Spontaneous Enhancement of Magnetic Resonance Signals Using a RASER

Abstract Nuclear magnetic resonance is usually drastically limited by its intrinsically low sensitivity: Only a few spins contribute to the overall signal. To overcome this limitation, hyperpolarization methods were developed that increase signals several times beyond the normal/thermally polarized signals. The ideal case would be a universal approach that can signal enhance the complete sample of interest in solution to increase detection sensitivity. Here, we introduce a combination of para‐hydrogen enhanced magnetic resonance with the phenomenon of the RASER: Large signals of para‐hydrogen enhanced molecules interact with the magnetic resonance coil in a way that the signal is spontaneously converted into an in‐phase signal. These molecules directly interact with other compounds via dipolar couplings and enhance their signal. We demonstrate that this is not only possible for solvent molecules but also for an amino acid.


Experimental Section
The experiments were performed using a Bruker 7 Tesla spectrometer Avance III HD equipped with a 5 mm probe with outer 1 H coil. The pH2 with 99% content was delivered by a custom-made parahydrogen generator from ColdEdge Technologies, Inc.. The hyperpolarization transfer occurs in the liquid state at high field inside the cryomagnet in a SPINOE fashion. The experiments are depicted in Figures 1 of the main text as schemes and detailed in the following. Samples containing various solvents and optionally 100 mM N-acetyl tryptophan were investigated and are listed in Table S1. The source of polarization was a reaction of pH2 with vinyl acetate-d6 at 100 mM concentration in the presence of 4 mM of Rh catalyst ([1,4-Bis(diphenylphosphino)butane] (1,5-cyclooctadiene)rhodium(I) tetrafluoroborate), unless otherwise stated. All samples were deoxygenated before the experiments by replacing oxygen with N2 gas. In the beginning of the experiment the samples were allowed to fully relax at 7 T. Then a gradient field causing 10 ppm of line broadening was applied and bubbling of pH2 at 8 bar pressure was started. The hydrogenation reaction was allowed to occur for 20 s at 320 K temperature to ensure complete hydrogenation of vinyl acetate-d6 into ethyl acetate-d6. Afterwards the pH2 pressure was released and nitrogen was bubbled for 2 s to stop further reaction and formation of bubbles in the sample. Subsequently, the gradient was switched off and 2 H radiofrequency decoupling applied. Together with an optional small B1 pulse of 0.5 s 2 H decoupling facilitates the stimulated emission of RASER [7b] which was allowed to happen for a total of 1 s. At this time point polarization transfer from positively polarized ethyl acetate-d6 to the solvent and solute in liquid begins in a SPINOE manner. The samples stayed inside the cryomagnet and spectra showing the effect are detected after variable delay with a 5° or 90° radiofrequency pulse. Optionally the polarization of ethyl acetate-d6 was inverted to negative with a 180° radiofrequency pulse after the RASER and before the SPINOE period. During the SPINOE period (D), a gradient field of 10 ppm was applied as well to prevent spontaneous emission, namely to suppress the RASER.
which yields Hereby, is a matrix of kinetic constants, is the relaxation matrix and ( ), ,0 and are vectors containing the system magnetizations at time point , at time point zero and at thermal equilibrium, respectively.

The vector
( ) contains the magnetizations of all given groups of chemically equivalent spins, which are given by with being the number of spins per volume, the gyromagnetic ratio of spins , ℏ Planck's constant divided by 2 , the spin quantum number of spins , the angular momentum quantum numbers of the 2 + 1 eigenstates of the Zeeman operator, and the populations of these respective eigenstates 2 . These populations can directly be related to the polarization , which for spin-1/2 simply (1 − ) 2 ) = ℏ 2 ( 4 ) For the following, we will decompose as = , ( 5 ) with as the molar concentration of molecules carrying the nuclei , as the number of chemically equivalent nuclei per molecule and as Avogadro's constant.
The vector of equilibrium magnetizations, is obtained from Boltzmann statistics, with the high temperature approximation given being valid under the experimental conditions Here, 0 is the magnetic flux density, is the Boltzmann constant and is the temperature.

Treatment of small flip-angle pulsing
During our experiments, the magnetizations , ( ) were probed by a series of small flip-angle pulses of = 5° in time intervals of 2 s. Since the scaling of the z-magnetization by these pulses is not negligible in our case, for each time point of the simulation, we scaled the magnetization of the previous point ( − Δ ) according to and then used the magnetization the equation of motion ( 2 ) to compute the current data point according to

Treatment of chemical exchange
The matrix contains the first-order rate constants of the chemical exchange happening in dynamic chemical equilibrium. For an exemplary reaction network involving the species A, B and C is given by ) , ( 9 ) and since we are in chemical equilibrium, where , is the equilibrium concentration of species .

Treatment of cross-relaxation
For our purpose, we explicitly include into only terms describing contributions from dipolar couplings to relaxation. We include the terms for intramolecular relaxation under isotropic molecular tumbling, and for intermolecular relaxation under isotropic diffusion, neglecting possible effects of transient complex formation in solution.
To this end, we express as the sum of the individual relaxation terms were contains the rates of relaxation due to the variation of intramolecular dipolar couplings by molecular reorientation, contains the rates of relaxation caused by variations of the intermolecular dipolar couplings by molecular diffusion and is a diagonal matrix containing phenomenological leakage rates * , which should take into account all spin-lattice relaxation due to all relaxation mechanisms not explicitly considered here.

Intramolecular relaxation due to dipolar couplings
For constructing the full matrix , the contributions from all intramolecular dipole-dipole interactions are collected. Pairwise interactions can be between nuclei belonging to the same group of chemically equivalent spins (case of like spins) or between nuclei belonging to two different groups and (case of unlike spins). We hereby follow a notation similar to that used in ref. 3 , with where , is the correlations time modulating the and interaction and where 〈 −6 〉 is the time-average over the inverse sixth power of the internuclear distance (assuming faster molecular tumbling than conformational interconversion) and where 0 is the vacuum permeability.
Interactions between nuclei belonging to the same group of chemically equivalent spins only contribute to the element with , , whereas interactions between nuclei of different groups and contribute to the elements , , and , according to , , = ( 0 + 2 1 + 2 ) , , , , , , The transition probabilities are hereby given by 2,3

Intermolecular relaxation due to dipolar couplings
The rate of intermolecular relaxation is modulated the mutual (translational) self-diffusion constants of the two molecules involved and by the so-called distance of closest approach of the two nuclei. We assume, that the mutual self-diffusion constants can be expressed by the self-diffusion coefficients and of the individual molecules according to = 1 2 ( + ).
The distance of closest approach is somehow vaguely defined, but a lower bound estimate can be obtained from the van der Waals radii of the atoms involved.
For constructing the matrix describing intermolecular relaxation due to diffusion, contributions from like spins and contributions from unlike spins need to be considered, as previously described for the intramolecular case.
In the limiting case, where 2 ≪ 2 , which usually is realized for low-viscosity solvents, the interaction between like spins once again only contributes to , and is given by 2

SPINOE for a two-spin system
Descriptions of cross-relaxation within two-spin systems are widespread amongst the literature. Particularly illustrative descriptions are given in chapter 5.1 of ref. 4 and in ref. 3 .

SPINOE driven by translational diffusion in an ideal two-spin system
The Solomon equations 5 for describing longitudinal relaxation of the longitudinal magnetizations and of a two-spin system, as the explicit form of equation ( 2 ) for two spins, are given by We will use as the hyperpolarized source of non-thermal magnetization and as the spin we are aiming to hyperpolarize.
Considering only intermolecular relaxation and no chemical exchange ( = ), we find where * and * are the rates of undefined leakage for and respectively.
(To avoid confusion, note at this point that the reverse relationship is obtained, when in equation ( 28 ) we are taking and as the z-components of the I and S spins as done in ref. 6 , instead of the taking and as the z-magnetizations of the two spin-groups.) For evaluating ( − ) in equation ( 3 ), we use where 1 and 2 are the eigenvalues of ( − ) and contains the corresponding eigenvectors. As solutions, we find which can be substituted into ( where ( ,0 − )⁄ = is the enhancement factor from hyperpolarization, achieved for spin S.
From ( 37 ) we find that when setting ,0 = the time for the maximum enhancement is

Size estimate of the SPINOE between protons
In the Solomon equations for the evolution of longitudinal magnetization (equation 28), we will describe the cross-relaxation rates by the terms for driven by translational diffusion (equations ( 26 ) and ( 27 )), as described in equation ( 30 ). This description is valid in the extreme narrowing limit for translational diffusion, which is the limit usually valid in low viscosity solutions, if at least one of the compounds is a small molecule (for further details, see chapter 2.4.2 of the SI). For a size-estimate of this cross-relaxation rate for a pair of 1 H-nuclei, let us start off with the system of hyperpolarized ethyl acetate-d6 in chloroform, discussed in Fig. 2 of the main article.
We assume that it is appropriate to combine the magnetizations of both 1 H on the ethyl acetate-d6 into , which should be a reasonably good approximation as long as the two protons do not feature significantly different intermolecular cross-relaxation rates to the CHCl3. For comparability with Fig. 2 we will therefore use = 0.2 and = 0.1 for CHCl3.
The distance of minimal approach is hard to define, but let's assume that a lower bound for /2 is given by the van der Waals radius of hydrogen (~ 1.2 Å).
With these values, we find ≈ * 2 * 10 −4 −1 −1 , where is the molar concentration of . From the smallness of this value, it can be seen, that for free diffusion of small molecules in low viscosity solutions, the SPINOE mediated by translational diffusion will be very weak. Even for concentrations in the molar range, the corresponding cross-relaxations and will usually be much smaller than the auto-relaxation rates and (as estimated from typical T1 for 1 H), and thus the expected transfer efficiencies will be low for small molecules in low-viscosity solutions.
Taking the example of a hyperpolarized spin S with ≈ (85 ) −1 as a value close to the average 1 H-T1 for the protons in the ethyl acetate-d6 used later as hyperpolarization source and assuming for spin I of the 1 H-T1 of CHCl3 measured ( ≈ (180 ) −1 ), we find the timepoint of maximum enhancement to be ≈ 120 (see equation ( 44 )). With this value we estimate that the maximum enhancement for CHCl3 that could be achieved with a theoretical 100% polarization on S at = 0 for this system is around 3, ≈ −77 (using equation S42), corresponding to a polarization of P = 1.7*10 -3 . Numerical simulations for a three-spin-1/2 system also considering inter-and intramolecular cross-relaxation for ethyl acetate yields 3, ≈ −73 (P = 1.6*10 -3 ), which is in reasonable agreement, given the drastic simplification introduced in the two-spin-1/2 treatment. Thus, for target compounds with very long 1 H-T1, such as chloroform, also notable enhancements can in principle be achieved.
When assuming ≈ (5 ) −1 as a more typical 1 H-T1 for a small organic molecule such as the N-acetyltryptophan also discussed, we find the timepoint of maximum enhancement to be ≈ 15 (equation S43). Using ≈ 2 * 10 −9 2 / , which probably is a more realistic estimate for the system discussed in Figure 4 of the main article, and again using = 0.2 and = 0.1 , the maximum enhancement we would expect for a system with such shorter 1 H-T1 to be around ε I,max ≈ -15 (P = 3.4*10 -4 ), when starting from 100% polarization on S.
Despite the relatively inefficient polarization transfer in low viscosity solvents through intermolecular NOE by translational diffusion, notable enhancements can be achieved with para-hydrogen enhanced substrates, due to the very high proton polarizations, that can be achieved these days.

Data Fitting
Fitting of the data shown in Figure 2 of the main article to equation ( 2 ) was performed using a custom script in Matlab® R2020b ((9.9.0.1467703) © 1984-2020 The MathWorks, Inc.).
We only analyzed the transfer of magnetization and the magnetization decay after conversion of the initial spin order into magnetization via the RASER. As a simplification, we assume that at the start of data acquisition, the PHIP reaction as well as the RASER have completely ceased and that only pure zmagnetization remains.
For all points of the experimental data, the simulated data was computed from the previous data point using equation ( 8 ), explicitly taking into account the magnetization scaling by the 5°-pulsing used.
Least squares fits were performed for three different datasets with initial concentration of vinyl acetate-d6 of 10mM, 100mM and 200mM.
For the fits shown in Figure 2, simulations were performed for a system of three groups of spin-1/2, namely the H of CHCl3, the methylene-H of ethyl acetate-d6 and the methyl-H of ethyl acetate-d6. Contributions from all other spins were condensed into the leakage rates * .
For each dataset, the distance of minimum approach was fitted as a global parameter condensing all possible within the system, to avoid linear dependencies. 3.8 ± 1.4 a : Product did not react to completeness. Used concentration measured after SPINOE experiment. b : Estimated from a structure optimized at B3LYP/6-31G(d,p) level, using Gaussian® 09 (rev C.01). 8