The effect of inter-compartmental water exchange on the apparent myelin water fraction in multi-exponential T₂ measurements of rat spinal cord

Abstract

The myelin water fraction (MWF) has been used as a quantitative measure of the amount of myelin present in tissue. However, recent work has suggested that inter-compartmental exchange of water between myelin and non-myelin compartments may cause the MWF to underestimate the true myelin content of tissue. In this work, multi-exponential T₂ experiments were performed in-vivo within the rat spinal cord, and a wide variation of the MWF (10–35%) was measured within four rat spinal cord tracts with similar myelin content. A numerical simulation based upon segmented histology images was used to quantitatively account for T₂ variations between tracts. The model predicts that a difference in exchange between the four spinal cord tracts, mediated by a difference in the average axon radius and myelin thickness, is sufficient to account for the variation in MWF measured in-vivo.

Introduction

The water proton NMR signal from white matter is known to exhibit two T₂ components—a fast-relaxing component that is believed to originate from the water within myelin, and a slow-relaxing component that is believed to originate from water in both the extra-and intra-axonal spaces (1,2). The myelin water fraction (MWF), defined as the fraction of water signal associated with the fast-relaxing T₂ component, has been found to decrease in demyelinating and dysmyelinating lesions in both white matter (3) & peripheral nerve (4) and found to correlate with optical densitometry of luxol fast blue stained white matter (5). These findings support the interpretation that MWF is a quantitative measure of myelin content in tissue, and has been used to surmise differences in myelin content in different white matter tracts of the human brain (6). A recent study in excised rat spinal cord, which showed MWF to vary substantially between spinal cord tracts with similar myelin volume fractions (v_m) (7), has raised questions about the interpretation of MWF in white matter. The proposed explanation for this lack of correlation between MWF and v_m was a variation in rate of water exchange between myelin and non-myelin tissue volumes across tracts, dictated by variation in axon radius and myelin thickness. This study, however, was not performed in-vivo and no quantitative model was presented to explain the postulated exchange-mediated changes in multi-exponential T₂ (MET₂) relaxation. The present work revisits MET₂ in the rat spinal cord and interprets in-vivo data with a histology-based quantitative model of water proton relaxation and diffusion.

Many previous computational models of MET₂ in tissue have been based on a finite number of discrete compartments (8–10) with exchange effects described by the Bloch-McConnell equations (11). One assumption in these models is that each compartment is instantaneously well mixed (i.e., that water diffusion is large enough to keep each pool in equilibrium with itself). Biases may exist in these compartmental models if diffusion is slow enough to keep compartments from being well mixed. Compartmental models also (typically) neglect the intrinsic heterogeneity of tissue, using only enough discrete compartments to account for the number of distinct tissue components and not factoring in variations in the sizes or shapes of these physical compartments throughout the volume of tissue being modeled. Finite difference and Monte-Carlo models, like those previously used in diffusion studies (12–14), do not require the assumption of well mixed compartments, and are naturally well-suited to account for variation in the shape and dimensions of tissue components, such as variation in axon radius and myelin thickness. The present study uses histology of the rat spinal cord to define a finite difference model of water proton exchange and transverse relaxation, and fits model parameters to explain the in-vivo observations of MET₂ relaxation in four spinal tracts of the rat spinal cord.

Methods

Histology

Histology sections stained with Toluidine blue were taken from a previous study of rat spinal cord (7). Within this analysis, effects of sample shrinkage due during fixation (4% gluteraldehyde, 0.5% paraformaldehyde), secondary fixation (osmium) and embedding (epoxy) were neglected; however, previous studies have shown less than a 10% decrease in cross-sectional area with this protocol (15,16). From the histology sections, digital images were acquired using oil immersion light microscopy (LM) at 75 nm × 75 nm resolution (higher resolution than in the previous study). LM images were collected from four spinal white matter tracts — vestibulospinal (VST), rubrospinal (RST), funiculus gracilis (FG), and dorsal corticospinal (dCST) tracts. Images were enhanced with an adaptive histogram equalization algorithm and smoothed with an edge preserving anisotropic diffusion filter. Anatomical boundaries were then defined by local curvature, and images were manually segmented into regions of myelin, intra-axonal space, and extra-axonal space. From each tract v_m, average myelin thickness, represented as $\overline{r_o-r_i}$, and average axon radius, r̄_i, were calculated, where the axon radius was measured inside the myelin. Also, since the signal from water is proportional to the axon area, an area-weighted average axon radius, 〈r_i〉_A, was also calculated as

$${\langle r_i\rangle }_A=\sum _n{r}_{in}·\pi r_{in}^2/\sum _n\pi r_{in}^2=2\sum _n{r}_{in}^3/\sum _n{r}_{in}^2$$ [1]

where r_in is the radius inside the myelin sheath of the n^th axon. An area-weighted myelin thickness, 〈r_o−r_i〉_A, was calculated by

$${\langle r_o-r_i\rangle }_A=\sum _n(r_{on}-r_{in})·(r_{on}^2-r_{in}^2)/\sum _n(r_{on}^2-r_{in}^2)$$ [2]

where r_on is the radius outside the myelin sheath of the n^th axon.

Magnetic Resonance Imaging

Animal studies were completed in accordance with the Vanderbilt University Institutional Animal Care and Use Committee. Five female Sprague Dawley rats (226 – 266 g) were anesthetized with isoflurane and imaged with a 9.4 Tesla 31-cm horizontal bore Varian (Palo Alto, CA) DirectDrive scanner, using a 38 mm Litz quadrature coil (Doty Scientific, Columbia SC) for RF transmission and reception. An external heater was used to maintain body temperature. A 1.5 mm thick slice transverse to the long axis of the rat cervical spinal cord was selected from scout images. Using an inversion-recovery prepared multiple spin echo imaging sequence, a total of 32 echoes were acquired with the first echo time at 7.4 ms, echo spacing of 9 ms through the first 24 echoes (optimized following methods in (17)) and 50 ms echo spacing for the final 8 echoes (18). Images were encoded using a 128 × 128 sampling over a 25.6 × 25.6 mm² field of view then zero-filled to 256 × 256 prior to reconstruction, resulting a nominal in-plane resolution of 100 μm in each direction. Inversion and refocusing were achieved with 500 μs duration 90_x-180_y-90_x composite hard pulses, with each refocussing pulse surrounded by a pair of amplitude-modulated spoiler gradients to remove signal from unwanted coherence pathways (19). Phase encoding for all echoes was achieved with one gradient pulse prior to the first echo. An inversion recovery time (TI) equal to 2 s resulted in the effective nulling of cerebrospinal fluid signal, which was necessary to avoid flow-induced ghosts of substantial magnitude in the late echo images. The repetition time (TR) was 6 s and 8 excitations were averaged, resulting in a total acquisition time of approximately 100 min and a mean signal-to-noise ratio (SNR) in the spinal white matter of 217 +/− 21 (first echo magnitude divided by the standard deviation of the background noise, where the standard deviation of the noise, σ, was estimated by the mean of the background signal divided by (π/2)^1/2).

Finite Difference (FD) Model

The Bloch-Torrey equations (20) classically describe the diffusion and relaxation of magnetization

$$\frac{d}{dt}X(\mathbf{r},t)=\nabla ·(D\nabla X(\mathbf{r},t))-\frac{1}{T_2}X(\mathbf{r},t)$$ [3]

where X=M(r,t)/M₀ is the mole fraction of water magnetization on the transverse plane, D is the diffusion coefficient, T₂ is a relaxation time constant, the magnetization, M, is the volume density of the net magnetic moment on the transverse plane, and M₀ is the volume density of the net magnetic moment in the longitudinal direction at thermal equilibrium, which is proportional to the proton density. Using standard approximations to differentiation

$$\frac{dX(x)}{dx}\approx \frac{X(x+\mathrm{\Delta }x)-X(x)}{\mathrm{\Delta }x}\approx \frac{X(x)-X(x-\mathrm{\Delta }x)}{\mathrm{\Delta }x}$$ [4]

and discretizing X in space (by Δx) and time (by Δt), the numerical solution of [3] (in one spatial dimension) can be found explicitly using Euler’s forward approximation at spatial index, j, and temporal index, n+1, by

$$X_j^{n+1}=X_j^n+c_{j\to j+1}(X_{j+1}^n-X_j^n)+c_{j\to j-1}(X_{j-1}^n-X_j^n)-\frac{\mathrm{\Delta }t}{T_{2j}}{X}_j^n$$ [5]

where c_j⃗j+1 is the jump probability from the spatial grid-point j to j+1. The jump probability is

$$c_{j\to j+1}=\frac{M_{0j+1}}{M_{0j}}\frac{\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}{\left(\frac{1}{2}\left(\frac{1}{D_j}+\frac{1}{D_{j+1}}\right)+\frac{1}{\mathrm{\Delta }x·P_B}\right)}^{-1}$$ [6]

where P_B is the compartment boundary permeability of water. If the neighboring grid-point is in the same compartment, the jump probability is simply

$$c_{i,e,m}=D_{i,e,m}\frac{\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}$$ [7]

where i, e and m designate the tissue compartment (i.e. intracellular, extracellular, or myelin). The MRI signal at a given echo time is given by

$$S_n(TE=n·\mathrm{\Delta }t)=g_n·\mathrm{\Delta }x·\sum _j{X}_j^n·M_{0j}$$ [8]

where g_n is a scaling factor related to coil sensitivity at the n^th spinal cord tract. The grid spacing, Δx, is chosen to resolve the simulation geometry. The time step, Δt, is then subject to the constraints

$$max(D_{i,e,m})\frac{\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}<\frac{1}{2}$$ [9]

$$\mathrm{\Delta }t=T_2$$ [10]

This 1-dimensional case is easily generalizable into multiple dimensions. Note that the 1/2 factor in [9] changes to 1/4 and 1/6 in 2- and 3-dimensions, respectively.

The segmented LM images of each white matter tract were used to define the geometry of the finite difference model. By constraining the geometry, the model was defined by 10 parameters: intrinsic water diffusion coefficient (D_i,e,m), transverse relaxation time constant (T_2i,e,m), equilibrium magnetization (M_0i,e,m), and compartment boundary permeability coefficient of water (P_B). For the purposes if this work, water diffusion coefficients of the intra- and extra-axonal compartments were fixed at D_i = 2.5 μm²/ms and D_e = 3.0 μm²/ms (large enough to make in intra- and extra-axonal space well mixed within the T₂-timescale). All tissue parameters were defined identically across spinal cord tracts, making the underlying geometry derived from the LM the only source of inter-tract signal variation.

Since each segmented LM image contains more than 100 axons, the LM images were deemed to be representative of axonal characteristics within the tract, and all boundaries of the geometry were made reflective. The finite difference calculations were implemented in C and CUDA, run on a Linux system with two NVIDIA GeForce GTX 295 dual GPU cards, which allowed simulation of transverse relaxation with the Bloch-Torrey equations from all four tracts in ≈ 60 seconds.

Bloch -McConnell (BM) Model

Transverse relaxation was also simulated by calculation of the Bloch-McConnell equations (11) for a three-compartment system. In matrix form,

$$\frac{d\mathbf{M}}{dt}=\mathbf{LM}$$ [11]

where

$$\mathbf{M}=\left(\begin{array}{l}{M}_i(t)\hfill \\ M_e(t)\hfill \\ M_m(t)\hfill \end{array}\right)$$ [12]

$$\mathbf{L}=\left(\begin{array}{ccc}-\left(1/T_{2i}+k_{im}\right)& 0& k_{mi}\\ 0& -\left(1/T_{2e}+k_{em}\right)& k_{me}\\ k_{im}& k_{em}& -\left(1/T_{2m}+k_{mi}+k_{me}\right)\end{array}\right)$$ [13]

where M_i,e,m is the magnetization within each compartment, and k_im, k_mi, k_em, and k_me are first order exchange rate constants between intra-axonal and myelin, and between extra-axonal and myelin compartments. Note that there is no direct exchange term between the intra- and extra-axonal compartments. Like the FD model, the BM model neglects any susceptibility differences between tissue compartments. The general solution for [11] is

$$\mathbf{M}(t)=exp(\mathbf{L}·t)\left(\begin{array}{c}{v}_i{M}_{0i}\\ v_e{M}_{0e}\\ v_m{M}_{0m}\end{array}\right)$$ [14]

where v_i,e,m is the compartmental volume fraction assuming hexagonal axon packing, exp is the matrix exponential function, and M₀ is the equilibrium magnetization from a uniform volume element. The detectable MRI signal from n^th tract is then

$$S_n(t)=g_n·(M_i(t)+M_e(t)+M_m(t))$$ [15]

where once again g_n is related to coil sensitivity. The BM model was implemented in MATLAB (The Mathworks, Natick MA).

Since we are attempting to model the role of exchange in MET₂ analysis, we must constrain the exchange rate constants within the spinal cord tracts analyzed. Hence, the exchange rate constants were defined as a function of water compartmental boundary permeability

$$k_{me}=P_B\frac{A_{me}}{V_m}$$ [16]

$$k_{mi}=P_B\frac{A_{mi}}{V_m}$$ [17]

where the surface area to volume ratio of the inner myelin barrier is

$$\frac{A_{mi}}{V_m}=\frac{2{\langle r_i\rangle }_A}{{({\langle r_i\rangle }_A+{\langle r_o-r_i\rangle }_A)}^2-{\langle r_i\rangle }_A^2}$$ [18]

That of the outer myelin barrier is

$$\frac{A_{me}}{V_m}=\frac{2({\langle r_i\rangle }_A+{\langle r_o-r_i\rangle }_A)}{{({\langle r_i\rangle }_A+{\langle r_o-r_i\rangle }_A)}^2-{\langle r_i\rangle }_A^2}$$ [19]

and 〈r_i〉_A & 〈r_o−r_i〉_A are taken from equations [1] and [2]. Exchange rate constants between the intra- & extra-axonal compartments and myelin are then

$$k_{im}=k_{mi}\frac{v_m}{v_i}\frac{M_{0m}}{M_{0i}}$$ [20]

$$k_{em}=k_{me}\frac{v_m}{v_e}\frac{M_{0m}}{M_{0e}}.$$ [21]

Note that, consistent with the FD model, M₀ is defined as a density within a uniform volume, and therefore this notation (Eqns. [11–21]) differs from the standard notation used to describe the BM model, which normally defines M and M₀ without assuming specific spatial volumes to which the signal is compartmented. The initial conditions (Eqn. [14]) and the relationship between exchange rates (Eqn. [20–21]) reflect that v_xM_x is proportional to the amount of signal within the x compartment.

Data Analysis

From each rat MRI image set, regions of interest (ROIs) were manually defined in the four spinal white matter tracts identified previously. To minimize bias due to Rician noise, the mean echo magnitudes were estimated from each ROI by maximum likelihood estimation of the signal magnitude (see eqn. [12] in (21)). These echo magnitudes were then fitted to the sum of 200 decaying exponential functions with time constants spaced logarithmically between 7 and 1000 ms, resulting in an estimated T₂ spectrum for each tract and each rat for in-vivo measurements. The fitting was implemented with a non-negative least squares algorithm with the addition of a minimum-curvature regularization (22). In order to compare similarly-smoothed spectra across tracts, the regularization parameter was set to the average of the regularization parameter values determined from preliminary fittings of the in-vivo measurements with regularization adjusted using the generalized cross-validation method (23).

From each in-vivo T₂ spectrum, the so-called myelin and other (intra- and extra-axonal) water components were distinguished by the local minimum between the two dominant T₂ spectral components, and three metrics were extracted. The MWF from each spectrum was calculated as the fraction of signal within the myelin water peak, the myelin water T₂ (MWT₂) and other water T₂ (OWT₂) were calculated as the first moment of their respective T₂ peaks. These metrics were tabulated (mean and standard deviation (SD)) across animals for each tract.

Following empirical evaluation of the T₂ spectra, the echo magnitudes were then also fitted to the FD and BM models of transverse relaxation. For each animal and model, three T₂ time constants (T_2i, T_2e, T_2m, constant across tracts), the scaling factor for each tract (g_n) and, for some models, an exchange-related parameter (D_m or P_B, also constant across tracts) were fitted by non-linear regression of the observed signal magnitudes to the model predictions for all four tracts simultaneously. The Bloch-McConnell simulations used P_B to dictate water exchange. For all simulations the compartmental equilibrium magnetizations were fixed at M_0m/M_0i = M_0m/M_0e = 0.5 (24). The finite difference simulation was run for all tracts in one of three conditions: i) no inter-compartmental water exchange, dictated by constraining D_m = 0 μm²/ms and P_B = 0 μm/ms (referred to as FD + NE); ii) diffusion-limited exchange, dictated by constraining P_B = 1 μm/ms (effectively infinite for the compartment geometries studied) and fitting for D_m (referred to as FD + D_m); iii) permeability-limited exchange, dictated by constraining D_m = 3.0 μm²/ms (large enough to allow rapid mixing of the myelin water for all tracts) and fitting for P_B (referred to as FD + P_B).

The reduced chi-square statistic (χ²) was computed for each fit as

$${\chi }^2=\frac{1}{d}\sum \frac{{(S-\widehat{S})}^2}{{\sigma }^2},$$ [22]

where S is the observed signal, Ŝ is the modeled signal, σ² is the noise variance, d is the degrees of freedom (total number of echoes from all four tracts minus the number of fitted parameters), and the sum is taken over the signal from each echo and each spinal cord tract. For each fit, this statistic was tested against a chi-square distribution to assess goodness-of-fit. In order to compare goodness-of-fit between the p^th and q^th models, an f-statistic was defined as

$$f=\frac{{\chi }_p^2}{{\chi }_q^2}.$$ [23]

The q^th model was deemed significantly better than the p^th model if the values of f from all 5 rats were greater than some cutoff value, f_c, such that the joint probability of this statistic was ≤ 0.05. That is,

$$P(f_1>f_c\cap f_2>f_c\dots f_5>f_c)=\prod _{n=1}^5(1-F_f(f_c))={(1-F_f(f_c))}^5=0.05,$$ [24]

where f_n indicates the f-statistic in Eq [23] evaluated for the n^th animal, and F_f(f_c) is the cumulative distribution of this statistic at critical value f_c.

Results

The segmented images from each tract are shown in Fig. 1 where white, gray and black represent the intra-axonal, extra-axonal, and myelin compartments, respectively. These images served as the basis of the tissue geometry used for the finite difference simulations. The myelin and axon characteristics derived from these images are summarized in Table 1 for each tract. The average myelin thickness and axon radius are presented as the intra-sample mean and standard deviation (SD), while v_m, 〈r_o−r_i〉_A, and 〈r_i〉_A, which are whole-tract measures, are presented without uncertainty, but we note that the inter-sample coefficient of variation within the rat population for histology-derived metrics of our previous study was <≈ 10% (7). It is clear from these data that while v_m was relatively constant across tracts, the axon radius and myelin thicknesses varied considerably.

Figure 1.

Segmented light microscopy images from four white matter tracts within the rat spinal cord: Vestibulospinal (VST), Rubrospinal (RST), Funiculus Gracilis (FG), and dorsal corticospinal (dCST). Scale bars = 10 μm, white = intra-axonal space, gray = extra-axonal space, black = myelin.

Table 1.

Quantitative axon tract characteristics derived from segmented light microscopy images. Results are given as mean (±SD, where applicable) within the axon population of the light microscopy images. Myelin volume fraction (v_m), mean myelin thickness ( $\overline{r_o-r_i}$), mean axon radius (r̄_i), area-weighted mean myelin thickness (〈r_o−r_i〉_A), and area-weighted mean axon radius (〈r_i〉_A).

Tract	vm	$\overline{r_o-r_i}$ (μm)	r̄i (μm)	〈ro−ri〉A (μm)	〈ri〉A (μm)
VST	0.54	1.25 (0.52)	1.37 (0.94)	1.49	2.63
RST	0.55	0.84 (0.33)	0.99 (0.52)	1.02	1.51
FG	0.54	0.59 (0.21)	0.87 (0.30)	0.69	1.07
dCST	0.51	0.38 (0.14)	0.56 (0.21)	0.43	0.72

An example T₂-weighted image (TE = 7.4 ms) is shown in Fig. 2 cropped to a size just larger than the spinal cord. Example ROIs corresponding to four white matter tracts are overlaid on the image: dCST = blue, FG = green, RST = red, and VST = teal. Fig. 3 shows the T₂ spectrum within each tract averaged across all 5 animals, where each color corresponds to a tract as indicated in Fig. 2. In contrast to the roughly constant v_m across these regions (Table 1 and (7,13)), the MWF varied from ≈10% to ≈35%. Fig. 4 shows the MWF (top panel), MWT₂ (lower panel, blue circles), and OWT₂ (lower panel, red triangles) (inter-sample mean ± SD) vs. myelin thickness across tracts. Consistent with our previous study of excised and fixed rat spinal cord (7), the MWF, MWT₂, and OWT₂ all tended to increase with myelin thickness. As apparent from Table 1, the same trends were present with respect to axon radius and area-weighted axon radius.

Figure 2.

Example T₂ weighted image (TR=6 s, TI = 2 s, TE = 7.2 ms) cropped to view the spinal cord. The bright butterfly-shaped region is gray matter, and the surrounding darker region is the white matter in which example ROIs of four tracts are drawn: dCST = blue, FG = green, RST = red, VST = teal. Scale bar = 1 mm.

Figure 3.

Population averaged T₂ spectra for the four spinal cord tracts indicated in Figure 2. There are large variations in the fractional area of the short-lived T₂ peak (i.e. myelin water).

Figure 4.

MWF as well as myelin water T₂ (MWT₂) and other water T₂ (OWT₂) values +/− SD plotted vs. the average myelin thickness within each spinal cord tract. The MWF, OWT₂ and MWT₂ tend to increase with increasing myelin thickness.

Fitted parameters (inter-sample mean ± SD) from each of the four simulation scenarios are given in Table 2 and two example fits are given in Fig. 5, cropped for ease of view to the first 24 echoes. As expected, all fits to the FD+NE model (no water exchange) failed the chi-square goodness-of-fit test at the 95% confidence level, while all other models exhibited a mix(across animals) of fits above and below the 95% confidence level. Comparing models by joint f-statistic test (described above) demonstrated that the FD models with exchange (FD+D_m and FD+P_B) were fitted significantly better than the BM model, which incorporated water exchange but not intra-compartment diffusion or intra-tract variation in axon/myelin dimensions. All of the models predicted a significant difference in T_2i and T_2e, but there was not a statistical difference in fits between FD-models with T_2i < T_2e compared with T_2i > T_2e.

Table 2.

Numerical fitting results and reduced chi-square error (χ²), given as mean (standard deviation) within the animal population. Two of the models (FD + D_m & FD + P_B) had two minima, distinguished by the compartment with long- vs. intermediate-lived T₂s, that were not statistically differentiable.

Model	T2i (ms)	T2e (ms)	T2m (ms)	Dm (μm2/ms)	PB (μm/ms)	χ2
FD + NE	39.7 (2.4)	77.9 (4.2)	20.2 (1.0)	0.0*	0.0*	2.87 (1.25)a
FD + Dm	81.5 (5.6)	35.5 (4.0)	10.9 (1.2)	0.0012 (0.0002)	1.0*	1.18 (0.40)
	40.7 (3.2)	102.6 (12.5)	14.7 (1.5)	0.0005 (0.0002)	1.0*	0.98 (0.25)
FD + PB	87.6 (7.0)	34.3 (3.8)	8.9 (1.5)	3.0*	0.0071 (0.0006)	1.43 (0.52)c
	38.2 (3.0)	127.0 (23.9)	14.3 (1.8)	3.0*	0.0040 (0.0012)	0.99 (0.28)
BM	67.2 (5.4)	43.9 (6.8)	14.0 (1.2)	n/a	0.0038 (0.0011)	1.51 (0.51)b

Fixed model parameters: M_0m/M_0e = M_0m/M_0i = 0.5, for the FD based models D_e = 3.0 μm²/ms and D_i = 2.5 μm²/ms;

^*indicates additional fixed parameters; significant difference in pair-wise comparisons (joint f-test, P<0.05) of χ² of

^aFD + NE vs. all other fits,

^bBM vs. all FD + exchange model fits, and

^cFD + P_B (T_2i>T_2e) vs. FD + D_m (T_2i<T_2e).

Figure 5.

Raw signal decay (circles) and two example fits: the finite difference model with exchange mediated by membrane permeability (FD + P_B, line), and a finite difference model with no exchange (FD + NE, dashed line). The plot was cropped to the first 24 echoes to magnify the difference in fit between the two models. Fitted parameters are given in Table 2.

Discussion

A recent study of water proton transverse relaxation in excised and chemically fixed rat spinal cord revealed a substantial variation in T₂ spectra between different white matter tracts with similar myelin content (7). By comparing observed T₂ spectra with histology, it was inferred by the authors that the variation in T₂ spectra was mediated by varying effects of water exchange between myelin and both axonal and extra-axonal spaces. It was further postulated that a similar effect could be expected in-vivo. Thus, the primary observation of this follow-up work is a similar variation between white matter tracts of T₂ spectral characteristics measured in-vivo. Indeed, Fig. 3, which shows the population-average in-vivo T₂ spectra from four spinal WM tracts, and Fig. 4, which shows MWF, MWT₂ and OWT₂ vs. mean myelin thickness, compare well qualitatively with a similar figures (cf 4 and 5, (7)) from excised spinal cord. As in the previous study, tracts with smaller axons and thinner myelin showed T₂ spectra shifted with peaks at lower apparent T₂s and with a smaller MWF, which is consistent with a greater effect of inter-compartmental water exchange on transverse relaxation. Quantitatively, the effect was even greater in the present study, with MWF as low as ≈0.10 in the dCST, compared to 0.19 in the dCST of excised spinal cord. The natural conclusion is that water exchange plays an even greater role in transverse relaxation in-vivo than in the previous study, which consequently indicates that the lower temperature of the ex-vivo work (≈18°C) reduced water exchange to a greater extent than it was increased by chemical fixation (25).

The relationship between these observations and the role of inter-compartmental water exchange in the transverse relaxation of water in cerebral white matter remains unclear. A previous study in human corpus callosum, which contains axons with sizes similar to the dCST (26), reported MWFs between 10–13% (6). While similar with the average MWF measured within the dCST in this work, ~10%, the corpus callosum is also thought to have lower myelin content than the spinal cord (27,28). Therefore, exchange properties of myelin water may be different within the brain. Future studies are being planned to explore the effects water exchange in cerebral white matter.

Computational Modeling

Given the known micro-anatomies of spinal WM tracts, computational models were employed in an attempt to quantitatively account for the T₂ variations with a physically realistic model of water diffusion and transverse relaxation. Two-dimensional models of WM micro-anatomy were built using a representative set of light microscopy images for each tract. Fig. 5 and Table 2 indicate that if intrinsic T₂ and diffusion characteristics for each tissue compartment are constant across tracts, a model without exchange does not predict the observed variations in T₂ across tracts. This is not surprising given the large variation of MWF observed across tracts in comparison to the relatively small variation in v_m. However, if water exchange is incorporated into the model, mediated either through a finite diffusion coefficient in the myelin compartment (FD+D_m model) or by defining a finite permeability at each compartment boundary (FD+P_B and BM models), reasonable model parameters can be found that predict the observed T₂ variation. The myelin water lifetimes, τ_m = 1/k_mi + 1/k_me,varied significantly between the tracks investigated in this work. Calculated based upon the mean P_B fit in Table 2 and area-weighted axon and myelin dimensions given in Table 1 with hexagonal packing, the myelin water lifetimes are 43, 69, 100, and 150 ms for the dCST, FG, RST, and VST, respectively.

The BM model, like the FD+P_B model, assumes boundary-mediated water exchange, but did not fit the data as well as the FD+P_B model, indicating that the intra-tract tissue heterogeneity accounted for in the FD models (but not the BM model) is a significant factor in the observed transverse relaxation. In comparing the FD+D_m and FD+P_B models, the lack of a significant difference in fits dictates that this study cannot distinguish whether water exchange in spinal white matter is mediated at the boundary of myelin or by the rate of water diffusion within the myelin. However, experiments in peripheral nerve, which has thicker myelin, may be more sensitive to this type of comparison. The permeability of water through a myelin boundary estimated from both the FD+P_B and BM fitting, P_B ≈ 0.004 – 0.007 μm/ms, is consistent with plasma membrane permeabilities estimated in previous studies, and is within the range believed to physiological relevant (29–32).

Myelin Water Diffusion

The value of D_m estimated in this work is more than three orders of magnitude smaller than the diffusion coefficient of water at body temperature (33), and two orders of magnitude smaller than the apparent diffusion coefficient (ADC) of myelin water previously estimated perpendicular to the axon orientation (34). However, this previously measured ADC originates from a combination of water diffusing in normal and tangential directions to membrane layers in myelin. In a circular axon, the myelin water ADC measured with low b-value can be shown equal the arithmetic mean of the water ADCs in the normal and tangential directions. In this work, D_m is the myelin water ADC normal to the myelin membranes, since it is in the normal direction that water must diffuse to enter or leave the myelin compartment. The value of D_m estimated in this work (0.005–0.0012 μm²/ms) corresponds reasonably with the diffusion coefficient of water measured through stacks of synthetic lipid bilayer near body temperature, ≈ 0.0017 μm²/ms (35).

Given D_m, it is possible to estimate the permeability of the lipid bilayers constituting the myelin. Extending the formula given of Tanner (36) used to calculate the effective diffusion coefficient across permeable barriers to one that accounts for a finite barrier thickness (35), one gets

$$\frac{{(a+L)}^2}{D_m}\approx \frac{a}{P_L},$$ [25]

where P_L is the permeability coefficient of water across the lipid bilayer, a is the spacing between the membranes, and L is the thickness of the bilayer. With D_m = 0.0009 μm²/ms (the mean of the two D_m values estimated in this work), the permeability of the individual lipid bilayers constituting myelin would be between 0.006 – 0.044 μm/ms, depending upon the value of the average membrane thickness and membrane spacing used in the calculation (37). This is within this range of membrane permeability believed to be physiologically relevant for a cellular lipid bilayer (29–32).

Three T₂ Components

All of the models predict a difference between the T₂ of the intra- and extra-axonal spaces. Within these models, no minima were found near T_2i ≈ T_2e. The compartmental T₂ values estimated within these models correspond well with the three T₂ components observed in peripheral nerve (38), where axons are much larger and myelin is much thicker thereby allowing for less water exchange between compartments. The observed short-, intermediate- and long-lived T₂ components within peripheral nerve are thought to originate from water within myelin, extra-, and intra-axonal space, respectively (39–41). A few studies have even hypothesized a difference in the T₂ of intracellular and extracellular space in gray matter (32,42). There is no direct evidence for a difference in the intra- and extra-axonal T₂ in white matter. Still, intrinsic differences in the compartmental T₂ may be present within white matter, since exchange and/or inadequate SNR could make these peaks indistinguishable under MET₂ analysis. When using the generalized cross-validation approach utilized in this work for regularization of T₂ spectra, simulations (not shown) indicated that a SNR of greater than 2000 would be required to distinguish the 3 components predicted by the finite difference model.

Conclusion

This study reports a variation of the MWF measured in-vivo between rat spinal cord tracts with a similar underlying myelin content. Further, numerical simulation suggests that exchange between underlying tissue compartments, mediated by a difference in the axon size and myelin thickness, plays a role in determining white matter T₂ characteristics and, in particular, the MWF. Within the model used, both exchange and anatomical heterogeneity was necessary to account for the variation of the observed MWF between white matter tracts in the rat spinal cord.