High-energy phosphate transfer in human muscle: diffusion of phosphocreatine

Abstract

The creatine kinase (CK) reaction is central to muscle energetics, buffering ATP levels during periods of intense activity via consumption of phosphocreatine (PCr). PCr is believed to serve as a spatial shuttle of high-energy phosphate between sites of energy production in the mitochondria and sites of energy utilization in the myofibrils via diffusion. Knowledge of the diffusion coefficient of PCr (D_PCr) is thus critical for modeling and understanding energy transport in the myocyte, but D_PCr has not been measured in humans. Using localized phosphorus magnetic resonance spectroscopy, we measured D_PCr in the calf muscle of 11 adults as a function of direction and diffusion time. The results show that the diffusion of PCr is anisotropic, with significantly higher diffusion along the muscle fibers, and that the diffusion of PCr is restricted to a ∼28-μm pathlength assuming a cylindrical model, with an unbounded diffusion coefficient of ∼0.69 × 10⁻³ mm²/s. This distance is comparable in size to the myofiber radius. On the basis of prior measures of CK reaction kinetics in human muscle, the expected diffusion distance of PCr during its half-life in the CK reaction is ∼66 μm. This distance is much greater than the average distances between mitochondria and myofibrils. Thus these first measurements of PCr diffusion in human muscle in vivo support the view that PCr diffusion is not a factor limiting high-energy phosphate transport between the mitochondria and the myofibrils in healthy resting myocytes.

phosphocreatine (PCr) serves as the main short-term energy reserve in muscle, heart, and brain. PCr generates ATP to fuel cellular processes, including ion transport and muscular contraction, via the creatine kinase (CK) reaction. The CK reaction reversibly transfers a phosphoryl group between PCr and ATP, with unphosphorylated Cr and ADP as the other reactants and k as the forward pseudo-first-order rate constant of the reaction: PCr + ADP + H⁺ ↔^k ATP + Cr_.

It was long ago hypothesized that the CK reaction serves as an intracellular energy shuttle, facilitating the transfer of high-energy phosphate from sites of de novo ATP creation in the mitochondria to sites of energy utilization, such as the myofibrils (4, 29, 45). The energy transfer mechanism in this PCr shuttle hypothesis involves 1) the creation of PCr at the mitochondria from the reverse CK reaction, 2) the diffusion of PCr to the point of use (including possible recycling through CK in the cytosol), and 3) the regeneration of ATP from ADP and PCr via the forward CK reaction to supply the needed energy (Fig. 1). Provided that the rate of ATP production via CK is manyfold higher than that generated by oxidative phosphorylation, the CK reaction could serve as a temporal-spatial energy buffer, supplying ATP when and where it is needed during periods of acute demand and/or stress.

Fig. 1.

Creatine kinase (CK) shuttle hypothesis. ATP is created in the mitochondria, and the reverse CK reaction transfers the phosphoryl group to creatine (Cr) to produce phosphocreatine (PCr). PCr then diffuses to the myofibrils, where it is converted back to ATP via the forward CK reaction to power muscle contraction. Cr diffuses back to the mitochondria to complete the cycle. D_Cr and D_PCr, diffusion coefficients for Cr and PCr.

Since PCr diffusion is central to the PCr shuttle hypothesis, knowledge of its diffusion coefficient (D_PCr) is key to full characterization of the shuttle, the application of reaction/facilitated diffusion models (19, 25, 29), and models of intracellular structures restricting diffusion (39). Indeed, the need for experimental measurements of metabolite diffusion was emphasized in a recent review (34). In tissue biopsies, PCr degrades in seconds, but PCr can be accessed in vivo by means of noninvasive phosphorus (³¹P) magnetic resonance (MR) spectroscopy (MRS). Because diffusion measurements are also possible with MR techniques that employ diffusion magnetic field gradients (41), noninvasive studies of metabolite diffusion are feasible, although they have been limited. Indeed, combining ³¹P-MRS with diffusion MR methodology has provided measures of D_PCr in excised strips of fish (19, 20, 24) and frog muscle (49, 50) and in the limbs of rats and rabbits (9, 32, 43). The average diffusion of PCr and Cr has also been measured by ¹H-MRS in isolated rat hearts (28). However, apart from a preliminary conference abstract (21), D_PCr has never been measured in human muscle.

In general, D_PCr will be anisotropic, for example, if cell shape and/or muscle fiber orientation imparts a preferred directionality to the diffusing species. Diffusion may also be restricted because of the presence of molecular-level barriers, such as organelles in the cytosol or cell membranes in living tissue that impede or restrict molecular motion after a certain distance is traversed (5, 9, 31, 32, 38, 39, 43). This restriction causes the apparent D_PCr to vary when it is measured at different diffusion times (t_diff).

In this work, we report the first ³¹P-MRS measurements of D_PCr in human calf muscle, including evidence for diffusion anisotropy, as well as restricted diffusion obtained by monitoring D_PCr as a function of t_diff. In the context of earlier ³¹P-MRS measurements of CK reaction kinetics in human calf muscle (7) and estimates of mitochondrial spacing in mammalian myocytes (17, 42, 44), knowledge about D_PCr provides important insights into the role that PCr diffusion could play in energy transfer via the putative PCr shuttle mechanism in human muscle.

METHODS

Diffusion MRS

The prevailing technique in MR encodes diffusion with a pair of magnetic field gradient pulses. This pulsed-field gradient method sensitizes the signal to the microscopic motion of the molecules, which translates into a phase dispersion, causing signal attenuation (41). This attenuation is modeled as a monoexponential decay

$$\mathrm{S}={\mathrm{S}}_0\text{exp}\hspace{0.17em}\left(-bD\right)$$ (1)

where S and S₀ are the diffusion-weighted and reference signals, respectively, D is the apparent diffusion coefficient, and b is a diffusion weighting factor determined by the gradient waveform. For two rectangular gradient pulses of strength G and duration δ, separated in time by Δ, and ignoring gradient rise time

$$b={\left(\gamma \mathrm{G}\delta \right)}^2\left(\Delta -\delta /3\right)$$ (2)

where γ is the MR gyromagnetic ratio.

Anisotropic diffusion is measured by repeating the MR diffusion measurements with the diffusion gradient applied along different directions. In addition, D may decrease as a function of t_diff = (Δ − δ/3) as a result of restricted diffusion. The mean distance to the barrier can be derived from a plot of D as a function of t_diff. In the case of muscle fiber, a cylindrical restriction gives rise to signal attenuation given by (9, 33, 43)

$$\text{log}\hspace{0.17em}\left(\mathrm{S}/{\mathrm{S}}_0\right)=-{\left(\gamma \mathrm{G}\delta \right)}^2\left(\Delta -\delta /3\right)D_{\mathrm{f}}$$ (3)

parallel to the cylindrical axis, and

$$\text{log}\hspace{0.17em}\left(\mathrm{S}/{\mathrm{S}}_0\right)=-2{\left(\gamma \mathrm{G}\right)}^2\times {\displaystyle \sum _{m=1}^{\infty }}\frac{2{\mathrm{C}}_m\delta -2+2e^{-{\mathrm{C}}_m\delta }+2e^{{\mathrm{C}}_m\Delta }-e^{{\mathrm{C}}_m\left(\Delta -\delta \right)}-e^{{\mathrm{C}}_m\left(\Delta +\delta \right)}}{{\mathrm{C}}_m^2{\alpha }_m^2\left(R^2{\alpha }_m^2-1\right)}$$ (4)

perpendicular to the long cylindrical axis. R is the radius of the cylinder, D_f is the unbounded diffusion coefficient, C_m = αm2D_f, and α_m represents the roots of J(α_mR) = 0, where J₁^′ is the derivative of the Bessel function of the first kind of order 1. While, in general, the orientation of the cylindrical compartment is unknown and may not coincide with any of the scanner's principal coordinate axes, the trace diffusion (D_tr), equal to the sum of the diffusion coefficients in three orthogonal directions, is rotation-invariant. Therefore, the average diffusion coefficient, D_av = D_tr/3, is fitted to the model represented by Eqs. 3 and 4. In practice, the infinite series of Eq. 4 was truncated to the first 20 terms, and a brute-force search in the two-dimensional space spanned by D_f and R (0.1 ≤ D_f ≤ 1.0 × 10⁻³ mm²/s and 1 ≤ R ≤ 100 μm) was performed to determine the best-fit parameters.

Our study design for measuring D_PCr in humans assumes the possibility of anisotropic and/or restricted diffusion: complete characterization of diffusion would require the measurement of the complete diffusion tensor using different diffusion gradient orientations and t_diff (26). To minimize losses in signal-to-noise ratio (SNR) during t_diff, diffusion MR experiments typically employ spin-echo or stimulated-echo pulse sequences. The stimulated-echo pulse sequence consists of three radio-frequency (RF) pulses. During the “mixing time” (TM) between the second and third pulses, the magnetization is in the longitudinal (z) direction and does not accumulate spin-spin relaxation (T₂) losses (Fig. 2). Thus this sequence permits measurements at long t_diff by increasing TM with minimal penalty in SNR for the diffusing metabolite. T₂ losses accumulate only during the echo period (TE), which is the sum of the periods between the first and second pulses and between the third pulse and signal acquisition (Fig. 2). Accordingly, TE is kept as short as possible. To enable D_PCr measurements with different t_diff up to ∼1 s for muscle PCr with T₂ of ∼300 ms and a longitudinal relaxation time (T₁) of ∼6–7 s at 3 T (12), we performed all experiments with the stimulated-echo pulse sequence. Experiments were repeated with the diffusion gradients directed along the three principal Cartesian axes to document anisotropy and determine the diagonal elements of the diffusion tensor, D_xx, D_yy, and D_zz.

Fig. 2.

Stimulated-echo depth-resolved surface coil spectroscopy (DRESS) pulse sequence for localized ³¹P diffusion spectroscopy. Diffusion gradients are shaded. RF, radio frequency; G_s, slice-selection gradients; G_d, diffusion gradients; TM, mixing time; TE, echo time, AQ, signal acquisition.

The main challenge with in vivo human ³¹P diffusion spectroscopy is low SNR because of the combined effect of the low sensitivity of ³¹P-MRS, the low concentration of PCr in muscle (∼25 μmol/g wet wt vs. ∼86 mol/kg for water protons), T₁ and T₂ relaxation effects, and the fact that diffusion weighting further attenuates the signal. Our first task was to develop a robust ³¹P diffusion MRS protocol that could be performed in an examination time suitable for human studies. We used a reduced receiver bandwidth, signal averaging, and depth-resolved surface coil spectroscopy (DRESS) (6) to localize the ³¹P-MRS signal of PCr to a relatively large, high-SNR, elongated slice compatible with the human calf.

Optimizing the D_PCr Measurements

The precision with which D_PCr can be measured depends on the SNR of the acquired PCr signals and on the selected b values and averaging scheme. For a given total scan time (T), we seek to optimize the diffusion MRS sequence repetition time (TR), the b values, and the fraction of time spent acquiring each b value.

To optimize TR for a partially saturated stimulated-echo experiment with three (π/2) RF pulses (3), we note, after averaging individual signals, that

$$\text{SNR}\propto \hspace{0.17em}\sqrt{\frac{\mathrm{T}}{\text{TR}}}{e}^{-\text{TM}/{\mathrm{T}}_1}{e}^{-\text{TE}/{\mathrm{T}}_2}\left[1-e^{-\frac{\text{TR}-\left(\text{TM}+\text{TE}/2\right)}{{\mathrm{T}}_1}}\right]$$ (5)

If TM and TE are much shorter than TR, the value of TR that maximizes the SNR (TR_opt) is TR

$${\text{TR}}_{\text{opt}}\approx 1.256{\mathrm{T}}_1$$ (6)

For muscle PCr with T₁ = 6–7 s, TR_opt = 7.5–8.8 s. If (TM + TE/2) ≈ TR, TR_opt is given by the solution to the following fixed-point problem

$$\text{TR}=\left[\text{log}\hspace{0.17em}\left(2\frac{\text{TR}}{{\mathrm{T}}_1}+1\right)+\frac{\text{TM}+\text{TE}/2}{{\mathrm{T}}_1}\right]{\mathrm{T}}_1$$ (7)

The solution progressively increases TR with increasing (TM + TE/2). However, for 50 ≤ TM ≤ 1,000 ms, optimizing TR using Eq. 7 results in 7.9 s ≤ TR_opt ≤ 11.0 s, which does not improve SNR by >3% compared with a fixed TR of 8 s. Therefore, TR = 8 s was used for all ³¹P studies.

In choosing b values, the optimum precision in D is arguably achieved by an MR experiment consisting of only two b values (13, 23). Devoting the total time for the experiment to averaging and improving the SNR of these two points is better than acquiring multiple b values with lower SNR. In the two-point experiment, one b value (b₀) is set to zero or very low (to dephase the transverse magnetization between stimulated-echo pulses) to maximize the SNR of the first signal measurement (S₀). Optimization then focuses on the high b value (b) with corresponding signal S. D obtained from the two measured signals is

$$D=\frac{1}{b}\left[\text{log}\hspace{0.17em}\left({\mathrm{S}}_0\right)-\text{log}\hspace{0.17em}\left(\mathrm{S}\right)\right]$$ (8)

For the case where b is unconstrained by practical system limitations (11, 22, 23, 48), the optimum value of b (b_opt) can be derived from the variance in D (σ_D²), given by an error propagation analysis of Eq. 8. For S₀ and S, each measured in a single acquisition

$${\sigma }_D^2=\frac{{\sigma }_{\mathrm{S}}^2}{b^2}\left\{{\left[\frac{\partial \text{log}\hspace{0.17em}\left({\mathrm{S}}_0\right)}{\partial {\mathrm{S}}_0}\right]}^2+{\left[\frac{\partial \text{log}\hspace{0.17em}\left(\mathrm{S}\right)}{\partial \mathrm{S}}\right]}^2\right\}=\frac{1}{b^2}\left[{\left(\frac{{\sigma }_{\mathrm{s}}}{{\mathrm{S}}_0}\right)}^2+{\left(\frac{{\sigma }_{\mathrm{s}}}{\mathrm{S}}\right)}^2\right]$$ (9)

where σ_s is the standard deviation in the noise of S₀ or S. If f is the fraction of the acquisition time spent acquiring the b₀ measurement and N is the total number of excitations in the combined experiment (the b₀ experiment + the b experiments), then

$${\sigma }_D^2=\frac{{\sigma }_{\mathrm{S}}^2}{b^{2}N{\mathrm{S}}_0^2}\left[\frac{1}{\mathrm{f}}+\frac{1}{\left(1-\mathrm{f}\right)e^{-2bD}}\right]$$ (10)

Denoting ψ = S₀/σ_s as the SNR per TR of the b₀ experiment and dividing by D yields the relative error in D or the reciprocal of the “diffusion-to-noise ratio” (DNR)

$$\frac{{\sigma }_D}{D}=\frac{1}{\text{DNR}}=\frac{1}{\sqrt{N}\psi bD}\sqrt{\left(\frac{1}{\mathrm{f}}+\frac{e^{2bD}}{\left(1-\mathrm{f}\right)}\right)}$$ (11)

Numerically minimizing this term gives the values of b and f that maximize the precision in D

$$b_{\text{opt}}=1.278/D\text{and}{f}_{\text{opt}}=0.218$$ (12)

With these values, the smallest error in D is

$${\frac{{\sigma }_D}{D}|}_{\text{opt}}=\frac{3.59}{\sqrt{N}\psi }$$ (13)

with TR = 8 s and a total scan time T = N·TR = 5 min, N ≈ 40, and precision in the calculated D is 14–23% for ψ = 2.5–4.

Unfortunately, as in the present case, the b_opt prescribed by Eq. 12 is often unachievable because of system limitations on G, TE, SNR, and/or eddy current artifacts. This constrains the choice to the highest practical b value (b_max). Minimization of Eq. 11 with respect to f with constant b = b_max shows that the experiment is optimized when

$$f_{\text{opt}}=\frac{1}{1+e^{b_{\text{max}\hspace{0.17em}}D}}$$ (14)

with an error in D given by

$${\frac{{\sigma }_D}{D}|}_{\text{con}}=\frac{1+e^{b_{\text{max}\hspace{0.17em}}D}}{\sqrt{N}\psi b_{\text{max}\hspace{0.17em}}D}$$ (15)

Compared with b_opt from Eq. 12, the error is now amplified by a factor

$$\beta =\frac{{\frac{{\sigma }_D}{D}|}_{\text{con}}}{{\frac{{\sigma }_D}{D}|}_{\text{opt}}}=\frac{1+e^{b_{\text{max}\hspace{0.17em}}D}}{3.59b_{\text{max}\hspace{0.17em}}D}$$ (16)

Figure 3 plots β as a function of the product bD. The curve is relatively flat around the optimal value of bD = 1.278, with the error doubling when b is reduced to 0.33/D.

Fig. 3.

Theoretical predictions of the factor β, by which the precision in the measurement of the diffusion coefficient D is degraded by use of a suboptimal b value with the optimal averaging scheme for the 2-point diffusion experiment, compared with use of the optimal b value and averaging scheme according to Eq. 12; β is given by Eq. 16. Values used for the experiments assuming D = 0.5 × 10⁻³ mm²/s (○) show a maximum loss in precision of β <1.5.

For the present study of human skeletal muscle, an expected D_PCr of ∼0.5 × 10⁻³ mm²/s based on animal studies (9, 32, 43) yields b_opt ≈ 2,600 s/mm² from Eq. 12. However, in practice, we were constrained in the worst case to b_max = 1,000 s/mm² at the shortest t_diff with TE = 80 ms and a maximum G = 31 mT/m. From Eq. 16 with bD = 0.5, this corresponds to a worst-case ∼1.5-fold increase in scatter in D compared with what would have resulted with the unconstrained optimum prescription of Eq. 12. Figure 3 also shows the values of bD used in the present study for the expected D_PCr of 0.5 × 10⁻³ mm²/s. In practice, the number of signal averages was increased for measurements of D_PCr at longer t_diff or TM values to compensate for T₁ losses and to maintain a comparable SNR and precision in D for all experiments, and experimental variations in f from f_opt affected the precision of D by <2%.

Experiments

All MR studies were done on a 3.0-T clinical MRI system (Achieva, Philips Healthcare, Best, The Netherlands) with a broadband MRS capability. All diffusion studies were preceded by scout ¹H-MRI, followed by main-field (B₀) shimming to optimize the field homogeneity in the region of interest (37).

Phantom studies.

The stimulated-echo DRESS diffusion pulse sequence (Fig. 2) was first validated with ¹H-MRS by measuring D for water in an agarose gel phantom doped with 100 mM Na₂PO₄ solution at room temperature using a ¹H transmit/receive circular 8-cm-diameter loop coil. Water has an isotropic unrestricted diffusion constant of ∼2.0 × 10⁻³ mm²/s at room temperature (30). D was measured along the three Cartesian axes, with TR/TE = 2,000/62 ms, b = 560 s/mm², b₀ = 10 s/mm², N/f = 2/0.5, and TM = 50, 100, 150, 200, 400, and 1,000 ms. Reference values for water diffusion were obtained by running a standard MRI spin-echo diffusion tensor imaging (DTI) protocol with a SENSE coil array (TR/TE = 2,000/62 ms, b = 560 s/mm², and N/f = 2/0.5) and calculating the average values of D_xx, D_yy, and D_zz in a central region of interest in the phantom.

To test whether anisotropic diffusion could be measured independent of the B₀ direction (z-axis) of the magnet and MR coils, a bundle of asparagus was scanned with the ¹H-MRS stimulated-echo DRESS diffusion protocol. Water diffusion in asparagus has been shown to be anisotropic (8). The scan parameters were the same as for the ¹H-MRS agarose gel experiment. After reorientation of the longitudinal axis of the fibers in the asparagus bundle parallel to each of the three principal Cartesian axes of the MRI scanner, the experiments were repeated using the scanner's whole body MRI coil and surface coils.

With ³¹P-MRS using a 17-cm transmit/8-cm receive ³¹P surface coil pair (12), the stimulated-echo DRESS diffusion protocol was next applied to measure the diffusion of the 100 mM P_i (D_Pi) present in a cylindrical gel phantom. D_Pi was measured as a function of t_diff = TM + 32 ms, with TM = 50, 100, 150, 200, 400, 700, and 1,000 ms, and diffusion gradients along the three Cartesian physical axes (TR/TE = 8,000/80 ms, b₀ = 10 s/mm², b = 1,000 s/mm², and N/f = 20/0.5 for TM ≤200 ms and 30/0.33 for TM >200 ms).

Human studies.

Eleven healthy volunteers [7 men and 4 women, mean 26 ± 4 (range 22–34) yr old] provided informed consent after receiving an explanation of the study and protocol and were enrolled in this study, which was approved by the Johns Hopkins Institutional Review Board on human investigation. Subjects were positioned supine, with the left calf muscle resting on the 17-cm transmit/8-cm receive ³¹P surface coil set, with the long axis aligned parallel with the z-axis of the scanner. The leg was immobilized, and subjects were instructed to remain still during the examination. After scout ¹H-MRI and shimming, the stimulated-echo ³¹P DRESS protocol was applied to localize the PCr signal to a 20-mm slice parallel to the coil at a depth of 30–60 mm from the coil surface. The RF pulses were optimized at the selected slice by application of a series of long-TR DRESS experiments with three to five different flip angles and interpolation of the results to determine the flip angle producing the maximum signal. The protocol parameters for D_PCr measurements were as follows: TR/TE = 8,000/80 ms; bandwidth = 500 Hz; TM = 50, 100, 150, 200, 400, 700, and 1,000 ms, except in three subjects, where the 700-ms TM scan was omitted because of scan-time constraints; G ≤31 mT/m for b = 1,000, 1,600, and 2,000 s/mm² for TM = 50, 100, and ≥150 ms, respectively; b₀ = 10 s/mm²; and N/f = 35/0.29, 40/0.25, 42/0.24, 47/0.26, 50/0.26, 55/0.27, and 55/0.27 for the seven TM values, respectively. The protocol was repeated with diffusion gradients directed along the three Cartesian axes to obtain D_xx, D_yy, and D_zz.

Data Analysis

MRS signals were quantified as peak areas using the circle-fit routine CFIT (14), and the apparent D was calculated from Eq. 8 for each gradient direction. D_av was calculated as follows: D_av = (D_xx + D_yy + D_zz)/3. The average diffusion distance was calculated from the Einstein-Smoluchowski equation (10): λ = 2(D_xx + D_yy + D_zz)t_diff 6D_avt_diff. The diffusion distance in the plane perpendicular to the muscle fiber was estimated as follows: λ_t = 2(D_xx + D_yy)t_diff. Paired t-testing was used to determine whether differences in D measured in each direction from the same subject were significant at P < 0.05.

D_av as a function of t_diff was fitted to the infinite-length, finite-radius cylindrical model for the restricting compartment given by Eqs. 3 and 4 (9, 33, 43) to obtain estimates of the radius of the cylindrical compartment and D_f. This analysis was performed with MATLAB (MathWorks, Natick, MA).

RESULTS

Phantom Studies

The D of water measured with stimulated-echo DRESS along the three directions at different t_diff is listed in Table 1. The D is isotropic and agrees with reported values for water measured at room temperature (30) and with the measurements obtained with diffusion tensor imaging (Table 1). The D of water measured in asparagus is reported in Table 2 and shows anisotropic diffusion, with a higher D along the fiber than in perpendicular directions. Orientation of the bundle of asparagus along the three axes did not significantly affect these measurements, confirming that neither the localization sequence nor the scanner biased the results in the B₀ or any other direction. Table 2 also shows that the same results are obtained with the stimulated-echo DRESS protocol whether the surface detection coil or the scanner's body MRI coil is used. The ³¹P diffusion measurements of P_i in the gel phantom are listed in Table 3. These results show no anisotropy or time dependence of D_Pi, indicating unrestricted isotropic diffusion, as expected.

Table 1.

Diffusion coefficient of water in inorganic phosphate agarose gel phantom

	D, ×10−3 mm2/s
	Dxx	Dyy	Dzz
Stimulated-echo DRESS
50 ms TM	2.06	2.04	2.11
100 ms TM	2.06	2.03	2.08
150 ms TM	2.04	2.01	2.06
200 ms TM	2.07	2.05	2.04
400 ms TM	2.08	2.05	2.03
1,000 ms TM	2.06	2.02	2.01
Mean ± SD	2.06 ± 0.01	2.03 ± 0.01	2.05 ± 0.04
DTI	2.02 ± 0.03	2.01 ± 0.03	2.02 ± 0.04

Values were obtained with stimulated-echo depth-resolved surface coil spectroscopy (DRESS) protocol and ¹H-MRS surface coil and with proton diffusion tensor imaging (DTI) using a conventional spin-echo sequence and the scanner's conventional SENSE ¹H phased-array coil. DTI values are means ± SD of values in a region of interest in the middle of the phantom. TM, mixing time; D_xx, D_yy, and D_zz, diagonal elements of the diffusion tensor.

Table 2.

Diffusion coefficient of water in a bundle of asparagus

	D, ×10−3 mm2/s
Fiber Direction	Dxx	Dyy	Dzz	D∥	D⊥
	Body coil
x	1.51	0.47	0.39	1.43 ± 0.08	0.50 ± 0.12
y	0.47	1.42	0.41
z	0.69	0.59	1.35
	Surface coil
x	1.50	0.38	0.36	1.53 ± 0.03	0.49 ± 0.15
y	0.42	1.52	0.42
z	0.62	0.72	1.56

Values were obtained with the stimulated-echo DRESS protocol and ¹H MRS surface coil and with the scanner's body coil. Diffusion experiment was repeated 3 times with the bundle oriented with the asparagus fibers parallel to the 3 Cartesian axes. Average of the diffusion coefficient (D) along (D_∥) and perpendicular to (D_⊥) fibers in the 3 sample positions are listed.

Table 3.

Diffusion coefficient of P_i in the agarose gel phantom

	DPi, ×10−3 mm2/s
TM, ms	Dxx	Dyy	Dzz
50	0.68	0.69	0.68
100	0.89	0.81	0.83
150	0.92	0.88	0.87
200	0.70	0.70	0.80
400	0.75	0.78	0.76
700	0.81	0.72	0.69
1,000	0.71	0.76	0.70
Mean ± SD	0.78 ± 0.10	0.76 ± 0.07	0.76 ± 0.07

Values were obtained with the stimulated-echo DRESS protocol and ³¹P-MRS surface coil set.

Human Studies

Figure 4 shows a representative set of spectra obtained from one volunteer. Comparable SNR is obtained at all TM values. The reduction in SNR with diffusion encoding along the z-axis is due to higher diffusion along that direction. Table 4 lists the mean D_av and diffusion distances in all subjects. Anisotropic diffusion is evident, with higher diffusion along the z-direction, substantially parallel to the fibers (Table 4), than in the x- or y-directions, at almost all diffusion times (P < 0.02 for D_zz vs. D_xx and D_zz vs. D_yy, except at TM = 700 ms, where P = 0.07).

Fig. 4.

Scout images [A; axial (top), sagittal (bottom left), and coronal (bottom right)] and a typical stimulated-echo DRESS data set (B) acquired from a 20-mm-thick slice in the left leg of a volunteer. Spectra show the PCr peak and are scaled identically. Spectra were zero-filled to 2,048 points, filtered with a 5-Hz exponential filter, zero-order phase-corrected, and displayed with the same scale in a 200-Hz window. S₀, reference signal; S_x, S_y, and S_z, signals in x, y, and z planes.

Table 4.

Diffusion coefficient of PCr in calf muscle

	DPCr, ×10−3 mm2/s
TM, ms	Dxx	Dyy	Dzz	Dav	λ, μm	λt, μm
50	0.47 ± 0.12	0.52 ± 0.20	0.88 ± 0.33	0.62 ± 0.17	17	13
100	0.46 ± 0.11	0.42 ± 0.09	0.63 ± 0.16	0.50 ± 0.10	20	15
150	0.44 ± 0.09	0.45 ± 0.14	0.65 ± 0.16	0.51 ± 0.10	24	18
200	0.47 ± 0.07	0.46 ± 0.12	0.60 ± 0.11	0.51 ± 0.07	27	21
400	0.43 ± 0.10	0.38 ± 0.07	0.56 ± 0.09	0.45 ± 0.07	34	26
700	0.38 ± 0.07	0.35 ± 0.08	0.47 ± 0.08	0.40 ± 0.06	42	33
1000	0.29 ± 0.10	0.31 ± 0.08	0.45 ± 0.07	0.35 ± 0.05	47	35

Values are means ± SD of 11 healthy volunteers obtained with the stimulated-echo DRESS protocol and ³¹P-MRS surface coil set. Average diffusion distance (λ) and approximation of average diffusion distance in the plane perpendicular to the muscle fiber (λ_t) were calculated using the Einstein-Smoluchowski equation.

Figure 5 shows D_av from all subjects as a function of t_diff. D_av decreases with t_diff, indicating restricted diffusion. The fit of D_av to the cylindrical model for restricted diffusion results in D_f = 0.69 × 10⁻³ mm²/s and a radius of the restricting compartment of 28 μm. The fitted model is also plotted in Fig. 5.

Fig. 5.

Average diffusion coefficient (D_av) of PCr in the calf muscle vs. diffusion time (t_diff). Each point is average of 11 independent measurements (except at t_diff = 0.732 s, where n = 8); error bars are SDs. Solid line is best fit of the data to the restricting cylindrical model (Eqs. 3 and 4).

DISCUSSION

PCr Diffusion Anisotropy and Restriction

The results reported here are, to the best of our knowledge, the first measurements of PCr diffusion in human muscle. They show anisotropy and restricted diffusion with D_PCr starting, on average, at ∼0.6 × 10⁻³ mm²/s and decreasing to ∼0.3 × 10⁻³ mm²/s as a function of t_diff ≤ 1 s (Fig. 5). This result is consistent with prior ³¹P-MRS findings in rat and rabbit muscle in vivo, which show D_PCr between 0.7 and 0.2 × 10⁻³ mm²/s over the same range of t_diff (9, 32, 43). All these results are a little higher than those for excised fish and frog muscle studied at 20–25°C with D_PCr between 0.4 and 0.2 × 10⁻³ mm²/s (19, 20, 24, 49, 50). While the latter measures could be confounded by PCr degradation in vitro, correction for temperature alone would raise these measures of D_PCr to the same range of 0.6 to 0.3 × 10⁻³ mm²/s, based on either direct experimental data (9) or an indirect empirical expression applied to the diffusion of PCr as a solute (See Eq. 5 in Ref. 47). The one preliminary value for D_PCr of 0.65 ± 0.04 × 10⁻³ mm²/s in human muscle reported in 2003 (21) was for a single, albeit unclear, diffusion time and a single gradient direction perpendicular to the fiber axis. This value is a little higher than the earlier results for transverse diffusion in mammalian tissue, as well as our own values of D_xx and D_yy in Table 4.

Similar to the animal studies that investigated anisotropic diffusion (9, 24, 43), we also found that PCr diffusion is 24–87% higher along the muscle fiber than in the transverse direction. However, in human calf muscle, our observation of dependence of D_PCr on t_diff is consistent with a ∼30-μm restricting compartment radius compared with an 8- to 11-μm radius reported in two of the prior animal studies that employed the same cylindrical diffusion model (9, 43). A 44-μm compartment size for restricted PCr diffusion measured parallel to the rat leg was reported in a third study (32). Our estimate of ∼30 μm for the cylindrical restricting compartment size results from a slower decrease in D_PCr with increasing t_diff in humans (Fig. 5) than in the animal studies that reported an 8- to 11-μm diffusion distance to the restricting barrier. Physiologically, this would be interpretable as a longer diffusion distance in human muscle than in the two animal studies, perhaps reflecting species differences of scale and/or mitochondrial density (17, 42, 44).

Our estimated diffusion distance of ∼30 μm is comparable to myofiber radii of 25–50 μm (1, 18, 27, 35), which would be consistent with the sarcolemma serving as the ultimate lateral diffusion barrier, since PCr is exclusively intracellular. However, the t_diff dependence of D_zz in Table 4 is evidence for some restricted diffusion, even along the fiber direction, which was also seen in the prior animal data (24, 32, 43). It has been suggested that PCr diffusion may be restricted by intracellular organelles in the sarcoplasm (9, 24), such as the mitochondria, the sarcoplasmic reticulum, and/or other microstructures (24, 39), rather than by the sarcolemma. If significant, this would be expected to result in diffusion distances that are significantly smaller than the fiber radius and those observed here.

At the short end of the distance scale, the measurement of restricted diffusion distances in the <10-μm range requires experiments with shorter t_diff than can be accommodated by b_opt or b_max for our studies. However, we note that diffusion for PCr as a solute in aqueous solution can be estimated from Eq. 5 in Ref. 47 and is 0.91 × 10⁻³ mm²/s at 37°C, with the assumption that water viscosity = 0.7 cP and density = 1.0 g/ml. This is close to measured values for diffusion of PCr in aqueous solution of 0.74 × 10⁻³ mm²/s (9) and 0.81 × 10⁻³ mm²/s (32). These values place an upper limit for the free isotropic diffusion of PCr in tissue, at least in the absence of any facilitating mechanisms (29). In particular, the fact that the measured values of D_zz are approximately equal to this upper limit at the shortest t_diff studied (Table 4) means that significant intracellular restrictions to diffusion in the direction parallel to the fibers at shorter t_diff cannot exist. However, the lower initial values for D_xx and D_yy than for free D_PCr in aqueous solution would suggest that additional restrictions to lateral diffusion might well be resolvable with shorter-t_diff experiments.

Implications for CK Reaction

The apparent, bulk-tissue, time-averaged pseudo-first-order reaction rate constant (k) for the CK reaction previously measured by noninvasive ³¹P-MRS in healthy human calf muscle was ∼0.27 s⁻¹ (7, 36). This leads to an apparent half-life for PCr in the CK reaction of log(2)/k ≈ 2.6 s. Extrapolation of the fit to the cylindrical diffusion model in Fig. 5 for long diffusion times yields D_av for PCr of ∼0.28 × 10⁻³ mm²/s at 2.6 s. Thus, in a time equal to the half-life of PCr in the CK reaction, PCr would diffuse an average distance of ∼66 μm. This distance is very long compared with a mitochondria-myofibril distance of up to ∼2 μm based on mitochondrial separations (17, 44). Table 4 shows that most of that distance is, in fact, traveled in ∼1 s. On the basis of an extrapolated D of 0.08 × 10⁻³ mm²/s at 2.6 s perpendicular to the long axis in the cylindrical model, the lateral component of the diffusion distance is ∼29 μm (which approximates the radius of the model cylindrical restriction). This is still large compared with the mitochondria-myofibril distance and is on the order of the typical radius of human muscle fibers (25–50 μm).

Therefore, these first measurements of PCr diffusion in human muscle appear to support the view that, on the time scale of CK reaction kinetics, PCr diffusion is not a factor limiting the supply of PCr to CK reactions occurring virtually anywhere in the myocyte, including the transfer of high-energy phosphate between the mitochondria and myofibrils in accordance with the PCr shuttle hypothesis. Such observations are similar to those from studies of isolated bullfrog muscle in which D_PCr = 0.28 × 10⁻³ mm²/s, diffusion distance of PCr = 57 μm, and lifetime PCr = 5.8 s (based on k = 0.17 s⁻¹) in CK (50). This previous study (50) it was concluded that the energy shuttle hypothesis is not obligatory for energy transport between mitochondria and myofibrils. Although it did not include direct measurements of D_PCr, another ³¹P-MRS study of high-energy phosphate diffusion in crab muscle concluded that the interaction between mitochondrial ATP production rates, ATP consumption rates, and diffusion distances is not particularly close to being limited by intracellular metabolite diffusion (25).

While our data support and extends this view to healthy human muscle at rest, it behooves us to also consider periods of peak energy demand for muscular contraction, wherein the PCr shuttle's putative role as a temporal-spatial energy buffer may come into play. A high jump, for example, requires in a period of ∼0.2 s a peak power that is some 15 times larger than the maximal aerobic energy capacity, which must therefore be supplied by PCr and CK (2). Even in this case, the assumption that D_av = 0.5 × 10⁻³ mm²/s for PCr at the shorter t_diff (Table 4) yields a diffusion distance of ∼26 μm in 0.2 s, which should not limit delivery of PCr over mitochondrial-myofibril distances of several micrometers in healthy muscle.

Study Limitations

In the present study, measurement of D_PCr and investigation of the effects of anisotropy and restricted diffusion in human muscle in vivo imposed a number of practical limitations. 1) Use of the stimulated-echo DRESS sequence, while providing a large voxel with high SNR, does result in poorly defined boundaries in the plane parallel to the surface detection coil. Nevertheless, the lower leg extended well beyond the detection coil without a significant change in muscle fiber orientation and yielded results consistent with prior animal studies (9, 32, 43). 2) We were unable to perform the diffusion sequence with the optimal diffusion parameter set (Eq. 12) because of system constraints on b. This resulted in some loss of precision for the shorter-t_diff measurements (Fig. 5). 3) Diffusion measurements are sensitive to motion and gradient eddy currents, which cause additional signal dephasing that is potentially reflected by artificially high D measurements. Our phantom studies (Tables 1–3) were designed to rule out eddy current effects, and subject motion was limited by immobilization of the leg. In the future, motion might also be corrected by phasing each of the N signals prior to averaging (15); unfortunately, the SNR of individual acquisitions was too low to allow this in the present study. 4) The protocol with seven t_diff measurement times and three gradient orientations takes ∼2 h to complete. Future studies would benefit from focusing on a particular t_diff and/or gradient orientation combination. 5) It was not possible to measure the diffusion coefficient of ATP (D_ATP) with the current protocol because of ATP signal loss associated with limitations in b, the long TE, and J coupling (Fig. 4).

Finally, it should be noted that PCr's participation in the CK reaction could affect D_PCr measurements to the extent that a fraction of the PCr signal generated from γ-ATP via the reverse reaction will have diffused for some portion of t_diff as ATP, and not as PCr. Because the MRS pulses are not spectrally selective, all species undergoing chemical exchange, both PCr and γ-ATP, are excited and “tagged” for diffusion. Given that CK is in equilibrium, the PCr signal depleted by the forward reaction is regenerated from γ-ATP via the reverse reaction. For a CK reaction rate k = 0.27 s⁻¹ (7, 36), ∼0.4 × 27% = 11% of the PCr will exchange with γ-ATP during a typical experimental t_diff of, e.g., 0.4 s. If it is assumed that the time at which γ-ATP is converted to PCr is distributed uniformly during t_diff, then 11% of the D_PCr measurement will comprise D_av (D_PCr + D_γ-ATP)/2, resulting in a 5% contamination of the D_PCr measurement by D_γ-ATP due to exchange. The contamination is proportionately less for shorter t_diff and, hence, is not significant over the range of t_diff that primarily determines the restricting compartment size or D_f.

Future Directions

It would be important to extend measurements of PCr diffusion in muscle to metabolic disorders involving energy transfer and to cytoarchitectural disorders, although the present work suggests that D_PCr would have to change rather dramatically to affect the spatial transfer of high-energy phosphate between mitochondria and myofibrils. In the myocardium, which has a higher density of mitochondria than skeletal muscle and a higher energy demand, significant reductions in the forward CK rate of ATP supply have been found in the failing human heart (40, 46). Although measuring diffusion in the heart is difficult because of its motion (16), elucidating whether reduced CK energy transfer in heart failure is due to altered PCr diffusion or other biochemical defects may provide useful insights for targeting therapy.

GRANTS

This study was supported by National Heart, Lung, and Blood Institute Grants R01 HL-056882 and HL-61912 and by funds from the Clarence Doodeman Endowment and the Russel H. Morgan Professorship.

DISCLOSURES

M. Schär is an employee of Philips Healthcare and is a visiting scientist at Johns Hopkins University.