³¹P Magnetic Resonance Fingerprinting for Rapid Quantification of Creatine Kinase Reaction Rate In Vivo

Abstract

Purpose

To develop a ³¹P spectroscopic magnetic resonance fingerprinting (MRF) method for fast quantification of the chemical exchange rate between phosphocreatine (PCr) and ATP via creatine kinase (CK).

Methods

A ³¹P MRF sequence (CK-MRF) was developed to quantify the forward rate constant of ATP synthesis via CK ( $k_f^{\text{CK}}$), the T₁ relaxation time of PCr ( $T_1^{\text{PCr}}$), and the PCr-to-ATP concentration ratio ( $M_R^{\text{PCr}}$). The CK-MRF sequence used a bSSFP-type excitation with ramped flip angles and a unique saturation scheme sensitive to the exchange between PCr and γATP. Parameter estimation was accomplished by matching the acquired signals to a dictionary generated using the Bloch-McConnell equation. Simulation studies were performed to examine the susceptibility of the CK-MRF method to several potential error sources. The accuracy of nonlocalized CK-MRF measurements before and after an ischemia-reperfusion (IR) protocol was compared to magnetization transfer (MT-MRS) method in rat hindlimb at 9.4 T (n=17). Reproducibility of CK-MRF was also assessed by comparing CK-MRF measurements to both MT-MRS (n=17) and four angle saturation transfer (FAST) methods (n=7).

Results

Simulation results showed that CK-MRF quantification of $k_f^{\text{CK}}$ was robust with <5% error in the presence of model inaccuracies including dictionary resolution, metabolite T₂ values, P_i metabolism, and B₁ miscalibration. Estimation of $k_f^{\text{CK}}$ by CK-MRF (0.38±0.02 s⁻¹ at baseline and 0.42±0.03 s⁻¹ post-IR) showed strong agreement with MT-MRS (0.39±0.03 s⁻¹ at baseline and 0.44±0.04 s⁻¹ post-IR). $k_f^{\text{CK}}$ estimation was also similar between CK-MRF and FAST (0.38±0.02 s⁻¹ for CK-MRF and 0.38±0.11 s⁻¹ for FAST). The coefficient of variation from 20-s CK-MRF quantification of $k_f^{\text{CK}}$ was 42% of that by 150-s MT-MRS acquisition and was 12% of that by 20-s FAST acquisition.

Conclusion

This study demonstrates the potential of a ³¹P spectroscopic MRF framework for rapid, accurate and reproducible quantification of chemical exchange rate of CK in vivo.

Graphical Abstract

A ³¹P spectroscopic pulse sequence based on magnetic resonance fingerprinting framework was designed to quantify the forward rate constant of ATP synthesis via creatine kinase. Simulation results demonstrated the robustness of this method to several sources of experimental errors. In vivo studies on rat hindlimb showed significantly higher reproducibility than conventional magnetization transfer method with only 13% of the acquisition time.

Introduction

Magnetic Resonance Spectroscopy (MRS) offers a unique opportunity to noninvasively characterize metabolic profiles due to its chemical specificity. Phosphorus-31 (³¹P) MRS has been used extensively to assess the energetics of living tissues^1–4. In addition to measuring tissue concentrations of phosphate metabolites, ³¹P magnetization transfer (MT) techniques using saturation transfer⁵ or inversion transfer^6,7 have been developed to quantify the phosphocreatine (PCr) synthesis rate via creatine kinase (CK) in heart, skeletal muscle, and brain^8–15. However, due to the low concentrations of phosphate metabolites, MT-MRS methods typically require long acquisition time to achieve adequate signal-to-noise ratio (SNR).

Shorter acquisition times have been achieved previously by using a combination of two strategies for CK measurements. The first strategy uses a reduced number of spectra acquired under partially relaxed conditions, such as in the Four Angle Saturation Transfer (FAST) method¹⁶, or the Triple Repetition Time Saturation Transfer (TRiST) method¹⁷. Reducing the number of acquired spectra is particularly useful in combination with chemical shift imaging (CSI) methods for spatially localized measurements. The second strategy further reduces the number of acquired spectra by assuming an intrinsic T₁ value for PCr. These methods include the Two Repetition Time Saturation Transfer (TwiST)¹⁸ and T₁ Nominal¹⁹ methods.

The Magnetic Resonance Fingerprinting (MRF) technique is a new framework to potentially reduce the time needed for quantitative measurements. MRF extracts parameter maps from signals generated using non-steady state pulse sequences in conjunction with Bloch simulations and pattern matching. This parameter estimation framework allows for greater flexibility in pulse sequence design than conventional methods and has been successfully utilized for efficient measurement of proton (¹H) relaxation times in the brain^20,21, abdomen²², and heart²³.

In this work, a new fast and robust ³¹P CK-MRF measurement strategy is proposed using the MRF framework. While the technique is based on the original MRF pulse sequence and dictionary matching method, several additional strategies to improve the sensitivity to chemical exchange via CK have been incorporated. First, a ³¹P bSSFP-type sequence²⁴ with a ramped flip angle series²⁵ was employed to improve SNR efficiency. Second, selective saturation pulses and excitation pulses were rapidly alternated to allow simultaneous encoding of chemical exchange and sampling of transient signal evolution. The accuracy and robustness of this method were first evaluated in simulation studies in which several potential error sources were assessed. In vivo studies on rat hindlimb were performed to validate the CK-MRF measurements against a conventional ³¹P magnetization transfer (MT-MRS) method. Finally, a reproducibility comparison between the proposed CK-MRF method and the MT-MRS and FAST methods was performed.

Materials and Methods

Pulse Sequence

A schematic of the CK-MRF pulse sequence is shown in Figure 1a. Following an inversion preparation, a bSSFP-type acquisition scheme for ³¹P spectroscopy was employed. The acquisition was made up of a total of 32 acquisition blocks. To enable accurate quantification of PCr and γATP signals, two types of acquisition blocks, ACQ-PCr and ACQ-γATP, were used to acquire signals from PCr and γATP, respectively. A 490-ms selective saturation block was used between two acquisition blocks, with the saturation frequency set either at the resonance frequency of γATP (SAT-γATP) or at the frequency contralateral to γATP (SAT-CNTL). The entire acquisition is comprised of two modules that employed SAT-CNTL and SAT-γATP, respectively, with the two acquisition blocks for PCr and γATP alternated eight times in each module.

Figure 1. Sequence design.

a. Schematic of the CK-MRF pulse sequence. ACQ (PCr) and ACQ (γATP) are acquisition blocks for PCr and γATP, respectively. SAT (CNTL) and SAT (γATP) are contralateral and γATP saturation blocks, respectively. ACQ (PCr) and ACQ (γATP) used Gaussian excitation pulses, while SAT (CNTL) and SAT (γATP) used continuous wave RF pulses. b and c. Pulse sequence diagrams for one block of ACQ (PCr) (b) and ACQ (γATP) (c). d. Timing and nominal flip angles of all excitation pulses. Blue and red colors indicate PCr and γATP excitation, respectively. Grey shaded areas indicate γATP saturation.

Diagrams of acquisition blocks for ACQ-PCr and ACQ-γATP are shown in Figures 1b and 1c. Each block consisted of a train of 10 linearly ramped-up and ramped-down excitations with alternating phase and a constant TR of 12.8 ms. Selective excitation was performed with a 4 ms Gaussian pulse centered at the resonance frequency of PCr and γATP, respectively. Following each excitation, 7.7 ms of FID signal was acquired with 256 data points, resulting in a spectral resolution of 130 Hz/pt.

The nominal flip angle, frequency, and timing of all 320 RF excitations for CK-MRF are shown in Figure 1d. The maximum flip angles in the 32 acquisition blocks were modulated by a sinusoidal envelope. The total time for one complete CK-MRF acquisition was 20 s. Multiple repetitions were acquired with no delay, with a single dummy acquisition preceding each series of repetitions.

Fingerprint Simulation

Signal time courses, i.e. the fingerprints, were simulated by solving the modified Bloch-McConnell equation^16,26 with two exchanging pools, i.e., PCr, and γATP. Details of the simulation methods are described in the Appendix. Simulation of a fingerprint required a total of nine input parameters: the forward rate constant of ATP synthesis via CK ( $k_f^{\text{CK}}$), the concentration ratio of PCr-to-ATP ( $M_R^{\text{PCr}}$), the T₁ and T₂ relaxation times and the resonance frequencies (chemical shift) of PCr and γATP ( $T_1^{\text{PCr}},T_1^{\gamma \text{ATP}},T_2^{\text{PCr}},T_2^{\gamma \text{ATP}}$, ω^PCr, ω^γATP), as well as B₀ distribution characterized by a linewidth, LW. Given a set of input parameters, spin evolution was simulated in sequential time steps, starting from a fully relaxed initial condition. Excitation was simulated by discretizing the Gaussian shaped RF pulse into 41 instantaneous complex rotations (Equations A7–9), with the magnitude of the rotation following a Gaussian envelope, and the total rotation summing to the nominal FA. The phase of each rotation was determined by the excitation frequency. γATP saturation was simulated by instantaneous saturation of γATP magnetization with no direct effect on PCr magnetization. The pulse sequence was simulated sequentially twice to account for the dummy scan. For each simulation, all FID signals were recorded from the bulk transverse magnetization, described as S(t) in Equation A12.

Dictionary Generation

The dictionary in the MRF framework is a set of fingerprints, or signal evolutions, for selected values of the tissue properties. Four tissue properties, namely $k_f^{\text{CK}},T_1^{\text{PCr}}$, ω^PCr, and $M_R^{\text{PCr}}$, were varied over their physiologically expected range to generate a dictionary for template matching. Specifically, 51 values of $k_f^{\text{CK}}$ ranging from 0.3 to 0.55 s⁻¹ with a resolution of 0.005 s⁻¹, 20 values of $T_1^{\text{PCr}}$ uniformly distributed from 2.8 to 4.7 s, 11 values for ω^PCr spanning from −15 to 15 Hz, and 49 values for $M_R^{\text{PCr}}$ uniformly distributed from 3.0 to 5.4 were simulated. It is worth noting that these ranges of parameter variations reflect physiological variations expected in the current study and may need to be expanded for other studies. ω^γATP for each entry was constrained to (ω^PCr − Δω^PCr−γATP), where Δω^PCr−γATP was the chemical shift difference between PCr and γATP and was fixed at 2.4 ppm. The remaining 4 parameters, $T_1^{\gamma \text{ATP}},T_2^{\text{PCr}},T_2^{\gamma \text{ATP}}$, and LW, were set at 120 ms, 16 ms, 0.8 s, and 15 Hz respectively based on pilot data. $T_1^{\gamma \text{ATP}}$ value was corrected to compensate for nuclear Overhauser effect (NOE)^6,27 from αATP and βATP.

Parameter Matching

Matching was performed in the frequency domain to reduce memory load. Each of the 320 FIDs, for all acquired data and simulated dictionary entries, was Fourier transformed, and the complex-valued data points corresponding to PCr or γATP resonances in these spectra were selected. This resulted in a time course signal, the fingerprint, showing the evolution of PCr and γATP peaks. The inner products between the normalized fingerprint and all entries of the normalized dictionary were computed. The dictionary entry that produced the largest magnitude of the inner product was considered to be the best match. From this match, the values of $k_f^{\text{CK}},T_1^{\text{PCr}}$, ω^PCr, and $M_R^{\text{PCr}}$ were derived. Subsequently, the spin density of γATP ( $M_0^{\gamma \text{ATP}}$) was computed as the scale factor between the fingerprint and its matched dictionary entry. All data processing and parameter estimation were fully automated.

Conventional MT-MRS

Conventional MT-MRS method was used for validation and comparison. MT-MRS acquisition consisted of 9 spectra: 7 spectra acquired after γATP saturation, with saturation times of 0.4, 0.9, 1.3, 2.2, 3.5, 5.3 and 7 s, respectively; a control spectrum with contralateral saturation; and a conventional spectrum without saturation¹⁹. All spectra were acquired using 90° hard pulses (130-μs duration) and a 16-s TR. Saturation was accomplished by applying continuous wave RF to the frequency of saturation. Post-processing consisted of 15 Hz line broadening, Fourier transform, and correction of phase and baseline. For phase correction, zero- and first-order phase correction values were obtained manually from spectrum with 12 signal averages and no γATP saturation. These phase correction values were applied to all the spectra acquired from the same animal. The phase correction was followed by automated baseline correction. The signal intensity from PCr was quantified by integrating the area under the PCr peak. Subsequently, $k_f^{\text{CK}},T_1^{\text{PCr}}$, and $M_0^{\text{PCr}}$ were determined by fitting the explicit solution of the Bloch-McConnell equation to the experimental data²⁸. Effects of RF spillover were corrected using the method outlined by Kingsley and Monahan²⁹.

FAST Method

The reproducibility and efficiency of the CK-MRF measurements were also compared with the FAST method. The FAST acquisition consisted of 4 spectra acquired with either 15° or 60° FA, and with either γATP or control saturation¹⁶. A BIR-4 pulse was used for excitation. All the spectra were acquired after five dummy scans with a TR of 1 s. Parameter estimation of $k_f^{\text{CK}},T_1^{\text{PCr}}$, and $M_R^{\text{PCr}}$ for FAST data was performed using the established closed-form derivation.

Simulations Studies

Performance of Parameter Quantification using Dictionary-Based Approach

Simulations were performed to assess the potential errors in parameter quantification caused by using a finite dictionary resolution. A total of 1000 sets of $k_f^{\text{CK}},T_1^{\text{PCr}}$, ω^PCr, and $M_R^{\text{PCr}}$ values were randomly generated with upper and lower bounds consistent with the dictionary’s ranges. The remaining 5 input parameters, i.e., ω^γATP, $T_2^{\text{PCr}},T_2^{\gamma \text{ATP}},T_1^{\gamma \text{ATP}}$, and LW were fixed, with ω^γATP constrained to (ω^PCr – 2.4) ppm. Test fingerprints were generated by simulation of the CK-MRF pulse sequence for each set of parameter values and were matched against the dictionary. The matching errors were calculated as the difference between the matched parameters and the parameters used in simulation.

Sensitivity of Parameter Quantification to Fixed Parameter Values

Since five parameters (Δω^PCr−γATP, $T_2^{\text{PCr}},T_2^{\gamma \text{ATP}},T_1^{\gamma \text{ATP}}$, and LW) were not varied in the dictionary, errors in their values may affect the accuracy of the matched parameters. Simulations were performed to assess how incorrect selection of the fixed parameters could influence the matched parameters. In addition to these five parameters, the impact of a constant B₁ scale error, resulting in a consistent error in flip angles, was also evaluated. First, a test parameter set (800, 3700, 16 and 120 ms for $T_1^{\gamma \text{ATP}},T_1^{\text{PCr}},T_2^{\gamma \text{ATP}}$, and $T_2^{\text{PCr}}$, −389 and 0 Hz for ω^γATP and ω^PCr, 0.42 s⁻¹ for $k_f^{\text{CK}}$, 4.1 for $M_R^{\text{PCr}}$, and 15 Hz LW, respectively) was chosen to evaluate how errors in the fixed parameters could alter dictionary matching results. To assess the impact of using the wrong values for fixed parameters, test fingerprints were generated by varying the values of a fixed parameter and subsequently matched to the dictionary generated using the test parameter set. The potential impact of each fixed parameter was assessed by calculating the difference between the matched parameters and the true test parameter values. Error calculations for $T_2^{\text{PCr}},T_2^{\gamma \text{ATP}},T_1^{\gamma \text{ATP}}$ were performed by varying each parameter between 50% and 150% of its test value. Error calculations for Δω^PCr−γATP used a range of −430 to −349 Hz. Error calculations for B₁ were performed by scaling all flip angles to between 50% and 150% of their nominal values.

In addition to error analyses, the effect of P_i-to-ATP exchange on parameter estimation, as well as the errors associated with using contralateral saturation to correct for the spillover effect of the γATP saturation on PCr, were also evaluated by simulation (See Supplemental Methods and Figures).

In Vivo Study Protocol

In vivo studies were performed to assess the accuracy of the CK-MRF measurements in rat hindlimb. Details of the experimental setup have been described previously³⁰. Briefly, three-month-old Sprague-Dawley rats (n=21) were anesthetized and positioned laterally in a cradle. The hindlimb was secured within a 15-mm ³¹P saddle coil placed within a ¹H volume coil and positioned at the isocenter of a 9.4T MRI scanner (Bruker Biospin Co., Billerica, MA). An air pressure cuff was placed proximal to the coil. Body temperature was maintained at above 35°C via a feedback control system (SA Instruments, Stony Brook, New York, USA). The respiratory rate was maintained between 45 and 60 breaths per minute by manually adjusting the anesthesia level. The animal protocol was approved by the Institutional Animal Care and Use Committee of Case Western Reserve University.

Following initial stabilization, shimming, and B₁ power calibrations, CK-MRF and MT-MRS data were acquired at baseline. Due to their substantially different acquisition times for a single-average dataset (20 s for CK-MRF versus 150 s for MT-MRS), 24 repetitions of CK-MRF data (8 min acquisition) and 4 repetitions of MT-MRS data (10 min acquisition) were acquired in an interleaved manner. A total of 96 repetitions of CK-MRF data and 12 repetitions of MT-MRS data were acquired. 14 rats subsequently underwent two rounds of ischemia/reperfusion (IR) before a second data collection session. Ischemia was induced by inflating the cuff to above 200 mmHg and lasted for 17 min. The ischemic period was followed by 17 minutes of reperfusion. At the end of the IR protocol, another session of data collection started, in which a total of 48 repetitions of CK-MRF data and 8 repetitions of MT-MRS data were acquired.

High SNR signals were obtained by averaging all repetitions from a data acquisition session in data analysis. For CK-MRF, this amounted to 96 and 48 averages for data acquired at baseline and post-IR, corresponding to a total acquisition time of 32 and 16 min, respectively. For MT-MRS, the number of signal averages was 12 and 8 at baseline and post-IR, corresponding to 30- and 20-min acquisition, respectively.

Three rats were used as the controls. They underwent the same data collection sessions, separated by a 70-min period, during which CK-MRF and FAST data were collected. An additional 4 rats were also used for the comparison between CK-MRF and FAST methods. Acquisition of 24 repetitions of CK-MRF data and 120 repetitions of FAST data were interleaved, corresponding to approximately 8-min acquisition for each method. A total of 96 and 480 repetitions were collected for the CK-MRF and FAST methods, respectively.

Statistical Analysis

All data processing, simulations, and statistical analysis were performed by custom-built software written in Matlab (Mathworks, Natick, MA). All results are presented as mean ± SD. Bland-Altman plot was used to compare CK-MRF and MT-MRS measurements with high SNR. The coefficient of variation (CV) of an estimated parameter, calculated as the ratio of the standard deviation to the mean, was used to assess the reproducibility. Two-tailed paired Student’s t-test was used to determine the significance of any observed differences between the CK-MRF and MT-MRS or FAST methods, or before and after IR. Two-tailed unpaired Student’s t-test were used to determine the significance of differences observed following IR compared to controls, as well as the differences between coefficients of variation in reproducibility study. A p value less than 0.05 was considered significant.

Results

Performance of Parameter Quantification using Dictionary Based Approach

Figure 2 shows the performance of the dictionary approach for parameter quantification. The average errors for matched parameters were 0.0014 s⁻¹ for the forward rate constant of CK ( $k_f^{\text{CK}}$), 0.03 s for T₁ of PCr ( $T_1^{\text{PCr}}$), 0.74 Hz for chemical shift of PCr (ω^PCr), and 0.014 for PCr-to-ATP ratio ( $M_R^{\text{PCr}}$). The worst case error for each parameter was 0.006 s⁻¹ for $k_f^{\text{CK}}$, 0.129 s for $T_1^{\text{PCr}}$, 1.55 Hz for ω^PCr, and 0.049 for $M_R^{\text{PCr}}$. With the exception of $T_1^{\text{PCr}}$, these errors were approximately one half of the dictionary step for each parameter.

Figure 2. Comparison of dictionary matched values with the ground truth values.

For each fingerprint generated using a random set of values for the four parameters to be determined, its dictionary matched parameter value for $k_f^{\text{CK}}$ (a), $T_1^{\text{PCr}}$ (b), ω^PCr (c), and $M_R^{\text{PCr}}$ (d) is plotted against its corresponding true value. Lines of identity are included in each plot.

Sensitivity of Parameter Quantification to Fixed Parameter Values

Figure 3 shows the effects of discrepancies in fixed parameters on the accuracy of the estimation of $T_1^{\text{PCr}},k_f^{\text{CK}},M_R^{\text{PCr}}$, and $M_0^{\gamma \text{ATP}}$. No error in ω^PCr was observed in any of the tested ranges. Discrepancies in $T_1^{\gamma \text{ATP}},T_2^{\gamma \text{ATP}}$, and B₁ resulted in errors with various degrees in the matched parameters. Specifically, overestimation of $T_1^{\gamma \text{ATP}}$ caused underestimation of $k_f^{\text{CK}}$ and $M_0^{\gamma \text{ATP}}$ and overestimation of $M_R^{\text{PCr}}$ (Figure 3a). Errors in $T_2^{\gamma \text{ATP}}$ led to less than 5% error in matched $k_f^{\text{CK}}$ for the range investigated but resulted in errors primarily in $M_R^{\text{PCr}}$ and $M_0^{\gamma \text{ATP}}$ (Figure 3b). A miscalibration in B₁ power resulted in significant errors in matched $T_1^{\text{PCr}}$ and $M_0^{\gamma \text{ATP}}$ but minimally affected $k_f^{\text{CK}}$ and $M_R^{\text{PCr}}$ (Figure 3c). Finally, errors in Δω^PCr−γATP, $T_2^{\text{PCr}}$ and LW caused relatively small errors in all matched parameters (Figure 3d–f).

Figure 3.

Errors in matched parameters $k_f^{\text{CK}},T_1^{\text{PCr}},M_R^{\text{PCr}}$, and $M_0^{\gamma \text{ATP}}$ caused by using values for fixed parameters that differ from those assumed when generating the dictionary: $T_1^{\gamma \text{ATP}}$ (a), $T_2^{\gamma \text{ATP}}$ (b), B₁ (c), $T_2^{\text{PCr}}$ (d), Δω^PCr−γATP (e), and LW (f).

Sensitivity of Parameters Quantification to P_i-to-ATP Exchange

Figure 1S shows the matching errors caused by using a two-pool exchange model. The effects of P_i-to-ATP exchange on the accuracy of the matched parameters were evaluated by increasing both the forward rate constant of P_i-to-ATP ( $k_f^{\text{Pi}}$, Figure 1Sa) and the P_i-to-ATP concentration ratio ( $M_R^{\text{Pi}}$, Figure 1Sb). At the current dictionary resolution, parameter matching using the two-pool exchange model produced no detectable errors until $k_f^{\text{Pi}}$ surpassed 0.17 s⁻¹. An increase in $k_f^{\text{Pi}}$ from 0.17 to 0.4 s⁻¹ resulted in errors in the estimation of $k_f^{\text{CK}},T_1^{\text{PCr}}$, and $M_R^{\text{PCr}}$ by −1.2, 1.2, and 2.7% respectively (i.e. one dictionary step). Similarly, increase in P_i-to-ATP ratio produced no detectable errors until it surpassed 0.35. As $M_R^{\text{Pi}}$ increased to 2.0, errors in $k_f^{\text{CK}},T_1^{\text{PCr}}$, and $M_R^{\text{PCr}}$ increased, but remained below 5%. No errors in ω^PCr were observed in any cases.

Sensitivity of $k_f^{\text{CK}}$ Quantification to Selective Saturation Power

The errors introduced by not fully modeling the spillover effect of the saturation pulses are shown in Figure 2S. Each investigated method showed a power-dependent error in $k_f^{\text{CK}}$ quantification. For CK-MRF and MT-MRS, an “optimal” saturation power was available that could obtain error-free quantification. Using a lower than “optimal” saturation power for both methods would lead to underestimation of $k_f^{\text{CK}}$, due to incomplete saturation of γATP. Using a higher than “optimal” saturation power led to overestimation of $k_f^{\text{CK}}$ due to incomplete correction of the spillover on PCr. Errors in $k_f^{\text{CK}}$ estimation by CK-MRF were comparable to conventional MT-MRS, while the FAST method showed a ~6% underestimation of $k_f^{\text{CK}}$ throughout the range of RF power investigated. MT-MRS using the Kingsley-Monahan correction method showed least errors comparing to the other methods.

In Vivo Results

Representative spectra acquired in vivo using CK-MRF and conventional MT-MRS methods are shown in Figure 4. The corresponding fingerprints and their matched dictionary entries are also shown. For a 96-average CK-MRF acquisition, SNR of PCr in Fourier-transformed spectra typically varied from 3 to 270 from the lowest to highest flip angles (from 2.6° to 82°), and SNR of ATP varied from 0.2 to 51. In comparison, for MT-MRS measurements, SNR of PCr from a 12-average acquisition was 294 and 118 in spectra acquired without saturation and with 7-s saturation, respectively. SNR of ATP was 42 in spectra acquired without saturation. With 96 signal averages, the average inner product between an acquired fingerprint and its matching dictionary entry was 0.998 ± 0.001. The inner product between a single-averaged fingerprint and its matching dictionary entry was 0.981 ± 0.008.

Figure 4. Representative in vivo CK-MRF spectra and fingerprints.

a and b. ³¹P spectra with 96 (a) and single (b) signal average(s). c and d. The corresponding fingerprints and their dictionary matches from 96 (c) and single (d) signal average(s). Blue and red data points represent PCr and γATP peaks, respectively. Relative timings for inversion and saturation pulses are indicated at the top. Dashed and solid arrows indicate contralateral and γATP saturation pulses, respectively. e and f. ³¹P spectra from 12 (e) and single (f) signal average(s) by MT-MRS. Results of the corresponding curve fitting to PCr signal are shown in the insets.

Comparison of In Vivo CK-MRF

Figure 5 shows the Bland-Altman comparison of $k_f^{\text{CK}},T_1^{\text{PCr}}$, and $M_R^{\text{PCr}}$ estimation obtained from high SNR data for both CK-MRF and MT-MRS, with group statistics summarized in Table 1. The use of CK-MRF led to a slight underestimation of $k_f^{\text{CK}}$ with a difference of (−3.6±4.8)% (p<0.05). Differences in $T_1^{\text{PCr}}$ and $M_R^{\text{PCr}}$ estimation were insignificant with values of (0.7±5.3)% and (−1.2±5.3)%, respectively.

Figure 5. Comparison between parameter measurements by CK-MRF and MT-MRS.

Bland-Altman plots are shown for $k_f^{\text{CK}}$ (a), $T_1^{\text{PCr}}$ (b), and $M_R^{\text{PCr}}$ (c). Squares indicate pre-IR data and stars indicate post-IR data. Solid lines indicate mean differences and dashed lines indicate limits of agreement.

Table 1.

Summary of Estimated Parameter Values Pre-IR and Post-IR (n=17)

		Acq Time (min)	Signal Averages	$k_f^{\text{CK}}$ (s−1)	$T_1^{\text{PCr}}$ (s)	$M_R^{\text{PCr}}$
Pre-IR	MT-MRS	30	12	0.39 ± 0.03	3.49 ± 0.11	4.22 ± 0.20
	CK-MRF	32	96	0.38 ± 0.02*	3.54 ± 0.11	4.16 ± 0.19
Post-IR	MT-MRS	20	8	0.44 ± 0.04†	3.48 ± 0.19	4.11 ± 0.30
	CK-MRF	16	48	0.42 ± 0.03*†	3.49 ± 0.13	4.07 ± 0.19†

^*p<0.05, MT-MRS versus CK-MRF

^†p<0.05, Pre-IR versus Post IR

Compared to baseline values, both methods detected a significant post-IR increase in $k_f^{\text{CK}}$ of ~13% (p<0.05). This increase was also significant compared to $k_f^{\text{CK}}$ changes observed in control animals. Additionally, a decrease in $M_R^{\text{PCr}}$ was observed in post-IR data from both CK-MRF and MT-MRS methods. However, only the differences detected by CK-MRF were found to be significant.

Reproducibility of In Vivo CK-MRF vs MT-MRS

Representative time courses of repeated parameter measurements by CK-MRF and MT-MRS during a data collection session at baseline are shown in Figure 6. The difference in measurement density for single-average data (Figures 6a–c) is due to the difference in acquisition time between the two methods (20 versus 150 s). Comparison between measurements made with approximately equal acquisition time (140 versus 150 s), i.e. seven averages for CK-MRF, are shown in Figures 6d–f. Parameter values obtained from 20-, 140-, and 150-s acquisition are presented in Table 2. These values did not differ significantly from those obtained from ~30-min acquisition (Table 1).

Figure 6. Repeated parameter measurements for reproducibility assessment.

Sequentially acquired CK-MRF (filled circles) and MT-MRS (open circles) measurements during a data collection session are shown. a–c. $k_f^{\text{CK}}$ (a), $T_1^{\text{PCr}}$ (b), and $M_R^{\text{PCr}}$ (c) estimation obtained from single-average data for both CK-MRF (20-s acquisition) and MT-MRS (150-s acquisition). d–f. $k_f^{\text{CK}}$ (d), $T_1^{\text{PCr}}$ (e), and $M_R^{\text{PCr}}$ (f) estimation obtained from 7 signal averages for CK-MRF (140-s acquisition) and single average for MR-MRS (150-s acquisition).

Table 2.

Estimated Parameters in Reproducibility Study

		Acq Time (s)	Signal Averages	$k_f^{\text{CK}}$ (s−1)	$T_1^{\text{PCr}}$ (s)	$M_R^{\text{PCr}}$
CK-MRF versus MT-MRS (n=17)	MT-MRS	150	1	0.40 ± 0.04	3.53 ± 0.44	4.22 ± 0.38
	CK-MRF	20	1	0.38 ± 0.03	3.57 ± 0.25	4.17 ± 0.25
	CK-MRF	140	7	0.38 ± 0.02	3.57 ± 0.13	4.16 ± 0.19
CK-MRF Versus FAST (n=7)	FAST	20	5	0.38 ± 0.11	2.63 ± 0.34	4.52 ± 0.85
	CK-MRF	20	1	0.38 ± 0.02	3.57 ± 0.21	4.20 ± 0.20

To separate the effects of inter-animal variations, the coefficient of variation (CV) for each parameter was calculated for each animal (Figure 7). Average CV of $k_f^{\text{CK}}$ from 20-s CK-MRF acquisition was 42% of the size of the CV from 150-s MT-MRS acquisition (p<0.05, Figure 7a). With equal acquisition time, the CV of $k_f^{\text{CK}}$ estimated by CK-MRF further reduced to 26% of the CV by MT-MRS. The CV of $T_1^{\text{PCr}}$ and $M_R^{\text{PCr}}$ estimated by CK-MRF were also significantly smaller than by MT-MRS for both 20-s and 140-s acquisition (Figure 7b–c). Finally, the CV of parameters estimated by CK-MRF was significantly smaller than that generated using FAST (p<0.05, Figure 7d–f). Specifically, CV of $k_f^{\text{CK}}$ from CK-MRF was 12% of that by FAST for 20-s acquisitions.

Figure 7. Reproducibility of parameter estimation.

Coefficient of variation (CV, defined as SD divided by the mean of the measurements) was compared between CK-MRF and MT-MRS methods (a–c) and between CK-MRF and FAST methods (d–f), for parameters $k_f^{\text{CK}}$ (a, b), $T_1^{\text{PCr}}$ (c, d), and $M_R^{\text{PCr}}$ (e, f). Acquisition times denoted represent time spent for a single measurement, including all signal averages but excluding dummy and control scans. Column height and error bars represent mean and SD of CV across animals. All comparisons are statistically significant (p<0.05).

Discussion

In this work, a novel ³¹P CK-MRF method to quantify CK rate based on the MRF framework has been proposed. Simulation studies were used to characterize the sensitivity of CK-MRF measurements to several potential sources of errors. The accuracy and reproducibility of the CK-MRF method was evaluated in vivo against a conventional MT-MRS method. Previously, it has been shown in ³¹P spectroscopic imaging that bSSFP-based approach offers a high gain of SNR per unit time with preserved coherence of magnetization²⁴. In the current study, a bSSFP-type pulse sequence was used with a unique RF and acquisition scheme. While the interleaved saturation and acquisition scheme gave rise to MT-modulated signal evolution, the MT effect was further enhanced by the use of bSSFP-like acquisition that allowed the history of signal evolution to be preserved throughout data acquisition. In addition, the combination of inversion recovery and saturation transfer also allowed greater separation between dictionary entries. As a result, the CK-MRF method showed improved acquisition time and reproducibility for quantification of CK rate, as demonstrated by in vivo experiments.

It is worth noting that acquisition time and reproducibility must be considered together when evaluating the efficiency of a method. In the current study, the CK-MRF method was able to quantify $k_f^{\text{CK}}$ in vivo in 20 s with a coefficient of variation of only 3.7%. Compared to the MT-MRS method, this demonstrated both an improvement in measurement reproducibility (CV for $k_f^{\text{CK}}$ of 3.7 versus 9% for CK-MRF and MT-MRS, respectively), as well as a reduction in acquisition time (20 s versus 150 s for CK-MRF and MT-MRS, respectively). Although acquisition parameters for MT-MRS used in the current study were not optimized, it is unlikely that its acquisition time can be significantly reduced without compromising the reproducibility. The reproducibility of CK-MRF was also benchmarked in the context of equal acquisition time. CK-MRF also showed significantly reduced CV in $k_f^{\text{CK}}$ quantification with the same acquisition time when compared to MT-MRS and FAST methods. Acquisition time for conventional MT methods can be further reduced by using a fixed $T_1^{\text{PCr}}$ value^18,19, which was not investigated in the current study. Future investigation comparing these approaches is warranted.

The CK-MRF method may provide the potential for spatially localized CK measurement with improved efficiency. Spatial localization is a long-standing challenge for MT-MRS methods due to the additional SNR penalty for spatial encoding. Using CK-MRF, more signal averages can be achieved within the same acquisition time. Furthermore, since CK-MRF data are acquired using spectrally selective excitation, it has the advantage of allowing spatial encoding to be performed using imaging methods rather than the more time-consuming CSI methods. A previous study using conventional MT-MRS method in combination with spectrally selective excitation has demonstrated feasibility of $k_f^{\text{CK}}$ mapping on a 7T clinical system in less than 60 min³¹. CK-MRF combined with spatial encoding may allow further reduction of the acquisition time.

The baseline $k_f^{\text{CK}}$ and PCr-to-ATP ratio ( $M_R^{\text{PCr}}$) obtained in this work are consistent with those reported previously³². The slight decrease in $M_R^{\text{PCr}}$ after two rounds of IR is also consistent with a previous study in pig muscle subject to three rounds of IR³³. In conjunction with the decrease in PCr, an increase in post-IR $k_f^{\text{CK}}$ was detected by both MT-MRS and CK-MRF methods. While this increase has not been reported previously, it might be a result of the decreased PCr concentration or concentration changes in other high-energy phosphate metabolites such as ADP. Previously, McFarland et al observed increased $k_f^{\text{CK}}$ with decreased PCr content in stimulated perfused soleus muscle³⁴, which was attributed to the concurrent increase in ADP concentration based on the Michaelis-Menten kinetics^35,36. However, studies using longer ischemic duration have observed decrease in both $k_f^{\text{CK}}$ and PCr concentration^3,37. The mechanisms and the relationship between ischemia duration and subsequent changes in creatine kinase kinetics need further investigation.

In the current study, several parameters, including $T_1^{\gamma \text{ATP}},T_2^{\gamma \text{ATP}},T_2^{\text{PCr}}$, Δω^PCr−γATP, linewidth, and B₁, used fixed values instead of being matched to reduce computation time and memory requirements. Insensitivity to these fixed parameters is a desirable quality for pulse sequence design for several reasons. First, it allows accurate measurement of the matched parameters without precise pilot measurements of these fixed parameters. Second, it allows a single dictionary to be computed for a particular study, despite minor inter-subject and inter-measurement variations such as in metabolite T₂ values³⁸. Robustness to B₁ errors is of particular importance as many ³¹P MRS studies are performed using surface coils with significant B₁ inhomogeneity. Our simulation results show that errors in $k_f^{\text{CK}}$ measurement was less than 3% in the presence of ±50% error in B₁ flip angle. Further, in vivo measurements of $k_f^{\text{CK}}$ also showed strong agreement with both MT-MRS and FAST measurements. However, our current study was performed using a saddle coil. Whether this robustness to B₁ miscalibration is sufficient for measurements using surface coils needs further investigation.

CK-MRF also showed tolerance to physiological variations such as P_i exchange rate with γATP. Simulations using values associated with resting rat skeletal muscle³⁹ showed no detectable errors. While P_i exchange rates with ATP may be higher in other organs or physiological conditions, simulations showed less than 5% error in $k_f^{\text{CK}}$ measurement even with over a fivefold increase in either the rate of P_i to ATP or the concentration of P_i compared to that of resting skeletal muscle.

A limitation of the current study is that the value of $T_1^{\gamma \mathit{ATP}}$ was fixed rather than matched to reduce the computation time and memory requirements during parameter estimation. Unlike MT-MRS method in which $T_1^{\gamma \mathit{ATP}}$ does not affect the kinetics of PCr signal, in CK-MRF method, errors in fixed parameter $T_1^{\gamma \mathit{ATP}}$ do have the potential to cause significant errors in $k_f^{\text{CK}}$ measurement. This error is approximately 30% the size of the error in $T_1^{\gamma \mathit{ATP}}$. Hence, variations in $T_1^{\gamma \mathit{ATP}}$ under pathophysiological conditions need to be carefully evaluated. In vivo measurement of $T_1^{\gamma \mathit{ATP}}$ is nontrivial, especially for the current CK-MRF method because the magnetization of αATP and βATP is not disturbed. Hence, the intramolecular NOE effects between γATP and αATP and between γATP and βATP also need to be considered. While this may be addressed by expanding the Bloch-McConnell equation to include these effects through additional exchanging pools⁶, substantial validation of such an approach would be needed to evaluate its potential accuracy and robustness.

Another limitation of the proposed CK-MRF method is that the data acquisition scheme using selective RF excitation only samples PCr and γATP magnetization. As a result, other information that are available in conventional ³¹P MT-MRS methods, such as P_i-to-ATP exchange, pH and PDE and ADP concentrations, cannot be obtained from CK-MRF data. While pH and metabolite concentrations can be obtained from a conventional ³¹P spectrum, the quantification of P_i-to-ATP exchange is much more challenging due to the low concentration of P_i. On the other hand, it has been shown that the P_i-to-ATP exchange is more relevant in diseases that affect skeletal muscle⁹. The development of a fast and robust MRF method that also enables the quantification of the P_i-to-ATP exchange should be a priority for future work.

Continuous wave RF pulses were used in the CK-MRF method to perform selective saturation. The spillover effect on PCr was partially accounted for by applying contralateral saturation pulses. Simulations show that the errors introduced by such an approach were power dependent, and the errors were comparable to those observed in conventional MT-MRS method. The observed 3.6% underestimation in $k_f^{\text{CK}}$ by CK-MRF is consistent with the bias expected between the two methods for a saturation power between 11 to 14 Hz. With improved mapping of the B₁ field, these errors may be correctable by using explicit methods²⁹, or by building saturation power into the dictionary as an additional fixed or matched parameter. Alternatively, continuous wave RF may be replaced by another saturation pulse such as BISTRO⁴⁰ with reduced spillover effect.

To reduce computation time and memory requirements, the current dictionary choice of matched parameters and parameter resolution was selected as an acceptable tradeoff between potential errors size and dictionary computation and memory size. Simulation results showed that the current dictionary resolution resulted in the nearest-neighbor match for matched parameters, with the exception of $T_1^{\text{PCr}}$, where the next-nearest-neighbor may also be matched. This result implies that the accuracy of parameter match can be further improved by using a dictionary resolution with a finer resolution.

In conclusion, this work is a first step towards quantifying phosphate metabolism using the MRF framework. CK-MRF was able to significantly increase measurement reproducibility compared to the conventional ³¹P MT-MRS method while exhibiting robustness to several error sources. The improvement in measurement efficiency may be leveraged in several ways, such as more accurate measurements made in equivalent experimental time, or reduced experimental time with equivalent measurement accuracy, allowing for new applications of ³¹P studies on tissue metabolism.

Abbreviations

³¹P

phosphorus-31

ACQ-PCr

acquisition block with PCr excitation

ACQ-γATP

acquisition block with γATP excitation

ADP

adenosine diphosphate

ATP

adenosine triphosphate

bSSFP

balanced steady-state free precession

CK

creatine kinase

CK-MRF

creatine kinase encoded magnetic resonance fingerprinting

CSI

chemical shift imaging

CV

coefficient of variation

FAST

four angle saturation transfer

FID

free induction decay

LW

linewidth

NOE

nuclear Overhauser effect

MRF

magnetic resonance fingerprinting

MT

magnetization transfer

MT-MRS

magnetization-transfer encoded magnetic resonance spectroscopy

PCr

phosphocreatine

P_i

inorganic phosphate

SAT-γATP

selective saturation block applied on γATP resonance

SAT-CNTL

selective saturation block applied contralateral to γATP resonance relative to phosphocreatine resonance

SD

standard deviation

TR

repetition time

Appendix

Signal evolutions in response to a pulse sequence were simulated using the Bloch-McConnell equations^16,26. The matrix form of the Bloch-McConnell equations can be expressed as:

$$\frac{d}{dt}M=AM+C$$ [A1]

where M is a vector that describes the evolution of the transverse and longitudinal magnetization. For two exchanging species PCr and γATP, M is defined as:

$$M={[\begin{array}{cccccc}{M}_x^{\text{PCr}}& M_y^{\text{PCr}}& M_z^{\text{PCr}}& M_x^{\gamma \text{ATP}}& M_y^{\gamma \text{ATP}}& M_z^{\gamma \text{ATP}}\end{array}]}^T$$ [A2]

C is a vector that describes the steady-state magnetization weighted by the longitudinal relaxation, i.e.,

$$C={\left[\begin{array}{cccccc}0& 0& \frac{M_0^{\text{PCr}}}{T_1^{\text{PCr}}}& 0& 0& \frac{M_0^{\gamma \text{ATP}}}{T_1^{\gamma \text{ATP}}}\end{array}\right]}^T$$ [A3]

where $M_0^{\text{PCr}}$ and $M_0^{\gamma \text{ATP}}$ are the steady-state magnetization and $T_1^{\text{PCr}}$ and $T_1^{\gamma \text{ATP}}$ are the longitudinal relaxation time for PCr and γATP, respectively. A is a matrix that describes the precession, relaxation, and magnetization exchange between PCr and γATP. For PCr and γATP with resonance frequencies ω^PCrand ω^γATP observed in a rotating frame of frequency ω, A matrix can be expressed as:

$$A=\left[\begin{array}{cccccc}-\frac{1}{T_2^{\text{PCr}}}-k_f^{\text{CK}}& \mathrm{\Delta }{\omega }^{\text{PCr}}+\mathrm{\Delta }{\omega }_0& & k_r^{\text{CK}}& & \\ -\mathrm{\Delta }{\omega }^{\text{PCr}}-\mathrm{\Delta }{\omega }_0& -\frac{1}{T_2^{\text{PCr}}}-k_f^{\text{CK}}& & & k_r^{\text{CK}}& \\ & & -\frac{1}{T_1^{\text{PCr}}}-k_f^{\text{CK}}& & & k_r^{\text{CK}}\\ k_f^{\text{CK}}& & & -\frac{1}{T_2^{\gamma \mathit{ATP}}}-k_r^{\text{CK}}& \mathrm{\Delta }{\omega }^{\gamma \text{ATP}}+\mathrm{\Delta }{\omega }_0& \\ & k_f^{\text{CK}}& & -\mathrm{\Delta }{\omega }^{\gamma \text{ATP}}-\mathrm{\Delta }{\omega }_0& -\frac{1}{T_2^{\gamma \text{ATP}}}-k_r^{\text{CK}}& \\ & & k_f^{\text{CK}}& & & -\frac{1}{T_1^{\gamma \text{ATP}}}-k_r^{\text{CK}}\end{array}\right]$$ [A4]

where $T_2^{\text{PCr}}$ and $T_2^{\gamma \text{ATP}}$ are the transverse relaxation time of PCr and γATP, respectively. Δω^PCr and Δω^γATP are the difference between the resonance frequency of PCr (ω^PCr ) and γATP (ω^γATP ) from the frequency of the rotating frame (ω.), i.e., Δω^PCr = ω^PCr − ω and Δω^γATP = ω^γATP − ω. Δω₀ is the frequency dispersion term used to account for B₀ field inhomogeneity. $k_f^{\text{CK}}$ and $k_r^{\text{CK}}$ are the forward and reverse exchange rate from PCr to γATP, respectively. Chemical equilibrium was assumed for all simulations such that:

$$k_r^{\text{CK}}=k_f^{\text{CK}}\frac{M_0^{\text{PCr}}}{M_0^{\gamma A\text{TP}}}$$ [A5]

Starting from a fully relaxed spin system, signal evolutions in response to a pulse sequence were simulated by iteratively solving for M in discrete time steps (Δt) using the discrete time solution for Equation A1:

$$M^{N+1}=e^{A\mathrm{\Delta }t}{M}^N+(e^{A\mathrm{\Delta }t}-I)A^{-1}C$$ [A6]

where M^N and M^N+1 are magnetization vectors before and after an iteration.

RF pulses were modeled as discretized instantaneous rotations about the x- and y-axis of the rotating frame:

$$M^{+}=R_x({\alpha }_x)R_y({\alpha }_y)M^{-}$$ [A7]

where M⁻ and M⁺ are the magnetization vector before and after the RF pulse, and R_x(α_x) and R_y(α_y) are the rotation matrix about the x- and y-axis with flip angles α_x and α_y, respectively, i.e.,

$$R_x({\alpha }_x)=\left[\begin{array}{cccccc}1& & & & & \\ & cos{\alpha }_x& -sin{\alpha }_x& & & \\ & sin{\alpha }_x& cos{\alpha }_x& & & \\ & & & 1& & \\ & & & & cos{\alpha }_x& -sin{\alpha }_x\\ & & & & sin{\alpha }_x& cos{\alpha }_x\end{array}\right]$$ [A8]

$$R_y({\alpha }_y)=\left[\begin{array}{cccccc}cos{\alpha }_y& & -sin{\alpha }_y& & & \\ & 1& & & & \\ sin{\alpha }_y& & cos{\alpha }_y& & & \\ & & & cos{\alpha }_y& & -sin{\alpha }_y\\ & & & & 1& \\ & & & sin{\alpha }_y& & cos{\alpha }_y\end{array}\right]$$ [A9]

Once M is solved, the FID signal, i.e., the transverse magnetization M_⊥, was computed as

$$M_{\perp }=\left[\begin{array}{c}{M}_{\perp }^{\text{PCr}}\\ M_{\perp }^{\gamma \text{ATP}}\end{array}\right]=\left[\begin{array}{c}{M}_x^{\text{PCr}}+i{M}_y^{\text{PCr}}\\ M_x^{\gamma \text{ATP}}+i{M}_y^{\gamma \text{ATP}}\end{array}\right]$$ [A10]

To account for the effects of B₀ inhomogeneity, simulations were repeated with 51 different values of Δω₀ ranging from −75 to 75 Hz with a uniform resolution of 3 Hz. The total signal, S(t), was calculated as the weighted sum of 51 simulations:

$$S(t)={\mathrm{\sum }}_{\mathrm{\Delta }{\omega }_0}W(\mathrm{\Delta }{\omega }_0)M_{\perp }(t,\mathrm{\Delta }{\omega }_0)$$ [A11]

where W(Δω₀) is the corresponding weight for a frequency component derived from a Lorentzian lineshape with a linewidth LW:

$$W(\mathrm{\Delta }{\omega }_0)=\frac{{\text{LW}}^2}{{(\mathrm{\Delta }{\omega }_0)}^2+{\text{LW}}^2}$$ [A12]