Andreas Korzowski^{1}, Sarah Neumann^{1}, Ludwig Dominik^{1}, Loreen Ruhm^{1}, Mark E. Ladd^{1}, and Peter Bachert^{1}

^{31}P
MRS allows the non-invasive observation of high-energy phosphate turnover *in vivo*. A model incorporating the
effects of exchange processes onto the transverse steady-state magnetization was
derived, which allows the direct estimation of relaxation rates from signal
intensities in MRSI datasets. Multiple ^{31}P MRSI datasets with
different excitation parameters of four healthy were evaluated.
Spatially-resolved turnover rates of high-energy phosphates could be estimated
and are in agreement with literature values proving feasibility of the proposed
approach.

Phosphorus
magnetic resonance spectroscopy(^{31}P MRS) allows the non-invasive
observation of high-energy phosphate turnover *in vivo*. Recently, Ren *et al.*
showed that the underlying exchange kinetics are described by a multi-pool
model^{1}, which can be analyzed by saturation transfer(ST)^{2} or
inversion transfer(IT) techniques. However, the prolonged signal
preparations and measurement durations of these experiments prohibit incorporation
into a MR spectroscopic imaging(MRSI) experiment for *in vivo* studies.

The purpose of
this study was to demonstrate that the general relaxation properties of a
coupled ^{31}P spin system can be estimated directly by the detected
steady-state magnetization that was generated in a ^{31}P MRSI
experiment.

In this study,
we focused on the chemical exchange between phosphocreatine (PCr), the γ-phosphate
of adenosine-5’-triphosphate (γ-ATP) and inorganic phosphate (P_{i}). Following
the description given in Ref.[1], the matrix solution to the Bloch-McConnell
equations is

$$M_z(t) = M_z^0-e^{At}(M_z^0-M_z(0)). [1]$$

For the assumed 3-pool model the actual and thermal z-magnetization vectors and the relaxation matrix are defined by

$$M_z = \begin{bmatrix}M_{PCr} \\M_{\gamma ATP} \\M_{P_i} \end{bmatrix},[2]$$

$$M_z^0 = \begin{bmatrix}M_{PCr}^0 \\M_{\gamma ATP}^0 \\M_{P_i}^0 \end{bmatrix},[3]$$

$$A = \begin{bmatrix} -( 1/T1_{PCr} + k_{PCr>ATP}) & k_{ATP>PCr} & 0 \\k_{PCr>ATP} & -( 1/T1_{\gamma ATP} + k_{ATP>PCr} + k_{ATP>Pi}) & k_{Pi>ATP} \\ 0 & k_{ATP>Pi} & -( 1/T1_{Pi} + k_{Pi>ATP}) \end{bmatrix}.[4]$$

1/T1 describes the intrinsic longitudinal relaxation
rates and k_{PCr>ATP} and k_{Pi>ATP} describe the pseudo first-order reaction rates of the creatine-kinase reaction and ATP hydrolysis. Repetitive
radio-frequency(RF) excitation with flip angles

$$FA = \begin{bmatrix}\alpha_{PCr} & 0 & 0 \\0 & \alpha_{\gamma ATP} & 0 \\ 0 & 0 & \alpha_{Pi} \end{bmatrix} [5]$$

and repetition time TR result in the following steady-state of transverse magnetization:

$$M_{xy}^{ss}=sin(FA)\cdot \left[1-cos(FA)\cdot exp(A\cdot TR)\right]^{-1} \cdot \left[1-exp(A\cdot TR)\right]\cdot M_z^0 , M_{xy}^{ss}=\begin{bmatrix}M_{xy,PCr}^{ss} \\ M_{xy,\gamma ATP}^{ss} \\ M_{xy,Pi}^{ss} \end{bmatrix}.[6]$$

Measured
transversal magnetization for different combinations of TR and FA are fitted to
two models employing Eq.[6]: the *3-pool
model* (coupled fit of PCr, γ-ATP and P_{i}), yielding the parameters of the relaxation matrix A; the *exchange-free model* (separate fit of
PCr, γ-ATP and P_{i}), yielding a diagonal matrix with apparent
relaxation rates R_{app}.

Comparison of
the measured values given in Tab.1 to values reported in Ref.[1] are in good
agreement. Nevertheless, the sensitivity to exchange processes of the proposed
approach is reduced compared to ST or IT techniques. To improve the robustness
of the *3-pool* fit, acquisition of
multiple FA and increased detection sensitivity for the P_{i} resonance
are required, as was deduced from simulations with synthetized noisy ^{31}P
MRS datasets.

The presented data
demonstrate that it is possible to probe relaxation mechanisms in a system of
coupled spins directly by the transversal steady-state magnetization of a MRSI experiment.
This allows spatially-resolved measurements of the exchange kinetics of
high-energy phosphates by ^{31}P MRSI without special signal
preparation in experimental durations feasible for in vivo studies.

1. Ren J, et al. Exchange kinetics by inversion transfer: Integrated analysis of the phosphorus metabolite kinetic exchanges in resting human skeletal muscle at 7 T. Magn. Reson. Med. 2015;73(4):1359–1369.

2. Shoubridge EA, et al. 31P NMR saturation transfer measurements of the steady state rates of creatine kinase and ATP synthetase in the rat brain. FEBS Lett 1982;140:289–292.

3. Vanhamme L, et al. Improved Method for Accurate and Efficient Quantification of MRS Data with Use of Prior Knowledge. J. Magn. Reson. 1997;129:35–43.

4. Naressi A, et al. Java-based graphical user interface for the MRUI quantitation package. Magn. Reson. Mater. Phys. Biol. Med. 2001;12:141–152.

Localized ^{31}P spectra of the human
calf muscle of volunteer #3 shown for five different repetition times TR=[0.3; 0.5; 0.7; 0.95; 1.8]s and one FA applied at f_{0} = 0 ppm (α_{PCr}
= 36°). Voxel 1 (V1) is localized in the *Tibialis
Anterior* muscle; voxel 2 (V2) is localized in *Gastrocnemius Mediale* muscle. The colored areas under the resonance
curves illustrate the signal intensities estimated by the AMARES algorithm which
are utilized in the fit of Eq.[6] in Fig.2.

Course of signal intensities for the PCr, γ-ATP
and P_{i} resonances evaluated with AMARES as illustrated in Fig.1
(circles; corresponding to areas under the curve). Eight ^{31}P EPSI
measurements with different combinations of TR and FA yielded (8 measurements ×
3 resonances) = 24 data points in total per voxel for volunteer #3. The fit of
signal intensities to Eq.[6] is shown for the *exchange-free* model (dashed lines) and the *3-pool* (solid lines).

Exemplary transversal slices of the
three-dimensional parameter maps (metabolite density ratios, T1 values)
calculated from ^{31}P EPSI datasets of
volunteer #3.

Exemplary transversal slices of the three-dimensional
reaction rate maps of volunteer #3 (a) and #4 (b). Displayed are the pseudo
first-order rate constants of the creatine-kinase reaction and ATP hydrolysis
(reverse rate).

Table 1: Mean
values and standard deviation of R_{app}, T1 and k for all volunteers across
a transversal slice through the calf muscle. The applied number of different FA
is given in parenthesis. No values k are reported for volunteer #2 due to weak
the convergence of the 3-pool model. For
comparison, values for T1 and k from Ref.[1] are denoted in the last row. Calculated
values T1(γ-ATP) include effects of intramolecular nuclear Overhauser effects to
neighboring ^{31}P nuclei within the ATP molecule as described in Ref.[1].