Band Inversion Amplifies ³¹P-³¹P Nuclear Overhauser Effects: Relaxation Mechanism and Dynamic Behavior of ATP in the Human Brain by ³¹P MRS at 7T

Abstract

Purpose

To develop an improved method to measure ³¹P nuclear Overhauser effect (NOE) for evaluation of ATP dynamics in terms of correlation time (τ_c), and contribution of dipole-dipole (DD) and chemical shift anisotropy (CSA) mechanisms to T₁ relaxation of ATP in human brain.

Methods

The NOE of ATP in human brain was evaluated by monitoring changes in magnetization in the β-ATP signal following a band inversion of all downfield ³¹P resonances. The magnetization changes observed were analyzed using the Bloch-McConnell-Solomon formulation to evaluate the relaxation and dynamic parameters that describe interactions of ATP with cellular solids in human brain tissue.

Results

The maximal transient NOE, observed as reduction in the β-ATP signal, was 24±2% upon band-inversion of γ- and α-ATP, which is 2-3-fold higher than achievable by frequency-selective inversion of either γ- or α-ATP. The rate of ³¹P–³¹P cross-relaxation (0.21±0.02 sec^-1) led to a τ_c value of (9.1±0.8)×10^-8 sec for ATP in human brain. The T₁ relaxation of β-ATP is dominated by CSA over DD mechanism (60% : 40%).

Conclusion

The band inversion method proved effective in amplifying ³¹P NOE and thus facilitating ATP τ_c and relaxation measurements. This technique renders ATP a potentially useful reporter molecule for cellular environments.

Introduction

The nuclear Overhauser effect (NOE), a phenomenon that describes magnetization transfer (MT) through nuclear cross-relaxation (1), is highly useful in a number of applications including improved sensitivity for ¹³C NMR, in solution chemistry for characterizing molecular and protein structures, and in MRI for generating image contrast (2-6). While all these applications involve magnetization transfer from ¹H polarization, several recent ³¹P MT studies demonstrated that in vivo ATP may give rise to detectable homonuclear NOE through ³¹P→³¹P transfer (7,8). This is indicated by the observation of β-ATP signal reduction upon RF-irradiation at γ-ATP, a long-known phenomenon previously either interpreted as ³¹P chemical exchange (9) or attributed to discrepancies between pulsing and slow irradiation MT experiments in activating bound pools of phosphate metabolites with very broad lines (10). But now it is recognized as a special type of ³¹P NOE due to inter-phosphorous cross-relaxation (11). Although the presence of ³¹P NOE in ATP may complicate the task of assessing tissue ATP flux using the widely-accepted ³¹P magnetization transfer (MT) approaches (8,12-14), NOE can be exploited to obtain information about cellular ATP dynamics in vivo (7). This may be clinically useful because the cellular environment and ATP dynamics may be altered by metabolic disorders including diabetes, traumatic brain injury and neurogenerative diseases such as Alzeimer's (15,16).

Therefore the main purpose of this study is to measure the correlation time (τ_c) of ATP in the human brain in vivo. The rationale for such an approach is that the rate of ATP ³¹P-³¹P cross-relaxation (s) is solely dependent on τ_c, since the inter-phosphorous distance r(P-P), the only other variable in s, is fixed in ATP molecule (17). To make NOE measurement accurate and the analysis reliable, a band inversion strategy is used to maximize the magnitude of the transient NOE. The capability of band inversion for amplifying a particular ³¹P MT effect within a complex MT network in vivo has recently been reported (12). As described by the ³¹P MT technique termed EBIT (exchange kinetics by band inversion transfer) for measuring ATP flux, a key concept in band inversion is to magnetically buffer a small pool of inverted short-T₁ magnetization (γ-ATP) by a large pool of co-inverted long-T₁ magnetization partner (PCr) so that the MT effect between these two pools (γ-ATP ↔ PCr) can replenish the targeted MT pathway (γ-ATP ↔ Pi). In current work the same principle is adopted to maximize NOE through γ- → β-ATP pathway by co-inverting γ-ATP (short T₁) with PCr as well as Pi (long T₁). To further enhance NOE at β-ATP, the co-inversion is expanded to also include α-ATP magnetization so that two separate cross-relaxation pathways, ³¹P_α → ³¹P_β and ³¹P_γ → ³¹P_β, are simultaneously activated by the band inversion, both contributing to the reduction of β-ATP signal through NOE (Figure 1). With this strategy, the dependence of transient NOE on inversion delay is measured to provide values for two basic parameters: 1) the rate of cross-relaxation (σ), from which ATP correlation time τ_c is derived; and 2) the relaxation time T₁, from which the contribution of dipolar and chemical shift anisotropy (CSA) mechanisms is quantitatively evaluated (see below).

Fig. 1.

(a) Diagram of ³¹P kinetic pathways describing nuclear Overhauser effect (NOE) and chemical exchanges in high energy phosphates in brain; (b) Scheme of the ³¹P band inversion sequence and (c) the profile of inversion band over a brain ³¹P spectrum. Abbreviation: Pi, inorganic phosphate; PCr, phosphocreatine; ATP, adenosine triphosphate; ADP, adenosine diphosphate; CK, creatine kinase; AK, Adenylate kinase.

Our recent study of ³¹P T₁ relaxation in vivo indicated that the trend of T₁ relaxation time decreases significantly in the order monophosphates < diphosphates < triphosphate (ATP), suggesting a major role of dipolar coupling in di- and triphosphates due to the presence of the P-O-P molecular fragment (18). According to the classic dipolar coupling model of the two spin-1/2 system (1), NOE is proportional to the rate of cross-relaxation (σ), given by the rate difference between double- and zero-quantum transitions (σ = W2 - W0), while NOE is inversely proportional to the total relaxation rate (1/T_1,tot), given by the sum of the rate of all transitions in the dipolar coupling system (1/T_1,tot = 1/T_1,dip = W0 + 2 W1 + W2). Thus the steady-state NOE under a pure dipolar mechanism is described by σ T₁ = (W2 - W0) / (W0 +2 W1 + W2). However, when a relaxation mechanism other than dipolar interaction exists, the total relaxation rate becomes larger and the NOE is reduced. For ATP, evidence from field-strength dependence of ³¹P relaxation indicates that the CSA mechanism can rapidly relax ³¹P resonances (19,20), suggesting that 1/T_1,tot = 1/T_1,dip + 1/T_1,CSA for ATP. However, it remains unknown to what extent the CSA mechanism affects ATP T₁. Therefore the second goal of this study is to dissect ATP ³¹P T_1,tot into CSA and dipolar contributions.

Theory

A simplified model that describes major ³¹P MT effects in vivo is shown in Figure 1a. The model is comprised of multiple MT patheways formed by seven spins including Pi, PCr, α-, β- and γ-ATP, plus α- and β-ADP. Among them, only the first five spins are directly detectable by in vivo ³¹P MRS under normal physiology, and therefore are considered for formulating the equations that describe the ³¹P MT effects in the human brain. It can be shown in matrix format (8) that, after a short B1 perturbation (such as an inversion), the normalized Z-magnetization $m\left(=M_Z/{\mathit{\text{M}}}_{\mathit{\text{Z}}}^{\mathit{\text{O}}}\right)$ for this spin system evolves with time by

$$m\left(t\right)=I-e^{At}\left(I-m\left(0\right)\right)$$ 1

in which I is a 5-element unity vector, m(0) is a 5-element vector representing the initial z-magnetization after perturbation at t = 0, the magnetization m and matrix A can be written as

$$m=\left[\begin{array}{l}{m}_a\hfill \\ m_b\hfill \\ m_c\hfill \\ m_d\hfill \\ m_e\hfill \end{array}\right]$$ 2

and

$$A=\left[\begin{array}{ccccc}-1/T_{1,a}-k_{ac}& 0& k_{ac}& 0& 0\\ 0& -1/T_{1,b}-k_{bc}& k_{bc}& 0& 0\\ k_{ca}& k_{cb}& -1/T_{1,c}-k_{ca}-k_{cb}& 0& -{\sigma }_{ec}\\ 0& 0& 0& -1/T_{1,d}& -{\sigma }_{ed}\\ 0& 0& -{\sigma }_{ce}& -{\sigma }_{de}& -1/T_{1,e}\end{array}\right]$$ 3

where a – e denote for Pi, PCr, α-, β- and γ-ATP, respectively, the forward and reverse first-order rate constant k (chemical exchange) and are related by $k_{ji}=k_{ij}\frac{M_i^o}{M_j^o}$

(i,j = Pi, γ-ATP or PCr, γ-ATP), and σ, the rate constant of ³¹P-³¹P cross-relaxation is assumed to be a constant within any two neighboring ATP spins (σ_γβ = σ_βγ = σ_αβ = σ_γα) and described by (21)

$$\sigma ={\left(\frac{D}{r}\right)}^6\left[-{\tau }_c+\frac{6{\tau }_c}{1+4{\omega }_o^2{\tau }_c^2}\right]$$ 4

In Eq. [6], r is the distance between two neighboring ³¹P spins within ATP (r_αβ = r_βγ = 2.95 Å, ref(17)), ω_o is the Larmor frequency (120.6 MHz for ³¹P at 7T), the constant D, ${\left[{\left(\frac{{\mu }_o}{4\pi }\right)}^2\frac{{\gamma }^4{\hslash }^2}{10}\right]}^{1/6}$

, is numerically equal to 33.93 Å s^-1/3, and τ_c is the rotational correlation time describing the dynamic property of ATP tumbling. Thus, by fitting the kinetic datasets m(t) ∼ t, quantitative information about ³¹P exchange kinetics and relaxation can be obtained including σ and T₁ (Eq. [1-3]). Then from the derived σ value, the cellular ATP dynamics information, in terms of rotational correlation time τ_c, can be obtained by Eq. [4].

Once τ_c is known, the contribution of dipolar and CSA mechanisms to T₁ relaxation can be evaluated (7). For β-ATP, the dipolar interaction from both α- and γ-ATP pathways has to be considered, thus its T₁ relaxation is given by

$$\frac{1}{T_1}=\frac{2}{T_{1,dip}}+\frac{1}{T_{1,\mathit{\text{CSA}}}}$$ 5

where

$$\frac{1}{T_{1,dip}}={\left(\frac{D}{r}\right)}^6{\tau }_c\left[1+\frac{3}{1+{\omega }_o^2{\tau }_c^2}+\frac{6}{1+4{\omega }_o^2{\tau }_c^2}\right]$$ 6

and

$$\frac{1}{T_{1,\mathit{\text{CSA}}}}=\frac{2}{15}{\omega }_o^2{\left(\mathrm{\Delta }\delta \right)}^2{\tau }_c\left[\frac{1}{1+{\omega }_o^2{\tau }_c^2}\right]$$ 7

where Δδ = δ_∥ − δ_(x022A5), characterizing the ³¹P chemical shift anisotropy (the difference in chemical shift for parallel and perpendicular orientations).

To derive the steady-state NOE formula for β-ATP, the expression of the specific Bloch-Solomon equation is required (from Eqs [1-3], 5^th row):

$${\dot{m}}_{\beta }\left(t\right)=\frac{1}{T_{1,\beta }}\left(1-m_{\beta }\left(t\right)\right)-\sigma \left(2-\left(m_{\alpha }\left(t\right)+m_{\gamma }\left(t\right)\right)\right)$$ 8

Eq. [8] contains two separate terms: the first term represents β-ATP T₁ relaxation (also known as auto-relaxation), and the second term represents the sum of β-ATP cross-relaxations from α- and γ-ATP. By definition, the steady-state NOE effect (η^ss) is given by

$$\eta {}^{SS}=m^{SS}-1$$ 9

where m^ss is the steady-state magnetization when the coupled signal is fully and completely saturated. For β-ATP, this means that its steady-state magnetization m_β^ss is achieved under the condition ṁ_β = 0 when both α- and γ-ATP signals are irradiated for a sufficiently long time (t_sat → ∞) to reach m_α → 0 and m_γ → 0, which leads to the following equation (from Eqs. [8] and [9]):

$${\eta }^{SS}=2\sigma {\mathrm{T}}_{1,\beta }$$ 10

However, if β-ATP steady-state is achieved by frequency-selective saturation of either α- or γ-ATP (i.e., ṁ_β = 0, and m_α → 0 or m_γ → 0), the value of η^ss (=2 σ T_1,β) is given by (from Eqs. [8] and [9])

$$\begin{array}{cc}{\eta }^{SS}=\frac{2\mathrm{\Delta }{m}_{\beta }}{1+\mathrm{\Delta }{m}_i}& \left(i=\alpha -\text{or}\gamma -\text{ATP for}\gamma -\text{or}\mathit{\alpha -ATP}\text{irradiation}\right)\end{array}$$ 11

where Δm is the irradiation-induced magnetization change at steady state (t → ∞).

Since inversion transfer rather than saturation transfer is used in the current work, a special magnetization state with ṁ_β (t_m) = 0 can be reached transiently at time point t = t_m during the process of inversion recovery, as the result of a balanced T₁ relaxation and cross-relaxation (Eq. [8]). At inversion delay time t_m, the steady-state NOE can be derived from the band inversion measurements by (Eqs. [8] and [10])

$${\eta }^{SS}=\frac{2\mathrm{\Delta }{m}_{\beta }}{\mathrm{\Delta }{m}_{\alpha }+\mathrm{\Delta }{m}_{\gamma }}$$ 12

where Δm_i is the magnetization change at t = t_m for spin i. This means that, for band inversion, the steady-state NOE can be determined from the magnetization curves at t = t_m without resorting to data fitting using software programs.

Methods

Human Subjects

The protocol was approved by the Institutional Review Board of the University of Texas Southwestern Medical Center. Prior to the MRS study, informed written consent was obtained from all participants. Eight subjects (4 male and 4 female), aged 36 ± 10 yr, BMI 23 ± 5, resting heart rate 71 ± 11, and peripheral capillary oxygen saturation (SpO₂) 97 ± 1%, participated in the study. All subjects were in good general health with no history of peripheral vascular, systemic, myopathic, cancer, psychiatric or neurodegenerative diseases. Heart rate and blood oxygen saturation levels were monitored during the scan. The study was well-tolerated by all subjects.

MRS Protocol

All subjects were positioned head-first and supine in the MRI scanner (7T Achieva, Philips Healthcare, Cleveland, OH), with the back of the head positioned in the center of the detection RF coil (Philips Healthcare, Cleveland, OH). The coil was a partial volume, double-tuned ¹H/³¹P quadrature TR coil consisting of two tilted, partially overlapping 10 cm loops. Axial, coronal, and sagittal T2-weighted turbo spin echo images were acquired for shimming voxel planning. Typical imaging parameters included field-of-view 180 × 180 mm (FOV), repetition time (t_R) 2.5 s, echo time (TE) 80 ms, turbo factor 15, in-plane spatial resolution 0.6 × 0.7 mm², slice thickness 6 mm, gap 1 mm, bandwidth 517 Hz, number of acquisitions (NA) one, and acquisition time 2.1 min. Second order ¹H-based automatic volume shimming was applied prior to ³¹P spectral acquisitions.

The pulse sequence for measuring NOE consisted of an adiabatic inversion pulse, followed by a variable post-inversion delay time t, a hard 90° readout pulse, and a recovery period with TR = 25 sec. A total of 8 delay times (t_D = 40, 140, 440, 1000, 1600, 2500, 5000 and 10000 msec) were used. The band-inversion pulse was a short trapezoid-shaped adiabatic pulse (pulse width pw = 38 msec, including 7 msec of pre- and post-ramp time), maximal B₁ 13 μT, inversion bandwidth of 2700 Hz, and centered at γ-ATP to invert all downfield ³¹P signals except β-ATP. The total data acquisition time was 20 min for 6 acquisition average. In a separate experiment, the inversion pulse was applied to upfield β-ATP signal while keeping all downfield ³¹P resonances un-inverted under otherwise the same conditions. The changes of γ- and α-ATP magnetizations upon β-ATP inversion and recovery were monitored. For comparison, a fully relaxed ³¹P spectrum was also acquired without pre-inversion pulse at TR = 25 sec.

Other ³¹P NMR parameters include sampling points 4 K, zero-filled to 8 K prior to FT, readout pulse dead-time (the delay prior to FID sampling for suppression of the broad background ³¹P peak) 0.6 ms, transmitter carrier frequency offset 50 Hz upfield from α-ATP signal. The chemical shifts of all ³¹P metabolites were referenced to PCr at 0 ppm. Gaussian apodization (6 Hz) was applied to each FID prior to Fourier transformation using the scanner software (SpectroView, Philips Healthcare).

³¹P Spectral Analysis

The frequency-domain ³¹P spectra were baseline corrected, and the intensity of each of the ³¹P peaks was fitted by a Voigt lineshape (a combination of Gaussian and Lorentzian lineshape) using ACD software (Advanced Chemistry Development, Inc., Toronto, Canada).

³¹P Data Fitting and Simulation

The experimentally acquired datasets m(t) ∼ t for Pi, PCr, α-, β- and γ-ATP were fitted according to the normalized magnetization solution for the Bloch-McConnell-Solomon equations (Eqs. [1-3]). The fitting was based on Matlab least-square function lsqcurvefit, with T₁ and σ as fitting parameters variables and kinetic constant k as prior knowledge parameters (k_Pi→γATP = 0.21 sec^-1 (18) and k_Pi→γATP = 0.29 sec^-1 (22)). For all fitting parameters, the outcomes from the fitting procedure were ensured to be within the pre-set upper and lower bounds. To examine the effect of SNR-based weighting on fitting results, a numerical “fitting” was also carried out using Excel Solver function with and without weighting factor applied to β-ATP magnetizations. The Excel numerical “fitting” was in fact based on a built-in iterative procedure to minimize the total difference between the calculated and the experimental m(t_i) values. The calculated m(t_i) values were obtained by

$$m\left(t_{i+1}\right)=m\left(t_i\right)+dm\left(t_i\right)/dt\mathrm{\Delta }t$$ 13

where Δt denotes stepwise change of inversion delay time in 0.01 sec, and the second term in Eq [13] denotes the corresponding magnetization change with the derivative dm(t_i)/dt described by the conventional Bloch-McConnell-Solomon formulism in normalized magnetization (m)

$$dm\left(t_i\right)/dt=Am+b$$ 14

where A is the spin matrix given by Eq.[3], and b is a 5-element vector given by

$$b=\left[\begin{array}{c}1/T_{1,\text{Pi}}\\ 1/T_{1,\text{PCr}}\\ 1/T_{1,\gamma }+\sigma \\ 1/T_{1,\alpha }+\sigma \\ 1/T_{1,\beta }+2\sigma \end{array}\right]\left[15\right]$$ 15

The first step of the numerical “fitting” was carried by calculating m(t₁) = m(Δt) = m(0) + dm(0)/dt * Δt. Note that only the time points that match the experimental conditions were used in the minimization procedure with weighting factor set to the corresponding experimental m(t_i) value for β-ATP and scaled by a constant which took into account the total weighting factors of the β-ATP dataset.

To examine the sensitivity of transient NOE effect (1 - m_βATP(t)) in response to variation in T₁ , σ and k, the m_βATP(t) values were evaluated over the entire experimental inversion delay range (t = 0 – 10 sec) with each parameter varied individually by 50% from a base dataset including k_Pi→γATP = 0.21 sec^-1(18), k_Pi→γATP = 0.29 sec^-1 (22), σ = 0.22 sec^-1, T₁ = 6.03, 5.06, 1.07, 1.06 and 0.95 for Pi, PCr, γ-, α- and β-ATP, respectively (fitted results, Table 1). Magnetization simulations were also carried under fully inversion condition (m(0) = -1.0) for all ³¹P spins but β-ATP which was un-inverted (m_βATP (0) = 1.0). The results were compared between conditions with and without Pi and PCr co-inversion (m_PCr (0) = -0.75 vs. 1.0, and m_Pi(0) = -0.58 vs. 1.0). For quantitative analysis of relaxation mechanism, simulated data were also generated for σ, 1/ T_1,dip, 1/ T_1,CAS , η^ss over the τ_c range of 10^-10 – 10^-5 sec by Equations [5-10].

Table 1.

³¹P T₁ relaxation time, NOE factor (η^ss), and contribution of CSA and dipolar mechanisms to T₁ relaxation in resting human brain at 7T (N = 8 subjects)^a

	T1 (s)	T1/T1,dip (s-1)	T1/T1,CSA (s-1)	NOE ηss b (s-1)
α-ATP	1.11 ± 0.16 (1.06)	0.24 ± 0.04	0.76 ± 0.11	-0.23 ± 0.06
β-ATP	0.90 ± 0.05 (0.95)	0.20 ± 0.02	0.60 ± 0.04	-0.38 ± 0.05
γ-ATP	1.08 ± 0.22 (1.07)	0.23 ± 0.05	0.77 ± 0.15	-0.23 ± 0.07
Pi	6.02 ± 0.47 (6.03)
PCr	5.03 ± 0.44 (5.06)
σ (31P-31P)	-0.21 ± 0.02 (-0.22) s-1

^aThe kinetic exchange rate constants k_Pi→γATP = 0.21 sec^-1 (18), k_Pi→γATP = 0.29 sec^-1 (22) were used as prior knowledge in the fitting.

^bThe steady-state NOE factor (η^ss) was evaluated by σT₁ for α- and γ-ATP, and by 2σT₁ for β-ATP (see text), respectively.

Statistical Analysis

All data are reported as mean ± standard deviation, calculated using Matlab.

Results

In vivo ATP NOE Effect

The 7T ³¹P spectra collected from the resting human brain (Fig. 2) show that, upon band inversion of α-, γ-ATP and other downfield resonances, a large MT effect is clearly detected at β-ATP as indicated by a reduction of 24 ± 2% its signal at -16.2 ppm (Fig. 2c). The band inversion is efficient, as measured by the initial magnetization m(0) value for PCr (-0.75), Pi(-0.58), γ-ATP (-0.53) and α-ATP (-0.57). In contrast, there is no detectable β-ATP reduction immediately after the band inversion (Fig. 2b), as compared to the fully relaxed ³¹P MR spectrum without inversion (Fig. 2a). The reduction of the β-ATP signal occurs after 1 sec of delay following the inversion during which α- and γ-ATP recovers toward their magnetization at equilibrium. Because of the absence of any RF pulsing in the delay period, the β-ATP reduction must be a result of magnetization transfer, rather than any off-resonance “spillover” artifact. The 1-sec delay after the band inversion provides the time necessary for the MT effect to buildup through NOE, by the transfer of magnetization from the inverted α- and β-ATP signals to the un-inverted β-ATP signal. The co-inversion of PCr and Pi is for further enhancing NOE since PCr and Pi, which are known to have long T₁ and chemically exchange with γ-ATP, can serve as “polarization reservoirs” to replenish the inverted γ-ATP magnetization.

Fig. 2.

Comparison of human brain ³¹P MR spectra acquired at 7T without (a), with (b) a band inversion of downfield resonances, and followed by 1 second of delay (c) prior to a readout MR pulse. Note that there is no detectable difference in β-ATP signal intensity between (a) and (b), and that the reduction in β-ATP signal upon inversion delay is due to transient NOE from the magnetization change in α- and γ-ATP resonances. Abbreviation: PE, phosphoethenolamine; PC, phosphocholine, GPE, glycerophosphoethanolamine; GPC, glycerophosphocholine; Pi, inorganic phosphate; PCr, phosphocreatine; ATP, adenosine triphosphate; NAD, nicotinamide adenine dinucleotide; UDPG, uridine diphosphate glucose.

T₁ and cross-relaxation σ

Mathematically, NOE is governed by two relaxation parameters: T₁ and σ (Eq. [10]). To quantify these two parameters, the value of β-ATP reduction is measured at varying delay time t (0 – 10 sec) under a constant TR (25 sec) as illustrated in Fig. 3. When the normalized β-ATP Z-magnetization m(t) is plotted against the delay time t, the curve clearly reveals an anticipated biphasic feature, characterized by a rapid signal reduction in the beginning (nearly linear in the first 0.5 sec), then followed by a rather slow recovery toward magnetization thermal equilibrium. The maximal intensity reduction occurs visually at 1 sec of delay time (Fig. 4). In contrast, the magnetization of those inverted signals recovers nearly exponentially, as depicted for α-, and γ-ATP.

Fig. 3.

A group-averaged ³¹P MR spectral series acquired from resting human brain at 7 T using a band inversion-recovery sequence at constant TR 25 sec, with 6 scan averages and varying inversion delay time t (n = 8 subjects). For comparison, the last trace of the spectral series represents a brain ³¹P MR spectrum acquired at TR 25 sec without applying the inversion pre-pulse (ATP fully relaxed).

Fig. 4.

(a) The change of α-, β- and γ-ATP ³¹P signal intensities at different inversion delay time t and (b) plots of normalized ³¹P magnetization against t for evaluation of ∼ross-relaxation rate constant σ and T₁ relaxation times (n = 8 subjects). The data were from the signal intensity measurements in the ³¹P spectra, and the solid curves represent the population-averaged data fitting based on a 5-pool MT model (see text). Note the biphasic curve of β-ATP magnetization versus t due to the change of contribution from cross-relaxation and T₁ relaxation (Eq. [8]).

Summarized in Table 1 are the results of ATP ³¹P T₁ and σ values (n = 8 subjects), obtained by curve fitting m(t) ∼ t datasets using Metlab lsqcurvefit function according to the Bloch-McConnell-Solomon formula for a five-pool exchange model (Eqs. [1-3] and Fig. 1). Prior knowledge of kinetic rate constants (k_Pi→γATP = 0.21 sec^-1, k_Pi→γATP = 0.29 sec^-1 previously established by band inversion transfer (18,22) and/or ST (22)) were used in the fitting since they are insensitive parameters for m_βATP(t) as illustrated in the simulated plots of m_βATP(t) ∼ t at varied k values (Figs. 5a and 5b). With these known k parameters, the fitted T₁ value is 6.02 ± 0.47 sec for Pi and 5.03 ± 0.44 sec for PCr, respectively (n = 8 subjects). In contrast, σ and T_1,β-ATP are the most sensitive among all fitting parameters in modulating β-ATP transient NOEs (Figs. 5c and 5d). The model fitting yielded well-defined values for these two parameters (σ = - 0.21 ± 0.02 sec^-1 and T_1,βATP = 0.90 ± 0.05 sec, n = 8 subjects). Overall, the fitted m(t) results agree reasonably well with the experimental data (the solid curves, Fig. 4b), and the averaged results from individual fitting (n = 8 subjects) are consistent with the results from fitting the averaged m(t) datasets (each magnetization scaled in reference to γ-ATP magnetization), as compared in Table 1. The difference in σ between these two fitting approach is minimal (- 0.21 vs. - 0.22 sec^-1). Furthermore, for the population-average data fitting using Excel Solver minimization function, the difference in resultant σ is negligible between numerical fittings with and without SNR-based weighting on β-ATP magnetizations.

Fig. 5.

Plots of simulated m(t) ∼ t data to illustrate the sensitivity of NMR parameters k_Pi→γATP (a), k_PCr→γATP (b), σ (c) and T_1,βATP (d) in modulating β-ATP transient NOEs. The red traces represents the base dataset with k_Pi→γATP = 0.21 sec^-1(18), k_Pi→γATP = 0.29 sec^-1 (22), σ = 0.22 sec^-1, T₁ = 6.03, 5.06, 1.07, 1.06 and 0.95 for Pi, PCr, γ-, α- and β-ATP, respectively (Table 1). The green and purple traces represent the effects of the change of examined parameters by ± 50% from the base values.

NOE factor η^ss

The η^ss calculated from the product of 2σ and T_1,βATP yields a value of 0.38 ± 0.03 for steady-state NOE factor for β-ATP (Eq. [10]). This matches well the result of - 0.36 ± 0.04, obtained by $\frac{2\mathrm{\Delta }{\mathit{m}}_{\beta }}{\mathrm{\Delta }{\mathit{m}}_{\alpha }+\mathrm{\Delta }{m}_{\gamma }}$ (Eq. [12]) from the experimental magnetization data Δm_β = 0.24 ± 0.02, Δm_α = 0.62 ± 0.03 and Δm_γ = 0.70 ± 0.03 measured at inversion delay time point t_m = 1.0 sec. As a comparison, the curve fitting by Eqs [1-3] yielded a maximal magnetization change of Δm_β = 0.26 at inversion delay time of t_m = 1.04 sec (Fig. 4b). The steady state NOE factor η^ss means that, if both α- and γ-ATP are irradiated to full saturation for a sufficiently long time, β-ATP magnetization will be reduced by 38% from its equilibrium state.

Rotational correlation time τ_c

The cross-relaxation rate constant σ is determined by two parameters, the rotational correlation time τ_c, and the internuclear distance r between two neighboring ³¹P spins involved in dipolar interaction (Eq. [4]). Since the ³¹P–³¹P distance r in ATP molecular is known (r _31P–31P ∼ 2.95 Å, ref(17)), it becomes straightforward to obtain τ_c from σ using Eq. [4]. This yields a τ_c value of (9.1 ± 0.8) × 10^-8 sec.

Dipolar and CSA contribution to T₁ relaxation

In contrast to cross-relaxation σ, the T₁ relaxation of ATP is contributed by both dipolar and CSA mechanisms (Eq. [5]). For β-ATP, its dipolar relaxation is contributed by two pathways: ³¹P_α-³¹P_β and ³¹P_γ-³¹P_β. Together, they account for 40% (2/T_1,dip = 0.44 sec^-1) of the T₁ relaxation rate measured experimentally (1/T₁ = 1.11 sec^-1). The remaining 60% of β-ATP T₁ relaxation is contributed by the CSA mechanism (Eq. [7]), with 1/T₁,_CSA = 0.67 sec^-1 (Table 1).

Profiling τ_c dependence of T_1,dip, T₁,_CSA and σ terms

To illustrate the dynamic feature of different relaxation mechanisms, Fig. 6a plots the τ_c dependence of 1/T_1,dip, 1/T₁,_CSA and σ over a large range of -Logτ_c (= 7 - 10) with the simulated data for β-ATP. Apparently, the τ_c profile is significantly different among these three terms. First, both T_1,dip, and σ terms vary rapidly with τ_c but toward opposite directions. Second, the CSA relaxation is bell-shaped with a maximum occurring at τ_c = 8.20 × 10^-9 sec (Fig. 6a). In addition, the contribution of CSA is greater than the dipolar mechanism under the short τ_c condition (τ_c < 1.1 × 10^-7 sec).

Fig. 6.

(a) Plots of simulated τ_c dependence of σ, 1/T_1,dip, 1/T₁,_CSA (a) and steady-state NOE (=2σT₁) (b) for β-ATP based on Eqs. [4], [5], [6] and [10] respectively (see Theory). The locations of τ_c for ATP in vivo and in vitro are marked by vertical dotted lines. Note the reduced NOE magnitude due to presence of CSA contribution in T₁ relaxation.

Fig. 6b compares the τ_c dependence of the values of σ T_1,dip and σ T_1,tot for β-ATP with the simulated data for β-ATP, which indicate that the presence of the CSA mechanism significantly reduces the measurable NOE. Furthermore, there is a linear relationship between the values of σ T_1,tot and -Logτ_c near τ_c = (9.1 ± 0.8) × 10^-8 sec for ATP in vivo (vertical line, Fig. 6b).

Discussion

Enhanced NOE by Band Inversion

The present study is the first investigation utilizing band inversion ³¹P MRS technique at 7T for quantitative measurement of ³¹P-³¹P ATP NOE in the human brain. Using this technique, transient NOEs can be amplified to enable reliable determination of σ-value for evaluation of cellular ATP dynamics through τ_c. Specifically, a wide-band adiabatic pulse was applied to invert ³¹P resonances in the chemical shift range of -7 ppm to 7 ppm (including Pi, PCr, α- and γ-ATP), while monitoring attenuation of the un-inverted β-ATP signal following post-inversion delay. The NOE as measured by the attenuation of the β-ATP signal is attributed to the ³¹P cross-relaxation between β-ATP and the inverted α- and γ-ATP signals, since in such pulsed MT experiment the potential contamination from β-ADP (which co-resonates with γ-ATP) transfer to β-ATP is negligible. This is because the ADP concentration is ∼ 2 order of magnitude smaller than that of ATP under normal physiology, and that the duration of the inversion MT pulse is too short to have an amplification effect on such a small pool. The presence of NOE between γ- and β-ATP has been confirmed previously by the observation of equal MT effect at β-ATP (induced by γ-ATP saturation) in the skeletal muscle of both wild-type and CK/AK mutant mouse models (7).

Using our band inversion strategy, the maximal β-ATP reduction observed for the normal human brain is 24 ± 2%, which is respectively ∼ 2- and 3-fold higher than achievable by frequency-selective inversion of α- and γ-ATP to the same levels (m_α(0) = -0.53 and m_γ(0) = -0.57). Despite its large bandwidth, the short inversion pulse led to a clear-cut transition between α- and β-ATP signals without direct RF spillover effect detectable at β-ATP (m_β(0) ∼ 1.0, Figure 2). In case when the β-ATP signal is partially inverted due to an imperfect pulse or a reduced frequency separation between α- and β-ATP signals as in lower Bo fields, one can simply correct such a RF spillover effect using a reduced m_β(0) value according to Eq. [1] to access the true transient NOEs accumulated during the period of inversion recovery. In this study, the enhanced transit NOE by the short-pulsed band inversion approach is mainly attributed to the simultaneous activation of two cross-relaxation pathways, α- → β-ATP and γ- → β-ATP (Fig. 1). In contrast, the conventional frequency-selective approach activates only one of the available pathways, rendering the other pathway a leaky source that drains the MT effect and consequently yields a reduced NOE.

The strategy to co-invert PCr and Pi together with α- and γ-ATP is to further enhance NOE at β-ATP since PCr and Pi are long-T₁ spins involving in chemical exchange with short-T₁ γ-ATP (Fig. 1). Upon inversion, PCr and Pi can serve as a “polarization reservoirs” for the inverted γ-ATP by prolonging γ-ATP recovery time, which helps the NOE to buildup between γ- and β-ATP. The increase in the NOE effect is ∼4% by co-inversion of PCr and Pi, as compared to that in the absence of such inversion (i.e. m_PCr (0) = -0.75 vs. 1.0, and m_Pi(0) = -0.58 vs. 1.0), estimated by simulation with Eqs. [1-3] and the fitted parameters (Table 1).

It should be pointed out that, compared to the EKIT (exchange kinetic by inversion transfer) method, which is based on frequency-selective inversion requiring stepwise frequency change (8), the current band inversion strategy is technically simple by virtue of using only a single short adiabatic pulse. Since the transient NOE by band inversion is significantly amplified, it simplifies the process of evaluation of ATP relaxation parameters (σ and T₁ as described in Eq. [3]), as compared to the EBIT method which depends on an additional step of profiling magnetization change against inversion frequency (8). In this sense the band inversion method facilitates a straightforward application of NOE in the evaluation of ATP dynamic behavior (τ_c) and ³¹P T₁ relaxation mechanisms (see below).

In vivo ATP is motion-restricted

In comparison to freely diffusing ATP in solution, which is characterized by a short τ_c of 3 ×10^-10 sec (22), in vivo ATP in the normal human brain tumbles two orders of magnitude slower (τ_c = 9.1 × 10^-8 sec), indicating a motion-restricted environment for ATP. The slowdown of ATP tumbling in vivo may be attributed to the fact that some of the ATP molecules are bound to cellular macromolecules such as catalytic enzymes (ATPases) which are known (at least a subset of them) to be integrated on cellular membranes for their functions and thus are immobilized. Provided that the τ_c value of the immobilized macromolecules is in the order of 10^-7 sec and that the bound ATP is characterized by the same τ_c value, then the relative population of the macromolecular-bound ATP is estimated in the order of a few percentage points under dynamic equilibrium, with the remaining majority in a fast tumbling state as in solution. In other words, the observed NOE for ATP in vivo may reflect a transferred NOE, a well-known phenomenon for solutions containing small ligands and macromolecules as with ATP and ATPases (23).

Absence of intramolecular ³¹P NOE for solutions of ATP

For ATP, lengthening τ_c is critical for the detection of its NOE in vivo. It has long been noted that there is no β-ATP attenuation associated with γ-ATP perturbation for freely-tumbling ATP molecules in solution (9). The lack of such an intramolecular NOE effect for solution ATP is a surprise at first sight, given the equal size of cross-relaxation rate (σ) and dipolar T₁ relaxation rate (1/T_1,dip), i.e., NOE parameter 2 σT_1,dip ≅ 2, for any τ_c shorter than 1.5 ×10^-9 sec (Fig. 6b, blue trace). This implies that likely another mechanism may also contribute to T₁ relaxation. Indeed, when the CSA mechanism is taken into account, the curve of 2σT₁ versus Log(τ_c) reveals an experimentally undetectable NOE effect (∼ 0) for ATP with τ_c < 3.8 ×10^-8 sec. This is exactly the case for unbound free ATP in solution (τ_c ∼ 3 ×10^-10 sec).

CSA is the dominant mechanism for ATP T₁ relaxation

In a previous human brain ³¹P NMR study at 7T (18), we observed that the apparent T₁ relaxation time (T_1,app) of phosphate metabolites follows the order of triphosphate (ATP: ∼ 1 – 1.7 sec) < diphosphates (NAD, UDPG: ∼ 2 – 3 sec) < monophosphates (PCr, Pi, PDEs and PMEs: ∼ 4 – 7 sec). It appears that the ³¹P dipolar relaxation, a mechanism that is not taken into account in an early ATP T₁ study (19), may be a major mechanism in T_1,app of di- and tri-phosphates, given the presence of ³¹P-³¹P dipolar coupling in these polyphosphates but absent in monophosphates. However, the quantitative T₁ and σ analysis in this work indicates that CSA contributes 60% more than dipolar mechanism for β-ATP T₁ rate (1/T_1,CAS : 2/T_1,dip ≅ 0.6 : 0.4). For α- and γ-ATP, the ratio of 1/T_1,CAS : 1/T_1,dip is 0.76 : 0.24 and 0.77 : 0.23, respectively (Table 1). This implies that the polyphosphate backbone generates a large electronic shielding anisotropy at all ³¹P spins of ATP in vivo. The presence of a dominant CSA contribution to brain ATP T₁ is consistent with the findings from the skeletal muscle ³¹P ST study (7) and from the ³¹P NMR studies of ATP solutions (19,24).

Linear relationship between NOE effect and Log(τ_c)

Interestingly, there is a linear dependence of the NOE (2σT₁) on Log(τ_c), as described by σT₁ = -0.552 Log(τ_c) – 4.078, for τ_c ranging (6.2 – 20) × 10^-8 sec (Fig. 6b). The dynamics of ATP molecules in the human brain fall well within this τ_c range (Fig. 6b). The linear relationship σT₁ ∼ Log(τ_c) plus the intrinsically high detection sensitivity of ATP renders the ³¹P NOE effect a useful probe for measuring the dynamics of ATP in cellular microenvironments. The potential utility of this probe is especially promising, given the importance of ATP in energy metabolism and the implication of impaired energy metabolism in numerous metabolic disorders including diabetes, insulin resistance and several brain diseases (15,16).

Alternative Approaches

It should be stated that, as a three-spin system, ATP offers multiple options for measuring homonuclear ³¹P NOE. Co-inverting α-, γ-ATP together with PCr and Pi while observing β-ATP appears to be the most efficient and technically simplest strategy. For single band inversion, one can also choose to observe γ-ATP while inverting α- and β-ATP. Given that the T₁ value of γ-ATP is longer than that of β-ATP, one would expect an even larger transient NOE if γ-ATP is observed. However, two chemical exchange pathways, i.e., γ-ATP ↔ PCr and γ-ATP ↔ Pi, work against such a strategy. Furthermore, it is technically difficult to take advantage of these two chemical pathways by simultaneously inverting both α- and β-ATP frequency band ( -7 – -17 ppm) together with Pi and PCr frequency band (6 – -1 ppm), which requires a dual band inversion.

An alternative approach to band inversion is frequency-selective inversion; this later approach is less efficient as previously shown in the skeletal muscle (∼10%). In the current brain study, frequency selective inversion of β-ATP led to a signal of 8% for γ-ATP and 12% for α-ATP at 1 sec of delay (data not shown), about half or less of the magnitude produced by the band inversion method used. Another approach is to use frequency-selective saturation transfer with long pulse or pulse train. While this conventional technique is effective at generating substantial NOE in animal models with long duration of saturation (typically 5-9 sec), the use of prolonged RF irradiation makes it prone to the potential risk of high SAR exposure in humans and unwanted off-resonance and other effects as discussed in the literature (10). These factors need to be carefully taken into consideration for its application in the human brain.

For ATP ³¹P T₁ relaxation, we analyzed the mechanism in terms of ³¹P-³¹P dipolar and CSA contributions. Additional mechanisms, such as ¹H-³¹P dipolar relaxation, may contribute, thus the CSA contribution evaluated by Eq. [5] may represent the upper limit. The presence of ¹H-³¹P dipolar relaxation has been documented in literature for monophosphates (such as PC, PE, GPC and PGE) as indicated by the observation of significant ¹H→³¹P NOE enhancements (25-27). In contrast, the ¹H→³¹P NOE enhancement is minimally detectable at ATP if any (28, 29). However, to rule out the possible contribution of ¹H-³¹P dipolar mechanism to ³¹P T₁ relaxation, a study at another magnetic field (e.g. 3T) is needed as advised by Eq. [7]. It should be noted that, even though the ¹H-³¹P dipolar mechanism may affect the CSA estimate, theoretically it does not affect the evaluation of the ATP dynamic parameter τ_c, which is derived solely from σ (Eq. [4]).

There were potential technical limitations in this study that have to be considered. First, although the band inversion method is able to amplify the transient NOE and thus it improves the quantitative evaluation of ATP relaxation and dynamic properties, the approach of using long TR (25 sec) with a large dataset (with 8 data points) is time consuming (20 min). To be practical for clinical application, a time-saving data acquisition scheme is needed for example by use of short TR with fewer data points. Second, while the high SNR ³¹P spectra acquired by using pulse-acquire sequence and a surface coil enable us to confidently determine minor changes in ³¹P signal intensity, any potential regional heterogeneity in metabolism and relaxation behavior might be lost by such acquisition strategy, this needs to be addressed in future studies, for example, by using a volume coil in combination with a suitable spatial encoding technique. Third, although the fitting curves are consistent with the observed trends of the ³¹P magnetization changes, not all data points are perfectly fitted. The minor deviations, especially for those data points at prolonged inversion delays, may reflect the limit of a simplified “chemical solution” model in presenting the in vivo reality. The later is apparently presented with tissue heterogeneity and cellular compartmentalization, a feature that is not taken into account by Eqs [1-3]. Therefore more robust models need to be explored in future studies to capture such in vivo feature for better description of the experimental data. Finally, it remains to be illustrated whether the dynamic behaviors of ATP is related to cellular activities of ATPase and creatine kinase. It would be helpful to measure τ_c and k_ex of ATP using further improved strategy suitable for measurements at rest and under stimulation toward wider applications (30-33).

In summary, we have demonstrated that band inversion is an effective strategy to generate large NOE suitable for quantitative evaluation of cellular ATP dynamics in terms of correlation time τ_c. This capability is attractive as an alternative to the conventional ST strategy. Given currently there is a lack of non-invasive means to report on cellular microenvironment and its modulation by diseases and injuries in human brain, ATP could serve as a molecular probe for reporting information about cellular microenvironment and also cellular energy metabolism.