Rapid detection of the presence of diffusion exchange

Abstract

Diffusion exchange spectroscopy (DEXSY) provides a detailed picture of how fluids in different microenvironments communicate with one another but requires a large amount of data. For DEXSY MRI, a simple measure of apparent exchanging fractions may suffice to characterize and differentiate materials and tissues. Reparameterizing signal intensity from a PGSE–storage–PGSE experiment as a function of the sum, b_s = b₁ + b₂, and difference b_d = b₂ − b₁ of the diffusion encodings separates diffusion weighting from exchange weighting. Exchange leads to upward curvature along a slice of constant b_s. Exchanging fractions can be measured rapidly by a finite difference approximation of the curvature using four data points. The method is generalized for non-steady-state and multi-site exchange. We apply the method to image exchanging fractions and calculate exchange rates of water diffusing across the bulk water interface of a glass capillary array.

Graphical Abstract

1. Introduction

Pulsed gradient spin echo (PGSE) NMR measurements of diffusion are well suited for characterizing material heterogeneity and tissue microstructure [1, 2, 3, 4, 5, 6]. A diffusion observation time sets a diffusive length scale to the interactions that spins experience. Thus, PGSE gives a somewhat blurred or averaged view of the microenvironments that are in a sample or a voxel (for MRI). diffusion is particularly sensitive to microenvironments that restrict or hinder the displacements of molecules during the observation time. The extent to which these hindrances or restrictions are unique within the material leads to separability of signal from molecules in different microstructures. diffusion exchange spectroscopy (DEXSY) experiments are able to measure movement of molecules from one environment to another, given molecules in those environments have different diffusivities [7, 8, 9, 10]. In the DEXSY experiment, shown in Fig. 1, diffusion is encoded with one PGSE block, the signal is stored during a mixing time, t_m, and diffusion is encoded again during a second PGSE block. t_m sets the timescale for spins to exchange between environments. The measurement and analysis is quite similar to those used in relaxation exchange spectroscopy (REXSY) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Traditionally, the signal from a matrix of (b₁,b₂) values from looping through the two PGSE blocks in a DEXSY measurement is fit with a two-dimensional (2-D) Fredholm integral of the first kind, known as the 2-D inverse Laplace transform (ILT), to produce a DEXSY exchange map [24, 7] (See Fig. 2 a and b). Spins that remain in the same microenvironment during both encoding periods will have D₁ = D₂, along the diagonal of the 2-D distribution. Spins that start and end in different microenvironments produce off-diagonal exchange peaks. For intermediate exchange rates, k ~ 1/t_m, the approach to a steady-state exchange can be observed by performing the DEXSY measurement at a number of mixing times.

Figure 1:

DEXSY pulse sequence. Two PGSE encoding blocks with b₁ and b₂ varied independently are separated by t_m. Gradients are pulsed for a time δ and the time between the leading edges of the gradients in a PGSE block define the diffusion observation time, Δ. Signal can be acquired in an echo or, as here, brought through an imaging block.

Figure 2:

(a) Simulated biexponential decay of normalized echo attenuation as given by Eq. (3). Parameters set to f_i,i = 0.32, f_e,e = 0.62, f_e,i = f_i,e = 0.03, D_i = 5.8 × 10⁻⁵, D_e = 2.1 × 10⁻³ mm²/s. (b) Example of typical result following 2D-ILT of time-domain data. Off-diagonal peaks in the D₁ − D₂ map indicate the presence of exchange. Exchanging fraction is calculated by integrating peaks. (c) Contour map of simulated data shown in (a). A slice of constant b_s = 1500 s/mm² is shown. (d) Normalized echo attenuation for the slice taken in (c) is plotted across all b_d values from [−b_s, b_s]. Curvature is clearly present. The shape of the curve is hyperbolic, following Eq. (5).

Intermediate exchange rates can also be measured with 1-D diffusion measurements by varying the observation time and fitting the data with the Kärger model [25], a method which is useful in studying adsorption in colloidal systems [26]. When using the Kärger model to measure exchange in systems where the differences in diffusivity come from restriction, the effects of exchange get mixed in with the effects of restriction [27]. With DEXSY, it becomes more feasible to independently probe exchange in heterogeneous materials; however, the large number of scans required to loop through b₁, b₂, and t_m limits its applicability. The number of scans can be reduced by constraining the 2-D fit by the marginal distribution (marginal distributions constrained optimization, MADCO), opening the possibility for in vivo DEXSY MRI [28, 29]. Certain situations may only require a measure of an apparent two-site exchange rate or the exchanging fractions at a given t_m rather than the detailed look at exchanging pools that DEXSY exchange maps provide. For such applications, it would be beneficial to sidestep the 2-D ILT. An approach taken by filter-exchange spectroscopy (FEXSY) is to use only a single, large b₁ value as a diffusion filter which attenuates the more mobile water. Then the signal at a number of b₂ and t_m values can be fit directly to obtain an apparent exchange rate [30, 31]. FEXSY has been shown to be effective at measuring intracellular–extracellular exchange rates and thus transmembrane permeability [30, 31, 32, 33]. These studies found intracellular–extracellular exchange rates from 1 to 2 s⁻¹, consistent with relaxation-based methods [34, 35]. Human in vivo filter-exchange imaging has shown exchange rates to depend on pathophysiology, and exchange rates have been shown to differentiate healthy and cancerous tissue [36, 37]. Considering FEXSY’s success, it seems worthwhile to investigate alternative ways of slicing a DEXSY data set.

Song and colleagues show that the symmetry of the 2-D exchange map about the diagonal (seen in Fig. 2) calls for a reparameterization of the data set to separate out the effect of exchange between compartments [21]. For DEXSY, the re-parameterization uses b_s = b₁ + b₂ to encode for diffusion, and b_d = b₂−b₁ to encode for exchange. Slices of the raw data set at a certain b_s value will be flat if there is no exchange and curved upward if there is exchange (see Fig. 2d). That observation is intriguing as it opened up the possibility for exchange-weighted imaging. In this paper, we dig deeper into this approach and come up with a method for measuring exchanging fractions. The method approximates the curvature $\left({\partial }^{2}I/\partial b_d^2\right)$ by using the finite difference method and, therefore, three data point along a slice of constant b_s and one data point at b_s = 0 to normalize are enough to measure exchanging fractions in a two-compartment system at steady state. We present the derivation for measuring exchanging fractions for a two-compartment as well as an n-compartment system. Exchange rates can be measured by repeating at multiple mixing times. The method is applied to image exchange at the interface between a glass capillary array and bulk water.

2. Theory

In a single PGSE NMR experiment, the incoherent random motion of self-diffusion leads to a Gaussian distribution of phase shifts following the second gradient pulse. The normalized echo attenuation, I, is given by

$$I\left(b\right)={\int }_0^{\infty }P\left(D\right)e^{-bD}dD,$$ (1)

where $b=q^2\left(\Delta -\frac{\delta }{3}\right)$, q is the wave vector amplitude of the gradient pulse given by q = γgδ, and P(D) is a probability distribution of diffusion coefficients describing local diffusivity. In the 2-D DEXSY experiment, Eq. (1) becomes

$$I\left(b_1,b_2\right)={\int }_0^{\infty }{\int }_0^{\infty }P\left(D_1,D_2\right)e^{-b_1{D}_1-b_2{D}_2}d{D}_{1}d{D}_2.$$ (2)

In the case of a DEXSY experiment on an exchanging two-compartment system with internal and external compartments denoted by i and e, respectively, I is given by Eq. (2) as a linear superposition of contributions from sub-ensembles:

$$I\left(b_1,b_2\right)=f_{e,e}{e}^{-b_1{D}_e-b_2{D}_e}+f_{e,i}{e}^{-b_1{D}_e-b_2{D}_i}+f_{i,i}{e}^{-b_1{D}_i-b_2{D}_i}+f_{i,e}{e}^{-b_1{D}_i-b_2{D}_e}$$ (3)

subject to conservation (f_e,e + f_e,i + f_i,i + f_i,e = 1) where f_e,e denotes the fraction of the ensemble residing in the external compartment during both encodings, f_i,e denotes the fraction residing in the internal compartment during the first encoding and in the external compartment during the second encoding, and similarly for the remaining fractions.

Eq. (3) assumes exchange between compartments occurs only during t_m and not during the PGSE blocks (requiring Δ ≪ 1/k, where k is the exchange rate). Significant exchange during the PGSE encoding blocks leads to the apparent, measured, exchange fractions and diffusion coefficients being different from the actual values (as has been discussed for REXSY [19]). Eq. (3) also assumes that the fluid in different environments have significantly different apparent diffusion coefficients. In the case that D_i < D_e due to water diffusion being restricted inside cells, the diffusive length scale of water must be greater than the cell diameter in the dimension of the applied gradients: $\sqrt{2D_0\Delta }>a$. Thus, we assume a²/2D₀ < Δ ≪ 1/k.Additionally, Eq. (3) does not account for relaxation by T₂ during the diffusion encoding periods and T₁ during the mixing time. Normalizing the DEXSY attenuation by the signal from a DEXSY acquisition performed using the same TE_PGSE and t_m but with b₁ = b₂ = 0 divides out the effect of relaxation when relaxation times are the same in each compartment. However, differences in relaxation times between compartments will result in apparent fractions differing from the true fractions, especially when TE_PGSE > T₂ or t_m > T₁. (Relaxation effects in DEXSY can be modeled similarly to how they have been modeled for REXSY [13, 15, 16, 17, 19].) With acknowledgement of these assumptions, we recognize that we are developing a method for measuring apparent exchange.

Eq. (3) can be rewritten in the alternative coordinate system

$$b_1=\frac{b_s-b_d}{2};b_2=\frac{b_s+b_d}{2}$$ (4)

yielding the expression

$$I\left(b_s,b_d\right)=f_{e,e}{e}^{-b_s{D}_e}+f_{i,i}{e}^{-b_s{D}_i}+e^{-b_s{D}_s}\left(f_{i,e}{e}^{-b_d{D}_d}+f_{e,i}{e}^{b_d{D}_d}\right)$$ (5)

where

$$D_s≔\frac{\left(D_e+D_i\right)}{2};D_d≔\frac{\left(D_e-D_i\right)}{2}.$$ (6)

Taking the second partial derivative of Eq. (5) with respect to b_d,

$$\frac{{\partial }^{2}I}{\partial b_d^2}=D_d^2{e}^{-b_s{D}_s}\left(f_{i,e}{e}^{-b_d{D}_d}+f_{e,i}{e}^{b_d{D}_d}\right).$$ (7)

In the case of steady-state exchange where $f_{i,e}=f_{e,i}=\frac{f}{2}$, Eq. (7) simplifies to

$$\frac{{\partial }^{2}I}{\partial b_d^2}=f{D}_d^2{e}^{-b_s{D}_s}\text{\hspace{0.17em}cosh}\left(b_d{D}_d\right).$$ (8)

Assuming D_i ≠ D_e, a plot of b_d vs. I at some fixed b_s has positive curvature ⇔ f > 0. Evaluating Eq. (8) at b_d = b and rearranging, an expression for f is obtained,

$$f=\left(\frac{{\partial }^{2}I}{\partial b_d^2}{|}_{b_d=b}\right)\frac{e^{b_s{D}_s}}{\text{cosh}\left(b_d{D}_d\right)D_d^2}.$$ (9)

The second derivative can be quantified from experimental data using finite difference approximations of the second derivative [38]. The second-order approximation using I measured at b_d values spaced evenly by Δb_d at fixed b_s is

$$\frac{{\partial }^{2}I}{\partial b_d^2}{|}_{b_d=b}\approx \frac{I{\left|{}_{b_d=b-\Delta b_d}-2I\right|}_{b_d=b}+I{|}_{{}_{b_d=b+\Delta b_d}}}{\Delta b_d^2}-\cancel{\frac{\Delta b_d^2}{12}\frac{{\partial }^{4}I}{\partial b_d^4}|{}_{b_d=b}\dots }$$ (10)

Higher order approximations or curve fitting may be used to minimize truncation error. The second-order finite difference approximation is presented and used here in the interest of simplicity and minimizing the number of necessary acquisitions. Eqs. (9) and Eqs. (10) demonstrate that f at some t_m can be measured with as few as three DEXSY data points (and one data point at b_s = 0 to normalize by) if a priori knowledge of the system is available (i.e., values for D_i, D_e).

The choice of b_s in this method is arbitrary. b_s should therefore be chosen to optimize the signal-to-noise ratio (SNR). The domain of possible b_d values at a given b_s is [−b_s, b_s] Assuming a second order approximation of ${\partial }^{2}I/\partial b_d^2{|}_{b_d=b}$ in a two-compartment steady-state system with b = 0, Δb_d = b_s, maximum signal (depth of curve) is achieved at

$$\begin{array}{l}\underset{b_s\in [0,\infty )}{\text{argmax}}I(b_s,\pm b_s)-I\left(b_s,0\right)\\ =\underset{b_s\in [0,\infty )}{\text{argmax}}f{e}^{-b_s{D}_s}\left(\text{cosh}\left(b_s{D}_d\right)-1\right)\\ =\left\{\text{ln}\left[\frac{D_s}{D_i}+\sqrt{{\left(\frac{D_s}{D_i}\right)}^2-1}\right]\frac{1}{D_d}\right\}\end{array}$$ (11)

In the case that D_e and D_i are well separated, the optimal b_s value fully attenuates extracellular signal while minimally attenuating the intracellular signal, similar to the optimal b₁ in FEXSY [30]. With D_e = 2.1×10⁻³ and D_i = 5.8×10⁻⁵ mm²/s the optimal b_s = 3.5×10³ s/mm². In choosing these experimental parameters (Δb_d = b_s, b = 0, b_s by Eq. (11)), we maximize sensitivity to exchange in the presence of noise at the cost of increased truncation error.

Up to now, this approach has been applied to a two-compartment system at steady state (f_i,e = f_e,i). The approach can also be used to interrogate exchange in n-compartment (n = 2, 3, …) systems at unsteady state (f_i,e ≠ f_e,i). Approximated curvature values in the b_s−b_d coordinate system permit calculation of exchanging fractions in any system. The derivation for the general case proceeds similarly. Let

$$I\left(b_1,b_2\right)=\sum _{i=1}^n\sum _{j=1}^n{f}_{i,j}{e}^{-b_1{D}_i-b_2{D}_j}$$ (12)

subject to conservation, $\left({\sum }_{i=1}^n{\sum }_{j=1}^n{f}_{i,j}=1\right)$. Rewriting Eq. (12) in the alternative coordinate system and taking the second derivative with respect to b_d,

$$\frac{{\partial }^{2}I}{\partial b_d^2}=\frac{1}{4}\sum _{i=1}^n\sum _{j=i+1}^n{e}^{-\frac{b_s}{2}\left(D_i+D_j\right)}{\left(D_i-D_j\right)}^2\times \left(f_{i,j}{e}^{-\frac{b_d}{2}\left(D_i-D_j\right)}+f_{j,i}{e}^{\frac{b_d}{2}\left(D_i-D_j\right)}\right).$$ (13)

Eq. (13) is similar to Eqs. (7) and Eqs. (8). Assuming D₁ ≠ D₂ ≠…D_n, a plot of b_d vs. I at some fixed b_s has finite positive curvature ⇔ ${\sum }_{i=1}^n{\sum }_{j=1}^n{f}_{i,j},i\ne j>0$. All exchanging fractions (f_i,j, i ≠ j) can be calculated by estimating ${\partial }^{2}I/\partial b_d^2$ using Eq. (10) at n(n − 1) unique b_d values, substituting each b_d and corresponding ${\partial }^{2}I/\partial b_d^2$ estimation into Eq. (13) and solving the resulting system.

Returning to a two-compartment steady-state system, exchange parameters can be determined assuming first-order exchange kinetics. If exchange in the system follows rate laws of the form d f_i,e/dt = k_i,e f_i,i − k_e,i f_e,i [14] with initial condition f_i,e(t = 0) = 0, the exchanged fraction as a function of time solves to [14]

$$f_{i,e}\left(t\right)=f_{e,i}\left(t\right)=\frac{f\left(t\right)}{2}=\frac{f_e{k}_{e,i}}{k_{e,i}+k_{i,e}}\left(1-e^{-\left(k_{i,e}+k_{e,i}\right)}{}^t\right)=\frac{f_i{k}_{i,e}}{k_{e,i}+k_{i,e}}\left(1-e^{-\left(k_{i,e}+k_{e,i}\right)}{}^t\right),$$ (14)

where k_i,e and k_e,i are rate constants, and f_e and f_i are equilibrium fractions. Fitting experimental f (t_m) values to the functional form $f\left(t_m\right)=\frac{\alpha }{k}\left[1-e^{-k{t}_m}\right]$ with free parameters k and α yields a characteristic exchange rate, k = k_i,e + k_e,i.

3. Materials and methods

The curvature method was used to image water exchange across the bulk water interface of a glass capillary array. A schematic is shown in Fig. 3. This system has been used previously as a DEXSY phantom for the MADCO method with the intention of modeling intraaxonal–extracellular water exchange in brain white matter [29].

Figure 3:

Diagram of water-filled glass capillary array [GCA (Photonis, Lancaster, PA)] used in MRI experiments. The nominal inner diameter of the capillaries is 5 μm. The open air ratio (OAR) is 0.55 [29]. The capillary region can be considered an internal region of restricted x-axis diffusion with D_x = D_i.

DEXSY MRI experiments were performed on a 7-T vertical wide-bore magnet (Bruker BioSpin, Germany) with an AVANCE III spectrometer, Micro2.5 microimaging probe and gradient set, GREAT 60 gradient amplifiers, and 20 mm rf coil. The DEXSY MRI sequence, implemented in ParaVision 5.1, involved a PGSE–storage–PGSE sequence tacked onto the front of a standard multi-slice multi-echo (MSME) imaging sequence (see Fig. 1). An inversion recovery MSME sequence was used to measure T₁ of water in the bulk and capillary regions of the phantom: T_{1 bulk} = 2.5 s and T_{1 capillary} = 1.7 s. Analysis was performed in MATLAB R2017b.

In the DEXSY block, pulsed gradients were trapezoidal with (effective Stejskal-Tanner [39] rectangular pulse lengths of) δ = 3 and Δ = 15 ms. The gradients were applied in the x-direction—perpendicular to the axis of the capillaries. Note that with D₀ = 2.1 × 10⁻³mm²/s, $\sqrt{2D_0\Delta }$= 8 μm, larger than the diameter of the capillaries, 5 μm. D_e and D_i were measured by using the DEXSY MRI sequence as a 1-D diffusion imaging measurement with b₁ = 0 and 47 b₂ values varied from 0 to 17100 s/mm². For exchange imaging, 12 mixing times ranging from 5 ms to 2 s were chosen. Gradient amplitudes for the first and second PGSE block were varied to define b_s and b_d based on (Eq. 4). Ten data points were obtained at each t_m. One point used b_s = 0. The other nine used b_s = 3000 s/mm² (chosen using Eq. 11) with b_d values spaced linearly from −3000 to +3000 s/mm². The DEXSY block used 50 μs rf pulses, PGSE echo time TE_PGSE = 22 ms, and crusher gradients during the storage time to dephase transverse magnetization. The storage pulses were cycled together in two steps, between ±x.

In the MSME block, sagittal anterior–posterior (AP) images were acquired, with slice thickness = 4 mm, field of view = 17.5 mm × 14 mm, resolution = 350 μm × 350 μm, and TE_MSME = 6.3 ms. The delay between scans was set to 3 s.

4. Results and discussion

A region of interest (ROI, shown in Fig. 4) was selected that contained three rows of voxels: row 1, bulk water, row 2, bulk water and capillary interface, and row 3, capillaries. Signal from voxels in row 2 were used to quantify exchange parameters (see Fig. 5). Signal intensity was normalized to a b_s = 0 acquisition. In the interest of demonstrating hyperbolic curve shape, nine (b_s, b_d) acquisitions were made at b_s = 3000 s/mm² for each t_m. Of the nine (b_s, b_d) acquisitions, a subset of three points were used to approximate curvature at b_d = 0 with (Eq. 10). All nine points were used to fit a second-order polynomial as an alternative approach to quantifying curvature.

Figure 4:

(a) Proton density weighted DEXSY image of the chosen ROI. Row 1 corresponds to bulk water, row 2 contains the bulk water and capillary interface, and row 3 contains capillaries. The proton density of the capillary region is lower due to space filled by glass (OAR = 0.55). Voxels are 350 μm×350 μm×4 mm (b) Image following b₁ = b₂ = 1500 s/mm² DEXSY MRI. Note that bulk water fully attenuates. (c) 1-D diffusion data for row 1 (blue circles), row 2 (yellow triangles), and row 3 (green squares). Solid lines are monoexponential (rows 1 and 3) and biexponential (row 2) fits. Row 2 shares a D_x = 2.1×10⁻³ mm²/s value with row 1 and a D_x = 5.8 × 10⁻⁵ mm²/s value with row 3, confirming that row 2 is a sum of capillary and bulk water.

Figure 5:

b_d vs. I curves at various mixing times for row 2 of the ROI. Points used for curvature calculation by finite differencing are labeled with red triangles. The dashed lines are results of the polynomial fits. As expected, degree of curvature increases as t_m increases. Curves are symmetric about b_d = 0 for t_m < 0.4 s, as expected for steady-state exchange. At higher mixing times, intensity is larger for the left-most point than for the right-most point because of longitudinal magnetization which relaxed during t_m and went through the rest of the sequence, thus having a b₂ dependence rather than a b_d dependence. Phase cycling the 90° s around t_m subtracts most of this contribution but seems to leave a slight bias.

f was calculated from the curves in Fig. 5 with (Eqs. 9) and (10) by using curvature values obtained from both finite difference and polynomial fit approaches. Parameters were D_e = 2.1 × 10⁻³, D_i = 5.8 × 10⁻⁵ mm²/s, b_s = 3000, b_d,i = 0, and Δb_d = 3000 s/mm². D_e and D_i were obtained from exponential fits of the 1-D diffusion image data (see Fig. 4 c). Exchange fractions calculated using both approaches are shown in Fig. 6. Both sets of data fit a first-order exchange model well. Exchange rates, k, are similar and agree with previous studies of the phantom [29]. The similarity in results from both approaches indicates second-order finite differencing provides a good curvature approximation. Measured f values agree with a simple Brownian motion model for exchange across the capillary interface. The simple Brownian motion model assumes that the interface is located along the center (175 μm from the bottom or top) of the voxels, whereas the interface is actually located approximately 100 μm above the bottom of the voxels in row 2. This does not lead to large deviations between measured and modeled f values because $\sqrt{2D_z{t}_m}<100$ μm for these t_m. The largest two t_m values were similar to T₁ capillary, yet measured f values do not deviate significantly from the fits. Although differences in T₁ between compartments are not accounted for in the method, apparent exchange fractions at these t_m do not seem to be impacted.

Figure 6:

Exchange fractions are plotted vs. t_m. Fits were performed to $f\left(t_m\right)=\frac{\alpha }{k}\left(1-e^{-k{t}_m}\right)$ using non-linear least squares. (a) Fit using curvature estimated from a second-order polynomial fit. Fitted parameters were α = 0.299s⁻¹, k = 1.13s⁻¹, R² = 0.972. (b) Fit using curvature estimated from second-order finite differencing with points labeled in Fig. 5. Fitted parameters were α = 0.309s⁻¹, k = 1.20s⁻¹, R² = 0.965. As expected, finite differencing results in a slightly larger value of fitted k due to truncation error. (c) Predicted f = f (t_m) from simple Brownian motion model: $f=\sqrt{MSD}/l=\sqrt{2D_z{t}_m}/l$ where MS D is mean squared displacement, l = 350 μm is the length of the voxel in the z direction, and D_z = 2.1×10⁻³ mm²/s is the unrestricted diffusion coefficient of water. Curve shape and magnitude agree with data in (a) and (b).

For the purpose of measuring k, the method is quite similar to FEXSY and likely shows no advantage from the standpoint of data reduction or accuracy. The uniqueness of the method is the ability to measure f from a single t_m. Exchange fraction images of the entire ROI for a low, intermediate, and high t_m (relative to 1/k) are shown in Fig. 7. The four data points needed to measure f with this method (given prior knowledge of D_e and D_i) can be compared with the number of points required to measure a DEXSY exchange map, e.g., b₁ × b₂ = 32 × 32 = 1024 [7], or 14 when MADCO is used [29]. Both measures (f and exchange map) are useful: an exchange map shows which pools communicate in a multisite system, and this information may be a needed first step in detailed studies. Other times, the value of f summarizes the pertinent exchange information. To image f in vivo, one needs to also image a 1-D distribution of diffusion coefficients, which requires invoking a distribution model and obtaining sufficient 1-D diffusion data to use that model. Alternatively, ${\partial }^{2}I/\partial b_d^2{|}_{b_d=0}$ from (Eq. 10) may display meaningful and unique exchange-weighted image contrast, truly requiring only 4 data points.

Figure 7:

Exchange fraction images of ROI. The rows above and below the interface show near-zero exchanging fractions, as expected. Exchanging fractions increase as expected for voxels containing the interface (row 2).

5. Conclusion

We have presented a method of acquiring images of apparent exchanging fractions from the curvature of DEXSY data along a slice of constant b_s = b₁ + b₂ as a function of b_d = b₂ − b₁.Re-parameterization using b_s and b_d separates the encoding of exchange from diffusion. We generalized the method for non- steady-state and multisite exchanging systems. We applied the method to image the exchange of water diffusing across the bulk water interface of a glass capillary array.

*Highlights.

A rapid DEXSY MRI measurement of exchanging fractions is proposed

The method uses four DEXSY data points at one mixing time

Exchange-weighted images and exchange rates are obtained