The extracellular space (ECS) in the central nervous system (CNS) comprises ~20% of total tissue volume, consisting of a jelly-like matrix in which neurons, glia, and blood vessels are embedded. Diffusion of solutes and macromolecules in the ECS is vital for many CNS functions, including extrasynaptic intercellular communication, delivery of substrates and drugs to cells, elimination of metabolites, and buffering of extracellular ions (1,2). Altered ECS diffusion produced by brain edema, astrogliosis, brain tumor, and ischemia is therefore potentially detrimental (3–8).

Most studies of ECS diffusion have used the tetramethylammonium (TMA⁺) technique, which involves pulsed iontophoretic introduction of TMA⁺ and microelectrode measurement of decreasing TMA⁺ concentration as TMA⁺ diffuses away from the injection site (9). However, the TMA⁺ method is largely restricted to brain slices, requires micropipette invasion of the recording site, and can only detect TMA⁺. We recently developed a spot photobleaching approach to overcome these limitations, in which the ECS is fluorescently stained for in vivo measurement of diffusion by photobleaching (3).

Here, we report a novel adaptation to the photobleaching method, called “elliptical spot photobleaching”, to measure direction-dependent diffusion coefficients in anisotropic environments. Elliptical spot photobleaching was applied to determine the relative contributions of ECS geometry (“tortuosity”) versus viscous interactions between the probe and extracellular matrix (ECM)/plasma membranes (“viscosity”) in hindering diffusion in ECS. The role of ECS tortuosity versus viscosity highlighted by Rusakov and Kullmann (10) has been widely discussed (1,9,11) and modeled mathematically (10,12), but has not been subject to experimental evaluation. Our approach to measure anisotropic diffusion, as diagrammed in Fig. 1 A, was to introduce directionality into the photobleaching measurement using an elliptical spot produced by cylindrical excitation optics. After photobleaching an elliptical spot, fluorescent molecules diffuse into the bleached area from the surrounding region. The time evolution of fluorophore concentration C(x, y, t) is described by the diffusion equation: C/t = D_x·²C/x² + D_y·²C/y², where D_x and D_y are fluorophore diffusion coefficients in the x- and y-directions, respectively. If diffusion is greater in the y-direction than the x-direction (D_y > D_x), then fluorescence recovery is faster when the long axis of the ellipse is oriented in the x-direction. Recovery is independent of ellipse orientation when diffusion is isotropic (D_x = D_y). This two-dimensional (2D) analysis ignores the contribution of diffusion along the z-direction (D_z) to fluorescence recovery. In the Supplementary Material, we solve the problem in three-dimensional (3D) and show that ignoring D_z does not appreciably influence the estimates of D_x and D_y.

FIGURE 1 Theory and apparatus for elliptical spot photobleaching. (A, top) Ellipse with radii wx and wy orientated along the x and y axes, respectively. After photobleaching fluorophores with diffusion coefficients Dx and Dy (Dx < Dy shown) enter the elliptical (more ...)

We measured the diffusion of an inert macromolecule in the ECS of spinal white matter, reasoning that ECS diffusion along axons should depend primarily on viscosity and not on tortuosity, whereas diffusion perpendicular to axons is reduced by both viscosity and tortuosity. Diffusion was also measured in the ECS of cerebral cortex, which is hindered by both viscosity and tortuosity. We found that both viscosity and tortuosity slow diffusion: viscosity reduces diffusion ~1.8-fold (compared to water); tortuosity reduces diffusion further by approximately fivefold across white matter tracts and by ~1.8-fold in CNS gray matter.

An analytical solution to the problem of relating diffusion coefficients to elliptical spot photobleaching data is derived below for 2D diffusion. A Monte-Carlo numerical solution for the 3D case revealed that the 2D analysis provides an excellent approximation to the 3D problem (see Supplementary Material and Fig. 3 A). We have therefore used the 2D analysis to interpret the data. Anisotropic diffusion in 2D is described by,

(1)

(2)

FIGURE 3 Determination of directional diffusion coefficients from fluorescence recovery curves. (A) Ratio of fluorescence recovery half-times with ellipse in orientation b (long axis in y-direction) versus orientation a (long axis in x-direction), as a function (more ...)

C(x, y, t) is obtained by integrating G(x, y, x′, y′, t, 0) over (x′, y′), with the initial distribution of the diffusing molecules C(x′, y′, 0),

(3)

We assume an initially uniform fluorophore concentration C₀ and that bleaching is an irreversible first-order reaction: dC/dt = −αI(x, y)C, where I(x, y) is the bleach intensity. Initial fluorophore concentration in the bleached area with bleach pulse of time duration T is thus: C(x, y, 0) = C₀exp[−αTI(x, y)]. For a uniform elliptic bleach spot, I(x, y) = P₀/(π·w_x·w_y) inside the ellipse and I(x, y) = 0 outside, where P₀ is the laser power, and w_x and w_y are radii of ellipse along the x and y axes, respectively. For a Gaussian elliptic beam profile, ), where w_x and w_y are the beam waists in x- and y-dimension, respectively. Parameter K = αTI(0, 0) described the extent of bleaching (14).

The time course of fluorescence recovery after photobleaching, F(t), is calculated by integration over (x, y) of the product of C(x, y, t) and probe beam intensity (q/A)I(x, y),

(4)

Parameter q is the product of the efficiencies of light absorption, emission, and detection, and A is the attenuation factor during observation of recovery. For display of recovery curves, the fractional f(t) was used: f(t) = [F(t) − F(0)]/[F(∞) − F(0)], where F(0) is initial fluorescence after photobleaching; F(0) = (qP₀C₀/A)exp(−K) for a uniform elliptic disk and (qP₀C₀/A)K⁻¹[1−exp(−K)] for a Gaussian elliptic beam, and F(∞) = qP₀C₀/A is fluorescence at full recovery (14). The f(t) for a uniform circular bleach spot of diameter 2w is,

(5)

For a uniform elliptical bleach spot, double integrals over the elliptical domain in Eqs. 3 and 4 were calculated by numerical integration. For efficiency in computation we consider the diffusion of bleached molecules out of the spot rather than the diffusion of unbleached molecules into the spot. The concentration C*(x, y, t) = C₀ − C(x, y, t) was computed numerically with C*(x, y, 0) = 0, reducing the double integral in Eq. 3 to zero outside of the ellipse. Software was written in FORTRAN and was run in a batch process using the GNU FORTRAN G77 compiler to generate f(t) and compute t_1/2 (time for 50% recovery) (Fig. 2 A). The computation was validated by comparing simulated f(t) for a circular spot with the analytical solution (Eq. 5), and showing insensitivity of computed f(t) to a fivefold increases in (x, y) and t resolution.

FIGURE 2 Computation of fluorescence recovery curves, f(t). (A) Computation involves numerical solution of the diffusion equation in anisotropic media with initial conditions established by bleaching an elliptical spot. See Theory section. (B) Calculated fluorescence (more ...)

Computed f(t) for different diffusion coefficients (D_x and D_y) and ellipse axial ratios allows determination of diffusional anisotropy (D_y/D_x) and absolute diffusion coefficients from measured recovery half-time t_1/2. Fig. 3 A shows ratios of recovery half-times () computed as a function of D_y/D_x for different ellipse axial ratios (with fixed 10-μm short ellipse diameter). As the ellipses become long and thin, approaches D_y/D_x because fluorescence recovery relies more on unidirectional diffusion. For more round ellipses, approaches unity as fluorescence recovery becomes more independent of ellipse orientation. All curves pass through (1,1) as expected. D_y/D_x can thus be deduced from t_1/2 ratios. Results in Fig. 3 A fitted well to the empirical equation: where γ = 0.527 for the 1:3 elliptical spot (used experimentally here). The maximum deviation of the fitted curve is <4% for 0.1 ≤ D_y/D_x ≤ 10. Fig. 3 B shows that the determination of absolute D_y and D_x from plots of Each curve is a rectangular hyperbola corresponding to specified D_y/D_x.

The computation above assumed a flat beam intensity profile in the elliptical area, whereas actual profiles were in-between flat-field and Gaussian distributions (not shown). To examine the sensitivity of the D_y/D_x computation to the assumed illumination profile, simulations were done for the 30 × 10 μm ellipse comparing Gaussian versus flat beam profiles. Fig. 3 A shows that simulation with the Gaussian versus flat beam profiles produced nearly identical relations between D_y/D_x and Fig. 3 A also shows results from a full 3D computation (open circles; see Supplementary Material) in which the diffusion equation in 3D was solved with anisotropic diffusion (for spinal cord, D_x = D_z) and reflective boundary conditions at the spinal cord surface, as well as experimentally measured 3D illumination profiles and detection efficiencies for our optical system. Computed for the 3D and 2D computations were in excellent agreement.

To simulate anisotropic diffusion, the FITC-dextran solution was sandwiched between a diffraction grating and coverslip, rendering the parallel grooves of the grating fluorescent (Fig. 4 B, top). Here, FITC-dextran diffusion occurs along but not across the grooves. Photobleaching was done using a circular spot, and an elliptical spot in three different orientations (a–c) as depicted. Fluorescence recovery was remarkably slowed when the long axis of the ellipse (orientation b) was parallel to the grooves of the grating (middle). The t_1/2 for fluorescence recovery was similar for ellipse b versus the circle (bottom), but 9- to 10-fold slowed for ellipse a versus ellipse b. These ratios agree with the theoretical ratios of and

The dorsal white matter tracts of the spinal cord consist mostly of densely packed myelinated axons orientated rostro-caudally (Fig. 6 A). Marked slowing in FITC-dextran fluorescence recovery was found when the long axis of the ellipse was positioned parallel (orientation b) versus perpendicular (orientation a) to the neuronal axons (Fig. 6 B, left), indicating faster diffusion in the rostro-caudal versus medio-lateral directions. We predicted that diffusional anisotropy would be even greater for a dye that is confined to the cytoplasmic compartment because diffusion can only occur along axons. Fig. 6 B (right) shows remarkable slowing of fluorescence recovery of cytoplasmic calcein when the long axis of the ellipse was oriented parallel versus across the tract. Ten minutes of anoxia (by anesthetic overdose) greatly impaired diffusion in the ECS (Fig. 6 A, left) both along and perpendicular to the neuronal fibers.

FIGURE 6 Anisotropic diffusion in spinal white matter tracts. (A) Spot photobleaching was performed on the dorsal columns viewed from the top, which consist of myelinated axons aligned rostro-caudally (y direction); top shows cord section immunostained for neurofilament (more ...)

Fig. 6 C summarizes t_1/2 data for FITC-dextran diffusion in the ECS of spinal cord, and calcein diffusion in the cytoplasm. Averaged t_1/2 ratios with the long axis of the ellipse orientated across versus along fibers are summarized in Fig. 6 D, shown with deduced diffusional anisotropy D_y/D_x. Also shown is FITC-dextran diffusion in isotropic solution (from Fig. 4 A) with D_y/D_x ~ 1, and under anisotropic conditions (from Fig. 4 B) with D_y/D_x → ∞. There was diffusional anisotropy in both cytoplasm (D_y/D_x = 11 ± 4) and the ECS compartment (D_y/D_x = 5 ± 2) of the dorsal white matter tracts. Cytoplasmic diffusional anisotropy was not infinite probably because of small imperfections in tract alignment, and diffusion in some astroglia aligned perpendicular to the neuronal fibers.

Absolute D_y and D_x were computed from elliptical spot photobleaching for the spinal cord dorsal white matter tracts by plotting D_x vs. t_1/2 curves corresponding to the D_y/D_x of the cytoplasmic and ECS compartments as explained above (Fig. 6 E). Diffusion of FITC-dextran along neuronal fibers was slowed 1.8-fold compared with diffusion in solution (at 37°C), whereas FITC-dextran diffusion across neuronal fibers was slowed by an additional 5.0-fold. In axonal cytoplasm, calcein diffusion along the fibers was 2.3 times slower than diffusion in water.

FIGURE 7 Isotropic diffusion in ECS of cerebral cortex. (A) Spot photobleaching was done on exposed parietal lobes, which consist of neuronal processes with no directional preponderance; top shows section immunostained for neurofilament. (B) Representative fluorescence (more ...)

Molecular diffusion in brain ECS is slower than diffusion in water (3,6,17). Here, a novel photobleaching technique was developed to resolve for the first time the relative contributions of the two major determinants of molecular diffusion in the ECS—viscosity versus geometry/tortuosity. Our method utilized an elliptical spot produced by cylindrical excitation optics to introduce directionality into the measurement. Diffusional anisotropy was deduced from differential rates of photobleaching recovery for parallel and perpendicular orientations of the long axis of the ellipse with respect to the direction of greatest diffusion. Measurement of direction-dependent diffusion was required to resolve viscous versus geometric factors in the highly anisotropic white matter of spinal cord, where diffusion along the tract axis is slowed mainly by viscosity.

The principal finding was that the viscosity modestly slows macromolecular diffusion in the ECS, ~1.8 times compared with diffusion in water. Because the major constituents of ECM (hyaluronic acid and hyaluronic acid binding proteins) are similarly distributed in gray versus white matter (18,19), we assumed that brain and spinal cord ECM hinder diffusion equally. In cerebral cortex, ECS geometry was found to slow diffusion ~1.8-fold, in agreement with theoretical models of packed cells, where geometry was predicted to hinder ECS diffusion 1.5- to 1.9-fold (10,12). The ECM in CNS is rich in glycosaminoglycans and hyaluronan but lacks fibrous components, such as collagen, which are common in peripheral tissues (20). In tumors, collagen but not hyaluronan or glycosaminoglycan content correlated with slowed macromolecular diffusion in ECS; collagenase treatment accelerated but hyaluronidase hindered macromolecular diffusion in tumor ECS (21,22). We propose that the ECM in the CNS has evolved to facilitate rather than hinder diffusion by forming a large-pore medium that maintains cell-cell spacing, which is essential for efficient cell-cell communication.

Diffusion measurements by spot photobleaching and fluorescence correlation spectroscopy (FCS) generally assume isotropic diffusion, although a few studies have considered anisotropy. An attempt was made to measure anisotropic diffusion in skeletal muscle using a rectangular window to follow fluorescence recovery after bleaching a large, circular spot (23); however, no anisotropic diffusion was found. FCS was used to investigate anisotropic diffusion of FITC-dextran in dendrites of mitral cells from Xenopus tadpoles (24). Slowed lateral diffusion was interpreted in terms of effects of an oriented dense intradendritic microtubule network. The cylindrical optics described here should also allow measurement of anisotropic diffusion by FCS utilizing autocorrelation functions for elliptical detection volumes.

Our 2D diffusion model yielding an analytical solution was validated by a 3D Monte-Carlo computation, which took into account diffusion in the third dimension with reflective boundary conditions at the surface of the spinal cord, as well as experimentally measured 3D profiles for illumination and detection efficiency. The 2D computation is an excellent approximation here because fluorophore diffusion from below is minimal in spinal white matter where D_z is small (~D_x). In addition, the confocal optics minimized the z-detection depth, and the objective and cylindrical lenses were chosen to achieve a near-constant cylindrical beam profile for many microns in the z-direction as verified experimentally (see Supplementary Material).

Diffusion measurements were done in spinal cord after laminectomy to expose intact dura, allowing excellent visualization of the intact and pulsatile spinal cord through the thin dural sac (Fig. 5). The hindpaw withdrawal reflexes were also present bilaterally, suggesting preserved functionality. In <10% of mice (not used for measurements) hemorrhage was seen on the surface of the spinal cord and the hindlimb pain reflexes became absent. Our technique does not allow the study of surgically inaccessible regions, such as the gray matter located deep in cord tissue, or the spinothalamic and corticospinal tracts that lie anterior and lateral. Deep structures can be accessed using slice preparations. However, we have consistently found faster dextran diffusion in vivo (3,4), compared with data from brain slices obtained using integrative optical imaging (17,25,26). Cell swelling markedly hindered dextran diffusion in vivo (3,4), but only mildly impaired diffusion in brain slices (25). These findings suggest that in slices, baseline ECS geometry is more restrictive than in vivo, which may be due to cellular damage during cutting.

The relative hindrances to ECS diffusion arising from ECS tortuosity versus viscosity were estimated assuming that molecules diffusing rostro-caudally in ECS of the dorsal columns do not circumvent axons, resulting in little or no geometric hindrance to their diffusion. This is supported by histological studies of rodent dorsal columns, which reveal densely packed, cylindrical neuronal fibers orientated parallel to the long axis of the spinal cord with diameters 0.5–2 μm (27,28), well below our circular (~10-μm diameter) and elliptical (~30- and ~10-μm diameters) spot sizes.

Diffusion of calcein inside axonal fibers was slowed ~2.3-fold compared with diffusion of calcein in water. Slowing of diffusion by 2.5- to 4-fold has been found for other fluorescein-sized small molecules in cytoplasm of various cultured cells (29) as reviewed by Verkman (30). Molecular crowding was shown to be the principal factor accounting for slowed molecular diffusion in cytoplasm, such that 10–15% crowding by proteins and other macromolecular solutes/structures is predicted to slow diffusion by two- to threefold. Binding of diffusing molecules to slowly moving cytoplasmic components can further slow diffusion, though generally by little for a small polar solute. In cytoplasm of CHO cells, calcein diffusion was slowed approximately fourfold compared to its diffusion in water (31). The lesser fold slowing of calcein diffusion by axonal cytoplasm compared with cytoplasm of round cells is probably due to the alignment of axonal microtubules and neurofilaments, and the low concentration of membraneous structures such as endoplasmic reticulum (32).

Attempts have been made to measure anisotropy with the TMA⁺ method by using different orientations of the iontophoresis and recording electrodes. However, anisotropies () were small; only 1.4 in rat corpus callosum in vivo (33), and 1.7 in ECS of rat spinal white matter in vitro (34). There are potential concerns with the application of the TMA⁺ method to study anisotropic diffusion, including direct invasion of the measurement site and the directional TMA⁺ iontophoresis pulse, both of which could perturb direction-dependent TMA⁺ diffusion. In vivo, cardiorespiratory pulsations of CNS tissue may further compromise the ability of the TMA⁺ method to detect anisotropic diffusion, whereas in brain slices differential transection of fiber tracts by the cutting procedure may introduce artifactual direction dependence. Also, computation of direction-dependent diffusion coefficients using the TMA⁺ method requires multiparameter curve fitting, including diffusion coefficients, extracellular volume fraction, and a correction factor for TMA⁺ partitioning into cells. Another method for studying anisotropic diffusion in vivo is diffusion tensor magnetic resonance imaging (35,36). Despite the advantage of noninvasiveness and the ability to image the entire neuraxis, tensor imaging is largely limited to the detection of proton/water diffusion and thus unable to distinguish between intracellular and extracellular compartments.

In summary, we have established and validated a minimally invasive elliptical spot photobleaching method to quantify anisotropic diffusion of fluorescent macromolecules in vivo. Unlike the TMA⁺ technique, our method can measure direction-dependent, intra- or extracellular diffusion of a wide range of biologically relevant molecules, such as DNA, proteins, drugs, or liposomes. Anisotropic diffusion or directed motion is anticipated in many biological systems, which have oriented assemblies of barriers, as shown here in spinal cord. Other examples include the layered arrangement of cells in the retina, cerebellar cortex, and striated muscle, as well as the realignment of cytoskeletal fibers, which occurs during cell migration and axonal sprouting. Elliptical spot photobleaching is thus applicable to measurements over distances of many cells, as well as for subcellular diffusion studies when a smaller spot is used.

An online supplement to this article can be found by visiting BJ Online at http://www.biophysj.org.

This work was supported by National Institutes of Health grants EB00415, DK35124, EY13574, HL59198, DK72517, and HL73856 (to A.S.V.). During his tenure in Dr. Verkman's laboratory, M.C.P. was an employee of St. George's, University of London, and was funded by a Wellcome Trust Clinician Scientist Fellowship.