1 INTRODUCTION

Pulse sequence design can be challenging due to both a complex theoretical description and hardware limitations. To address these problems, optimization-based approaches have been introduced.1-5 However, many operate by optimizing a unitary transformation, which manipulates the spin system to the desired state, even in the presence of experimental limitations.3, 4, 6 A caveat is that the operator must be known in advance and optimization is restricted to simple design scenarios.6 For the more general case of state-to-state optimization, Khaneja et al.1 introduced an optimal control-based technique, which has found many applications.7-12 The limitation of this approach is that the optimization can get trapped in local minima,1 and the gradient of the cost function can be difficult to calculate. Another constraint is that the time-steps be constant,1 imposing a further restriction on the process. As a potential solution for nonconvex-constrained optimization problems, a genetic algorithm was proposed,13 which is a stochastic solver working with a population of solutions. Convergence to a global optimum is possible with no initial guess and convexity requirements on the cost function. A further advantage is that a cost-function gradient does not need to be calculated, yielding a simple framework that can incorporate any constraint and can be used to compose arbitrarily complex cost functions. Veglia et al2 introduced a genetic algorithm-based approach for optimized spin dynamics, but this optimizes only the phases of a fixed number of pulses and delays between the pulses in order to avoid the need for spin dynamics simulations. Here we present a more general approach, which includes an efficient method for spin dynamics simulations and incorporates frequency selectivity. The optimization process was used here to design improved pulse sequences for transferring polarization from hyperpolarized X nuclei to spin coupled protons with the aim of increasing the sensitivity of detection of hyperpolarized 13C and 15N-labeled metabolites.14-16 Given the reciprocity of this polarization transfer process, we used thermally polarized instead of hyperpolarized metabolites and compared the performance of the optimized pulses to the conventional insensitive nuclei enhanced by polarization transfer (INEPT) pulse sequence17 for transferring polarization from protons to X nuclei in [1-13C]lactate, [2-13C]lactate, and [15N2]urea. The former shows weak J-coupling, for which relaxation losses are mainly responsible for decreased sensitivity. The latter 2 are examples of strong J-coupling where the validity of the product operator formalism, which assumes instantaneous rotations, is compromised; and for [2-13C]lactate, the several ms long adiabatic pulses used in experiments with surface coils in vivo to improve B1 field uniformity cannot fit into the ideal time window of the polarization transfer block. With [15N2]urea, we also demonstrate the feasibility of partial transfer of polarization from the hyperpolarized 15N nucleus to the spin-coupled protons.15 Finally, we design excitation pulses that result in the selection of a single component of the 13C and 15N multiplets in [2-13C]lactate and [15N2]urea respectively, which eliminate J-coupling–associated image artefacts, and compare the results obtained with the solution proposed in Ref. [18].

2 THEORY

The optimization treats the pulse sequence as a shaped pulse with arbitrary amplitude, phase, and duration at each pulse point on each frequency channel. The cost function contains the spin dynamics simulation and a subsequent Bloch simulation to yield the desired frequency selectivity.

2.1 Encoding the solutions

The shaped pulse at each frequency (e.g., 13C and 1H) is discretized to N number of time-steps. During each time-step, every pulse has an amplitude (), duration (), and phase () (Figure 1). For delays, the RF amplitude of the time-step is set to 0. The only restriction is the number of time-steps. The minimum length of the time-steps can be constrained by the time resolution of the waveform-generator and the amplitudes of the RF pulses by the performance of the RF amplifier and transmit coil. Duty cycle constraints can also be incorporated as a penalty term in the cost function. Including the length of the pulse points in the optimization variables increased the flexibility of the algorithm; however, the final pulses had to be adjusted to match the RF raster time, which was 4 µs. This entailed rounding of the time-steps to integer multiples of the raster time and a slight correction in the corresponding amplitude. This had no noticeable effect on pulse performance. Alternatively, the time-steps can be constrained during the optimization to only take values that are integer multiples of the raster time.

Parameters used in pulse sequence optimization (A). The parameters are the amplitude (B1), phase (Φi) and the length (τi) of the individual pulse points. A shaped pulse is played out on the RF channel(s), which steer the initial density operator () to the target density operator (). In a heteronuclear polarization transfer experiment, there are two RF channels: one for proton and one for the X-nucleus (B). Flowchart for calculation of the cost function (C). Φi, phase; τi, length; B1, amplitude 2.2 Smoothness and amplitude constraints on shaped pulses Because RF coils and amplifiers cannot produce arbitrarily high B1 fields or produce an instantaneous change in pulse amplitude, the amplitudes of the shaped pulses were truncated to a user-defined maximum value and then projected onto a set of waveforms with prescribed upper bounds on the first and second derivatives for each iteration of the optimization. The result is a smoothed equivalent, which is the closest to the original input waveform in the function norm defined in Equation 1. This is a convex problem that can be solved efficiently with accelerated proximal gradient descent.19 (1)

This projection does not add a noticeable increase to the run time of the cost function evaluation (Figure 2) and gave smooth changes in spin dynamics with respect to B1 deviations and frequency offsets. This allowed the use of very coarse B1 and frequency offset grids and accelerated considerably solver iterations. The values of α and β were selected empirically and had no noticeable effect on the runtime.

(A) Runtime of the cost function evaluation for 5 coupled spins as a function of the number of RF pulse points. The additional runtime for implementation of the smoothness constraint (B) and the Bloch simulation for enforcing frequency selectivity (C) was negligible in comparison (6–10 ms and 200–400 ms, respectively). The runtime analysis for the Bloch simulation also illustrates the speed of the proposed method for pulse sequence design when there is no spin–spin coupling

2.3 Cost function and spin-dynamics simulations The spin system Hamiltonian contains interaction terms for the nuclear spins with the external magnetic field and with each other. By adding RF pulses, a general effective Hamiltonian can be simulated. In Liouville-space, the equation of motion for the density operator () in the presence of relaxation can be written as20: (2)where is the free-evolution Hamiltonian superoperator; are the RF field Hamiltonian superoperators corresponding to the different frequency channels and coordinate axes (d is the overall number of degrees of freedom, therefore for 2 RF channels ); and is the thermally corrected21, 22 relaxation superoperator. With a discretized shaped pulse, when the Liouvillian of the individual pulse points is time-independent, the evolution of the density operator is given by the N propagator through the N time-steps: (3) The density operator at the end of the pulse sequence is: (4)

The value of the cost function can be obtained by comparing the final density operator, which is the result after the propagator train, to the preferred density operator. The preferred density operator reflects the desired result, for example, that all of the available magnetization is on the proton x-axis.

The pulse sequence has to be robust to the effects of B0 and B1 field inhomogeneities. Therefore, the cost function is evaluated over a range of frequency offsets and B1 amplitudes. The final fitness of the solution is the weighted average of these cost function values for which the weights account for the width of the resonance and the expected B1 variation over this range of frequencies. These weights were assigned according to Gaussian distributions.

2.4 Spin systems The Hamiltonian of the [1-13C]lactate spin-system in the presence of RF irradiation in the doubly rotating frame, with magnetic equivalence of the 3 methyl protons, is: (5)where is the sum of the individual spin operators, and and denote operators for the carbon and proton spins, respectively. The spin system in [2-13C]lactate is the same as for [1-13C]lactate except that the J-couplings and relaxation times differ and the methine proton is on resonance; therefore, the off-resonance term is . In [15N2]uream there are 2 pairs of protons, each pair coupled to one 15N nucleus. Therefore, the Hamiltonian of the spin system in the presence of RF irradiation in the doubly rotating frame is: (6)

The dual-tuned surface coils used enabled simultaneous pulsing on both frequency channels; therefore, the RF pulse terms for both protons and X-nuclei are present in .

2.5 Approximation used in the spin dynamics simulations Simulation of the spin dynamics scales exponentially with the number of spins and quickly becomes computationally intensive and often unfeasible. Numerous approaches have been introduced to tackle this problem.23-28 These build mostly on the inherent symmetries of the spin system and reduce the matrix size by means of state space restriction methods. Simulation of the 5 spins in [1-13C]lactate yields a Liouvillian matrix with dimensions of 1024 × 1024. Exponentiation of such a matrix is computationally expensive. Repeating it at every pulse sequence element for every member of the genetic algorithm population in every iteration makes the optimization unfeasible, even when Krylov propagation29 is used. State space reduction methods are designed to reduce significantly the dimensionality but are more efficient for larger spin systems. Instead, the following approximation was used: (7)

The error of this approximation for noncommuting operators has an upper bound,30 and the discrepancy is small provided that the relaxation rateseffective rotation field. For the optimization process, close approximation is not required if the monotonicity of the cost function is preserved. This means that the location of the minima are preserved, and a minimizer of the transformed cost function is also a minimizer of the original cost function. Splitting of the Liouvillian is especially useful for hyperpolarized experiments, for which the thermal correction in the relaxation superoperator becomes redundant,14, 15, 31 yielding a diagonal matrix when the phenomenological relaxation model is used. Therefore, the approximate spin dynamics can be simulated in Hilbert space, which reduces the matrix size to 32 × 32 and, in the uncorrelated random fields relaxation regime, the exponentiation of the relaxation matrix can be performed by using a matrix composed of the exponentials of the entries. The situation is similar for thermally polarized samples with long T1 times, for which signal recovery during the RF pulse sequence can be neglected. This approximation reduced the computational time by more than a hundredfold. This means that the relaxation superperator had diagonal entries containing only the relaxation rates. Higher-order spin terms were assumed to relax with the sum of the individual relaxation rates. For example, the time required for 1 iteration of the solver for the [1-13C]lactate spin system decreased from > 500 s to ~2–5 s (Figure 2A). The polarization transfer sequences required approximately 1000 solver iterations, resulting in a total runtime of approximately 1 h. The J-coupling artefact-free excitation pulse required approximately 10 min. The convergence criterion was met when the decrease in the cost value was < 1% over the course of 50 solver iterations. If a specific problem needs higher performance, or to test optimality, the solver can be restarted; the output solution can be used as 1 member of the initializer population, which is otherwise a completely random sample. In off-line optimization-based design processes, it is often acceptable to allow several days of run time.6, 32

2.6 Validation of the simulations

The spin dynamics in the cost function of the optimization and the predicted transfer values were simulated with in-house MatLab (version 2020a; Mathworks, Natick, MA) scripts. The peak transfer values were confirmed using SpinDynamica (www.spindynamica.soton.ac.uk) in Wolfram Mathematica (version 11; Wolfram Research, Inc, Champaign, IL). When relaxation losses were estimated, the PhenomenologicalRelaxationSuperoperator function was used with the indicated T1 and T2 relaxation times.

2.7 Achieving frequency selectivity Following the spin dynamics simulation, a subsequent Bloch simulation was inserted to yield the desired frequency selectivity. This is different from the robustness to frequency offset, which aims to yield the same action on the resonance of interest over a range of frequency offsets, whereas the frequency selectivity aims to prevent unwanted excitation of other resonances. This selectivity is achieved by prescribing frequency regions by means of a weight function (W) with ideally 0 excitation, and the deviation from this ideal excitation profile is used to calculate a penalty term (Equation 8) for the final cost according to the norm defined in Equation 1. (8)

This penalty term is indispensable where multiple resonances are of interest, especially with nonrenewable hyperpolarized signals. For example, in an experiment for which exchange of hyperpolarized 13C label between [2-13C]pyruvate and lactate is monitored, the lactate pulse should yield 0 excitation in the neighborhood of the pyruvate resonance in order to preserve the stored polarization in pyruvate. The Bloch simulation increased the total time of the cost function evaluation by 1%–3% (Figure 2C).

The frequency offsets and relative B1 values (relative to the nominal B1 value) used in the optimization process were [−200 Hz, 200 Hz] and [0.1, 3.3], with step-sizes of 100 Hz and 0.8, respectively. The cost function values were weighted according to Gaussians, with a FWHM of 400 Hz for the frequency offsets and a FWHM of 2 for the relative B1 amplitudes.

3 METHODS

Experiments were performed at 7 Tesla (Agilent, Palo Alto, CA) using home-built dual-tuned 2-cm diameter 1H-13C and 1H-15N transmit–receive surface coils. Phantom experiments were performed using thermally polarized 5M [1-13C]lactate, 2M [2-13C]lactate, and 1M [15N2]urea solutions. Spectra were acquired into 1024 complex points with a spectral bandwidth of 4006 Hz. In the polarization transfer experiments, the reference experiment, acquired without polarization transfer, used a 2 ms 90° type-4 B1 -insensitive rotation (BIR4) adiabatic excitation pulse immediately followed by signal acquisition.

3.1 Adiabatic INEPT pulse sequence

The adiabatic INEPT pulse sequence used for proton to carbon polarization transfer in the [1-13C]lactate phantom is described in Ref. [14]. The transfer block employed 4 ms 90° sinc pulses and 10 ms hyperbolic secant (HS8) adiabatic refocusing pulses. The delay between the center of the first and second 90° pulses was 121.95 ms, and the delay between the center of the second 90° pulse and the start of acquisition was 47.78 ms.

3.2 BINEPT pulse sequence

The pulse sequence was implemented as described in Ref. [33]. The transfer block starts with 2 ms BIR4 pulses segmented into 0.5 ms–1 ms–0.5 ms subpulses. The 3 subpulses were separated by two 2.78 ms delays. The segmented pulse was followed by an evolution period of 2.78 ms, with a 2 ms 180° BIR4 pulse in the middle to refocus anti-phase magnetization.

3.3 Partial transfer of polarization from hyperpolarized 15N to protons

The [15N2]urea sample contained 45.9 mg [15N2]urea, 2.31 mg OXO63 trityl radical, 62.8 mg 2H2O, and 55.4 mg glycerol, and 37.5 mg was polarized for 3 hours in a Hypersense DNP Polarizer (Oxford Instruments, Abingdon, UK). Dissolution was performed in 6 mL 2H2O heated to 180°C and pressurized to 10 bar.34 The pulse sequence used for 1H detection of the hyperpolarized [15N2]urea employed a variable power radiofrequency pulses with optimized relaxation delays (VAPOR) water-suppression block in the vendor-provided PRESS sequence. Delays and RF amplitudes were optimized online with the built-in module of the water suppression block. The hyperpolarized sample was injected into a 2 mL Eppendorf-tube next to a thermally polarized 1M [15N2]urea phantom used for center frequency calibration. The dynamic acquisition was started immediately after injection.

3.4 J-coupling modulation-free excitation of thermally polarized [2-13C]lactate and [15N2]urea

Forty-five degree excitation pulses were designed to eliminate J-coupling modulation in [2-13C]lactate (13C doublet peak) and [15N2]urea (15N triplet peak). Spectra were acquired into 1024 complex points with a spectral width of 4006 Hz. The spectra were compared to those obtained using a 100 μs hard pulse with matching flip angle.

3.5 Imaging a [2-13C]lactate phantom

Single-shot 3D non-Cartesian images were acquired using a 1-cm diameter 13C transmit–receive surface coil (Agilent, Palo Alto, CA). Images were acquired using a multi-spin echo pulse sequence.32 The FOV was 3.2 cm with a nominal image resolution of 2 mm. The excitation pulse was either a 2 ms 45° BIR4 pulse or the optimized pulse designed for elimination of J-coupling modulation. The 7 uniformly spaced spin echoes were generated using 12 ms hyperbolic secant (HS8) adiabatic refocusing pulses without slice selection, with a TE of 25.4 ms. The refocusing pulses were phase-compensated.35 Readout band-width was set to 250 kHz. The sampled k-space was gridded to a twofold oversampled 32 × 32 × 32 3D Cartesian grid using a Kaiser-Bessel function with density pre- and postcompensation, followed by 3D inverse Fourier-transform and deapodisation. The same pulse sequence with a 2 ms 90° BIR4 excitation pulse was used for T2 measurements with the readout gradients switched off. At each of the 7 spin echoes, 3157 complex points were acquired; and after Fourier-transform, the 7 peaks were fitted to a monoexponential decay function.

3.6 Dynamic acquisition of J-coupling modulation-free spectra in vivo

A series of 13C spectra were acquired from the brain of a BALB/c nude mouse (Charles River Laboratories, Wilmington, Massachusetts, US) following injection of 300 μL of 80 mM hyperpolarized [2-13C]pyruvate. The acquisition started immediately after the start of injection. The experiment was carried out in compliance with a project and personal licenses issued by the Home Office, UK, and was approved by the Cancer Research UK, Cambridge Institute Animal Welfare and Ethical Review Body. The [2-13C]pyruvate contained 44 mg [2-13C]pyruvic acid (CIL, MA), 15 mM OX063 (GE Healthcare, Chalfont St. Giles, Buckinghamshire, UK), and 1.4 mM gadoterate meglumine (Dotarem; Guerbet, Roissy, France) and was polarized for 1 h. The sample was then rapidly dissolved in 6 mL buffer containing 40 mM HEPES, 94 mM NaOH, and 30 mM NaCl 100 mg/L EDTA heated to 180 °C and pressurized to 10 bar.

4 RESULTS 4.1 Polarization transfer from proton to carbon in [1-13C]lactate

Polarization is transferred from the methyl protons to the 13C nucleus (J = 4.1Hz). However, the methine proton is also coupled to both the methyl protons (J = 7 Hz) and to the carboxyl carbon (J = 3.2 Hz). The simulation was insensitive to the values of the 1H and 13C T1 relaxation times; therefore, published values were used: s (at 7 Tesla),36 s (at 4.7 Tesla).37 T2 relaxation times were measured using a multi-spin echo sequence,32 giving values of s and ms. The target state for the optimization was to place the 13C magnetization in-phase in the x–y plane immediately after polarization transfer. The resultant spectrum was compared to spectra obtained without polarization transfer and using the pulse sequence described in Ref. [14], which employed adiabatic refocusing pulses in a refocused INEPT experiment (Figure 3B). The SNR of the optimized experiment was 1.41 ± 0.19 times higher compared to detection without polarization transfer and 1.29 ± 0.17 times higher than in the INEPT experiment. The simulated SNR improvement of the optimized sequence relative to detection without polarization transfer was 1.42.

(A) Optimized pulses for transfer of 1H polarization to 13C in [1-13C]lactate. (B) Comparison of the spectrum acquired with the optimized sequence to the spectra acquired with the adiabatic insensitive nuclei enhanced by polarization transfer (INEPT) sequence described in Ref. [14] and by detection without polarization transfer (direct 13C detection). The latter spectra are shifted by ±1000 Hz. (C) Optimized pulses for transfer of 1H polarization to 13C in [2-13C]lactate. (D) Comparison of the spectrum acquired with the optimized sequence to that acquired without polarization transfer (direct 13C detection). (E) Optimized pulses for transfer of 1H polarization from the pairs of 1H nuclei to the 15N nuclei in [15N]urea. (F) Comparison of the spectrum acquired with the optimized sequence to that acquired using the B1 -insensitive heteronuclear adiabatic polarization transfer (BINEPT) sequence and by detection without polarization transfer (direct 15N detection). The latter spectra are shifted by ±1000 Hz 4.2 Polarization transfer from proton to carbon in [2-13C]lactate

Polarization is transferred from the methine proton to the 13C nucleus (J = 140 Hz). However, the methyl protons are also coupled to both the methine proton (J = 7 Hz) and to the C2 carbon (J = 4.2 Hz). Because use of the exact relaxation time values is not critical in cases in which the J-coupling constant is large, the simulation was run with the 1H and 13C T2, and the 1H T1 relaxation times used for [1-13C]lactate. The 13C T1 ( was set to 7.2 s.38 The target for the optimization was to place the 13C magnetization in-phase in the x–y plane immediately after polarization transfer. The resultant spectrum was compared with the spectrum obtained by detection without polarization transfer and with the conventional INEPT sequence, which used 1-ms hard pulses (Figure 3D). Adiabatic INEPT sequences could not be used due to the relatively long adiabatic pulses (2–10 ms) when compared to the short interpulse delays in the polarization transfer sequence, which are the result of the large coupling constant between the methine proton and C2 carbon. The SNR of the optimized experiment was 3.81 ± 0.83 times higher compared to detection without polarization transfer and 1.91 ± 0.67 higher compared to the INEPT sequence. The theoretical maximum for the transfer of proton polarization to 13C is , when neglecting the other much weaker couplings and relaxation. The insensitivity of the 13C signal to variation in the 13C and 1H B0 and B1 fields in the optimized experiment was demonstrated by sweeping the 13C and 1H center frequencies and changing the transmitter gains. The results obtained were compared with those obtained using the hard pulse INEPT sequence (Figure 4).

Simulated and measured signal intensity of the 13C resonance in the [2-13C]lactate phantom after transferring polarization from the spin coupled C2 proton. The results are normalized to the equilibrium 13C polarization. Results obtained using the hard pulse INEPT sequence (A,C,E,G) and the optimized pulse (B,D,F,H) are compared. A,B,E and F show simulated profiles and C,D,G and H show those determined experimentally. A–D show the effects of varying the 13C offset and transmitter gain and E–H show the effects of varying the 1H offset and transmitter gain. The data were acquired at 7T with the dual-tuned 1H-13C surface coil

4.3 Polarization transfer from proton to nitrogen in [15N2]urea

Polarization is transferred from the 2 pairs of magnetically equivalent protons to each of the two 15N nuclei (J = −90 Hz). The optimization used the measured values for 1H and 15N relaxation times from Ref. [15], s, ms, s, and s. The SNR of the 15N triplet with the optimized sequence was 1.66 ± 0.20 and 9.96 ± 1.83 times higher than that obtained using the B1-insensitive heteronuclear adiabatic polarization transfer (BINEPT) pulse sequence and detection without polarization transfer, respectively (Figure 3F). The optimized sequence better preserved the multiplet structure, which may be lost in the BINEPT sequence due to anti-phase magnetization resulting from the relatively long adiabatic pulses. The optimized sequence required a more than twofold lower peak B1 compared to the BINEPT sequence.

4.4 Polarization transfer from hyperpolarized nitrogen to protons in [15N2]urea

In order to allow serial acquisition of spectra, polarization has to be transferred from the hyperpolarized 15N nuclei to the coupled protons in discrete packets.15 Simulations showed ΔB0 and ΔB1 robustness comparable to the IRRUPT pulse sequence,15 which is a partial transfer version of the BINEPT sequence (Figure 5A). Signal was observable for up to 24 s (12 repetitions) even though injection of the hyperpolarized urea solution significantly degraded B0 homogeneity (Figure 5C). The required B1 amplitude for this optimized sequence was approximately twofold lower than that needed for the IRRUPT sequence.

(A) Simulated efficiency of partial polarization transfer from 15N to 1H in [15N2]urea, defined as Ptransfered/(1−Ppreserved). (B) Evolution of the 1H and 15N magnetizations. (C) Time course of polarization transfer from hyperpolarized 15N to protons in a [15N2]urea phantom. The broad urea peak is the result of the degraded magnetic field homogeneity caused by injection of the hyperpolarized [15N2]urea solution. The residual water peak is the result of inadequate water suppression

4.5 Pulses immune to the effects of J-coupling in [2-13C]lactate and [15N2]urea

An optimized 13C pulse (Figure 6A) designed to yield a 45° excitation of the upfield peak in the [2-13C]lactate doublet (Figure 6B) and minimal excitation of the pyruvate and pyruvate–hydrate resonances (Figure 6C) took only 9.76 ms. The algorithm finds the shortest pulse that minimizes signal loss but can still perform selection of the singlet component over the prescribed range of resonance frequency offsets. The pulse, which is applied at the center frequency of the 13C doublet, steers the 13C magnetization to a final state, where there is an equal mixture of in-phase and anti-phase 13C magnetization. A better selectivity was observed between the frequency offsets of interest at +J/2 = 70 Hz and −J/2 = −70 Hz when compared with a 32 ms hard pulse and a 10 ms sinc pulse applied at the resonance frequency of 1 of the components of the doublet (+J/2 = 70 Hz), as described in Ref. [18] (Figure 7). A similar excitation pulse was designed for J-coupling artefact-free imaging of the triplet resonance from [15N2] urea. The analytical derivation is complicated even for a system of 2 coupled spins18 and does not hold for a system of 3 or more spins. For the numerical optimization approach, it is sufficient simply to define the desired state of the spin system immediately after excitation as that containing a singlet peak. This is a mixture of in-phase (e.g., etc.) and certain anti-phase (e.g., , etc.) magnetization terms so that the other components of the multiplet cancel. This is trivial to find and constitutes the input for the optimization. For example, the desired state is for the 13C doublet case and for the 15N triplet. The coefficients are arbitrary real numbers such that obeys density matrix normalization rules. They also indicate that the final phase of the spectrum relative to the receiver is unimportant, but it needs to be same for the in-phase and anti-phase terms so that the respective components of the multiplets cancel. It is important to note here that the anti-phase terms relax according to the sum of the relaxation rates of the individual spin operator components. The optimization used the T1 and T2 values for [2-13C]lactate and [15N2]urea given above. The optimized excitation pulse for [15N2]urea and the resultant singlet spectrum are shown in Figure 6D,E, respectively. The pulse was longer, at 13.84 ms, reflecting the smaller J-coupling and therefore slower spin evolution. The 13C pulse was used for single-shot 3D imaging of the lactate phantom using a multi-spin echo readout.32 The image acquired with the optimized excitation pulse showed only readout point spread function-associated artefacts, whereas the image acquired using a BIR4 excitation pulse introduced blurring and image distortion originating from J-coupling dependent modulation of the 13C signals (Figure 8). A series of 13C spectra acquired from the rat brain following intravenous injection of hyperpolarized [2-13C]pyruvate using the same optimized pulse contained only the upfield peak of the [2-13C]lactate doublet, which was observable for up to 40 s. The spectra showed no detectable pyruvate or pyruvate-hydrate signal (Figure 9).

(A) An optimized 45° 13C excitation pulse for artefact-free imaging of [2-13C] lactate (J = 140 Hz). (B) The resultant spectrum compared with that acquired with an equivalent hard pulse. The latter was shifted by 1000 Hz. (C) The Bloch-simulated 13C z-magnetization immediately after the pulse. The dashed lines indicate the ±200 Hz region containing the [2-13C]pyruvate resonance, where the pulse gave minimal excitation. (D) An optimized excitation pulse for [15N2]urea. (E) The resultant spectrum compared with that acquired with an equivalent hard pulse, which showed the proton-coupled triplet. (F) The Bloch-simulated 15N z-magnetization immediately after the pulse

(A) Simulated frequency profile of the 9.76 ms optimized RF pulse designed to give J-coupling artefact-free excitation of [2-13C]lactate when the pulse is applied exactly on-resonance at the center frequency of the 13C doublet. The profile is compared to that of a +J/2 = 70 Hz off-resonant 32 ms hard pulse and a 10 ms sinc pulse. (B) The same profiles measured experimentally. (C) 13C spectra acquired from [2-13C]lactate with the optimized pulse at different offset values of the doublet center frequency. (D) 13C spectra acquired from [2-13C]lactate with the 10-ms sinc pulse applied at +70 Hz from different offset values of the doublet center frequency

Single-shot 3D imaging of a [2-13C]lactate phantom. (A) and (D) are central sagittal and axial 1H images respectively acquired using a fast spin echo sequence. (B), (E) The corresponding 13C images acquired with the pulse sequence described in Ref. [