23:971.

13. Swanson SD, Malyarenko DI, Fabiilli ML, Welsh RC, Nielsen J-F, Srinivasan A. Molecular, dynamic, and structural origin of inhomogeneous magnetization transfer in lipid membranes. Magn. Reson. Med. 2017;77:1318.

14. Mchinda S, Varma G, Prevost VH, et al. Whole brain inhomogeneous magnetization transfer (ihMT) imaging: Sensitivity enhancement within a steady-state gradient echo sequence. Magn. Reson. Med. 2018;79:2607.

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Figures

Fig1. A) Energy diagram of a single spin ½ subject to Zeeman interaction. |α> and |β> are the Zeeman eigenstates. The corresponding spectral representation is a singlet. B) Energy diagram of a pair of magnetically equivalent spins subject to Zeeman and dipolar interactions. The singlet-triplet basis states (i.e. |Tis> and |S0>) are the energy eigenstates of the system. The NMR spectrum is a doublet corresponding to slightly different energy level transitions. The dipolar splitting is induced by the secular component of the dipolar Hamiltonian, while the time-dependent part is responsible for relaxation processes.

Fig2. The spin-½ pair model is described by two constants of motion: the inverse Zeeman and dipolar temperatures. The Zeeman and dipolar orders result from the corresponding energy level population differences A). Ideal cases of pure Zeeman B) and pure dipolar C) orders are shown in terms of energy level populations and corresponding NMR lineshapes. The signature of dipolar order is an asymmetric spectrum. dipolar order can be viewed as an additional degree of freedom in the spin system that represents the differential magnetization of each peak of the doublet, whereas Zeeman order represents the total magnetization of the doublet.

Fig3. Isotropic lineshape averaging based on the singlet and doublet models (9). The singlet model accounts for dipolar broadening through relaxation mechanisms only whereas the doublet model also includes a dipolar splitting term. Elementary lineshapes corresponding to a single RDC vector orientation are displayed in A) and B) and the corresponding spatial averaging in C) and D). Under strong broadening conditions such as typically observed for biological tissues and lipid systems both models are virtually undistinguishable and lead to the well-known Super-Lorentzian lineshape.

Fig4. Relaxation of Adamantane measured with a Jeener-Broekaert experiment performed on a 400MHz NMR spectrometer. At time zero the system is prepared in a pure dipolar order state (bottom-left curve) and shows the characteristic asymmetric dispersion-like spectrum. As the relaxation period increases the system relaxes towards thermal equilibrium, reaching a pure Zeeman state for (infinitely) long durations. Spin-lattice relaxation of the dipolar order follows the dipolar relaxation time T1D whereas the Zeeman order relaxes according to T1. Adamantane does not show the idealized doublet structure depicted in Fig.2, because of strong broadening due to neighboring spins, but the dipolar order can still be revealed.

Fig5. Thermodynamic representation of Zeeman and dipolar orders according to the Provotorov theory A). Dipolar and Zeeman orders are coupled under off-resonance RF irradiation. Dipolar order can be viewed as an additional degree of freedom acting against RF saturation of Zeeman order. For a broad Super-Lorentzian lineshape, the frequency dependency of the dipolar order contribution to RF saturation effects RDD B) reveals interesting features of the NMR spectrum, that can be viewed as the underlying dipolar splitting signature of the resonance line. RDD can be related to the ihMT technique which aims at isolating dipolar order contribution to MT by combining different frequency irradiation patterns .

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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