Long-lived polarization protected by symmetry

Recent work has shown that hyperpolarized magnetic resonance spectroscopy (HP-MRS) can trace in vivo metabolism of biomolecules and is therefore extremely promising for diagnostic imaging,^1–5 the study of metabolic pathways and kinetics⁴ or even the study of protein dynamics.⁶ The primary advantage of HP-MRS over other molecular imaging modalities such as PET/CT, is that, in addition to anatomic localization, HP-MRS can report on biochemical transformations and their kinetics. Nevertheless, general application of HP-MRS is hindered by a fundamental limitation: the signal lifetime for hyperpolarization, which is dictated by the spin-lattice (T₁) relaxation, is typically on the order of seconds.

Symmetry-protected states with no dipole allowed transitions^7,8 offer a solution to this challenge.^9–17 Spin population on these states can have lifetimes dramatically longer than T₁. A prototypical example is para-hydrogen, with the nuclear spins in the singlet state |$S \equiv (\alpha \beta - \beta \alpha)/\sqrt 2$|$S≡(αβ−βα)/2$⁠; parahydrogen can be separated from the other three states (the orthohydrogen states) and can persist for weeks at room temperature.¹⁸ About a decade ago, long lived singlet states were found and accessed on pairs of chemically inequivalent spins^9,10 with a chemical shift difference (Δω) much larger than their mutual J-coupling, and signal lifetimes of up to 25 minutes were observed.¹⁹ However, to maintain long-lived singlet states in these systems, the chemical shift difference has to be suppressed either by shuttling the sample into low field⁹ (∼mT) or by strong spin-locking,¹⁰ neither of which is particularly appealing for in vivo MRI experiments. Systems without a chemical shift difference can support long lived states at high magnetic fields without any additional manipulations. The first direct access to a singlet between chemically equivalent spins (other than parahydrogen) was achieved using the reversible hydration of diacetyl (CD₃¹³C=O¹³C = OCD₃).¹⁷ No spin locking was required to sustain the long-lived carbon singlet state at high magnetic fields; however, the chemical transformations limited the generality of this technique. Newer approaches in two-spin systems include a singlet state between a single pair of nearly equivalent spins²⁰ (Δω ≪ J), where Pileio et al.²¹ demonstrated the so-called “MSM” (Magnetization to Singlet to Magnetization or M2S-S2M) pulse sequence to achieve interconversion between bulk magnetization and the singlet state population (Figure 2(a)). An alternative sequence called “SLIC” (Spin Lock Induced Crossing) was demonstrated by DeVience et al.²² to achieve the same purpose using continuous wave (CW) irradiation.

This paper examines larger spin systems (multiple spin pairs) which as we detail later provide considerable advantages over two-spin systems. With modified parameters,^23–25 both M2S and SLIC can also excite long-lived states between chemically equivalent (Δω = 0) yet magnetically inequivalent spins and create long-lived polarization protected by symmetry (what we call here LOLIPOPS). The simplest example is the AA′XX′ 4-spin system, where A spins are low-γ nuclei such as ¹³C or ¹⁵N providing the core lifetime enhancement, and X are surrounding protons. We observed long-lived signals that arises from states such as the “singlet-singlet” (⁠|$SS \equiv {\textstyle{1 \over 2}}( {\alpha \beta - \beta \alpha })_A ( {\alpha \beta - \beta \alpha })_X $|$SS≡12(αβ−βα)A(αβ−βα)X$⁠) which has no dipole-allowed transitions to the other states.²⁶ Here we develop several themes. We show that this approach is generally extendable to larger spin systems (i.e., AA′X_nX_n′ with increasing n), detail how the disconnected state changes with increasing n, and explain how the pulse sequences access the long lived states in larger spin systems. We also show what fraction of the polarization can be stored with increasing n, including the surprising result that from a signal-to-noise ratio (SNR) perspective, adding more protons can actually improve the signal and the robustness of pumping, even though the long-lived state derives its longevity from the low-γ A component. Finally, we also explore the behavior of these systems upon introduction of a small chemical shift difference.

To address these questions, we first review how to access LOLIPOPS in AA′XX′ 4-spin systems with two examples, diacetylene (DIAC, Figure 1(a)) and a newly studied molecule, the ¹⁵N¹⁵N′HH′ 4-spin system 3,6-dichloro-¹⁵N₂-pyridazine (DCP, Figure 1(b)). In either DIAC or DCP LOLIPOPS can be accessed from either A(¹³C in DIAC, ¹⁵N in DCP) or X (¹H in DIAC and DCP) polarization, yet proton excitation results in a significant signal enhancement because of the difference in gyromagnetic ratio (as previously shown in other molecules).²⁴ In DIAC the resonance conditions for (singlet)_A-(singlet)_H or the (singlet)_A-(triplet)_H are very similar, but in DCP they are significantly different, so each can be selectively populated, and either gives long-lived magnetization.

Next, we use group theory to identify all disconnected states in AA′X₂X₂′ 6-spin systems. From this analysis, two distinct resonance conditions to access the LOLIPOPS are found analytically and verified by simulation and experiment on ¹³C₂-diphenyl acetylene (DPA, Figure 1(c)). Then, extension to even larger spin systems (AA′X₃X₃′) is made with numerical simulation and demonstrated using the 8-spin system in 2,3-dimethylmaleic anhydride (DMMA, Figure 1(e)).

Finally, the behavior of LOLIPOPS with a chemical shift difference is exemplified by the 6-spin system in ¹³C₂-meta methyl diphenyl acetylene (mDPA, Figure 1(d)), where the chemical shift difference breaks the symmetry but the important properties allowing for polarization transfer are retained. Measured parameters (e.g., J-couplings and chemical shift difference) for all compounds are summarized in Table I.

The energy levels in a typical AA′XX′ 4-spin system have been thoroughly discussed in previous studies^23,25,27 but are reviewed here to facilitate analytical treatment of larger spin systems. The Hamiltonian for such a 4-spin system where A and X are different nuclei is

where S denotes the low-γ A nucleus (¹⁵N or ¹³C) and I denotes the high-γ X nucleus (¹H); ω_A and ω_X are the respective Larmor frequencies. Equation (1) assumes “weak coupling” between the A and X spins, appropriate when |ω_A − ω_X| ≫ J_AX and valid in all cases we will discuss in this paper; if A and X are the same nucleus, strong coupling can be reintroduced by intense rf irradiation, and then additional terms are capable of contributing to the evolution.²⁸ In DCP (Figure 1(b)) J_AX (=0.69 Hz) and J_AX′ (=0.15 Hz) are the two “between-pair” J-couplings connecting one proton and one nitrogen whereas J_AA (=−24 Hz) and J_XX (=9 Hz) are the “in-pair” J-couplings. For further analysis the between-pair J-coupling interactions are best divided into their sum and their difference, and we obtain

It is the H_ΔJ term that breaks the magnetic equivalence and is used to access LOLIPOPS. As shown by Pople et al. in 1957,²⁷ the Hamiltonian of Eq. (2) can be block-diagonalized by choosing symmetry-adapted basis functions.^27,29,30 For describing the dynamics during the MSM sequence, we assemble the 16 nuclear states by combining the singlet state |$S = (\alpha \beta - \beta \alpha)/\sqrt 2 $|$S=(αβ−βα)/2$ and triplet states |$T_1 = \alpha \alpha,\,T_0 = (\alpha \beta\break - \beta \alpha)/\sqrt 2,\,T_{ - 1} = \beta \beta $|$T1=αα,T0=(αβ−βα)/2,T−1=ββ$ on the AA′ spin pair with the singlet and triplet states on the XX′ spin pair. An example is the “(singlet)_A − (singlet)_X” state (⁠|${\textstyle{1 \over 2}}( {\alpha \beta - \beta \alpha } )_A( {\alpha \beta - \beta \alpha })_X $|$12(αβ−βα)A(αβ−βα)X$⁠). The assignment to an irreducible representation depends on the molecule and its point group (or more generally the permutation group).³¹ In DCP (C_2v symmetry) the (singlet)_A as well as the (singlet)_X are antisymmetric with respect to rotation about the principle C₂ axis (z-axis) and thus are of B₂ symmetry. To assign the correct irreducible representation to the combined (singlet)_A − (singlet)_X state we use the product table of C_2v and we assign B₂ × B₂ = A₁ symmetry. In contrast, the “(singlet)_A − (triplet)_X” state, ST₀, has B₂ symmetry (B₂ × A₁ = B₂), as does the T₀S state. The mixing rules state that there is no mixing between states with different symmetry, so there is no off-diagonal term that connects SS and ST₀.

Next, let us consider specifically the matrix elements of H_ΔJ, which breaks the magnetic equivalence within the AA′ and XX′ pairs.

These matrix elements are only non-zero for states with anti-aligned spins, that is, the S and T₀ states of proton or nitrogen such that m₁ + m₂ = m₃ + m₄ = 0. (Consider the S and T₀ states for the AA′ spin pair, for example, ⟨T₀|S_z1 − S_z2|S⟩ = ⟨T₀|T₀⟩ = 1.) Accordingly, the Hamiltonian in Eq. (1) can be divided into sub-matrices confined to a given symmetry and projection number Σm = 0. The important pairs of states that qualify are the SS state coupled to the T₀T₀ (“triplet-triplet”) state, both of A₁ symmetry, and the ST₀ and T₀S (“triplet-singlet”) states, both of B₁ symmetry.^23,27. Finally, the two important sub-matrices of the Hamiltonian in Eq. (1) are

where ΔJ_AX = J_AX − J_AX′. Here and in later matrices, we label the corresponding irreducible representation in the upper left corner.

The goal is to drive transitions between these states in order to access the long lived states SS and ST₀ (singlet on the A pair). ΔJ_AX can drive these transitions if we can compensate for the diagonal terms J_AA ± J_XX. In the MSM sequence (Figure 2(a)), this is achieved with a multiple echo pulse train where the frequency of the 180° pulses matches the resonance frequency of these two-level systems, which is dictated by the J-couplings.

Specifically, as shown previously^23,24 the resonance condition is reflected by the inter-pulse delay, τ in the multiple echo pulse train and the number of echo pulses, n (Figure 2(a)). They are different for accessing the SS (Eq. (6)) and ST₀ (Eq. (7)) states in DCP.

Note that Eqs. (6) and (7) are equivalent when J_XX is zero or negligible relative to J_AA, a relatively common situation, as is the case in 2,3-¹³C₂-diacetylene (DIAC, Figure 1(a))^23,32 and all previous demonstrations.^23,24 However, this is not generally valid. For instance, DCP (Figure 1(b)) has J_XX = 9 Hz and J_AA = −24 Hz, giving rise to different resonance conditions. Accordingly, separate measurements can be made to determine relaxation lifetimes (T_S) of the ST₀ (Figure 3(a)) and SS (Figure 3(b)) states, respectively. However, the signal produced from either state (5%–8% of total ¹⁵N magnetization, Figure 3) is only half as large as what can be observed with DIAC where access to both states is synchronized, so polarization transfer from proton magnetization to the long-lived states is advantageous, and this can be achieved by implementing the M2S part of the sequence on the proton channel ((M2S(¹H)-S2M(¹⁵N)).²⁴ Note that a tenfold increase is expected based on the ratio of gyromagnetic ratios of ¹H and ¹⁵N, which is approximately achieved for the ST_o state compared with a MSM measurement with only ¹⁵N pulses (M2S(¹⁵N)-S2M(¹⁵N), Figure 3), although a smaller gain (about 3-fold signal enhancement) is seen for the SS state. We believe that the reason is that ¹H polarization is exposed to more relaxation during the M2S pulse sequence (for DCP, the duration of M2S pulse sequence is (1.5 × 96 × 15.2 ms)_ST ≅ (1.5 × 44 × 33.3 ms)_SS = 2.2 s). The measured T_S for SS and ST₀ are both around 37 s while the nitrogen T₁ at the same field strength is about 15 s. This indicates that the long-lived nature of such states arises from the localized ¹⁵N₂-singlet and is little affected by the proton states. Also note that at short times (short τ_r, the delay between M2S and its inverse S2M) the decays are dominated by fast equilibration within the ¹⁵N triplet state manifold, reducing the signal in the beginning; only after that is the long-lived component observed.

An alternative and rather elegant pulse sequence to access the long-lived states is the SLIC sequence.^22,25 This method uses continuous irradiation at a B₁ power that matches the diagonal terms of Eqs. (4) and (5), i.e., J_AA + J_XX to access the SS state or J_AA − J_XX to access the ST₀ state. As shown in the previous study,²⁵ the convenient basis for SLIC is the X basis for the irradiated spins and the singlet-triplet (ST) basis for the other pair. For a given spin pair the X basis consists of

The system Hamiltonian, including a CW B₁ field (assumed to be along the x axis) can be found by combining the states of AA′ in the X basis with those of XX′ in the ST basis. The two relevant sub-matrices that contain states SS and ST₀ are

These matrices allow us to give a relatively simple description of how SLIC works. As noted earlier, it is only the term H_ΔJ (Eq. (3)) which breaks the magnetic equivalence of the A spins or X spins-if H_ΔJ = 0 no terms interconvert singlet and triplet. For the M2S sequence, we noted that H_ΔJ|T₀^A⟩ ∝ |S^A⟩ and in fact the term creates useful couplings, for example, between S^AS^X and T₀^AT₀^X. Those two states are not degenerate, so the sequence of precisely timed 180° pulses is needed to prevent the energy difference between the two states from averaging away the effect. In the SLIC case, note that H_ΔJ|X₋₁^A⟩ ∝ |S^A⟩ and so in this case as well H_ΔJ creates useful couplings, for example, between S^AS^X and X₋₁^AT_o^X. However, a fundamental difference in the SLIC case is that the X₋₁^AT_o^X state can be shifted by a weak irradiation field, and thus choosing the value correctly permits resonant transfer.

Specifically, if we choose ω₁ = 2π(J_AA + J_XX) then we induce resonance between the first (S^AS^X) and the last state (⁠|$X_{ - 1}^A T_0^X $|$X−1AT0X$⁠) in the matrix of Eq. (9). If we choose ω₁ = 2π(J_AA − J_XX) then we induce resonance between the first (⁠|$S^A T_0^X $|$SAT0X$⁠) and the last state (⁠|$X_{ - 1}^A S^X $|$X−1ASX$⁠) in the matrix of Eq. (10). The off diagonal element πΔJ_AX/√2 now converts this last element into singlet population at a frequency of ΔJ_AX/√2. Singlet state populations can of course also be induced by choosing ω₁ = −2π(J_AA − J_XX), in which case population on state|$X_1^A T_0^X $|$X1AT0X$is converted into the S^AS^X state population. Finally, note that the duration of the SLIC pulse to create population inversion is given as|$\tau _{SL} = 1/(\Delta J_{AX} \sqrt 2)$|$τSL=1/(ΔJAX2)$⁠.

An important observation that was not discussed in the previous paper²⁵ arises when the CW-B₁ field and the preceding 90_y excitation pulse are applied to X spins. Then AA′ spins should be in the ST basis whereas XX′ spins are in the X basis. For the same Hamiltonian we obtain the following sub-matrices that have the same off-diagonal element, πΔJ_AX/√2:

and

Note in Eqs. (9) and (11), spin A and X swap roles, yet S^AS^X is the same state. Therefore, the same fraction of X (instead of A) magnetization is transferred into the S^AS^X state according to Eq. (11). In contrast, in the asymmetric manifold, the SLIC sequence populates either S^AT₀^X when A spins are irradiated (Eq. (10)) or T₀^AS^X when X spins are under irradiation (Eq. (12)). Assuming only states with a singlet component on the AA′ spin pair are long-lived; the net effect is that a smaller fraction of X spin magnetization is stored as long-lived polarization compared with the fraction of A spin magnetization that can be stored by SLIC irradiated on A. This “asymmetric” nature of SLIC with respect to A and X spins is distinct from the “symmetric” M2S sequence, which can transfer equal fraction of A or X spin magnetization into the A spin singlet in an AA′XX′ 4-spin system. As discussed later, this gives rise to different conversion efficiencies and ultimate different detection SNR of the two sequences with all AA′X_(n)X_(n)′ spin systems.

Symmetry effects can also be applied to identify disconnected states in an AA′X₂X₂′ spin systems. ¹³C₂-DPA, which is of D_2h symmetry, can be modeled as such a spin system (Figure 1(c)) with two carbons (¹³C₂) and four protons (¹H₄). The other protons are neglected for they have negligible couplings to the ¹³C spins; this assumption will be justified later. To further simplify the derivation, we have assumed zero J-coupling between protons on opposite aromatic rings (J_XX = 0), which synchronizes resonance conditions for multiple 2-level systems (i.e., same τ and n for MSM or same ω₁ for SLIC). The remaining J-coupling between protons on the same ring is denoted as J_gauche.

First, we treat carbons and protons separately to find the symmetry-adapted basis for each species. As before, for the ¹³C pair, we use the ST basis with states T₊₁, T₀ and T₋₁ of A_g-symmetry and the singlet of B_1u-symmetry in the D_2h point group of DPA (see Fig. 1). On the other hand, the four protons form 16 states in total. To build up the proton states we use a ST basis for the individual spin pairs on either side of the molecule coupled by J_gauche. The combination of the individual singlet and triplet states results in the 16 desired states which we characterize and sort according to their symmetry as listed in Table II.

Subsequently, the 4 carbon states (S, T₊₁, T₀, and T₋₁) and the 16 proton states are directly combined to ultimately form the disconnected energy subspaces and symmetries are assigned based on the point-group multiplication of D_2h symmetry. The same rules applied to the 4-spin system are still valid (mixing can only occur between states with the same symmetry as well as the same total projection numbers ∑m_H and ∑m_C). Previous experiments^23,24 relied on the assumption that DPA supported two-level subspaces similar to those found in the 4-spin systems (Eqs. (4) and (5)) without describing their specific nature which can now be accomplished with the machinery introduced here. The states we are seeking are interconnected by πΔJ = π(J_AX − J_AX′)(where J_AX and J_AX′ denote the near and far carbon-proton couplings, respectively). The task at hand is to find non-zero matrix element for ΔJ in the developed basis. Following a procedure identical to the one used in going from Eq. (1) to (2) and using the spin notation of Figure 1(c), we rewrite the terms containing the AX couplings of the AA′X₂X₂′ Hamiltonian as

Next, we single out |$H_{\Delta J} = {\textstyle{1 \over 2}}\Delta J_{AX} ( ( {S_{z1} - S_{z2} } )( I_{z3} + I_{z4}$|$HΔJ=12ΔJAX((Sz1−Sz2)(Iz3+Iz4$ −I_z5 − I_z6)) being the term breaking the magnetic equivalence and find its matrix elements given by

As before, states of differing symmetry do not interconnect and the matrix elements of S_z1 − S_z2 vanish unless the spins are pointed in opposite directions therefore the ¹³C₂ state can only be S or T₀. In addition, the total projection number of the four protons (∑m_H), which decomposes into m₁₂ + m₃₄ (total projections of spins 1, 2 and spins 3, 4), can only take values of ±1 (where either m₁₂ or m₃₄ is ±1 and the other is zero) or 0 (when m₁₂ = ±1 and m₃₄ = ∓1). We finally find 20 combinations of proton and carbon states (S and T₀) that satisfy all stated rules. They form ten isolated 2-level systems listed in Table III. For example, the Hamiltonian matrix of one of these systems is

Equation (15) is isomorphic to the Hamiltonian in Eqs. (4) and (5), assuming J_HH = 0. Note that the coupling between protons on the same ring (J_gauche) has no effect on these systems since it shifts the two states identically. This is one reason why neglecting the couplings to the more distant protons in DPA is reasonable; the other is that this simpler model agrees extremely well with experiment.

State functions are written as combination of proton and ¹³C states, e.g., |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ + 1} T_0 + T_0 T_{ + 1} })T_0 $|$12(T+1T0+T0T+1)T0$ has ¹H₄ in state |${\textstyle{1 \over {\sqrt 2 }}}| {T_{ + 1} T_0 + T_0 T_{ + 1} }\rangle $|$12|T+1T0+T0T+1⟩$ and ¹³C₂ in state |T₀⟩. The irreducible representations (Γ) of each 2-level system as well as their sum of proton projection number (∑m_H) are also provided. Long-lived signal comes from the states with ¹³C₂ spins in the singlet |S⟩, which are shown in the right column. All eight 2-level systems that have∑m_H = ±1 are accessed by the MSM sequence with regular resonance condition.

Population differences can be driven between the two states by the MSM pulse sequence. For example, from Eq. (15) the state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} })S$|$12(T−1T0−T0T−1)S$ can be accessed, which has a carbon singlet component and hence is a source of long-lived signal. According to Table III, for eight of the ten 2-level systems, (16 states) all states have ∑m_H = ±1. All of them can be accessed with MSM using the same resonance condition shown in Eqs. (6) and (7). Thus, a quarter of total spin states (16 out of 64) in a 6-spin system are used to store bulk magnetization, the same as in a 4-spin system (4 out of 16 states), and the interconversion efficiency is the same for 4-spin and 6-spin systems between carbon bulk magnetization and the long-lived signal.

Interestingly, in the AA′X₂X₂′ 6-spin system, there exists a second type of 2-level sub-space that has the following structure:

where the two states are interconnected by double the frequency found in the other eight pairs (off-diagonal element is 2πΔJ rad/s or |$2 \times {\textstyle{{\Delta J} \over 2}} = \Delta J$|$2×ΔJ2=ΔJ$Hz) and both states have ∑m_H = 0 instead of ±1. This 2-level system in B_1u (and the ∑m_H = 0 state in A_g, see Table III) are not seen with MSM using the optimized transfer condition for the other states, as their double-speed evolution makes the net effect of the MSM sequence vanish, but can be seen under other circumstances.

The two different sets of states can also be found with the detected signal from the SLIC sequence. As for the AA′XX′ system we express the disconnected states of the irradiated spin species (¹³C in this case) in the X basis.²⁵ For example, the sub-matrix of the Hamiltonian that contains the long-lived state written out in equation¹⁵ is, for SLIC,

and the sub-matrix for the state written out in equation¹⁶ is

Equation (17) is isomorphous with the matrix for a 4-spin system derived above in Eq. (11) (assuming J_XX = 0).²⁵ Again J_gauche shifts all levels identically and therefore has no effect on the system. Irradiation on carbon spins with ω₁ = ±2πJ_CC puts state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} } )S$|$12(T−1T0−T0T−1)S$ in resonance with state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )X_{ - 1} $|$12(T−1T0+T0T−1)X−1$ or |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 + T_0 T_{ - 1} } )X_1 $|$12(T−1T0+T0T−1)X1$ and population on |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_0 - T_0 T_{ - 1} })S$|$12(T−1T0−T0T−1)S$ (denoted as p_s) is created at a frequency of |$ \pm \Delta J/\sqrt 2 $|$±ΔJ/2$⁠. To a good approximation, |$p_s ( {\tau _{SL} }) \break \propto - {\textstyle{1 \over 2}} + {\textstyle{1 \over 2}}{\rm cos}( {\sqrt 2 \pi \Delta J\tau _{SL} })$|$ps(τSL)∝−12+12 cos (2πΔJτSL)$⁠,²⁵ where τ_SL(Figure 2(b)) is the duration of the CW pulse. However, a new manifold in the 6-spin system is also put to resonance by the same irradiation (ω₁ = ±2πJ_CC). Specifically, in the Hamiltonian sub-matrix that contains state |${\textstyle{1 \over {\sqrt 2 }}}( {T_{ - 1} T_{ + 1} + T_{ + 1} T_{ - 1} })S$|$12(T−1T+1+T+1T−1)S$ of B_1u-symmetry shown in Eq. (18) the singlet population is created twice as fast (at a frequency of |$ \pm \sqrt 2 \Delta J$|$±2ΔJ$⁠) as that created in Eq. (17). Therefore, for any given τ_SL, population induced on the overall ¹³C₂-singlet state (denoted as P_S) is a combined effect of both resonance conditions, which can be approximated as

where the factors of 8 and 2 come from eight long-lived states that are accessed with the normal condition (Eqs. (17)) and two accessed with Eqs. (18) at double the frequency. This is supported by Figure 4, where the singlet state population after the first CW is simulated³³ and also measured against the first CW duration, τ_SL.