On the spin order transfer from parahydrogen to another nucleus

Journal of Magnetic Resonance 225 (2012) 25–35

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On the spin order transfer from parahydrogen to another nucleus q Sébastien Bär, Thomas Lange, Dieter Leibfritz, Jürgen Hennig, Dominik von Elverfeldt, Jan-Bernd Hövener ⇑ Medical Physics, Department of Radiology, University Medical Center Freiburg, Germany

a r t i c l e

i n f o

Article history: Received 19 April 2012 Revised 30 August 2012 Available online 15 September 2012 Keywords: Hyperpolarization Parahydrogen PASADENA PHIP 13 C

a b s t r a c t The hyperpolarization of nuclear spins holds great potential e.g. for biomedical research. Strong signal enhancements have been demonstrated e.g. by transforming the spin order of parahydrogen (pH2) to net polarization of a third nucleus (e.g. 13C) by means of a spin-order-transfer (SOT) sequence. The polarization achieved is vitally dependent on the sequence intervals, which are a function of the J-coupling constants of the molecule to be polarized. How to derive the SOT sequence intervals, the actual values for molecules as well as the (theoretical) polarization yield and robustness, however, are not fully described. In this paper, (a) we provide the methods to obtain the SOT intervals for a given set of Jcoupling constants (i.e. of a new hyperpolarization agent); (b) exemplify these methods on molecules from literature, providing the hitherto missing intervals and simulated polarization yield; and (c) assess the robustness of the sequences towards B1 and J-coupling errors. Close to unity polarization is obtained for all molecules and sequences. Furthermore, the loss of polarization caused by erroneous B1 and Jcoupling constants is reduced by choosing the channel and phase of some pulses in the SOT sequences appropriately. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction The potential of magnetic resonance (MR) in medicine nowadays goes far beyond the imaging of anatomic morphology, but extends to the non-invasive mapping of functional parameters in vivo, such as perfusion [1,2] or oxygen consumption in the brain [3,4]. This is achieved despite the inherent insensitivity arising from the fact that only a minuscule fraction of all nuclear spins contribute to the MR signal. In the thermal (i.e. Boltzmann) equilibrium, this fraction, called polarization (P), depends on the gyromagnetic ratio of the nuclear spin, the external static magnetic field (B0) and the temperature. Even if the strong magnetic fields of modern systems are combined with the most abundant and strongest (stable) nuclear moment (hydrogen 1H), the polarization hardly exceeds a few ppm at in vivo temperatures (10 ppm/30 ppm at 3 T/9.4 T respectively, at room temperature). In other words, MR is very insensitive, detecting only approximately three 1H spin in a million per Tesla and less for other nuclei, which, in addition, are generally much less abundant. For this very reason, powerful methods successfully employed in other scientific fields cannot be translated to routine medical

q Part of this work was supported by the Innovationsfonds Baden-Würtemberg and the Academy of Excellence of the German Science Foundation. ⇑ Corresponding author. Address: Breisacher Straße 60A, 79106 Freiburg, Germany. Fax: +49 761 270 38310. E-mail address: [email protected] (J.-B. Hövener).

1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2012.08.016

application. This includes the chemical analysis of small volumes by spectroscopy at concentrations <1 mM and real-time metabolic imaging. Various techniques have been devised to access this potential [5–9]. These so-called hyperpolarization methods circumvent the thermal distribution, achieving strong non-equilibrium polarizations approaching the order of unity (e.g. Phyp = 10% corresponding to a signal enhancement of 114,500 compared to the 13C polarization at B0 = 1 T). Some of these techniques are based on parahydrogen (pH2), the spin-singlet isomer of molecular hydrogen, which was discovered in the early 20th century [10–16]. pH2 is a highly ordered quantum state, but has spin 0, i.e. is generally MR-invisible. Dedicated methods, e.g. by means of r.f. pulses or field-cycling, transform this order to an observable nuclear polarization [8,17–20]. With the exception of signal amplification by reversible exchange (SABRE) [21,22] and field-cycling methods [18,23], these techniques rely on a spin-order-transfer sequence (SOT) composed of evolution intervals and r.f. pulses tailored to the J-coupling between the hydrogen nuclei originating from pH2 and the nucleus to be polarized (e.g. 13C, Fig. 1). Here, we investigate the SOT sequences developed by Goldman et al., Kadlecek et al., and Haake et al. [8,17,19]. Goldman’s and Kadlecek’s SOT sequences have been employed for in vivo 13C imaging using the agents succinate (SUC) and hydroxyethylpropionate (HEP) [20,24–26]. To our knowledge, Haake’s sequence has been used for in vitro NMR only.

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S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

Fig. 1. Scheme of the hyperpolarization process for [1-13C 2,3-2H]succinate. (a) Molecular parahydrogen ( ) is added to [1-13C 2,3-2H]fumarate by catalytic hydrogenation. (b) A pulse sequence is applied transferring the spin order Via J-couplings to large, non-equilibrium polarization on the labeled carbon site (c). : deuterium, : oxygen, : hydrogen, : carbon, : 13C.

The polarization yield depends on the right choice of the SOT sequence parameters. Kadlecek et al. [19] derived explicit formulas for the SOT sequence parameters, which can be readily applied to arbitrary molecules. No such formulas are available for Goldman’s and Haake’s sequences [8,17,20]. While the theory was laid out in detail, the adaptation of their methods to other molecules of interest is not straightforward. For the sequence most frequently used in vivo, Goldman’s, the intervals are reported for HEP only. Furthermore, the susceptibility to experimental imperfections has not yet been investigated for any sequence. This is an important aspect since coupling constants of molecules are only known with limited accuracy and B1 inhomogeneity was recognized to cause polarization loss before [25]. Therefore, the goal of this publication is threefold: (a) to provide a method to derive the optimal intervals for Goldman’s and, with some restrictions, for Haake’s SOT (Eqs. (1) and (8)), (b) to provide the intervals and (theoretical) polarization of previously employed molecules using the results of (a). Where available, we use published values to verify our results (Tables 1–3), (c) to assess the robustness of all three SOT sequences towards experimental imperfections (Tables 1–3). Assuming errors in J-couplings and B1, we simulate the impact on the polarization yield and compare it to experimental literature values. 2. Theory As the underlying theory of each SOT sequence was described in depth by the respective author, we only repeat the details required for the simulations here. For more details, the reader is referred to the original papers [8,17,19,20]. 2.1. The model of the spin system

Kadlecek et al. assumed a magnetic field low enough (mT) to neglect chemical shift differences between the (former para-)1H nuclei, resulting in weak heteronuclear interaction between 1H and X (J13, J23), and strong homonuclear 1H–1H coupling (J12) [17,19]. At high field (T), Haake et al. assumed weak 1H–X and 1H–1H couplings [8]. Refocusing pulses were applied during intervals of free evolution. 2.3. The density matrix When an unsaturated precursor molecule is being hydrogenated with pH2, the formally isolated spin-singlet system of pH2 is incorporated into a new network of molecular interactions. It is assumed that initially the singlet spin order is preserved and available within the molecule. The evolution of the singlet order is governed by the newly formed Hamiltonian and thus depends on the magnetic field. As the individual molecules are hydrogenated at different time points (and thus have evolved for different times before the SOT), after a while, coherence will be lost without further measures. For hyperpolarization at low field, Goldman and Johannesson [17] and Kadlecek et al. [19] assumed that the full singlet-state density matrix is preserved. This is achieved by applying decoupling pulses throughout the hydrogenation reaction until the SOT is played out:

rstrong ¼ bI 1bI 2 pH ¼ ðbI 1xbI 2x þ bI 1ybI 2y þ bI 1zbI 2z Þ ðstrong H—H coupling; e:g: B0  mTÞ Haake et al. [8] suggested to start from the high-field, timeaveraged density matrix. This is obtained when the experiment is carried out at high field without decoupling pulses during the hydrogenation reaction, prior to the SOT. Only the longitudinal spin order is preserved, the transversal components are averaged out [8]:

All simulations were performed using the product operator formalism [27,28]. The hyperpolarization agents were approximated as a three-spin-½ system, consisting of two hydrogen nuclei originating from pH2 (nucleus 1: three bonds apart from X, nucleus 2: two bonds apart from X), and one nucleus of isotope X, which is to be polarized (heteronucleus 3, e.g. 13C, 15N, 29Si, Fig. 1). The J-coupling constants for the spin systems under investigation were obtained from literature (J12, J13, J23) [8,17,24,29,30].

rweak ¼ bI 1zbI 2z ðweak H—H couplings; B0  TÞ pH

2.2. The Hamiltonians

To obtain the intervals of Goldman’s sequence, analytical expressions were derived using his geometric picture. Kadlecek et al. reported formulas to calculate the intervals in the original publication [19]. For PHINEPT+, the intervals were obtained by maximizing the signal equation given in [8].

The liquid state, isotropic Hamiltonian in the rotating frame was used to evolve the spin system (AA0 X, with chemically but magnetically non equivalent 1H nuclei). Goldman et al. and

where bI is the angular momentum operator, and matrices indicated in bold. 3. Methods 3.1. Sequence intervals

S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

27

Fig. 2. Schematic representation of the SOT sequences suggested by Goldman, Kadlecek and Haake (PHINEPT+) [8,17,19]. Enhanced heteronuclear polarization is available after the last r.f. pulse in each sequence. Refocusing pulses during the free evolution intervals are marked in black, while functional SOT inversion pulses are hollow. Haake’s sequence was modified to obtain longitudinal spin order (l-PHINEPT+). We found that choice of channel (1H or X, here: 13C) and phase (x, y) of functional SOT 180° pulses do not change the polarization yield under ideal conditions, but may reduce the effect of B1- or J-coupling errors.

3.2. Spin order transfer and polarization yield

To estimate the robustness of the SOT sequences, the effect of a ±10% deviation from the optimal value of the

To simulate the effect of a SOT sequence on a spin system, the corresponding density matrix (rstrong or rweak pH , respectively) was pH subjected to the SOT pulse sequences composed of pulses and intervals of free evolution (Fig. 2). Following the original publications, r.f. pulses were assumed as ideal operator rotations, relaxation and other effects such as the phase of low-field pulses [31,32] were neglected. To demonstrate the effect of maladjusted intervals on the polarization yield, the intervals of the sequences were varied while keeping the J-coupling of the molecule constant.

on the polarization yield was assessed. A variation of 10% corresponds to previously reported errors of B1 [25] and pH-dependence of J-coupling constants [24]. As the combination of both may cancel or amplify the polarization loss, we report the largest loss, i.e. the worst combination.

3.3. Susceptibility to experimental errors

3.4. Verification of obtained intervals, polarization yield and experimental imperfections

The polarization obtained in experiments using Goldman’s and Kadlecek’s SOT was reported to be one order of magnitude less than the predicted theoretical values of unity [20,24,25]. This loss was attributed to imperfections of the actual spin order transfer and relaxation during transport of the hyperpolarized sample, as well as to non-MR based effects such as incomplete hydrogenation, 13 C labeling and pH2 enrichment. Which errors are likely to occur? The J-coupling constants of the agents are decisive for the polarization yield. They are prone to errors, too, as they have to be determined experimentally, and strongly depend on the chemical environment (e.g. pH, Coulomb effects, counter ions, etc.). For example, it was reported that appropriate pH is decisive for successful polarization transfer of succinate [24]. Furthermore, some flip angle error is likely, in particular because the setups used before provide a limited B1 and B0 homogeneity [25,33,34].

To verify the intervals and polarization calculated with our methods, we compared our results to numerical quantum mechanical (q.m.) simulations and results published elsewhere [17,20,24,25]. For the q.m. simulations, the intervals were varied until the maximal polarization was found (using an optimization algorithm). For Goldman’s sequence, HEP and a fictitious molecule (with given J-coupling constants), intervals were reported elsewhere [17] and used for comparison. The effect of experimental errors was compared to data available for succinate [24,25]. For some cases, numerical optimization of the evolution intervals lead to increased polarization or decreased duration. These values are marked accordingly. All calculations and simulations were performed using a numerical computing environment (MATLAB R2009A, The Mathworks Inc., Nantucket, USA). To exemplify our methods, to calculate the intervals and to assess the performance and robust-

(a) J-couplings, (b) flip angles and (c) combination of both

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S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

Fig. 3. Stick models of 13C and 29Si hyperpolarization agents formed upon para-hydrogenation: [1-13C 2,3-2H]succinate (a, SUC [24]), hydroxyethylpropionate (b, HEP [17]), trifluorobut-2-enoate (c, TIFBU [30]), trimethylvinylsilane (d, TMVS, [8]), [2-(2-methoxyethoxy)ethyl] malate (e, MEPA1, MEPA2, [29]), 2-(2-methoxyethoxy)ethyl acrylate (f, BIMAC [29]). The asterisk indicates the spin label.

ness of the sequences, eight molecules were chosen from literature (Fig. 3), which have been used in the context of parahydrogen and hyperpolarization before [8,17,24,29,30]. This includes succinate (SUC), the only biologically relevant pH2 agent which has been applied in vivo. With the exception of the molecule HEP, neither intervals nor polarization yield have been reported, making the exact reproduction of the experiments difficult. 4. Results For all sequences, the polarization yield was strongly dependent on the intervals, as exemplified on hyperpolarization agent succinate (Fig. 4): For Kadlecek’s and Goldman’s sequence, the loss in polarization due to maladjusted intervals was stronger than for Haake’s. This illustrates the need for methods to obtain accurate SOT intervals, which are described in the following. 4.1. Intervals and polarization yield of Goldman’s SOT This sequence was successfully employed for the hyperpolarization of HEP and succinate and subsequent in vivo imaging [24,26].

It is composed of three evolution intervals ðtG1 ; tG2 ; t G3 Þ intersected by pulses for molecules with 30° < hG < 60° [20] where  23 j hG ¼ arctan ba , where a ¼ jJ13 J , b = J12 (for more details, see the 2 original publications [17,19,20]). For molecules with hG > 60°, two more pulse evolution cycles are required, one before the basic sequence (tG0  180 , preparation cycle, repeated n1 times, Fig. 2) and one after ðtG4  bn2 , pumping cycle with variable flip angle bn2, repeated n2 times, Eq. (3)). All example molecules found in literature are in the range of 42.2° 6 hG 6 73.2°. The underlying theory, based on a fictitious spin and B0, was laid out in detail in the original publications and shall not be repeated here [17,20]. Simple equations to calculate the free evolution intervals t Gi , though, as well as the actual intervals for molecules other than HEP were not published, neither in the original or following work. Obtaining the parameters is tedious; one may work through the theory or revert to simulations, e.g. numerically optimizing the intervals in product operator simulations. To facilitate this process, we provide general equations to calculate the free evolution intervals (Eqs. (1) and (2)). This method was derived considering Goldman’s geometric representation of a fictitious spin.

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S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

Fig. 4. Simulated 13C-polarization yield for [1-13C, 2,3-2H]succinate at pH 3 as a function of free-evolution intervals for l-PHINEPT+, Goldman’s and Kadlecek’s SOT: the polarization decreases drastically when wrong intervals are applied. Simulation details: For Goldman’s sequence, tG1 and t G2 were varied, while tG3 was kept constant at optimum value. The analytical (circle) and numerical optimized intervals (asterisk) are indicated.

Using the intervals provided by Eqs. (1)–(3), the simulations yielded a polarization close to unity (>95%) for all regarded molecules, after max. six pumping and two preparation cycles. The polarization yield is independent of whether the inversion pulses are applied on X or 1H. To verify our methods, we compared our results to the intervals and polarization yield reported by Goldman and Johannesson [17]: all published numbers were reproduced (for HEP and a fictitious molecule). Note that the pumping cycles n2  (t4  180°) become relevant for molecules with large and small hG, > 60° and < 30° (Fig. 6, thick and thin dashed blue line, with and without pumping cycles, respectively). Especially for small hG < 10°, tG4 is long (100 ms) and the number of pumping cycles n2 required to reach unity polarization is a few thousand. This is of no concern for the eight molecules regarded here, which are in an intermediate to large range of hG. For simulations spanning the entire range of hG (Fig. 6), though, we decided to apply no more cycles if either P did not increase by more than 1% per cycle, n2 = 10 (Fig. 6, solid black line) or P > 0.87 (Fig. 6, dashed blue line). As a result, the sequence always remains shorter than 1 s (Fig. 6, green line), but the polarization is compromised for small and large hG (Fig. 6, blue line).

t G1

2

a2  b ¼ arccos 2pX 2ab

t G2 ¼

1

1 2pX

t G0;3;4 ¼

arccos

a2 M 2 a2 M 22 þ

! 2 a2  b arccos ! 2pX 2a2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 hG 660 1 3a2  b ! arccos  2pX 2a

aN2 þ bN 1 bN2  aN 1 !

X2 M 21

!

hG 660

1

p 2pX ð1Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 where a ¼ b = J12, X ¼ a2 þ b , hG ¼ arctan ba and N and M depend on the number of preparation cycles (n1) required: jJ13 J 23 j , 2



N1





Re



2

b  a2 þ i  2ab

!ðn1 þ1Þ

¼ in tG1 Im N2 X2 2 3 aN 2 þbN1   M1 2b 4 ¼ M¼   5 in t G2 G 1 M2  bN22aN X  sin 2pX  t 1 N¼

ð2Þ

The flip angle b for the pumping cycles for molecules with hG < 30° is given by:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin ð2hG Þ  cos2ðn2 1Þ ð2hG Þ bðn2 Þ ¼ arctan 1  cos2ðn2 1Þ ð2hG Þ

ð3Þ

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S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

Fig. 5. Simulated polarization yield of l-PHINEPT+ SOT as a function of the molecular J-coupling constants using starting values (Eq. (8), A) and numerically optimized (B) evolution intervals. For an intermediate range of J-coupling constants, the intervals obtained by Eq. (8) are close to optimum (C). Note, J13 was kept constant at 5 Hz.

For molecules with hG < 60°, the phase of the pulse of the pumping cycle alternates:

bðn2 Þ ¼ ð1Þ

ðn2 1Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin ð2hG Þ  cos2ðn2 1Þ ð2hG Þ arctan : 1  cos2ðn2 1Þ ð2hG Þ

Fig. 6. (a) Polarization (thick dashed blue line) obtained by Goldman’s sequence as a function of hG. For large hG, both preparation (n1, solid black line) and pumping cycles (n2, dotted red line) are required, small hG necessitate only the latter to achieve high P. Without these cycles, high P is obtained only for an intermediate hG (thin blue). The duration of the sequence decreases with increasing hG (dash-dot green line). (b) When a 10% error of the J-coupling constants and B1 is assumed (keeping all other parameters constant), the overall polarization yield decreases, especially for large hG. Simulation details: pumping cycles were applied until either P did not increase by more than 1%, n2 = 10 or P > 0.87. Coupling constants J13, J23 of succinate at pH 3, J12 varied to match hG.

Table 1 Intervals, polarization yield and robustness of Goldman’s spin-order transfer sequence with the functional inversion pulses applied to 1H or 13C. The polarization yield under ideal conditions, a ±10% B1-, ±10% J-coupling deviation and the worst combination of both is evaluated. The loss is reduced when the inversion pulses are applied on 1H. The phase of the 180° pulses and sign of the B1 deviation have no effect. If reported in the original publication, the pH is provided. For the abbreviation of molecules, see Fig. 3. Molecule

pH

J13, J23, J12 (Hz)

hG (°)

Goldman Evolution intervals (ms)

P

P(B1 ± 10%), P(J ± 10%), P((J and B1) ± 10%), 180° pulses on 13C

P(B1 ± 10%), P(J ± 10%), P((J and B1) ± 10%), 180° pulses on 1H

–

24, 2.5, 12

hG = 42.2

t G1 ¼ 13:18; t G2 ¼ 21:38; t G3 ¼ 27:97; tot: 62:53

1

0.89, 0.92, 0.82

0.92, 0.92, 0.86

1.8

7.15, 5.84, 7.45

hG = 48.9

t G1 ¼ 27:85; t G2 ¼ 36:67; t G3 ¼ 50:59; tot: 115:11

0.99

0.89, 0.90, 0.80

0.92, 0.90, 0.84

3

7.15, 5.82, 7.41

hG = 48.8

t G1 ¼ 27:87; t G2 ¼ 36:84; t G3 ¼ 50:78; tot: 115:49

0.99

0.89, 0.91, 0.80

0.92, 0.91, 0.84

8.7

6.61, 4.2, 6.62

hG = 50.8

t G1 ¼ 33:96; t G2 ¼ 41:53; t G3 ¼ 58:51; tot: 134

0.98

0.88, 0.89, 0.79

0.91, 0.89, 0.83

13.3

6.32, 4.13, 6.68

hG = 52.0

t G1 ¼ 35:54; t G2 ¼ 41:19; t G3 ¼ 58:96; tot: 135:69

0.97

0.88, 0.88, 0.78

0.90, 0.88, 0.82

HEP [17]

7

7.24, 5.62, 7.57

hG = 49.6

t G1 ¼ 28:28; t G2 ¼ 36:2; tG3 ¼ 50:34; tot: 114:82

0.99

0.89, 0.90, 0.80

0.92, 0.90, 0.83

MEPA 2 [29]

–

15.8, 2.5, 12.6

hG = 54.0

t G1 ¼ 20:8; tG2 ¼ 21:71; tG3 ¼ 32:11; tot: 74:62

0.95

0.86, 0.86, 0.77

0.88, 0.86, 0.79

MEPA 1 [29]

–

10, 1.8, 12.6

hG = 64.9

t G0 ¼ 35:94  1; tG1 ¼ 20:50; t G2 ¼ 32:08; t G3 ¼ 35:94; tG4 ¼ 35:94  2; tot: 196:34

0.96

0.73, 0.62, 0.44

0.79, 0.62, 0.47

TIFBU [30]

–

8.4, 0.8, 12.5

hG = 73.0

t G0 ¼ 38:27  2; tG1 ¼ 20:88; tG2 ¼ 36:18; tG3 ¼ 38:27; t G4 ¼ 38:27  6; tot: 401:49

0.96

0.58, 0.11, 0.01

0.79, 0.11, 0

TMVS [8]

–

15.3, 6.5, 14.6

hG = 73.2

t G0 ¼ 32:79  2; tG1 ¼ 18:10; tG2 ¼ 30:76; t G3 ¼ 32:79; t G4 ¼ 32:79  6; tot: 343:97

0.96

0.58, 0.11, 0.01

0.79, 0.11, 0.01

Table 2 Intervals, polarization yield and robustness of Kadlecek’s SOT as a function of phase and channel of the functional inversion pulses. The polarization yield under ideal conditions, with a ±10% B1 deviation, ±10% J-coupling deviation and the worst combination of both is evaluated. The polarization is independent of the sign of the B1 deviation. Note: Values obtained using 13C pulses around the x-axis are indicated in italic fonts. For the abbreviation of molecules, see Fig. 3. Intervals tK2a and t K2a are applied twice. 1 2 Molecule

pH

J13, J23, J12 (Hz)

hG (°)

Kadlecek Evolution intervals (ms)

P

P(B1 ± 10%), P(J ± 10%), P((J and B1) ± 10%), 180° pulses on 13CY(13CX)

P(B1 ± 10%), P(J ± 10%), P((J and B1) ± 10%), 180° pulses on 1H

BIMAC [29]

–

24, 2.5, 12

hG = 42.2

¼ 5:72; tK2a ¼ 24:13; tot: 59:71 tK2a 1 2

1

0.96 (0.83), 0.94 (0.94), 0.84 (0.78)

0.94, 0.94, 0.88

SUC [24]

1.8

7.15, 5.84 7.45

hG = 48.9

¼ 15:8; tK2a ¼ 46:4; tot: 124:4 tK2a 1 2

1

0.92 (0.82), 0.91 (0.91), 0.76 (0.75)

0.94, 0.91, 0.86

3

7.15, 5.82, 7.41

hG = 48.8

¼ 15:77; t2K2a ¼ 46:55; tot: 124:63 tK2a 1

1

0.92 (0.82), 0.91 (0.91), 0.77 (0.75)

0.94, 0.91, 0.86

8.7

6.61, 4.2, 6.62

hG = 50.8

¼ 20:11; t2K2a ¼ 54:06; tot: 148:34 tK2a 1

1

0.91 (0.82), 0.90 (0.90), 0.74 (0.74)

0.94, 0.90, 0.85

13.3

6.32, 4.13, 6.68

hG = 52.0

¼ 21:52; t2K2a ¼ 54:64; tot: 152:33 tK2a 1

1

0.90 (0.83), 0.89 (0.89), 0.73 (0.74)

0.95, 0.89, 0.84

HEP [17]

7

7.24, 5.62, 7.57

hG = 49.6

¼ 16:34; t2K2a ¼ 46:33; tot: 125:34 tK2a 1

1

0.92 (0.82), 0.90 (0.90), 0.76 (0.75)

0.94, 0.90, 0.85

MEPA 2 [29]

–

15.8, 2.5, 12.6

hG = 54.0

¼ 12:99; t2K2a ¼ 29:81; tot: 85:59 tK2a 1

1

0.89 (0.83), 0.88 (0.88), 0.71 (0.73)

0.95, 0.88, 0.84

MEPA 1 [29]

–

10, 1.8, 12.6

hG = 64.9

¼ 29:81; t2K2a ¼ 29:01; tot: 117:63 tK2a 1

1

0.81 (0.86), 0.78 (0.78), 0.61 (0.67)

0.96, 0.78, 0.74

TIFBU [30]

–

8.4, 0.8, 12.5

hG = 73.0

tK2b ¼ 25:38; t2K2b ¼ 42:79; t4K2b ¼ 28:43  1; tot: 193:2 1

1

0.54 (0.70), 0.50 (0.50), 0.26 (0.31)

0.94, 0.50, 0.42

TMVS [8]

–

15.3, 6.5, 14.6

hG = 73.2

tK2b ¼ 22:42; t2K2b ¼ 36:45; t4K2b ¼ 24:37  1; tot: 166:48 1

1

0.54 (0.70), 0.50 (0.50), 0.26 (0.31)

0.93, 0.50, 0.42

S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

BIMAC [29] SUC [24]

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S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

0.93, 0.95, 0.90

0.91, 0.92, 0.87

0.90, 0.91, 0.87

0.93, 0.94, 0.89

0.90, 0.91, 0.87

0.89, 0.90, 0.86

0.86, 0.87, 0.83

0.88, 0.89, 0.85

0.98

0.95

0.95

0.97

0.95

0.94

0.90

0.93

tlPHþ ¼ 70:84; tlPHþ ¼ 38:61; tot: 109:45 1 2

tlPHþ ¼ 79:75; t lPHþ ¼ 48:31; tot: 128:06 1 2

tlPHþ ¼ 69:84; t lPHþ ¼ 38:96; tot: 108:8 1 2

¼ 34:82; t lPHþ ¼ 29:7; tot: 64:52 t lPHþ 1 2

¼ 43:7; t lPHþ ¼ 46:45; tot: 90:15 t lPHþ 1 2

¼ 46:21; t lPHþ ¼ 58:3; tot: 104:51 t lPHþ 1 2

¼ 34:27; t lPHþ ¼ 23:94; tot: 58:21 t lPHþ 1 2

0.66, 0.59, 0.56

0.93, 0.95, 0.90

0.69

0.98

¼ 25:46; t lPHþ ¼ 60:44; tot: 85:9 t lPHþ 1 2

tlPHþ ¼ 70:63; tlPHþ ¼ 38:52; tot: 109:15 1 2

tlPHþ ¼ 78:69; t lPHþ ¼ 46:69; tot: 125:38 1 2

P(B1 ± 10%), P(J ± 10%), P((J and B1) ± 10%)

Kadlecek and coworkers [19] derived differential equations describing the evolution of the spin system and used a geometric picture similar to Goldman’s, which is mainly different by the choice of axes. The analytical expressions for the evolution intervals for five different cases are given in [19] (dependent on hK, related to hG through hG = 90°  hK). For the sake of consistency, we use hG. Two cases apply for the molecules considered here: Again, a relatively ‘‘simple’’ sequence (case 2a, 35.3° < hG 6 65.7°, Eq. (4), Fig. 2) and an extended version with a repetition cycle (case 2b, hG > 65.7°, Eq. (5), Fig. 2). The sequence for case 2b includes two preparation cycles, which are identical and repeated n3-times until the desired polarization is reached (tK2b 3 , preparation cycle, Fig. 2). For the molecules regarded here, maximal polarization is achieved already at n3 = 1. For the sake of consistency, we rewrite the equations provided by Kadlecek et al. in the nomenclature used above: Case 2a:

t 1K2a ¼

! pffiffiffi 2 sinðhG Þ  cosðhG Þ arccos 1  pffiffiffi 2pX 2 cosðhG Þ  sinð2hG Þ 1

1 2pX

arccos 

cosðhG Þ 2

1 þ sin ðhG Þ

npffiffiffi 2 sinðhG Þ

! o

  pffiffiffi   K2a K2a  1  cos 2pX  t 1 þ 2 sin 2pX  t 1 þ cosðhG Þ

0.84, 0.78, 0.74

0.76, 0.66, 0.63

0.70, 0.57, 0.54

0.68, 0.53, 0.50

0.93, 0.92, 0.88

0.90, 0.88, 0.84

0.93, 0.94, 0.89

0.90, 0.88, 0.84

0.15, 0.01, 0.01

t 2K2a ¼ 0.93, 0.94, 0.89

P(B1 ± 10%), P(J ± 10%), P((J and B1) ± 10%)

Optimized evolution intervals (ms)

Popt

4.2. Intervals and polarization yield of Kadlecek’s SOT

ð4Þ

0.80

0.74

0.88 ¼ 39:22; t lPHþ ¼ 22:94; tot: 62:15 tlPHþ 1 2

0.72 ¼ 45:98; t lPHþ ¼ 27:32; tot: 73:3 tlPHþ 1 2

¼ 54:05; tlPHþ ¼ 42:37; tot: 96:43 tlPHþ 1 2

0.97

¼ 58:48; t lPHþ ¼ 54:35; tot: 112:83 tlPHþ 1 2

0.94 ¼ 84; t lPHþ ¼ 47:85; tot: 131:85 tlPHþ 1 2

¼ 71:43; t lPHþ ¼ 38:88; tot: 110:31 tlPHþ 1 2

0.98

0.94

¼ 71:97; t lPHþ ¼ 38:55; tot: 110:52 tlPHþ 1 2

¼ 83:16; t lPHþ ¼ 46:25; tot: 129:41 tlPHþ 1 2

0.16

0.98

tlPHþ ¼ 39:6; tlPHþ ¼ 18:87; tot: 58:47 1 2

¼ 71:71; t lPHþ ¼ 38:49; tot: 110:2 tlPHþ 1 2

l-PH INEPT+

Starting evolution intervals (ms)

Pstart

Case 2b: 1 t K2b ¼ arccos 1 2pX  1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffi 2 sinðhG Þ þ ð1Þn3  sinð2n3 hG Þ  2  cosðhG Þ2  cosðhG Þ  cosð2n3 hG Þ C B B1  C pffiffiffi @ A 2 cosðhG Þ  sinð2hG Þ t K2b ¼ 2 þ

p 2pX

1 2pX

t K2b 3

¼

arctan

cosðhG Þ  sinð2pX  tK2b 1 Þ sinð2hG Þ  sinðhG Þ þ cosð2pX  tK2b 1 Þ  cosðhG Þ  cosð2hG Þ

!

p

2pX arctanðcscðhG ÞÞ K2b t4 ¼ 2pX ð5Þ

15.3, 6.5, 14.6

15.8, 2.5, 12.6 –

–

7,24, 5,62, 7,57

10, 1.8, 12.6

6.32, 4.13, 6.68 13.3

–

8.4, 0.8, 12.5

4.2, 6.61, 6.62 8.7

–

7.15, 5.82, 7.41 3

–

24, 2.5, 12

7.15, 5.84 7.45

–

1.8

pH

J13, J23, J12 (Hz)

where

a¼

jJ 13  J 23 j ; 2

TMVS [8]

MEPA 1 [29]

TIFBU [30]

MEPA 2 [29]

HEP [17]

BIMAC [29]

SUCC [24]

b ¼ J 12 ;

X¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ b ;

hG ¼ arctan

  b a

and the preparation cycles are repeated n3-times. This number can be obtained by increasing n3 while in Eq. (6) holds true:

hG 6 arctan Molecule

Table 3 Polarization yield and robustness of l-PHINEPT+ obtained using the starting values (Pstart) and optimized (Popt) intervals. The polarization yield under ideal conditions, with a ±10% B1 deviation, ±10% J-coupling deviation and the worst combination of both is evaluated. The polarization is independent of the sign of the B1 deviation. The best solution is printed in bold. For the abbreviation of molecules, see Fig. 3.

32

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2  sinð2hG Þ þ ð1Þðn3 Þ ½cosð2n3 hG Þ  sinð2n3 hG Þ  2sec2 ð2hG Þ  1 pffiffiffi 2 ð6Þ

At ideal conditions, maximum yield is predicted (P = 1) for all hG, and neither channel (1H, X) nor phase (x, y) of the effective inversion pulses affect the polarization.

S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

4.3. Intervals and polarization yield of l-PHINEPT+ In contrast to the previous sequences, PHINEPT+ achieves high X-nuclei polarization with a starting density matrix after hydrogenation given by rweak ¼ bI 1zbI 2z and generates only transverse spin pH order. However, transverse magnetization is short-lived and therefore not suitable for common hyperpolarization experiments, where the transfer to the detection system is of the order of few seconds or metabolic reactions are observed. By adding another 90°-pulse to the end of PHINEPT+, long-lived longitudinal polarization is obtained (l-PHINEPT+, Fig. 2). When the sequence was applied previously, a range rather than specific values were used for the intervals [35]. No polarization yields or expressions to obtain optimal SOT parameters are described. The sequence parameters tlPHþ can be obtained by quani tum mechanical simulations or by maximizing the signal equation (Eq. (7), [8]).



 lPHþ

pJ12 t1

      sin B2ptlPHþ  cos AptlPHþ  sin Bpt lPHþ 2 1 1 4       þ sin A2pt lPHþ  cos BptlPHþ  sin Apt lPHþ ð7Þ 2 1 1

S ¼ I3x

sin

23 23 with A ¼ J13 þJ and B ¼ J13 J 2 2

2 ð2  jJ 12 j þ jJ 13 j þ jJ 23 jÞ 1 ¼ 2  ðjJ 13 j þ jJ 23 jÞ

tlPHþ ¼ 1 tlPHþ 2

ð8Þ

As starting values for the optimization of Eq. (7), we found relatively compact expressions which, for an intermediate range of J-couplings, already yield polarization Pstart within 1% to the optimum Popt (Eq. (8), Fig. 5). Note, since the starting density matrix is different to Goldman’s and Kadlecek’s, hG is not an appropriate parameter to characterize molecules for l-PHINEPT+. In all but one case (molecule BIMAC), P > 90% is obtained by lPHINEPT+. There are no published values to compare to, but we found that maximization of Eq. (7) leads to the same intervals and polarizations as q.m. simulations. 4.4. Robustness of Goldman’ SOT For hG < 60°, i.e. most of the molecules regarded here, Goldman’s sequence is relatively robust (Fig. 6b, dashed blue line). A simulated ±10% error in J-coupling constants or flip angles causes a polarization loss of approx. 10% only. When both effects are combined, the polarization still remains P > 78% (Table 1).

33

Molecules with hG > 60°, though, are rather susceptible to Jcoupling errors. This is the case for TIFBU and TVMS where simulations yielded P < 1% when both errors are present. Interestingly, we find that the susceptibility to flip angle variation is reduced if the functional 180° pulses are applied on 1H instead of 13C (Table 1). The experimental data available [25,26] describes the polarization loss of SUC as a function of B1 deviations. 33 s after the SOT, the polarization was quantified to Pt=33s = 6.5%. This was estimated to a polarization right after the SOT of Pt=0 = 15.4% using a monoexponential T1 decay. The loss of polarization relative to this optimum was measured to be (6–25)% for a (6–20)% deviation of flip angles, which is of the same order of magnitude as the simulated values reported here.

4.5. Robustness of Kadlecek’s SOT Again, experimental errors have a relatively small effect on the polarization of molecules of intermediate 35.3° < hG 6 65.7°. A strong loss, though, is observed for molecules with hG > 65.7°, and in particular for molecules with hG < 14.4° where the polarization is essentially 0 in the presence of B1 and J-coupling errors (Fig. 6b). For the molecules regarded here, the polarization decreases to P  0.26 minimum (Table 2). We found that the polarization loss caused by J-coupling and B1 deviations depends on the channel and phase of the inversion pulses: For hG = 35.3°, 13Cy inversion pulses provide the highest polarization in the presence of J-coupling and B1 errors (Fig. 7, indicated by asterisk). This does not hold true, though, for all hG. In fact, 13 Cy inversion pulses seem to be more sensitive to flip angle variations than 13Cx or 1H inversion pulses (Table 2): 13Cx or 1H inversion pulses provide optimal polarization for 14.4° < hG < 35.3° and hG > 35.3°, respectively. Especially for hG > 65.3°, the advantage of 1 H inversion pulses is significant (Fig. 7).

4.6. Robustness of l-PHINEPT+ While the maximum polarization is a few percent lower than for the other sequences, l-PHINEPT+ is more robust towards B1 and J-coupling errors (Fig. 8, right): A 10% error causes only a 5% decrease in polarization. For the worst combination, the polarization loss is 8%, with the exception of BIMAC (loss = 12%) (Table 3). Contrary to the other sequences, P > 50% is maintained in the presence of errors for all molecules regarded.

Fig. 7. Simulated polarization of Kadlecek’s SOT for different inversion pulses when B1 errors only (a) and together with J-coupling errors (b) are present. By choice of phase and channel (13Cx, 13Cy, 1H), the polarization loss induced by the errors was reduced. Note that the phase of the 1H pulses does not affect the polarization. Without errors, P = 1 is achieved for all hG. The vertical lines separate sequence regimes introduced by Kadlecek, the asterisk indicates case 1.

34

S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

Fig. 8. Polarization as a function of erroneous flip angles and J-coupling constants for Goldman’s, Kadlecek’s and modified Haake’s (l-PHINEPT+) sequence exemplified for the hyperpolarization agent HEP. The effect of both parameters is shown in a 2D representation (top); the effect of the individual parameters is plotted below.

5. Discussion

5.3. Robustness of the sequences

5.1. Sequence intervals

For the molecules regarded, Kadlecek’s sequence is more robust towards B1- and J-coupling errors when compared to Goldman’s sequence. Regarding the full range of hG (for a fictitious molecule based on succinate), though, Goldman’s sequence provides greater robustness compared to Kadlecek’s sequence for small hG (Figs. 6 and 7). Both sequences suffer from strong polarization loss for hG > 80°. Interestingly, the robustness of Kadlecek’s sequence improves when the appropriate inversion pulses are chosen: 1H for hG > 35°; 13Cy for hG = 35° (used in the original sequence [19]); 13 Cx for hG > 35°. l-PHINEPT+ is less susceptible to B1 errors when applied to the eight molecules. B1 errors may be of lesser issue for l-PHINEPT+, as the sequence contains fewer pulses and is usually applied on commercial NMR systems with precise, automatic flip angle calibrations and probes with high r.f. homogeneity.

The analytical expressions presented allow the adaptation of Goldman’s SOT (Eq. (1)) and l-PHINEPT+ (Eq. (8)) to any new hyperpolarization agent. Previously, product operator simulations and optimizations were required to tailor the SOT to a new molecule. The parameters obtained for Goldman’s sequence were validated by simulations and correspond to the values published elsewhere (for HEP and a fictitious succinate incl. pumping phases in [17]). For l-PHINEPT+, the equations presented (Eq. (8)) were derived to provide starting values for numerical optimization of the signal equation given in [8] (Eq. (7)). As it turns out, for some cases, these values are close to the optimum (Fig. 5).

5.2. Polarization yield

5.4. Other considerations

The explicit parameters and polarizations for eight previously used hyperpolarization agents are provided, including the frequently used agents SUC and HEP (Figs. 4 and 8). Again, the values found correspond to those published [17]. The juxtaposition of molecules and sequences allows to choose the optimal combination (Tables 1–3). All sequences efficiently convert the spin order of parahydrogen to longitudinal polarization of >0.9 for molecules of intermediate hG, like succinate and HEP (Figs. 4 and 8). In general, the polarization obtained with Kadlecek’s sequence is unity, slightly surpassing the others, while l-PHINEPT+ is shortest, and most robust. For most molecules, Goldman’s sequence is 10% shorter than Kadlecek’s.

Disregarding experimental errors, all values reported are close to P = 1. This has not been achieved experimentally so far. Reported polarization yields range from much less of 1% to max. 50% [8,20,24,34,36]. The experimental imperfections considered here account for a similar range of a few percent up to one order of magnitude. Other factors contributing to the loss include incomplete hydrogenation, incomplete isotope labeling, non-unity pH2 enrichment and relaxation (e.g. in between SOT and signal detection). Quantification directly after the SOT may elucidate this matter further, as demonstrated recently at a field of 48 mT [37]. How valid are our simulations, i.e. how valid is the product operator formalism employed? Product operators are an established tool for quantum mechanical simulations, but do not account for

S. Bär et al. / Journal of Magnetic Resonance 225 (2012) 25–35

effects like chemical exchange or relaxation. To what extent the assumption of negligible relaxation holds true is hard to assess. Super-operators allow to incorporate relaxation and exchange, but do not help much because the life times at a few mT or of intermediate spin states like KySx [20] during a SOT are generally unknown. We measured the accessible longitudinal and transversal 1 H relaxation of tap water at B0 = 1.8 mT to T1  2.4 s, T 2  8 ms and T2  0.9 s. It is known, though, that singlet states may have a much longer life time, as pointed out by Carravetta et al. [38]. Therefore, the simulated polarization is rather an upper limit, which will be lowered by relaxation. Relaxation is of particular interest for molecules with small and large hG. To obtain high polarization, such molecules require preparation and pumping cycles, which prolong the sequence. During these cycles, relaxation causes polarization loss. One has to consider carefully, whether it is beneficial to add another pumping cycle to the sequence in order to increase polarization by a few percent. For example, when using Goldmans sequence, the polarization for the molecule TIFBU increases with each cycle of t G4 ¼ 38 ms from P = 0.56 to 0.72, 0.82, 0.88, 0.92, 0.94, 0.96, 0.97, 0.98. 6. Conclusion In this contribution, we present the tools to obtain the optimal intervals for Goldman’s and Haake’s sequences to transfer the spin order of parahydrogen to hyperpolarize another nucleus. For eight molecules, previously unknown intervals and polarization yield are provided. These values were confirmed by numerical simulations and, where available, values published elsewhere. Kadlecek’s sequence provided the highest polarization at ideal conditions, slightly surpassing Goldman’s and Haake’s, though at the cost of a prolonged sequence duration in most cases. Overall, the high-field method l-PHINEPT+ proved to be most robust towards simulated B1 and J-coupling errors. For molecules with small and large hG, the low-field sequences suffer from almost complete polarization loss. We found that the robustness of these sequences was improved by choosing the phase and channel of the pulses appropriately. These results overcome the requirement to run quantum mechanical simulations to obtain the intervals for Goldman’s SOT sequence for X nuclei hyperpolarization using parahydrogen. Lowering this hurdle may be an important step for hyperpolarization to leave its niche and to enter its fields of future application. Acknowledgments Jan-Bernd Hövener wishes to thank the Academy of Excellence of the German Research Foundation and the Innovationsfonds Baden-Württemberg. References [1] D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, M. Laval-Jeantet, MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders, Radiology 161 (1986) 401–407. [2] B.R. Rosen, J.W. Belliveau, D. Chien, Perfusion imaging by nuclear magnetic resonance, Magn. Reson. Q 5 (1989) 263–281. [3] J. Pekar, L. Ligeti, Z. Ruttner, R.C. Lyon, T.M. Sinnwell, P. van Gelderen, D. Fiat, C.T. Moonen, A.C. McLaughlin, In vivo measurement of cerebral oxygen consumption and blood flow using 17O magnetic resonance imaging, Magn. Reson. Med. 21 (1991) 313–319. [4] T. Arai, K. Mori, S. Nakao, K. Watanabe, K. Kito, M. Aoki, H. Mori, S. Morikawa, T. Inubushi, In vivo oxygen-17 nuclear magnetic resonance for the estimation of cerebral blood flow and oxygen consumption, Biochem. Biophys. Res. Commun. 179 (1991) 954–961. [5] J. Brossel, A. Kastler, Comptes Rendus 229 (1949) 1213. [6] A. Abragam, M. Goldman, Principles of dynamic nuclear-polarization, Rep. Prog. Phys. 41 (1978) 395–467. [7] M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids, Oxford University Press, Oxford, 1970.

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