Bogdan A.
Rodin
ab,
James
Eills
*cd,
Román
Picazo-Frutos
cd,
Kirill F.
Sheberstov
cd,
Dmitry
Budker
cde and
Konstantin L.
Ivanov
*ab
aInternational Tomography Center SB RAS, Novosibirsk, Russia
bNovosibirsk State University, Novosibirsk, Russia. E-mail: ivanov@tomo.nsc.ru
cJohannes-Gutenberg University, Mainz 55099, Germany. E-mail: eills@uni-mainz.de
dHelmholtz Institute Mainz, GSI Helmholtzzentrum für Schwerionenforschung, 55128 Mainz, Germany
eDepartment of Physics, University of California, Berkeley, CA 94720-7300, USA
First published on 22nd March 2021
The field of magnetic resonance imaging with hyperpolarized contrast agents is rapidly expanding, and parahydrogen-induced polarization (PHIP) is emerging as an inexpensive and easy-to-implement method for generating the required hyperpolarized biomolecules. Hydrogenative PHIP delivers hyperpolarized proton spin order to a substrate via chemical addition of H2 in the spin-singlet state, but it is typically necessary to transfer the proton polarization to a heteronucleus (usually 13C) which has a longer spin lifetime. Adiabatic ultralow magnetic field manipulations can be used to induce the polarization transfer, but this is necessarily a slow process, which is undesirable since the spins continually relax back to thermal equilibrium. Here we demonstrate two constant-adiabaticity field sweep methods, one in which the field passes through zero, and one in which the field is swept from zero, for optimal polarization transfer on a model AA′X spin system, [1-13C]fumarate. We introduce a method for calculating the constant-adiabaticity magnetic field sweeps, and demonstrate that they enable approximately one order of magnitude faster spin-order conversion compared to linear sweeps. The present method can thus be utilized to manipulate nonthermal order in heteronuclear spin systems.
Dissolution dynamic nuclear polarization (d-DNP) is currently the favored method to hyperpolarize biomolecules,10,11 but this method is cumbersome and comes with high associated costs. An appealing alternative is parahydrogen induced polarization (PHIP)12,13 which is a chemical method renowned for its low cost and accessibility. There are several molecules that have been hyperpolarized via PHIP and applied for in vivo studies.14–19 The source of nonthermal spin order in PHIP experiments is the singlet order of parahydrogen20 (pH2, molecular hydrogen in the nuclear spin-singlet state). Although pH2 does not have a magnetic moment and is thus NMR-silent, upon symmetry breaking (i.e. by rendering the two protons chemically or magnetically inequivalent) the nonthermal singlet order can be converted into observable NMR signals, which are strongly enhanced compared to those under equilibrium conditions. The first step for hydrogenative PHIP is a catalytic hydrogenation reaction (addition of H2 to a suitable substrate, usually one with an unsaturated C–C bond). When the two pH2-nascent protons occupy inequivalent positions in the reaction product the symmetry is broken, and NMR signal enhancements can be obtained. The magnetic interaction that induces symmetry breaking is typically a chemical shift difference, or inequivalent J-couplings to a third nucleus.
A common step in PHIP is transferring nonthermal spin order from the source spins – here the pH2-nascent protons – to target spins of choice, which are more suitable for NMR detection for various reasons (longer relaxation times, higher spectral resolution, lower background signals). A number of methods have been developed to transfer the pH2 spin order to various heteronuclei, via rf pulse methods at high field,14,21–34 or through coherent spin mixing under zero- to ultralow-field (ZULF) NMR conditions.35–42 In the ZULF regime, Larmor frequencies are small, and nuclear spins belonging to different isotopic species become “strongly coupled” – that is the difference in Larmor frequencies becomes comparable to the spin–spin couplings. Under these conditions, coherent exchange of polarization among the spins becomes possible.
A number of polarization-transfer techniques exploiting ultra-low magnetic field manipulations have been developed, for example one can:
(1) Perform the reaction with pH2 at ultralow magnetic field to induce spontaneous polarization transfer.40
(2) Perform the reaction with pH2 at high field, nonadiabatically drop to ultralow or zero field, and adiabatically return to high field.36–38 In this work we refer to this process as a field sweep from zero field (FS-f-Z). In previous literature this is commonly referred to as a magnetic field cycle.
(3) Perform the reaction with pH2 at high field, then adiabatically invert the magnetic field, passing through zero field.41,43 In this work we refer to this process as a field sweep through zero field (FS-t-Z).
All NMR methods using adiabatic variation of the spin Hamiltonian are confronted with a common problem: adiabatic processes are by definition slow, and spin relaxation can be significant. Relaxation of hyperpolarized samples is generally detrimental as it gives rise to irreversible decay of the nonthermal spin order back to thermal equilibrium. It is therefore desirable to use the fastest possible adiabatic variation without disturbing the adiabatic nature of the process.44–47 Solutions have been proposed such as “fast” adiabatic processes given by optimal control theory48 or by varying the Hamiltonian Ĥ(t) such that the effective adiabaticity parameter is constant at all times.49 The latter approach, constant-adiabaticity, is easy to implement and to adapt to specific molecular cases.
In this work we demonstrate constant-adiabaticity ultralow magnetic field manipulations to transfer proton singlet order into 13C magnetization in PHIP-polarized [1-13C]fumarate. We form hyperpolarized fumarate by chemical reaction of para-enriched hydrogen with an acetylene dicarboxylate precursor molecule50 (see Fig. 1). The protons are initially in the singlet state, and are scalar-coupled to the 13C spin in the carboxylate position (we work at natural 13C abundance). In the case of [1-13C]fumarate, the J-coupling between the protons is significantly larger than the proton-carbon J-couplings; this is referred to as the “near-equivalence” regime. As a consequence, the proton singlet state is close to an eigenstate, and significant state mixing which allows for polarization transfer occurs only at well-defined magnetic fields, ±B(i)LAC, corresponding to the i-th level anti-crossings (LACs) of the spin system.41 Here we specifically investigate two ZULF methods to perform polarization transfer: a field sweep from zero field, which uses a magnetic field variation from zero to Bmax, and a field sweep through zero field, which uses a magnetic field variation from −Bmax to Bmax. For the case of [1-13C]fumarate, Bmax is a few μT, which is considerably higher than the LAC fields, B(i)LAC. For both FS-f-Z and FS-t-Z experiments we derive constant-adiabaticity magnetic field profiles, B(t), and compare the performance with linear (uniform) field variations.
Ĥ(t) = ĤZ(t) + ĤJ, | (1) |
ĤZ(t) = −B(t){γI(Î1z + Î2z) + γSŜ3z}, | (2) |
ĤJ = 2πJ12(Î1·Î2) + 2πJ13(Î1·Ŝ) + 2πJ23(Î2·Ŝ), | (3) |
STZ = {|S12〉,|T12+〉,|T120〉,|T12−〉} ⊗ {|α3〉,|β3〉}. | (4) |
(5) |
By plotting the eigenvalues of STZ′ as a function of magnetic field as shown in Fig. 2, it can be seen that there are a number of energy level-crossings. On close inspection, one can see that in four places the crossings are in fact avoided; the inequivalence in proton–carbon couplings acts as a small perturbation which lifts the degeneracy of the crossing states, and the level crossings are turned into LACs. The positions of the LACs have been determined previously:41
(6) |
(7) |
At LACs, coherent spin mixing comes into play, which can be exploited in polarization-transfer experiments. For instance, adiabatic passage through a LAC gives rise to swapping of the populations of the unperturbed STZ′ states (commonly termed diabatic states). The reason is that the populations follow the instantaneous eigenstates, i.e., the eigenstates of the full Hamiltonian (defined in the presence of the perturbation terms). Hereafter, by adiabatic variation of the Hamiltonian Ĥ(t) of a spin system under study we mean that its eigenstates |ψi(t)〉 vary with time so slowly that the state populations have sufficient time to adjust to such changes (populations “follow” the time-dependent states). The focus of this work is to optimize adiabatic passage through LACs through the use of constant-adiabaticity field profiles to minimize the passage time.
(8) |
(9) |
In order to determine optimized B(t) ramps, we impose the condition that the general adiabaticity parameter is equal to a constant value, ξ(t) = ξ0, during the variation. Before proceeding, we introduce a few improvements for calculating constant-adiabaticity B(t) profiles.
First of all, to transfer populations between the diabatic states we use LACs, but there are also many level crossings in the system which occur between two states of degenerate energy when there are no perturbation terms to induce state mixing. Level crossings of a pair of levels occur when the Hamiltonian has a block-diagonal structure and the two states belong to different blocks. In the case under study, the block-diagonal structure of the Hamiltonian is dictated by the fact that the z-projection of the total spin is conserved, since the commutator [Ĥ, Î1z + Î2z + Ŝz] is zero. We exclude level crossings from consideration since, as follows from eqn (9), calculation of the |i〉ξij|j〉 parameter meets certain difficulties (the numerator and denominator tend to zero). Although this uncertainty in calculating ξij can be resolved analytically, numerical calculation of the adiabaticity parameter becomes problematic. Hence, we need to evaluate the ξij parameters only for the states belonging to the same blocks and eqn (9) can be modified as follows:
(10) |
Second, we take into account that in the case of [1-13C]fumarate prepared via PHIP, to a good approximation only the |Sα〉′ and |Sβ〉′ states are populated, which is only two states out of eight. To adiabatically manipulate the spin order, it is sufficient to consider only mixing of these states with other states belonging to the same blocks in the Hamiltonian. Specifically, the block of spin states characterized by the angular momentum projection on the field axis m = 1/2 comprises three states |T0α〉′, |Sα〉′ and |T+β〉′. If initially only the |Sα〉′ state is populated, spin mixing only in pairs of states |Sα〉′ ↔ |T0α〉′ and |Sα〉′ ↔ |T+β〉′ is important. To take this into account, eqn (10) should be modified as follows:
(11) |
With these two considerations, we can calculate the optimized B(t) profile. The time derivative of the Hamiltonian (1) is:
(12) |
(13) |
In the FS-f-Z experiment, the field is then rapidly (nonadiabatically) dropped to zero, which preserves the populations of |Sα〉′ and |Sβ〉′, and then abiabatically increased to exchange the populations of |Sα〉′ and |T+β〉′, but leave the population of |Sβ〉′ unchanged. At the end of the process the |T+β〉′ and |Sβ〉′ states are populated, and hence the 13C spin is hyperpolarized. In this experiment only one LAC is relevant: the LAC at B(1)LAC.
In the FS-t-Z experiment, after the hydrogenation step the field is reversed rapidly (nonadiabatically) to −2 μT (although the hydrogenation could be done at −2 μT and this step skipped) which preserves the populations of |Sα〉′ and |Sβ〉′, and then increased adiabatically through zero to +2 μT. The population in |Sα〉′ ends in |T+β〉′, and the population in |Sβ〉′ ends in |T0β〉′. At the end of the process, the |T+β〉′ and |T0β〉′ states are populated, and hence the 13C spin is hyperpolarized. In this experiment three LACs are relevant: LACs occurring at the fields −B(1)LAC, B(1)LAC and B(2)LAC. Note that because −B(2)LAC is not part of the adiabatic pathway, the resulting field profile is slightly asymmetric with respect the center point.
In Fig. 2, we also show the constant adiabaticity B(t) sweeps for both cases. One can see that the proposed algorithm dictates a slow increase of the field at the LACs, whereas away from the LACs field sweeping can be fast.
All chemicals were purchased from Sigma Aldrich. A solution of 50 mM monopotassium acetylene dicarboxylate, 100 mM sodium sulphite and 7 mM ruthenium catalyst [RuCp*(CH3CN)3]PF6 in D2O was prepared by dissolving the solids by heating and sonication. The sodium sulphite was added to increase the rate of reaction as discussed in ref. 52 and 53. The pH of the solution was adjusted to pH 10 with NaOD to further improve the rate of reaction. Oxygen was removed from the solution by bubbling nitrogen through for 5 min. 300 μL of this precursor solution was used for each experiment.
All experiments were performed at natural 13C abundance, but in experiments we specifically observe the [1-13C] isotopomer and generally refer to this molecule in the text.
The NMR experiments were performed in a 1.4 T 1H-13C dual resonance SpinSolve NMR system (Magritek, Aachen).
Parahydrogen at >98% enrichment was generated by passing hydrogen gas (>99.999% purity) through an Advanced Research Systems (ARS, Macungie, USA) parahydrogen generator operating at 25 K.
For ultralow-field experiments, a magnetic shield (MS-1F, Twinleaf LLC, Princeton, USA) was used to provide a 103 shielding factor from the laboratory field. Static internal magnetic fields for shimming were produced using built-in Bx, By, and Bz coils (on a flexible PCB), powered with computer-controlled DC calibrators (Krohn-Hite, model 523, Brockton, USA), providing three-axis field control to provide an additional order of magnitude in field reduction. The residual field in the shield was on the order of nanotesla, as verified with a fluxgate magnetometer. The time-dependent applied magnetic fields were generated with a Helmholtz coil (70 mm diameter) with 10 turns on each side, wound with 0.5 mm diameter copper wire onto a 3D-printed former, with current supplied by a power amplifier (AE Techron 7224, Elkhart, USA). The magnetic-field profiles were generated using a data acquisition card (NI-9263, National Instruments, Austin, USA) with 10 μs time precision.
Low-pressure/vacuum J. Young NMR tubes held in the ZULF chamber and 1.4 T SpinSolve NMR spectrometer were connected with polytetrafluoroethylene (PTFE) tubing (1/16 in. O.D., 0.5 mm I.D.), as shown in Fig. 3. Gas and liquid flow were controlled by pneumatically actuated valves (Swagelok, Frankfurt, Germany). Sample hydrogenation was followed by shuttling into the SpinSolve by reversing the gas flow. The sample transport was performed with nitrogen gas (any unreactive gas could be used) and took 2.5 s. In order to prevent the sample from passing through any fields that could lead to undesired state-mixing during sample transport, a penetrating solenoid was used to provide a guiding field during transit out of the magnetic shield. To avoid having bubbles in the detection region after sample transport, 100 μL of acetone was placed in the SpinSolve tube at the start of each experiment. This mixed with the fumarate solution after shuttling, and served to reduce the surface tension and viscosity of the D2O solvent. The experimental apparatus is shown in Fig. 3.
At the start of the experiment the sample was in the ZULF chamber in a 5 mm NMR tube, in a +2 μT (chosen as a relatively low field that is still high enough for the Hamiltonian eigenstates to be, to a good approximation, the STZ′ basis states) field provided by the Helmholtz coils, and parahydrogen gas was bubbled in at 7 bar for 30 s. After a 1 s delay to allow the sample to settle, a field manipulation was applied using the Helmholtz coils. After the field sweep, the solenoid guiding field was switched on to provide a +20 μT field, and nitrogen gas at 7 bar was used to shuttle the sample into the SpinSolve NMR spectrometer. After a 1 s delay for the sample to settle, a 90 pulse was applied followed by data acquisition. A simplified event sequence is shown in Fig. 4.
After the hyperpolarization had fully relaxed, a thermal equilibrium 1H NMR spectrum was acquired on each sample to quantify the concentration of fumarate formed. To calculate the polarization levels reported in Fig. 4(c and d), the hyperpolarized results were compared to a thermal equilibrium 13C NMR spectrum of an isotopically-enriched 500 mM [1-13C]fumarate standard in D2O.
To perform the spin dynamics simulations, firstly the density matrix is projected onto the eigenbasis of the Hamiltonian (1) defined at B = 2 μT and all off-diagonal elements (coherences) of the density matrix are removed, since they are averaged out upon continuous production of polarized molecule by the hydrogenation reaction. The resulting density matrix describes the spin system immediately following the hydrogenation step. Next, we numerically solve the Liouville–von Neumann equation with the time-dependent Hamiltonian (1), where B(t) corresponds to the magnetic field profile. Finally, we extract the expectation value of S-spin polarization from the final density matrix. Relaxation was not included in the simulations.
The experimental results are generally in good agreement with the simulations, but the 13C signals for the FS-t-Z experiments are notably lower than in the FS-f-Z experiments, with FS-f-Z showing ∼15% higher transfer efficiency. This is not intrinsic to the methodology, since both methods can lead to >97% 13C polarization in this molecular system. We believe the lower efficiency of the FS-t-Z is predominantly for two reasons. Firstly, in the FS-t-Z experiment, adiabatic passage through three LACs is necessary for polarization transfer, whereas for the FS-f-Z experiment only one LAC is used. Imperfections in the adiabatic passages will therefore compound, and be more detrimental in the FS-t-Z experiment. Secondly, and likely more importantly for most experimental cases, the requirement to pass through zero magnetic field for the FS-t-Z experiment can cause significant loss of polarization if there are residual magnetic fields along other axes. This introduces additional undesirable LACs which can lead to the populations being diverted from the desired transfer path. In Fig. 5 we show how the final 13C polarization for both FS-f-Z and FS-t-Z experiments with the sweep in the z-axis depends on the presence of a transverse field in the x/y-plane. We now use Bz to indicate a B field applied along the z-axis.
Despite shorter sweeping times when using a constant-adiabaticity profile, these methods are more sensitive to magnetic field offset in the field sweep axis (z) than the linear profiles. This is because the constant-adiabaticity profiles are designed around the knowledge of LAC fields, and if there is a magnetic field offset or inhomogeneity across the sample, the slow part of the constant-adiabaticity field ramp will not match the LAC field. The dependence of the constant-adiabaticity FS-f-Z and FS-t-Z conversion efficiencies on magnetic field offset is shown in Fig. 6. A Bz offset on the order of 100 nT is sufficient to reduce the transfer efficiency by >10%. The case is worse for the FS-t-Z experiment, which requires three LACs, compared to just one for the FS-f-Z experiment. We expect this situation can be improved by designing pseudo-constant-adiabaticity profiles to be close to constant-adiabaticity, but made to be more robust with respect to Bz field offset/inhomogeneity by broadening the LAC field condition.
In these experiments the FS-f-Z experiment performs better than the FS-t-Z. However, there are certainly experimental cases in which the FS-t-Z might be preferred. One example is in a system in which the rapid (nonadiabatic) initial field drop of the FS-f-Z is inconvenient or not possible, such as when instead of varying the magnetic field applied to a static liquid, a static magnetic field spatial profile is constructed and the liquid flows through to achieve the desired polarization transfer. Another example of when the FS-t-Z experiment may be preferred is for molecular systems outside the near-equivalence regime (i.e. when |J12| < |δJ|, in our case J12 = 15.7 Hz and δJ = 3.4 Hz). In these cases, the proton singlet state is no longer close to an eigenstate at zero field, and the nonadiabatic field switch at the start of the FS-f-Z experiment can cause the population differences to be converted into coherences which rapidly dephase. This can be avoided by using the FS-t-Z protocol which does not require nonadiabatic field changes.
For this work [1-13C]fumarate was chosen as a model system because it is a 3-spin system in the near-equivalence regime (J12 ≫ |J13 − J23|). This is the relevant regime for many PHIP-polarized molecules for which this inequality is similar or even more pronounced, with some examples being ethyl propionate,38 tetrafluoropropyl propionate,16 and allyl pyruvate.54 The greater this inequality, the greater the difference in time required for a linear sweep profile over its constant adiabaticity analog. On the other hand, [1-13C]fumarate is not an ideal molecule for the demonstration since the proton singlet lifetime and the 13C T1 are both relatively long,52 meaning there is little difference in the 13C polarization that can be reached whether using a constant adiabaticity or a linear sweep. This would certainly not be the result for molecules with more rapidly relaxing hyperpolarized spins, which is often the case when additional protons are present in the molecules.
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