Rapid, $B_1$-insensitive, dual-band quasi-adiabatic saturation transfer with optimal control for complete quantification of myocardial ATP flux

Purpose: Phosphorus saturation-transfer experiments can quantify metabolic fluxes non-invasively. Typically, the forward flux through the creatine-kinase reaction is investigated by observing the decrease in phosphocreatine (PCr) after saturation of $\gamma$-ATP. The quantification of total ATP utilisation is currently under-explored, as it requires simultaneous saturation of inorganic phosphate (Pi) and PCr. This is challenging, as currently available saturation pulses reduce the already-low $\gamma$-ATP signal present. Methods: Using a hybrid optimal-control and Shinnar-Le-Roux method, a quasi-adiabatic RF pulse was designed for the dual-saturation of PCr and Pi to enable determination of total ATP utilisation. The pulses were evaluated in Bloch equation simulations, compared with a conventional hard-cosine DANTE saturation sequence, before application to perfused rat hearts at 11.7 Tesla. Results: The quasi-adiabatic pulse was insensitive to a $>2.5$-fold variation in $B_1$, producing equivalent saturation with a 53% reduction in delivered pulse power and a 33-fold reduction in spillover at the minimum effective $B_1$. This enabled the complete quantification of the synthesis and degradation fluxes for ATP in 30-45 minutes in the perfused rat heart. While the net synthesis flux ($3.3\pm0.4$ mM/s, SEM) was not significantly different from degradation flux ($8\pm2$ mM/s) and both measures are consistent with prior work, nonlinear error analysis highlights uncertainties in the Pi-to-ATP measurement that may explain the possible imbalance. Conclusion: This work demonstrates a novel quasi-adiabatic dual-saturation RF pulse with significantly improved performance that can be used to measure ATP turnover in the heart in vivo.


Introduction
The heart is a metabolic omnivore with the greatest energy requirement, per gram of tissue, of any organ in the body, in order to power contraction and maintain cellular membrane potentials. This requirement is met by sophisticated metabolic machinery, culminating in a high rate of adenosine triphosphate (ATP) hydrolysis and regeneration that is mediated through the "energy buffer" and subcellular transportation system that shuttles phosphocreatine (PCr), a moiety that forms the primary energy reserve of the heart [1,2], through the enzyme creatine kinase (CK) to regenerate ATP. This is represented by the coupled reactions: where k f and k r represent the pseudo first-order forward and reverse reaction rates through the CK reaction, and k f and k r represent the effective reaction rates for the conversion of ATP to adenosine diphosphate (ADP) and inorganic phosphate (Pi). Together these coupled reactions determine how much ATP is synthesised and utilised in the heart. This reaction scheme (1) has been extensively studied in the heart using Phosphorus Magnetic Resonance Spectroscopy ( 31 P-MRS) saturation transfer approaches.
Such methods permit the quantification of the rate constants for the CK reaction itself and the net rate of ATP hydrolysis and synthesis as a whole. [3] These rates are typically determined by saturating or inverting spins of one/or more exchange partner/s, e.g., γ-ATP, for a period of time, and observing the subsequent depletion in signal of the other exchange partner, e.g., PCr. If metabolite concentrations are known or measured, the total metabolic flux through any given part of the system can be calculated.
Metabolic dysregulation plays a key role in common heart diseases, [4]and the forward CK rate constant k f has been associated with cardiac metabolic health, with reductions in k f reported in conditions ranging from obesity [5,6] to heart failure and myocardial infarction. [7,8] Thus far, this is the only rate constant currently routinely measurable in the human heart in vivo. The direct quantification of both sets of forward rate constants (Eq. (1)) is challenging in clinical settings, due to the inability to reliably quantify the myocardial Pi signal in the presence of overlapping signals from diphosphogylcerate compounds originating in the blood pool for the forward saturation transfer experiment; and the typically low signal-to-noise ratio (SNR) of Pi achievable in healthy tissue in clinically feasible scan times.
The reverse experiment is much more complex, as it requires the simultaneous saturation of PCr and Pi for varying durations. The determination of the total forward and reverse flux of ATP are obtained by fitting the acquired signals to solutions of the Bloch-McConnell equations, [3,9]. Therefore, this experiment is rarely if ever performed in humans. Whilst the SNR of γ-ATP is typically worse than that of PCr, the dual-saturation experiment has the potential to report on the biologically-key energetic process: the net turnover of the whole cellular ATP pool, beyond that involved in the CK flux reaction alone.
Additionally, k f and the forward CK reaction is technically easier to perform than the k f /k r experiment requiring only narrow single-band saturation pulses with a sharp stop-band. Narrow bandwidth pulses can be made longer and tailored to better minimise Gibbs ringing than the larger-bandwidth shorter pulses that are required to saturate both PCr and Pi simultaneously. Furthermore, the latter are typically associated with higher peak B 1 fields, increased power requirements for the non-proton RF amplifier and higher rates of RF specific absorption in the body (i.e. higher SAR).  [10] Attempting to design "perfect" saturation pulses that would avoid the need for control experiments is therefore desirable for both single-band and dual-band saturation transfer experiments. Furthermore, at high fields in the complex electrodynamic environment of the heart and in clinical research employing 31 P-MRS surface coils, B 1 is not uniform. It is therefore desirable to design adiabaticity into the RF pulse to provide tolerance to B 1 inhomogeneity to aid its application.
In this work, we demonstrate a hybrid optimal-control RF pulse design scheme, with alterations to the alpha and beta polynomials after Shinnar-Le Roux (SLR) transformation that optimise the passband and stop-band homogeneity, while achieving low sensitivity to nonuniformity in the RF excitation field, B 1 . The novelty of this design approach is that it combines the accuracy of computationally-challenging optimal-control methods which excel at constrained problems, with the speed and analytic underpinning of the SLR approach. Specifically, we present a highly uniform and temporally long dual-band saturation pulse that simultaneously saturates PCr and Pi, and eliminates the need to obtain control data within each transfer experiment. Whilst other adiabatic saturation pulses have been proposed in the context of 31 P-MRS for applications including the CK-flux measurement (e.g. [9]), we believe that this approach is the first multi-frequency selective quasi-adiabatic pulse designed for saturation transfer. We demonstrate the new pulse in a saturation transfer experiment on the Langendorff retrograde perfused rat heart at 11.7 T, to measure the energetic status of the heart as a whole. This set-up is of particular value in drug studies on disease models, where the workload of the heart, oxygenation status, and pressure afterload can be independently controlled rapidly, repeatably, and with or without pharmacological alteration.

Theory Pulse design
The extreme requirements of uniform saturation, low off-resonance spillover, and a relative degree of insensitivity to B 1 pose a set of unique challenges to the design of multi-frequency saturation RF pulses.
In comparison to low flip-angle RF pulse design, the problem of designing efficient saturation pulses is complicated by the fact that the RF is not purely the Fourier transform of the desired excitation profile. A number of methods have been proposed to function in this regime. Most famous is the SLR approach, [11] which relies on an analytic transformation of the RF excitation into a domain which permits phrasing of the design problem as matrix algebra, which can be solved with recursive and computationally efficient filter design algorithms such as Parks-McClellan. [12] The SLR algorithm reduces the problem of pulse design to that of finding two polynomials, A(z) and B(z) where z = e −iω , which may be reversibly transformed to find an RF pulse [13]. It is also possible to transform between the coefficients of the A and B polynomials and their equivalent frequency representation via a z-transformation, which is related to the discrete Fourier transformation. The latter, in turn can be Although SLR neglects relaxation terms and, conventionally, considerations relating to B 1 homogeneity, it has been widely adopted and remains a "gold standard" for RF pulse design. It has been shown to be possible to expand the SLR design framework to generate adiabatic pulses, that is, pulses that are insensitive to a range of B 1 values above a certain threshold. This is desirable, both for the heart at ultrahigh fields and for in vivo studies employing surface coils.
Briefly, [14] the process functions by manually imposing a quadratic phase in the B(ω) frequency domain, modulating the linear phase generated by a filter designed with the specified single-band pass/stop-band requirements. The A(ω) polynomial is designed under a least-phase constraint, to yield a minimum-energy (least SAR) pulse with a quadratic phase variation. Not all quadratic phase RF pulses are adiabatic, but those with a resulting frequency ramp that is slow enough to satisfy the adiabatic condition will be so. It remains uncertain how to directly specify multiple frequencies for excitation or saturation within this framework, as it requires root-flipping procedures to minimise the peak amplitude of a fixed duration pulse, or equivalently, reducing its duration with a fixed peak amplitude. This process involves replacing the selected roots of its A(z) and/or B(z) polynomials with the inverses of their complex conjugates. The best pattern of flipped roots produces the most uniform distribution of RF energy in time, imitating a quadratic phase pulse. [15] Owing to its non-linear nature, the process of root flipping is not directly compatible with an imposition of a complex phase in the B(ω) domain.
An alternative framework for designing RF pulses in MR based on optimal-control considers the Bloch equations without exchange, and minimises an integrated metric for the difference between their current and the desired solution. Typically this is performed with a cost term based on the square of the RF pulse amplitude incorporated as an SAR penalty. Under these conditions, the design problem falls into the analytical framework of linear quadratic control. Assuming one spatial degree of freedom -a slice selection gradient along a slice axis z) -the Bloch equations in the rotating frame can be written as given where u(t) = u x (t), u y (t) describes the normalised, non-dimensional RF pulse and the linearised Bloch rotation matrix A and longitudinal relaxation vector b are given by with G z (t) an applied slice-selection gradient. [16] Within the framework of optimal-control, we seek an RF pulse u(t) of a pre-specified duration T u that minimises a chosen cost function after it is played (at t > T u for time T = T u + where 1) to generate a numerically-determined solution M (T, z) as compared to a desired predefined solution, M d (z), within a finite computational domain (z ∈ [−a, a]). Including the quadratic cost function to accommodate the practical limitations on RF amplifier power and SAR limitations, the one-dimensional pulse design problem is Here λ effectively acts as a regularisation term that relates the competing goals of pulse fidelity and SAR reduction. This minimisation problem is then typically approached via numerical methods. However, optimal-control algorithms remain far more computationally expensive than the matrix approach used by SLR transformation, and convergence is slow with most naïve gradient-descent or quasi-Gauss Newton approaches.
Whilst Eq. (5) is modifiable to problems beyond one-dimension, the extension of this method into ≥ 2 dimensions becomes challenging in practice. [17] The already-slow convergence can become unacceptable with increasing dimensionality, for example, by modifying Eq. (4) to include off-resonance effects or chemical exchange, or Eq. (5) to include information about B 1 variation. Whilst examples of successful applications exist [18], multidimensional optimal-control methods generally present as highly computationally expensive minimisation problems that have limited their application in the design of adiabatic RF pulses that require tradeoffs between frequency selectivity, adiabaticity and pulse power. [19] For the one-dimensional RF pulse design problem, Aigner et al. [16] recently showed that the analytic calculation of a second-order Hessian matrix of partial derivatives acting at a location in a direction can permit the development of a globally convergent trust-region conjugate-gradient Newton method with quadratic convergence, to yield simultaneous multislice imaging pulses with excellent slice profiles. We note that with ideal gradient waveforms, the analytical treatment of simultaneous multislice excitation is directly analogous to that of the multi-band saturation needed for PCr and Pi saturation in the present application. Furthermore, an ability to excite PCr and Pi in antiphase analogous to a CAIPIRINHA-based simultaneous multislice excitation, [20], could provide a simple alternate phase-cycling scheme to improve saturation efficiency.

Pulse design
This work combines the SLR and optimal-control approaches described above to provide dual-band saturation with minimal ripple for simultaneous PCr/Pi saturation. The new pulses are given a degree of adiabaticity by the imposition of a quadratic phase in the A, B polynomial domain. The general approach can be summarised as 1. Highly optimised optimal-control pulses are developed subject to hardware constraints, using the analytically enhanced optimal-control framework [16]. The initial condition is either an appropriate SNEEZE RF pulse [21,22] for single-band saturation, or a hard-cosine pulse for dual-band saturation.
Appropriate T 1 and T 2 values are included explicitly in this framework for each metabolite, as the optimal-control framework explicitly includes relaxation in pulse design. The result is a non-adiabatic pulse u 0 (t).
2. Obtain the resulting A and B polynomials for each pulse, that is ( 3. Obtain the discrete Fourier transform of the coefficients b of the B polynomial, which would be equal to the linear phase profile of the pulse traditionally generated via filter-design methods in the conventional SLR approach, say F(b) = F filter (ω).
4. Impose a quadratic phase in the frequency domain: that determines the rate of quadratic phase cycling across, described in detail elsewhere. [14] 5. Inverse the process to obtain an updated set of B polynomial coefficients: 6. Obtain the optimised waveform with a degree of adiabaticity imposed, This approach is computationally quick, and produces a far simpler optimisation problem than using a full optimal-control framework for the design of adiabatic pulses with multiple excitation bands. The imposition of quadratic phase on the B polynomial does not affect the spectral bandwidth(s) of the pulse, but increases its overall duration and decreases the effective peak B 1 value.
We used the above protocol to design a novel quasi-adiabatic dual-band saturation RF pulse for the 31 P saturation transfer experiment. This was a dual-band excitation with a 950 Hz gap between saturation bands, a 150 Hz FWHM of each pole. The pulse was 25 ms in length comprised of 2500 points, and k = 4.1 × 10 −6 , with a nominal B 1 of 1.3 µT. The optimal-control algorithm was initiated with a hard-cosine pulse and a target magnetisation consisting of two slabs filtered with a 2.5 Hz transition width convolved with a Gaussian kernel. The optimal-control methods were otherwise as described previously, [16] with a maximum relative passband excitation, that is, at 0 Hz, of 10 −7 . The simulation was allowed to converge before the manual imposition of quadratic phase, which took about 4 hours on a MacBookPro15,3 laptop computer (2.9 GHz Intel Core i9 processor). The resulting pulse is shown in Figure 1A, together with its spatial and spectral response after a 90°pulse with subsequent crusher ( Figure 1B) and implementation into a saturate/crush/excite sequence ( Figure 1C).
Following design, the immunity of the pulse to B 1 variation was quantified via Bloch equation simulations, and the saturation efficiency compared to conventional DANTE-based trains [23] with hard-cosine pulses to produce dual-band excitation at the requisite frequencies. Specifically, a 19 ms hard-cosine pulse with a 500 µs nutation frequency was chosen to provide an appropriate control, in keeping with previous work. [3] In order to produce saturation, as with the new dual-band pulses, this was repeated numerous times back-to-back in order to produce the DANTE chain. The total duration of saturation required is determined by the T 1 of the molecule in question and typically 3 − 5 × T 1 is necessary for adequate saturation. Therefore, for subsequent relative power calculations determined from the integral of B 2 1 , the total saturation duration was kept constant, and B 1 set to the minimum B 1 to produce effective saturation.

Experimental methods
All experiments were performed on a vertical-bore 11.7 T spectrometer (Magnex Scientific / Varian DDR2) at a constant 37.0 ± 0.1°C, with temperature maintained by blowing heated air from a Bruker

MR Protocol
The measurement protocol for assessing total myocardial energetics was comprised of: (i) the acquisition of localisers; (ii) shimming; (iii) a 31 P spectrum based Ernst-angle frequency adjustment; (iv) acquisition of a 5-minute fully-relaxed 31 P spectrum; (v) single-band and dual-band saturation-transfer experiments; and (vi) acquisition of three one-minute fully-relaxed spectra to obtain the volume of the internal PPA phantom to use as a concentration reference (see below) and for the fully-relaxed metabolite signals at the end of the perfusion experiment. The entire protocol took 30-45 minutes, including preparation and insertion of the perfused heart into the MR system.
The saturation pulse was developed as a "module" that preceded a simple hard-pulse-acquire spectroscopic readout. This was applied (protocol step v) under fully-relaxed conditions (TR 10 s, 16 averages, 90°flip angle, 10 kHz bandwidth, 2048 complex points) to enable absolute quantification based on the internal PPA phantom and volume, and the weight of the heart. We note that the presence of the phantom inside the LV ensures that the phantom signal is detected with substantially the same sensitivity as that of the myocardium. Trains of either single-band or dual-band saturation pulses of total duration [0, 0.15, 0.275, 0.575, 2.25, 4.575] s, were applied, followed by all-axes gradient crusher pulses, the hard excitation, and FID readout. After acquisition of fully-relaxed, and selectively saturated spectra, absolute quantification (protocol step vi) was performed by adding two known volumes of PPA (typically ∼ 50 µl) to the internal PPA phantom and obtaining two separate fully-relaxed spectra. This process accounts for variability in heart sizes, and hence LV balloon volumes since the balloon is inflated to a diastolic pressure that ensures retrograde perfusion via the coronary sinus and arteries during diastole.

Data Processing
Spectra were fitted in the time domain via the OXSA implementation of the AMARES algorithm [24] given prior knowledge of the expected location of the peaks of phenyl-phosphoric acid, intracellular and extracellular phosphate, phosphocreatine, the two phosphodiesters glycerophosphocholine (GPC) and glycerophosphoethanolamine (GPE), nicotinamide adenine dinucleotide (NAD/NADH; modelled as a single peak), and ATP, including J-coupling. A linear correction factor was determined from the fully-relaxed spectra with different PPA volumes in protocol step (vi) to obtain the volume of the 100 mM PPA present during the saturation transfer experiments (step v). Absolute metabolite concentrations were calculated using an intracellular volume fraction for healthy myocardium of 52% and tissue specific gravity of ∼ 1.05 g/ml. [25] Saturation transfer data were fitted to integrated solutions of the Bloch-McConnell equations via a bounded nonlinear least-squares method, in the form where τ = 1 T 1 + k; and where k = k f if Y is the the amplitude of PCr, k f of Pi, or (k r + k r ) if it is γ-ATP following dual saturation. For this, a global optimisation algorithm was used with the linear constraint that τ − k ≥ 0 and all parameters ≥ 0. [26] All values reported are mean ± SEM.

Results
Bloch equation simulations revealed that the designed dual-band quasi-adiabatic pulse demonstrates a far greater immunity to B 1 variation, with the optimised dual-band pulse featuring a dramatic increase in immunity to variation compared to the conventional hard-cosine pulses when used either for 90°excitation ( Figure 2A, B) or in a DANTE chain (C, D). As pulse power increases further, the behaviour of the new pulse remains benign, with minor variations in the total effective saturation occurring within the designed saturation bandwidth, rather than between and beyond them. This reflects on the optimisation process which provides both very flat passbands and immunity to B 1 variation. The pulse frequency modulation (∂φ/∂t) has three linear but interleaved frequency ramps with constant offsets, characteristic of adiabatic pulses. [14] Compared to an equivalent hard-cosine DANTE chain saturating both PCr and Pi with > 99% saturation efficiency on both resonances, the pulse required a 70% higher peak B 1 (0.32 µT vs 0.55 µT) but possessed a B 1 rms that was 50% lower: 0.0516 µT for the hard-cosine vs 0.0275 µT. This corresponded to a lower integrated power deposition (P ∝ Tu 0 |B 1 (t)| 2 dt) over the duration of the longest DANTE saturation chain considered, i.e. 4.58 s: the deposited power of the new pulse is just 53.4% of that for the hard-cosine pulse. Similarly, at the frequency offset corresponding to γ-ATP (i.e. ∼ −1000 Hz compared to the mid-point of Pi and PCr), the designed pulse featured a relative excitation of ∼ 10 −9 in comparison to 10 −3 for a conventional hard-cosine pulse. This improvement in selectivity, by six orders of magnitude, translates into a dramatic reduction in the degree of spill-over following saturation by a DANTE chain. At the frequency of γ-ATP, a 10% erroneous saturation occurs if the hard-cosine pulse is played at or above a peak B 1 of 0.63 µT, whereas the designed novel pulse does not reach that point until peak B 1 exceeds 4.2 µT. This therefore gives it a greatly expanded range in which B 1 variation does not significantly affect the γ-ATP signal; the saturation remains ≤ 1% until peak B 1 > 1.5 µT, yielding a ≥ 2.5-fold effective immunity to B 1 variation. When played at the minimum effective B 1 , spillover at γ-ATP is therefore ∼ 0.03% compared to 1% for the hard-cosine pulse, 33-fold lower.
The predicted frequency-domain response was verified with the PPA phantom ( Figure 3A). A control experiment performed by shifting the frequency of the saturation pulse shows ( Figure 3B) excellent agreement between the predicted experimental pulse behaviour.
Single-band and dual-band saturation experiments performed on n = 4 naïve Wistar rat hearts show that that no measurable saturation correction is needed when the pulse is applied off-resonance, as illustrated in Figure 3C.
The new saturation transfer protocol also permitted the absolute quantification of all major phosphorus compounds of interest. Multiplying the reaction rates by the corresponding concentrations, yielded the total ATP synthesis and degradation fluxes for the heart in approximately 30-45 minutes, a duration compatible with perfused heart experiments. The mean ATP concentration was 7.65 ± 0.50 mM and the mean PCr concentration was 11.00 ± 0.54 mM ( Table 1). The synthetic rate constants k f and k f were 0.31 ± 0.04 s −1 and 0.28 ± 0.08 s −1 , respectively with a degradative rate of (k r +k r ) = 1.07 ± 0.07 s −1 . The corresponding synthesis and degradation fluxes for ATP were thus 3.32 ± 0.39 mM/s and 8.05 ± 1.80 mM/s. Heart function did not decrease significantly throughout the duration of perfusion.

Discussion
The assessment of the rate constants of ATP synthesis and degradation have a long history in 31 P spectroscopy. While the use of saturation transfer experiments can provide quantitative measurements of k f , k f and k r and k r , it is the forward CK flux reaction k f that has seen most attention in the heart, in part owing to the poor SNR of cardiac 31 P-MRS wherein PCr, having the highest SNR, provides the most accessible readout in the saturation transfer experiment in clinical settings. Nevertheless, alterations in cardiac k f have been shown to be present in both humans and animals, in diseases such as diabetes, [27] heart failure, [28] and in myocardial infarction. [7] However, the role of the reverse synthesis flux, (k r + k r ) has not been studied in humans or extensively in experimental heart models, in part because of the challenging nature of the experiment.
Here we have demonstrated the development of a quasi-adiabatic dual-band saturation pulse that is able to efficiently and simultaneously saturate both Pi and PCr with a ≥ 2.5-fold immunity to B 1 at ultrahigh field, and hence permit the determination of (k r + k r ). The constraint on B 2 1 applied in the optimal-control scheme used for designing pulses, reduces SAR for practical pulses that can be used in animal models and adapted for human studies at 3 T and 7 T systems within the same framework. Compared to an optimised single-band saturation pulse broadly saturating both PCr and Pi, the dual-band pulse is significantly longer with a lower B 1 rms , sharper transition bands, and minimised off-resonance spill-over irradiation.
The use of dual-band saturation permitted the rapid quantification of the total ATP degradation flux from a fully-relaxed 31 P spectrum, and one additional saturation transfer experiment. The increased temporal resolution potentially afforded by the omission of the control saturation data allows for dynamic experiments and extends the viability of the perfused heart model, wherein cardiac performance slowly deteriorates after a ∼ 2 hour survival window. We note that the designed quasi-adiabatic multiple-band saturation or excitation pulses could be incorporated into more widely used saturation schemes, such as FAST, [29] TRiST, [2] STReST, [30] and TWiST, [31], and find other applications beyond 31 P MRS.
Examples could include simultaneous multi-slice imaging with a degree of immunity to B 1 variation; the selective excitation of metabolites in hyperpolarised metabolic imaging; and spatial tagging of multiple arteries or planes in the arterial spin labelling experiment.
The results of the experimental studies are in agreement with previously-reported values of k f : our 0.30 ± 0.04 s −1 compares to 0.35 ± 0.06 s −1 in the in vivo control rat heart [27] and 0.32 ± 0.05 s −1 in the healthy human heart [9]. For concentrations, we have [PCr] = 11.03 ± 0.54 mM vs. 11.50 ± 1.01 mM [25] and [ATP] = 7.65 ± 0.51 mM vs. 6.75 ± 0.50 mM in perfused rat heart. Similarly, our reverse rates of (k r + k r ) = 1.07 ± 0.25 s −1 agree with a value of 1.15 ± 0.19 s −1 measured by Spencer et al. using hard cosine saturation pulses. [3] Nevertheless, the mean value of the total flux of ATP synthesis determined with conventional single-band saturation transfer experiment is roughly a half that of the degradation flux as measured by the dual-band experiment. Although the difference is not statistically significant, this result is troublesome. If it were real, it would suggest that the heart was in the process of dying and [ATP] would not have been detectable in the three PPA calibration scans at the end of the experiment. In the steady-state, it is expected that these fluxes remain balanced up until the point of death, consistent with the common observation that [ATP] is effectively constant in vivo and in situ, although a gradual decline with time is seen in Langendorff preparations. It is possible that the inequality observed in mean flux may result from nonlinear error propagation in both saturation transfer schemes. In particular, whilst k f can be determined accurately owing to the larger size of the PCr peak, the determination of k f is much more challenging owing to the smaller size of the intracellular Pi peak. The addition of phosphate buffer to the perfusate increases the size of this peak via the slow intracellular import of phosphate making it easier to measure, but not fundamentally changing the kinetics through ATP-synthase. Consequently, although our determined with σ k f etc for the rate constants determined by a curve-fitting regime, then Taylor expansion for error propagation for the synthesis (Synth) and degradation (Deg.) flux shows, given Numerically, the estimated uncertainty in Eq. (7) is significantly larger than that in Eq.

Conclusion
This work demonstrates a novel approach for designing and an application of a quasi-adiabatic dualfrequency saturation pulse, suitable for measuring forward and reverse metabolic reaction rates. The design employs a hybrid optimal-control and SLR transformation to achieve sharp frequency selection and a degree of B 1 insensitivity suitable for ultrahigh field NMR in experimental animals and the human heart at ≥ 7 T. The resultant pulse was exceptionally flat, with no experimentally detectable "spillover". It was incorporated into a protocol to assess the complete high-energy phosphate compounds and energetic status in the perfused heart at 11.7 T, providing measures in good agreement with the literature. We hope this technology will be useful in future studies intent on quantifying the total flux of ATP turnover in the heart.   A comparison of the designed quasi-adiabatic optimal control pulse when compared to the conventional hard-cosine approach. After a single excitation is played (A, B) the designed pulse exhibits a significantly decreased degree of erroneous excitation outside of the designed spectral passband as shown by the base-ten logarithm of transverse magnetisation. As a consequence, when integrated into a DANTE saturation chain, the novel pulse is effectively immune to plausible variation in B 1 , in stark comparison to a hard-cosine pulse (C, D), which functions as designed within a comparatively small window of acceptable B 1 given a desire to minimise unwanted saturation of γ-ATP.   Figure 2: A comparison of the designed quasi-adiabatic optimal control pulse when compared to the conventional hard-cosine approach. After a single excitation is played (A, B) the designed pulse exhibits a significantly decreased degree of erroneous excitation outside of the designed spectral passband as shown by the base-ten logarithm of transverse magnetisation. As a consequence, when integrated into a DANTE saturation chain, the novel pulse is effectively immune to plausible variation in B 1 , in stark comparison to a hard-cosine pulse (C, D), which functions as designed within a comparatively small window of acceptable B 1 given a desire to minimise unwanted saturation of γ-ATP. Example phantom saturation data on resonance (i.e. fully-saturating the phantom peak), off-resonance between the saturation bands (i.e. at 0 Hz, and a separately acquired control acquisition with no dual-band saturation pulse played. B: The resulting behaviour as a function of saturation power is identical over an experimental factor of two variation in B 1 . C: In the perfused heart, the performance of the saturation pulse is the same, with complete saturation of intracellular inorganic phosphate and PCr, and no detectable alteration in the β-ATP peak. Simultaneous and complete saturation of phosphocreatine and the intracellular and extracellular inorganic phosphate peaks was observed, with concomitant decreases due to saturation transfer in ATP, shown with both individual AMARES fitting components and fit residuals. Saturated peak intensities were either numerically found to be zero, or the standard deviation of the noise floor, shown by a dotted red line. C, D: The time-evolution of the dual-band saturation transfer experiment shows, as expected, approximately exponential behaviour.  left-ventricular developed pressure; RPP: rate-pressure product).