Estimation of the magnitude of quadrupole relaxation enhancement in the context of magnetic resonance imaging contrast.

Magnetic Resonance Imaging (MRI) is one of the most powerful diagnostic tools providing maps of 1H relaxation times of human bodies. The method needs, however, a contrast mechanism to enlarge the difference in the relaxation times between healthy and pathological tissues. In this work, we discuss the potential of a novel contrast mechanism for MRI based on Quadrupole Relaxation Enhancement (QRE) and estimate the achievable value of QRE under the most favorable conditions. It has turned out that the theoretically possible enhancement factors are smaller than those of typical paramagnetic contrast agents, but in turn, the field-selectivity of QRE-based agents makes them extremely sensitive to subtle changes of the electric field gradient in the tissue. So far, QRE has been observed for solids (in most cases for 14N) as a result of very slow dynamics and anisotropic spin interactions, believed to be necessary for QRE to appear. We show the first evidence that QRE can be achieved in solutions of compounds containing a high spin nucleus (209Bi) as the quadrupole element. The finding of QRE in a liquid state is explained in terms of spin relaxation theory based on the stochastic Liouville equation. The results confirm the relaxation theory and motivate further exploration of the potential of QRE for MRI.

Magnetic Resonance Imaging (MRI) provides spatially and temporally resolved maps of 1 H relaxation times of the inside of human bodies. The need to enlarge the difference (contrast) in the relaxation times between healthy and pathological tissues has given rise to a development of paramagnetic contrast agents causing Paramagnetic Relaxation Enhancement (PRE) effects. [1][2][3] Paramagnetic contrast agents are complexes of transition or rare-earth metal ions.
Usually Gadolinium based chelates are used in clinical practice. [3][4][5][6][7] The 1 H relaxation enhancement is achieved due to very strong magnetic electron-proton dipole-dipole interactions between the electron spin of the paramagnetic species and neighbouring proton spins. As the ratio between the electron and proton gyromagnetic factors is very large (about 656 times larger than the proton gyromagnetic factor) and the 1 H relaxation rate depends on a square of the electron gyromagnetic factor, one could expect a very large PRE. In practice, the observed 1 H relaxation enhancement is much smaller (at least at lower magnetic fields) due to fast electron spin relaxation (caused by Zero Field Splitting interactions) that acts as an additional (besides the molecular motion) source of modulations of the electron-proton dipole-dipole interaction. 1,2,[8][9][10][11][12][13][14][15] In this work we demonstrate the prospective of a novel contrast mechanism for MRI, referred to as Quadrupole Relaxation Enhancement (QRE). [16][17][18][19][20][21][22][23][24][25] QRE is a complex, quantummechanical phenomenon which, in the context of MRI, is sometimes regarded as a counterpart of PRE, although this analogy is not fully justified. QRE leads to a frequency specific relaxation of 1 H spin-lattice relaxation (referred to as quadrupole peaks) originating from dipole-dipole interactions between protons ( 1 H nuclei) and nuclei possessing a quadrupole moment. As a potential alternative to PRE this approach opens several desirable possibilities which are interesting in terms of molecular imaging with MRI. Although there is a clear theoretical foundation and the effect has been shown for 14 N of the amide groups of proteins in low-field MRI experiments, 18 no attempts have been made so far to exploit QRE for extrinsic MRI contrast agents. In this context it is worth to point out that actually the results available for solids are also mostly limited to 14 N. [16][17][18][19][20][21][22][23][24] Considering QRE as a mechanism of novel extrinsic MRI contrast agents it is very important to stress that while PRE is observed in the whole frequency range as a result of a strong proton-electron dipole-dipole coupling, in the case of QRE the 1 H relaxation enhancement appears only at selected frequencies. The frequency positions of the QRE peaks can be altered by subtle changes in the EFG offering the possibility to activate or inactivate QRE contrast. This offers a high potential for the design of frequency selective MRI contrast agents in the context of molecular imaging.
Wigner rotation matrices of the  angle.
When a nucleus possessing a quadrupole moment is placed in an external magnetic field, its energy level structure is determined by a superposition of the Zeeman and quadrupole interactions. The total Hamiltonian describing the energy levels is given as: Hamiltonian from its averaged value (a fluctuating part of the dipole-dipole coupling). The residual dipolar coupling provides a pathway for polarization (magnetization) transfer between the participating nuclei. When the 1 H transition frequency matches one of the S spin frequencies, the 1 H magnetization can be taken over by S spin nucleus that manifests itself as a frequency specific decay of the 1 H magnetization. This effect is referred to as polarization transfer [30][31][32][33][34] and can be quantitatively described by the time -independent Liouville equation.
The fluctuating part of the dipole-dipole interaction causes 1 H spin-lattice relaxation. The relaxation also becomes faster at the magnetic fields at which the transition frequencies match.
The 1 H and S spin transitions are coupled and when the frequencies match the S spin transitions enhance the efficiency of the 1 H transitions and hence the 1 H spin-lattice relaxation. 20,21,33,34 This effect is referred to as QRE.

QUADRUPOLE RELAXATION ENHANCEMENT IN LIQUIDS
The critical point is that to think about exploiting the QRE effect in MRI one must be able to observe a frequency specific 1 H spin-lattice relaxation enhancement in solution of 209 Bi containing species, for the solvent protons -in analogy to paramagnetic contrast agents. In solution the 209 Bi containing species undergoes rotational dynamics that modulates the orientation of the EFG tensor with respect to the external magnetic field. When the dynamics is very fast the quadrupole interaction gets averaged out from the perspective of the reference frame determined by the direction of the magnetic field and, in consequence, it acts as a relaxation mechanism for the quadrupole nucleus. In such a case quadrupole peaks do not exist -the 1 H relaxation is affected by the quadrupole spin relaxation acting as an additional source of modulations of the 1 H-209 Bi dipole-dipole coupling. When the rotational motion occurs on an intermediate time scale, the quadrupole interaction neither contributes to the energy level structure (creating the already described picture of QRE in solids) nor acts as a relaxation mechanism. In such a case the influence of the quadrupole spin dynamics on the 1 H relaxation cannot be any more just directly related to the concept of matching transition frequencies. A quantitative description of the 1 H spin-lattice relaxation for arbitrary motional conditions has been provided in Ref. 20,21 The approach is based on the stochastic Liouville equation. The theory has been outlined in Appendix A, on the basis of Ref. 1,2,[8][9][10][11][12][13][19][20][21][35][36][37][38] Let us quantitatively compare the PRE and QRE effects. When the motion responsible for the modulations of the inter-spin dipolar coupling (for instance rotation of the molecule carrying both spins) is fast (much faster than the relaxation itself), the corresponding 1 H spin-lattice relaxation rates originating from the I-S dipole -dipole coupling are given as: 1,2,39-41   The paramagnetic centre can be immobilized, for instance by binding to large protein molecules that ensures slow tumbling in water. One distinguishes between inner-sphere and outer-sphere PRE. Inner -sphere PRE is associated with water molecules forming a coordination sphere around the species carrying the paramagnetic centre and rotating with it as a whole body. 1,2,[8][9][10][11]15 There are three factors modulating the proton spin -electron spin dipole-dipole coupling as seen in Eq. (4). The first one is the rotational dynamics, but it can be, in fact, neglected as it is purposely slow. The next factor is the electron spin relaxation which is caused by fluctuations of Zero Field Splitting (ZFS) interaction around its averaged (residual) value. The fluctuations are caused by internal dynamics of the molecule containing the lanthanide ions. The third factor is the exchange lifetime of water molecules between the coordination sphere and bulk water.
The electron spin relaxation is very complex. The main reason for that is the energy level structure of the electron spin. The energy level structure is determined by a superposition of the electron spin Zeeman coupling and the ZFS interactions. For slow molecular tumbling one can assume that the principal axis system of the ZFS interaction is fixed with respect to the direction of the external magnetic field leading to an orientation dependent energy level structure (the overall PRE effect is a result of averaging over all molecular orientations). This complex energy level structure combined with the high spin quantum number leads to multi-exponential electron spin relaxation characterised by several, field dependent relaxation rates.      Appendix B, that also includes the relaxation data for pure THF.

IV.2. Sample preparation
As synthetic concept, Grignard type reaction starting from BiCl3 and the ArMgBr was containing molecules that in this case works to our advantage.

VI. CONCLUSIONS
Aiming at exploring the potential of the concept of the QRE based contrast mechanism for MRI, a thorough analysis of the achievable QRE has been presented. It has been concluded that the concept of QRE-based contrast agents including nanoparticles as carriers of 209 Bi nuclei is promising. One should be able to obtain sufficiently large QRE effects provided the following conditions are fulfilled:  209 Bi nuclei are placed on the surface of the nanoparticles to ensure a short distance to the solvent (water) protons. When talking about 'surface' one is not restricted to the outer shape of the particle. The effective surface could, e.g., be significantly enlarged by using "sponge-like" structures.
The 2 q S operators are related to the   S T q 2 components as: As the spin and spacial variables cannot be separated in the stochastic Liouville approach one constructs a basis being an outer product of the spin and rotational variables: 1 The 1 H spin-lattice relaxation rate is given as a matrix product: