Effects of Barrier-Induced Nuclear Spin Magnetization Inhomogeneities on Diffusion-Attenuated MR Signal

Abstract

The spatial distribution of the transverse nuclear spin magnetization, appearing in a single compartment with impermeable boundaries in a Stejskal-Tanner gradient pulse MR experiment, is analyzed in detail. At short diffusion times the presence of diffusion-restrictive barriers (membranes) reduces effective diffusivity near the membranes and leads to an inhomogeneous spin magnetization distribution (the edge-enhancement effect). In this case, the signal reveals a quasi-two-compartment behavior and can be empirically modeled remarkably well by a biexponential function. The current results provide a framework for interpreting experimental MR data on various phenoma, including water diffusion in giant axons, metabolite diffusion in the brain, and hyperpolarized gas diffusion in lung airways.

Numerous studies of the incoherent displacement (diffusion) of water in brain tissue and other biological systems have provided important information on structure and function (see, e.g., special issues of NMR in Biomedicine (1,2) and numerous references therein). Because of hardware restrictions, the diffusion times (Δ) in such experiments are 20 ms or longer. Based on the typical apparent diffusion coefficient (ADC) of water in tissue (ADC ~1 μm²/ms), the characteristic diffusion distance (a_c = [ADC · Δ]^1/2) traveled by a water molecule during time Δ is 4–5 μm, which implies numerous encounters with cell boundaries. The MR diffusion signal is thus greatly influenced by boundaries to displacement.

The ADCs obtained in such experiments, and the anisotropies of those ADCs, are averaged over thousands of cells contained in an imaging voxel. To obtain information on single-cell structure, studies have been performed on excised single-cell giant squid axons (3), excised lamprey spinal cord (4), and the Xenopus oocyte (5). One advantage of these model systems is their large diameter (~300 μm for the giant squid axon, ~40 μm for the Mauthner axon in the lamprey spinal cord, and ~1000 μm for the oocyte). Thus, a characteristic free diffusion distance that is short compared to the cell diameter can be achieved. A number of important observations have resulted from these single-cell studies. Takahashi et al. (4) measured a signal from a voxel in the center of the axon and found no diffusional anisotropy, whereas Beaulieu and Allen (3) measured the signal from the whole axon and observed a slight anisotropy. Sehy et al. (5) demonstrated a minimal anisotropy but strong biexponential signal behavior from the intracellular space of the oocyte, which is rich in microscopic lipid droplets.

At a different extreme of diffusion MR studies, lung imaging with hyperpolarized gases also suffers from the effects of numerous encounters with boundaries, albeit on a larger scale than that associated with liquid water displacement. The typical size of restrictions in healthy lungs is about 0.3 mm. Given that the diffusion coefficient of ³He gas in air is about 0.09 mm²/ms, a diffusion time Δ less than ~1 ms would be required to achieve an a_c smaller than the compartment size. The theory of diffusion MR signal attenuation in lungs, which quantifies the effect of displacement boundaries, was recently developed based on a model of uniformly oriented, alveolar-covered lung airways (compartments) (6).

To better understand the characteristics of the diffusion-attenuated MR signal arising from the complex structure of whole tissue, it behooves us to appreciate the characteristics of the signal arising from a single compartment or cell. As a first step, a theory that deals with the spatial distribution of magnetization inside an idealized cell-like single compartment (not only the net signal) should be developed. It is well known from MRI studies (7–12) and a direct solution of the Bloch-Torrey equation for a Stejskal-Tanner experiment (13) that the presence of diffusion-restrictive barriers reduces effective diffusivity near the barriers and leads to an inhomogeneous magnetization distribution (the edge-enhancement effect). An inhomogeneous distribution of the transverse magnetization similar to that obtained in Ref. 13 was recently confirmed by numerical simulations in Ref. 14. The physical origin of this phenomenon is rather simple: again considering an idealized single compartment, for sufficiently short diffusion times, when the characteristic diffusion length a₀ = (D₀Δ)^1/2 is small compared to the system size a, the diffusion of molecules located far from the boundaries can be considered as unrestricted, and the decay of the transverse magnetization of these molecules in a Stejskal-Tanner gradient pulse MR experiment can be described by the effective “free” diffusion constant D₀. The diffusion of molecules located near the boundaries (at distance ~[D₀Δ]^1/2) is restricted due to encounters with the boundaries. Here, the net MR signal decay can be described by the ADC value D₁, D₁ < D₀. Note that the effective intracellular “free” diffusion constant D₀ may be substantially smaller than the “true free” diffusion constant due to the presence of intracellular barriers (e.g., elements of the cytoskeleton, mitochondria, and endoplasmic reticulum, etc.). Because both the characteristic cell size a and the diffusion length of interest a₀ are much bigger than an effective length scale associated with the intracellular barriers, the intracellular media can be considered as homogeneous, and can be described by an effective “free” diffusion coefficient D₀.

In the present communication, we provide a detailed theoretical analysis of the MR signal in two models of a single-compartment system with an impermeable boundary. In particular, we show that under certain conditions, the spatial distribution of the transverse magnetization can be substantially inhomogeneous and result in a biexponential behavior of the net signal, as well as cause a spatially dependent anisotropy of the diffusion coefficient.

METHODS

Our analysis is based on a solution to the Bloch-Torrey equation for the local transverse magnetization σ(r) in the presence of a constant field gradient. Such a solution has been found in a “close-to-analytic” form in the framework of the random-walk approach (13). The solution is expressed in terms of the expansion into a series of eigenfunctions of the diffusion equation (with specific boundary conditions), with coefficients presented in the form of matrix products (the main formulas are sketched in the Appendix, and details can be found in Ref. 13). This formalism makes it possible to obtain the spatial distribution of the local transverse magnetization σ(r), as well as the net signal S, with any prescribed accuracy. In the following, we use this method to calculate the local transverse magnetization and the net signal.

It should be noted that a matrix dimensionality M appearing in the solution is determined by the number of eigenfunctions involved in the eigenfunction expansion. To describe magnetization inhomogeneities over a characteristic size scale l that is much smaller than the system size a, high-order eigenfunctions must be involved (with ~a/l ≫ 1 nodes). Thus, if we are interested in the fine-scale structure of the magnetization distribution in the case of a₀ ≪ a (short diffusion time), the required matrix dimensionality turns out to be very high and the numerical calculations become time-consuming. In consideration of this, we restricted our numerical calculations to M = 125, which yields an accuracy of 0.5% (or better) for a₀/a ≤ 0.0125 in the one-dimensional (1D) model and a₀/a ≤ 0.1 in the 2D model.

The detailed spatial structure of the magnetization distribution depends on the specific pulse sequence used, and particularly on the b-value. For the standard Stejskal-Tanner pulse sequence with a rectangular bipolar gradient waveform, the latter is equal to

$$b={(\gamma G\delta )}^2\left(\mathrm{\Delta }-\frac{\delta }{3}\right),$$ [1]

where γ is the gyromagnetic ratio, G is the field gradient strength, δ is the pulse duration, and Δ is the diffusion time (time between gradient pulse centers). In the following section we consider two limiting cases of such a sequence: 1) a Hahn spin echo, when Δ = δ, b = (γG)²Δ³/3 (corresponding results will be referred as SE); and 2) a narrow pulse approximation, when δ ≪ Δ, b = (γGδ)²Δ (corresponding results will be referred as NP).

RESULTS

Diffusion Between Two Infinite Planes

We start from 1D diffusion between two impermeable infinite planes separated by a distance 2a. This model allows us to clarify the major physical effects of restricted diffusion. In some cases, such a model can be applied to describe the diffusion-attenuated MR signal from extracellu-lar space, because the distance between cells is typically much smaller than the cell itself. In Fig. 1 we show the spatial distribution of the magnitude of the local transverse magnetization |σ(x)|, following a standard Tanner-Stejskal pulse sequence. The field gradient is applied perpendicular to the boundary planes and the b-value is fixed, bD₀ = 1. The different curves in Fig. 1 correspond to different values of the ratio α = a₀/a = (D₀Δ)^1/2/a that result from different diffusion times Δ. As we see, for α < 0.3 there is a substantial area in the center part of the system where the quantity |σ| is small and flat, whereas at the edges of the compartment it is substantially higher (the edge-enhancement effect). For α < 0.3, the values of |σ| in the center (σ₀) and the edges (σ₁) are independent of α (see Fig. 2). The size of the region where |σ(x)| ≈ σ₀ increases with decreasing α, and the transition from σ₀ to σ₁ becomes more abrupt. The magnetization in the center of the 1D system, where diffusion can be considered as unrestricted, is completely determined by the b-value: σ₀ ≃ exp(−bD₀) ≈ 0.368 for bD₀ = 1. Therefore, if a voxel size is chosen to be small enough so as to cover only this central part of the 1D system, the MR signal from this voxel decays as a monoexponential with ADC ≈ D₀. The magnetization in voxels near the edges can be described by a net ADC D₁ < D₀. Note also that in this regime, α < 0.3, molecules located near one boundary do not encounter the second boundary, and therefore the value of D₁ is independent of the system size. Therefore, the value of σ₁ in this regime is also determined only by the b-value: i.e. for bD₀ = 1 we obtain σ₁ = 0.66 for SE and σ₁ = 0.71 for NP.

FIG. 1.

The amplitude of the local magnetization |σ| as a function of the reduced coordinate x/a in the 1D model for the SE (solid lines) and NP (dashed lines) signals at different values of the parameter α = a₀/a = (D₀Δ)^1/2/a (identified by numbers near the corresponding curves). The b-value is fixed, bD₀ = 1.

FIG. 2.

The amplitude of the local magnetization |σ| in the 1D model as a function of α = a₀/a at two characteristic points: in the center, σ₀ = |σ(0)|, and at the edge, σ₁ = |σ(x = a)|. The b-value is fixed, bD₀ = 1. Solid line: SE signal; dashed line: NP.

As Δ → 0 (a₀/a → 0), the diffusion of practically all of the spins is observed as unrestricted, and the central area of the system, corresponding to the fast-diffusion component, covers essentially the whole system. With increase in Δ (the ratio a₀/a increases), the central region narrows, whereas the region corresponding to the slow-diffusion component broadens. As the number of spins encountering the boundaries increases with time as Δ^1/2, it should be expected that the “volume fraction” of the slow-diffusion component also scales as Δ^1/2 (see below). For a ≤ 3a₀, the transition between the center and edge regions becomes smooth, and therefore the concept of a division of the system into two components or two “quasi-compartments” becomes meaningless, although for a > 2a₀ the value of σ₁ remains practically unchanged. For a < 2a₀, when all of the spins encounter the boundaries, the local magnetization becomes uniform and increases with increasing a₀/a (the motional narrowing regime).

Because for a₀/a ≪ 1 the transverse magnetization spatial distribution is strongly inhomogeneous (quasi-two-compartment), one might expect that the net signal from the whole system can be approximated in the form of the biexponential function

$$\stackrel{\sim }{S}=\zeta exp(-b{D}_1)+(1-\zeta )exp(-b{D}_2),$$ [2]

where ζ is the “volume fraction” of the molecules near the boundaries. In Table 1 we provide the results of fitting the net signal (as a function of b-value in the interval b ∈ [0, 2]), calculated in the 1D model of restricted diffusion, to the function (Eq. [2]) for different α = a₀/a (fitting parameters: D₁, D₂, and ζ). The biexponential function (Eq. [2]) fits the signal S(b) extraordinarily well, with statistic χ² < 10⁻⁹. According to these results, with α decreasing, the parameter D₁ monotonically increases and then saturates at D₁ ≈ 0.22D₀ for the SE signal and D₁ ≈ 0.30D₀ for the NP signal, the parameter D₂ monotonically tends to D₀, and the amplitude ζ monotonically decreases. The fitting parameters D₁ and D₂ can be associated with the ADCs of the slow- and fast-diffusion quasi-compartments discussed above. It can be readily seen from the data in Table 1 that for sufficiently small α (i.e., for sufficiently short diffusion time Δ), this dependence is linear:

Table 1.

Fit of the Net Signal S(b) to the Bi-Exponential Function [2], 1D Model

a0/a	ζ		D1		D2
	SE	NP	SE	NP	SE	NP
0.3	0.121	0.109	0.114	0.206	0.801	0.844
0.25	0.122	0.106	0.151	0.236	0.849	0.880
0.2	0.110	0.095	0.175	0.257	0.890	0.912
0.15	0.090	0.078	0.193	0.272	0.924	0.939
0.1	0.064	0.055	0.207	0.284	0.954	0.962
0.05	0.034	0.029	0.218	0.294	0.979	0.982
0.02	0.014	0.012	0.222	0.299	0.992	0.993

$$\zeta \simeq k_1\alpha ,$$ [3]

where k₁ ≃ 0.69 is for the SE signal and k₁ ≃ 0.59 is for the NP signal. We can interpret the fitting parameter ζ as the “volume fraction” of the slow-diffusion quasi-compartment. From the data in Table 1, we also find that the fitting parameter D₂, which is interpreted as the ADC of the fast-diffusion quasi-compartment, for small α tends to the free diffusion coefficient D₀ according to the linear law:

$$D_2\simeq D_0·(1-k_2\alpha ),$$ [4]

where k₂ ≃ 0.42 is for the SE signal and k₂ ≃ 0.35 is for the NP signal.

For α > 0.3, when division of the system into two quasi-compartments becomes meaningless, fitting of the signal to the biexponential function also fails.

Another description of the MR diffusion signal is based on a traditional evaluation of the overall system net ADC, D̄, according to an assumed monoexponential (free diffusion) b-dependence of the signal:

$$S=S_{0}exp(-b\overline{D}).$$ [5]

However, restrictions cause the signal to depend on the b-value in a non-monoexponential manner. The ADC, D̄, thus introduced by Eq. [5] depends not only on the ratio α = a₀/a, but (because Eq. [5] is the incorrect model function) also operationally on the b-value as well. For a fixed b-value (bD₀ = 1), the dependence of D̄ on α is shown in Fig. 3 (curves 1). This graph plots the quantity, r⁻¹ = D̄/D₀, which is reciprocal to the anisotropy of the diffusion coefficients, r = D₀/D̄ (recall that the ADC for the field gradient applied parallel to the planes coincides with the free diffusion coefficient D₀). As expected, for short times Δ, when α ≪ 1 and the major population fraction of molecules diffuse freely, D̄ is close to the free diffusion coefficient D₀ and r ≈ 1. Note that D̄ is significantly smaller than D₀ even for rather small values of a₀/a (e.g., for a₀/a = 0.1, in the SE signal D̄/D₀ = 0.88, which corresponds to an anisotropy r = 1.13). With α increasing, D̄ decreases and tends to zero at long times Δ (r →∞), when α ≫ 1 (the motional narrowing regime).

FIG. 3.

The global ADC D̄(curves 1) and “local” ADC D̃ in the center region (curves 2) normalized to D₀ as functions of α = a₀/a in the 1D model. The b-value is fixed, bD₀ = 1. Solid line: SE signal; dashed line: NP.

Recall that the signal discussed immediately above corresponds to the net signal from the entire system, and thus the ADC D̄ (Eq. [5]) characterizes the system as a whole. However, if the voxel in an imaging experiment is small and covers only a part of the system, the ADC calculated by means of the expression similar to Eq. [5] (say, D̃) can be substantially different from D̄. Obviously, such a “local” ADC depends on a voxel’s size and its position with respect to the compartment’s boundaries. In particular, if the voxel covers only the central part of the system, where the transverse magnetization spatial distribution for sufficiently small α is practically uniform, σ(x) ≃ σ₀ (see Figs. 1 and 2), the “local ADC” is

$$\stackrel{\sim }{D}\simeq -\frac{1}{b}ln{\sigma }_0.$$ [6]

The dependence of D̃ from Eq. [6] on α = a₀/a is shown in Fig. 3 by dashed lines. Like the global ADC D̄, the “local” ADC D̃ is practically equal to D₀ for small α and tends to zero in the motional narrowing regime when α ≫ 1. In contrast to the global ADC D̄, the quantity D̃ practically coincides with D₀ over a broad range of α, especially for the NP signal (up to α ~1). This implies that in high-spatial-resolution MR experiments with a small voxel covering only the central part of the single compartment system, the calculated ADC should be equal to D₀ and the system should exhibit no diffusion coefficient anisotropy(r = 1). Whereas the global ADC for the same values of b and α differs from D₀ and the anisotropy is observed, r ≠ 1.

Diffusion Within a Cylinder

Next, we consider a model in which the diffusion of water molecules is restricted inside an infinitely long cylinder of radius a. If the field gradient is applied perpendicular to the cylinder axis, the problem can be considered as two-dimensional (2D) and the local magnetization is a function of the 2D radius-vector ρ. Figure 4 shows the radial (a) and angular (b) dependence of |σ| for bD₀ = 1 (SE signal) and different values of the parameter α = a₀/a (the azimuth angle ψ is reckoned from the direction of the field gradient). The radial dependence of |σ|, which is displayed in the direction of the field gradient (ψ = 0), is similar to that in the 1D model discussed above: for small α, there is a characteristic region of uniform magnetization in the central part of the cylinder where diffusion is practically unrestricted and |σ| ≃ σ₀ ≈ exp(−bD₀), whereas at the circumference of the cylinder the value of |σ| is substantially higher (the edge-enhancement effect). With increasing α, the central region of uniform magnetization decreases and the transition in |σ| between the central region and the circumference becomes smoother. At α ~0.5, the magnetization becomes practically homogeneous across the radial dimension for ψ = 0. The angular dependence of |σ| is shown at the boundary defined by the circumference ρ = a. As we can see, the decrease in transverse magnetization is strongest for ψ = π/2, where diffusion along the gradient direction is the least restricted (i.e., the restrictive boundary is parallel to the gradient direction).

FIG. 4.

The distribution of the amplitude of the transverse magnetization |σ| in the 2D model for the SE signal at different values of the parameter α = a₀/a (identified by numbers near the corresponding curves). The b-value is fixed, bD₀ = 1. a: Radial distribution. ρ/a is a reduced distance from the center, and the radial dependence is shown along the radius parallel to the gradient, ψ = 0. b: Angular distribution on the circumference, ρ = a. The azimuth angle ψ is reckoned from the direction of the field gradient.

The dependence of |σ| for the SE and NP signals on the parameter α (bD₀ = 1, fixed) is shown in Fig. 5 at three characteristic positions: in the center, σ₀ = |σ(ρ = 0)|, at the pole, σ₁ = |σ(ρ = a, ψ = 0)|, and at the equator, σ₂ = |σ(ρ = a, ψ = π/2)|. For α ≫ 1 (the motional narrowing regime), all of the curves practically coincide and |σ| tends to one with increasing diffusion time. In the short-time regime, α ≪ 1, the magnetization in the center, σ₀, and on the equator, σ₂, tends to the value characteristic of unrestricted diffusion, exp(−bD₀) = 0.368. The magnetization at the pole, σ₁, tends to the same values σ₁ = 0.66 for SE and σ₁ = 0.71 for NP as in the 1D model discussed above. The behavior of the quantities σ₁ and σ₂ can be readily explained: At very short times, Δ → 0, the diffusion of molecules near the boundary can be considered as that in a half-infinite space restricted by the tangential plane. Therefore, at the pole, where the field gradient is perpendicular to the tangential plane, diffusion in the gradient direction can be considered in the same way as the diffusion of molecules near the edge in the 1D model. At the equatorial point, the gradient is parallel to the tangential plane; thus, diffusion in the gradient direction is unrestricted, and σ₂ → σ₀.

FIG. 5.

The amplitude of the local magnetization in the 2D model as a function of α = a₀/a at three characteristic points: in the center, σ₀ = |σ(ρ = 0)|, at the pole, σ₁ = |σ(ρ = a, ψ = 0)|, and at the equatorial point, σ₂ = |σ(ρ = a, ψ = π/2)|, as a function of the parameter α. The b-value is fixed, bD₀ = 1. Solid line: SE signal; dashed line: NP.

In Table 2 we summarize the results of fitting the net signal in the 2D model to the biexponential function (Eq. [2]). As in the 1D model, in the small-α regime, when there is a region with homogeneous magnetization (i.e., at α ≤ 0.3), the signal can be empirically modeled by the function (Eq. [2]) extremely well (with χ² < 10⁻⁹), and the general behavior of the fitting parameters D₁, D₂, and ζ is similar to that in the 1D model. In particular, for α ≤ 0.2, the “volume fraction” ζ demonstrates an approximately linear proportionality to α. An exact linear dependence could be achieved for α < 0.1; however, as mentioned above, we restricted our calculations to the matrix dimensionality M = 125, which enabled us to achieve an accuracy of 0.5% in the 2D model for α > 0.1 (vs. α > 1/80 in the 1D model).

Table 2.

Fit of the Net Signal S(b) to the Bi-Exponential Function [2], 2D Model

a0/a	ζ		D1		D2
	SE	NP	SE	NP	SE	NP
0.3	0.0685	0.0642	0.105	0.207	0.730	0.791
0.25	0.087	0.079	0.185	0.277	0.796	0.842
0.2	0.092	0.082	0.238	0.323	0.854	0.886
0.15	0.083	0.073	0.276	0.355	0.902	0.923
0.1	0.064	0.055	0.311	0.378	0.942	0.954

In Fig. 6 we show the dependence of the total system ADC D̄ on the parameter α, calculated from the net signal S for bD₀ = 1 according to Eq. [5] (curve 1), as well as the quantity D̃ calculated by Eq. [6] (curve 2). The behavior of these quantities in the 2D model is similar to that in the 1D model: In the short-time regime, when α ≪ 1, D̄ and D̃ tend to the free diffusion coefficient D₀; D̃ approaches D₀ for a substantially broader range of α; in the motional narrowing regime, α ≫ 1, D̄ and D̃ tend to zero.

FIG. 6.

The ADC D̄(curve 1) and “local” ADC D̃ in the center region (curve 2) normalized to D₀ as a function of α = a₀/a in the 2D model. The b-value is fixed, bD₀ = 1. Solid line: SE signal; dashed line: NP.

Small-b Limit

If the b-value is small enough, bD₀ ≪ 1, both the bi- and monoexponential functions reduce to a simple linear function of b. In this regime there is a relationship between D̄ and the parameters of the biexponential function [2]:

$$\overline{D}\simeq D_2+\zeta (D_1-D_2),\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }b{D}_0\ll 1.$$ [7]

For small α, when Eqs. [3] and [4] are valid, the ADC is then a linear function of α:

$$\overline{D}\simeq D_0·\left[1-\alpha \left(k_1+k_2-k_1\frac{D_1}{D_0}\right)\right],\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }b{D}_0\ll 1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\alpha \ll 1.$$ [8]

For an ID case, using the values of the numerical coefficients k₁ and k₂, we obtain:

$$\overline{D}=D_0·(1-k_3\alpha ),\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }b{D}_0\ll 1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\alpha \ll 1,$$ [9]

where k₃ ≈ 0.95 for the SE signal and k ≈ 0.76 for the NP signal. The latter result practically coincides with the 1D analog of the short-time expansion of the ADC obtained by Mitra et al. (15) for NP, and by De Swiet and Sen (10) for SE. In our notation, the ADCs obtained in those studies can be written in a form similar to Eq. [9], with the coefficient $k_3^{\prime }$ rather than k₃:

$$\begin{array}{ll}{k}_3^{\prime }=\frac{4}{3{\pi }^{1/2}}\approx 0.75,\hfill & \text{NP}\hfill \\ k_3^{\prime }=\frac{32(2\sqrt{2}-1)}{35{\pi }^{1/2}}\approx 0.94.\hfill & \text{SE}\hfill \end{array}$$ [10]

Note that in the 1D case, the surface-to-volume ratio appearing in the general results of Refs. 10 and 15 is equal to 1/a. The numerical coefficient in the corresponding equations of those works should be tripled because in the system under consideration, the direction of the field gradient with respect to the boundaries is fixed and angular averaging is absent. Hence, the comparison of our results with those obtained in Refs. 10 and 15 enables us to infer that the short-time behavior of the ADC, similar to Eq. [9], is strictly valid under the additional condition bD₀ ≪ 1.

DISCUSSION

It is well known that the presence of barriers leads to a non-monoexponential b-value dependence of the MR signal. For instance, as shown in the 1D case of restricted diffusion by Cory and Garroway (16), the signal for the Stejskal-Tanner-pulse sequence with Δ ≫ a²/D₀ is equal to S = S₀ · [sin(qa)/qa]², q = γGδ. This expression is obviously nonexponential in b-value. Based on the exact solution to the Bloch-Torrey equation in the 1D model obtained by Stoller et al. (17), De Swiet and Sen (10) demonstrated that in the short-time interval, when a₀ ≪ a, there exist two different regimes in the dependence of the signal on the field gradient G: the signal crosses over from nearly free diffusion for small gradients when −ln(S/S₀) ~G² ~b to a so-called localization regime when −ln(S/S₀) ~G^2/3 ~b^1/3 for large gradients. The existence of these two regimes was experimentally confirmed in the model system by Hurlimann et al. (18). Mitra and Sen (19) demonstrated that the deviation in the diffusion propagator from a Gaussian due to the presence of barriers also leads to the deviation of signal behavior from monoexponential dependence on the b-value.

Our analysis revealed specific conditions under which the general non-monoexponential behavior of diffusion-attenuated MR signal manifests itself as a biexponential function. The current study presents a simple rationale for the biexponential behavior of the MR diffusion signal in a single cell with impermeable membranes. The behavior is caused by the inhomogeneous spatial distribution of the transverse magnetization at sufficiently short diffusion times. Obviously, this cause of “biexponentiality” of the signal (and non-monoexponentiality, in the more general case) is only one among many others that might be invoked (20 –26). Moreover, it can be expected to contribute significantly only under a rather strong condition: the diffusion time Δ must be short enough (or the system size a must be large enough) to satisfy the inequality a₀ = (D₀Δ)^1/2 ≪ a (see Figs. 1 and 4a, and Tables 1 and 2). This condition is met, for instance, in the giant squid axon. In the human brain, a typical axonal radius a is <1–2 μm; hence, for the water diffusion coefficient D₀ ~1 μm²/ms, a very short diffusion time Δ ~1 ms would be required for this effect to be significant. Typical diffusion times used in MR experiments on the human brain are much longer than 1 ms. For instance, for diffusion time Δ ~20 ms, only cells larger than 15–20 μm satisfy the condition a₀ < 0.3a. Also, for some brain metabolites (NAA, Cr, and Cho), the diffusion coefficient is much smaller than that of water (D₀~0.15 μm²/ms) (27). Therefore, for the same diffusion time Δ ~20 ms, the characteristic diffusion length is a₀ ~2 μm, and the inequality a₀ ≪ a can be satisfied for a > 6 μm.

In this study, we made an assumption that the single compartment’s boundaries are impermeable. In previous experiments on giant axons (3), an excised axon was immersed in paraffin oil to prevent water evaporation. Under these conditions, the membrane is obviously impermeable. Our theory can be also applied to lung airways (the alveolar-covered surface is impermeable for diffusing ³He atoms) and to some metabolites in the CNS (e.g., NAA), the diffusion of which is restricted within the cells. Note, however, that our results are also valid when the boundary membranes are permeable but the diffusion time Δ is much shorter than the characteristic time t_μ for exchange between, for example, intra- and extracellular compartments. This exchange time can be defined in terms of the membrane’s permeability μ (the dimensionality of μ is cm/s): t_μ = D₀/μ². If Δ ≪ t_μ, membrane permeability does not affect the inhomogeneous distribution of magnetization in the single compartment or cell, and the signal remains biex-ponential (if, of course, the other criteria discussed above are met).

In an experimental study of the Mauthner axon in the lamprey spinal cord (diameter ~40 μm), Takahashi et al. (4) analyzed ADC by means of high-resolution MRI with a voxel size of ~19 μm and found an absence of diffusion coefficient anisotropy for a diffusion time Δ = 11 ms. This result is in agreement with our theory, because for D₀ ≈ 0.98 μm²/ms and a = 40 μm (4), α ≈ 0.16 and the radial magnetization distribution has a clearly visible flat interval (see Fig. 4a). If the voxel covers only the center part of the axon, the measured ADC is the “local” ADC (in our terminology) and D̃ ≈ D₀. In another experimental study by Beaulieu and Allen (3) on the giant axon in the squid (diameter 2a ~300 μm), the parameter α was also small: diffusion time Δ ≈ 30 ms, D₀ ≈ 1.6 μm²/ms, and α = (D₀Δ)^1/2/a ≈ 0.047. However, an anisotropy of the diffusion coefficients r ≈ 1.2 was found. This result is also in agreement with our theory because in Ref. 3, the “global” ADC D̄ was measured, which is smaller than D₀, even for small α (see Fig. 6). Note also that according to Table 2, the volume fraction of the slow-diffusion quasi-compartment ζ at α = 0.047 should be smaller than 0.05, which is also in agreement with experimental data in Ref. 3.

APPENDIX

A method of solving the Bloch-Torrey equation for the magnetization distribution σ(r,t), as proposed in Ref. 13, is based on the random-walk approach for describing spin’s diffusion. From a mathematical point of view, it is similar to the multiple-pulse approximation proposed by Cal-laghan (28) for calculating the net signal. This method is based on dividing the trajectory of a spin into N small time intervals Δt, t = N · Δt, and using the propagator P(r, r′, Δt), which is a solution to the diffusion equation with boundary conditions specified for the geometry of a system, for describing the spin’s diffusion on each time interval. By making use of this method, we can derive expression for spin magnetization distribution σ(r,t) resulting from a Stejskal-Tanner experiment:

$$\begin{array}{c}\sigma (\mathbf{r},t)=\mathbf{F}\left(\frac{\mathbf{Q}}{2}\right)·\widehat{\mathbf{T}}(\mathbf{Q},\mathrm{\Delta }t)·\widehat{\mathbf{\Lambda }}(\mathbf{\Delta }-\delta )·{\widehat{\mathbf{T}}}^{†}(\mathbf{Q},\mathbf{\Delta }t)·{\mathit{\varphi }}^{†}\left(\mathbf{r},\frac{\mathbf{Q}}{2}\right)\\ \widehat{\mathbf{T}}=\widehat{\mathbf{\Lambda }}(\mathbf{\Delta }t)·{[\widehat{\mathbf{U}}(\mathbf{Q})·\widehat{\mathbf{\Lambda }}(\mathbf{\Delta }t)]}^{N-1}\end{array}$$ [11]

where Δt = δ/N, Q = γGΔt, the row vectors F and ψ, the matrix Û {U_kk′} and the diagonal matrix Λ̂ = diag{Λ_kk} are defined by their components

$$F_k(\mathbf{Q})=\frac{1}{V}{\int }_{V}d\mathbf{r}\hspace{0.17em}{u}_k(\mathbf{r})exp(i\mathbf{Qr}),$$ [12]

$${\phi }_k(\mathbf{r},\mathbf{Q})=\frac{1}{V}{u}_k(\mathbf{r})exp(i\mathbf{Qr}),$$ [13]

$${\mathrm{\Lambda }}_{kk}(\tau )=exp(-{\lambda }_k\tau ),$$ [14]

$$U_{k{k}^{\prime }}=\frac{1}{V}{\int }_{V}d\mathbf{r}\hspace{0.17em}{u}_k^{\ast }(\mathbf{r})u_{k^{\prime }}(\mathbf{r})exp(i\mathbf{Qr}),$$ [15]

where V is the system volume, ^† means the Hermitian conjugation, and u_k(r) and λ_k are the eigenfunctions and eigenvalues, respectively, of the diffusion equation. For the 1D geometry discussed in the main text,

$$u_k(x)={\eta }_{k}cos\frac{\pi kx}{2a},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\lambda }_k=\frac{D{\beta }_k^2}{a^2},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\beta }_k=\frac{\pi k}{2}$$ [16]

where the normalization factors are η₀ = 1 and η_k = √2 for k ≠ 0. For the 2D geometry,

$$u_k(\mathbf{\rho },t)\equiv u_{n\mu }(\mathbf{\rho },t)={\eta }_{n\mu }{J}_n\left({\beta }_{n\mu }\frac{\rho }{a}\right)exp(in\psi ),\hspace{0.17em}{\lambda }_k\equiv {\lambda }_{n\mu }=\frac{D{\beta }_{n\mu }^2}{a^2}$$ [17]

where the radius ρ and polar angle ψ define the 2D radius-vector ρ in the polar coordinate system, J_n(x) are the Bessel functions (n = 0, ±1, ±2, …), and β_nμ (μ = 0, ±1, ±2, …) are nonnegative roots of the equation J′_n(x) = 0. The normalization factors η_nμ are equal to

$${\eta }_{ns}={\left[\left(1-\frac{n^2}{{\beta }_{n\mu }^2}\right)·J_n^2({\beta }_{n\mu })\right]}^{-1/2}.$$ [18]

A numerical evaluation of the matrix form in Eq. [11] can be performed with any matrix-handling tool, such as Mathematica or MatLab. For this purpose, one should choose an appropriate time step Δt and a number of eigenfunctions involved (the dimensionality M of the matrices Û and Λ̂). As the matrix elements Λ_kk (Eq. [14]) decrease exponentially with λ_k, it is possible to restrict M and N to some comparably small value determined by the precision criteria.