# High b-value q-space analyzed diffusion-weighted MRS and MRI in neuronal tissues a technical review

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NMR IN BIOMEDICINE NMR Biomed. 2002;15:516–542 Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/nbm.778 Review Article High b-value q-space analyzed diffusion-weighted MRS and MRI in neuronal tissues ± a technical review Yoram Cohen1* and Yaniv Assaf1,2 1 School of Chemistry, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel 2 Wohl Institute for Advanced Imaging, Tel Aviv Sourasky Medical Center, Tel Aviv, 64239, Israel Received 16 May 2001; Revised 25 December 2001; Accepted 10 February 2002 ABSTRACT: This review deals with high b-value q-space diffusion-weighted MRI (DW-MRI) of neuronal tissues. It is well documented that at sufficiently high b-values (and high q-values) neuronal water signal decay in diffusion experiments is not mono-exponential. This implies the existence of more than one apparent diffusing component or evidence for restriction. The assignment of the different apparent diffusing components to real physical entities is not straightforward. However, the apparent slow diffusing component that was found to be restricted to a compartment of a few microns, if originating mainly from a specific pool and if assigned correctly, may potentially be used to obtain more specific MR images with regard to specific pathologies of the CNS. This review examines the utility of analyzing high b-value diffusion MRS and MRI data using the q-space approach introduced by Callaghan and by Cory and Garroway. This approach provides displacement probability maps that emphasize, at long diffusion times, the characteristics of the apparent slow diffusing component. Examples from excised spinal cord, where the experimental conditions for which the q-space analysis of MR diffusion data was developed can be met or approached will be presented. Then examples from human MS patients, where q-space requirement for the short gradient pulse is clearly violated, are presented. In the excised spinal cord studies, this approach was used to study spinal cord maturation and trauma, and was found to be more sensitive than other conventional methods in following spinal cord degeneration in an experimental model of vascular dementia (VaD). High b-value q-space DWI was also recently used to study healthy and MS diseased human brains. This approach was found to be very sensitive to the disease load in MS, compared with other conventional MRI methods, especially in the normal appearing white matter (NAWM) of MS brains. Finally, the potential diagnostic capacity embedded in high b-value q-space analyzed diffusion MR images is discussed. The potentials and caveats of this approach are outlined and experimental data are presented that show the effect of violating the short gradient pulse (SGP) approximation on the extracted parameters from the q- space analysis. Copyright 2002 John Wiley & Sons, Ltd. KEYWORDS: diffusion MRI; high b-value DWI; q-space diffusion; white matter disorders; multiple sclerosis; MS; spinal cord trauma; neuronal degeneration; neuronal maturation INTRODUCTION MRI.2 However, following the seminal papers of Le Bihan3,4 and the demonstration regarding the high In the last decade, diffusion has become an important sensitivity of diffusion-weighted MRI (DWI) for early contrast mechanism, particularly in MRI of the CNS.1 detection of stroke by Moseley et al. in 1990,5 DWI Interestingly, diffusion-weighting as a contrast mechan- attracted much more interest.6–10 After the high sensi- ism did not play an important role in the early days of tivity of DWI to experimental acute stroke was demon- *Correspondence to: Y. Cohen, School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Email: ycohen@ccsg.tau.ac.il Contract/grant sponsor: United States–Israel Binational and Foundation 97-00346. Abbreviations used: ADC, apparent diffusion coefficient; CNS, central nervous system; CSF, cerebral spinal fluid; DW-MRS, diffusion weighted magnetic resonance spectroscopy; DWI, diffusion-weighted imaging; DTI, diffusion tensor imaging; EAE, experimental allergic encephalomyelitis; EAN, experimental allergic neuritis; EM, electron microscopy; EPI, echo planar imaging; FA, fractional anisotropy; FLAIR, fluid attenuated inversion recovery; FT, Fourier transform; MR, magnetic resonance; MRI, magnetic resonance imaging; MRS, magnetic resonance spectroscopy; MS, multiple sclerosis; MT, magnetization transfer; NAWM, normal appearing white matter; PNS, peripheral nervous system; ROI, region of interest; SD, standard deviation; SNR, signal to noise ratio; SGP, short gradient pulse; SP-SHR, stroke-prone spontaneous hypertensive rat; TE, time to echo. Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 517 strated in hundreds of studies,11,12 DWI developed, CNS in particular, will then be described briefly, despite its high sensitivity to motion, into a viable followed by the extension of this application to diffusion clinical routine practiced in many medical centers.13 In MRI. A few examples for the application of this approach addition, DWI and diffusion-weighted magnetic reso- for analyzing high b-value DWI data will be presented for nance spectroscopy (DW-MRS) were used extensively demonstrating the potential of this approach in extracting with the aim of unraveling the origin of the reduced structural and physiological information on neuronal apparent diffusion coefficient (ADC) of water in systems. Examples both of high b-value q-space analyzed ischemic tissue, compared to normal brain tissue.14,15 diffusion MR images of excised organs, where the short DWI was then used to study, inter alia, brain tumors,16 gradient pulse (SGP) approximation of the q-space head trauma,17,18 experimental allergic encephemyelitis theory can be met or approached, and of in vivo human (EAE),19 spreading depression,20 multiple sclerosis21 and studies, will be presented. Finally, a short discussion will other CNS pathologies.22 Diffusion anisotropy, at least in deal with the potentials and limitations of the q-space the CNS, was documented at the relative early stages of approach in the context of high b-value DWI of neuronal DWI,23,24 but the important development was the tissues. The potential diagnostic capacity embedded in introduction of diffusion tensor imaging (DTI) by Basser the apparent slow diffusing component will be discussed et al.25 DTI provides a means to separate and quantify the briefly at the end of this review. isotropic and anisotropic components of the diffusion tensor and is the basis for MR fiber mapping and MRI tractography.26,27 Recently, DTI has increasingly been used to study different neurological disorders.28–30 High b-value diffusion-weighted MRS and MRI of Until recently, in DWI and DTI studies of neuronal water in neuronal tissues tissues, the experimental data was nearly always analyzed using the well-known Stejskal–Tanner equa- Once the signal decay is not mono-exponential and eqn. tion.31 This equation assumes the existence of a single (1) cannot be used to fit the data it seems that the simplest isotropic and unrestricted diffusing component and is approach is to use a multi-exponential function as shown given by: in eqn. (2) X n E=E0 exp g 2 2 2 =3D exp bD 1 E=E0 Ai exp g 2 2 2 =3Di i1 where E/E0 is the normalized signal attenuation, g is the Xn Ai exp bDi 2 gyromagnetic ratio, g is the pulsed gradient amplitude, i1 is the pulsed gradient duration, D is the time separation between the leading edges of these gradients and D is the In eqn. (2), Ai is the relative fraction of molecules diffusion coefficient. In this equation the term D /3 having the diffusion coefficient Di, n represents the represents the effective diffusion time (for the simple number of components and all other parameters have the case of rectangular pulse gradients) while the b-value same meanings as in eqn. (1). However, it should be represents the overall diffusion weighting in the experi- noted that the use of this equation implies the existence ment.3,4 As most of the DWI and the DTI studies were of several water populations that exhibit free diffusion performed with relatively low b-values (b < 1500 and are in the slow exchange regime. Fitting the data s mm 2), only a single ADC was detected for water in with the simplest case of eqn. (2), i.e. a bi-exponential neuronal tissues. function, means that the system can be described by two During the past few years several groups have shown free diffusing components with a slow exchange that, at sufficiently high b-values, the water signal decay between them. Here, three independent parameters are in diffusion experiments in neuronal tissues is not used to obtain the fitting. Therefore, it is clear that the mono-exponential.32–40 Once the experimental data use of this fitting function is coupled to a certain model cannot be analyzed by eqn. (1), it is necessary to decide that does not necessarily describe the system accurately. on new approaches of analyzing the diffusing data. The However, the bi-exponential fit has been used more different apparent components should then be assigned frequently, partially because the effect of exchange to real physical entities, if these components indeed could be introduced to this model.43 Until recently, most represent such separate entities. Finally, verification of the non-mono-exponential water signal decays in should be attempted as to whether the different neuronal tissue were fitted with a bi-exponential identified components provide more specific structural function,32–40 in analogy to what was proposed for and physiological information concerning the investi- water diffusion in cells.44 It was therefore tempting to gated sample. assign to the two major components to intra- and In this review, we will give a brief description of the q- extracellular water. However, in some of these studies it space approach.41,42 The recent applications of q-space was pointed out that the relative populations of the diffusion MRS on biological samples in general, and of different components do not match those of the intra- Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

518 Y. COHEN AND Y. ASSAF and extracellular spaces,34 if one accepts the assumption that the low ADC of water should be attributed to water in the intracellular space. Others have used more elaborate models that include both exchange and geometrical factors of the systems.35 In addition, it is important to note that NMR diffusion experiments, such as the pulsed gradient spin-echo (PGSE), measure the mean displacement that the observed molecules perform during the diffusion time and not their diffusion coefficients per se. Therefore, to minimize the effect of restriction one should aim at performing diffusion measurements with very strong pulse gradients in which sufficient diffusion weighting is achieved when the gradient duration () and the diffusion time (D) are both very short (1 ms or less). For example, for a compartment of 2–4 mm and a self-diffusion coefficient of 1 mm2 ms 1 (1 10 5 cm2 s 1) the diffusion time should be significantly shorter than 2– Figure 1. Simulated echo attenuation for spins trapped 8 ms, in order to minimize the effect of restriction. It between parallel planes separated by a distance of 2a. Open should be noted that with such diffusion times the effect circles represent data obtained with a diffusion time (D) of 0.2a2/D; solid circles represent D of 0.5a2/D; open of exchange might also be largely suppressed. Under squares represent D of 1.0a2/D; solid squares represent D these experimental conditions, one may approach the of 2.0a2/D; and the continuous line represent in®nite situation in which differentiation between compartments diffusion time (reproduced by permission of Academic Press based on the difference in their intrinsic diffusion from Callaghan47) coefficients is possible. There, the relative populations of the diffusing components should reflect the real populations of each compartment after correcting the signal intensity for relaxation effects (T1 and, more diffusion time. Callaghan and his group demonstrated importantly, T2 effects). However, for extracting struc- that, by using the ‘reciprocal spatial vector’, q, defined as tural information from MR diffusion experiments, (2p) 1gg, instead of the b-value, it is possible to extract diffusion should be studied at the other extreme; namely, structural information on (pseudo)-periodic samples. in the long diffusion time limit.41,42 The long diffusion According to this approach the echo attenuation in time limit is satisfied when D a2/2D, where D is the NMR diffusion experiments relates to the displacement diffusion time, D is the diffusion coefficient and a is the probabilities, using the reciprocal spatial vector q dimension of the compartment in which the diffusion according to eqn. (3), takes place. At such a long diffusion time the molecules Z are able to explore the compartment in which they diffuse E q Ps R; exp i2q RdR 3 and hence their diffusion characteristics should, in principle, report on the compartment size and geometry. However, it should be noted that, under these experi- where ED(q) represents the echo decay as a function of q, mental conditions, both restriction and exchange might R is the net displacement vector (R = r r0), and be important, making the analysis of the diffusion data Ps(R,D) is the displacement probability. For molecules more complex. One way to analyze such data is to use the trapped in a compartment in which diffusion is restricted q-space analysis.41,42 to a particular geometry (i.e. spherical, cylindrical) the displacement probability function may (at long D) relate to the size and shape of the compartment in which the q-Space analysis of NMR diffusion experiments diffusion occurs. These structural parameters will be reflected by diffraction peaks in the signal decay, as was Since excellent comprehensive and authoritative reviews predicted and found in porous materials.47–49 Figures 1 of the q-space theory of NMR diffusion experiments have and 2 show simulations of the signal decay E(q) as a appeared in recent years,45,46 only a brief description function of qa for diffusion between two perfectly addressing practical issues relevant to the specific reflecting rectangular barriers separated by a distance a. application of q-space DWI in the CNS will be given Pictorially, one can state that when a2 is small compared here. As stated previously, in NMR diffusion experiments to the distance that molecules having a diffusion one tags the observed spins at two time points, and the coefficient, D, can travel during the diffusion time D, echo decay in these experiments depends on the net molecules will be ‘reflected’ from the barriers. Therefore, displacement of the observed molecules during the these molecules will have a greater probability of Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 519 Figure 2. Echo attenuation calculated from Monte Carlo simulations for spins trapped between perfectly re¯ecting rectangular barriers. The line legends are for in a2/D units. Note that as becomes bigger the diffraction peak moves to the right, meaning that the extracted distance for in between the plates becomes smaller. Reprinted with permission from Coy A, Callaghan PT. Pulsed gradient spin echo nuclear Figure 3. Diffusion signal decay curves for water in magnetic resonance of molecules diffusing between partially suspensions of human erythrocytes as a function of the q- re¯ecting rectangular barriers. J. Chem. Phys. 1994; 101: value at different hematocrit. The hematocrit values are in 4599±4609. Copyright 1994, American Institute of Physics decreasing order from the top of the ®gure, starting from a value of 93%, followed by 83, 73, 63, 47, 42, and ending at 25% at the bottom of the ®gure (reproduced by permission of Wiley-Liss, Inc. from Kuchel et al.50) returning to the position they occupied during the first pulse gradient. Consequently, the signal decay due to diffusion as a function of q will not decrease mono- tonically. As expected, the simulations in Fig. 1 show restricted diffusion in the cells and to pore hopping in diffraction peaks and demonstrate that, when the the extracellular space.52 These experiments showed diffusion time becomes shorter and D is no longer larger that even in biological systems that are known to be than a2/2D, the diffraction peaks disappear.47 These heterogeneous, structural information can be extracted, results demonstrate that, for obtaining structural informa- although the diffraction peaks were not as sharp as in tion from NMR diffusion data the diffusion time, D, porous materials, probably due to the effects of should be longer than a2/2D. exchange and cell size variability. Subsequently, Kuchel Figure 2 shows simulations similar to those in Fig. 1. showed, also by simulations, that when exchange is Here, the signal decay is plotted as a function of qa for included the simulated curves showed much less different values of . This figure shows that, when the pronounced diffraction peaks, which resembles the pulse gradient duration increases and the SPG experimental curves.53 approximation no longer holds, the diffraction peaks Peled et al. showed that when the sample is move to lower values of qa. Under these experimental heterogeneous, consisting of cells of different sizes, even conditions, the values that are extracted from the q- though they are all cylindrical, the diffraction phenomena space analysis, values that should reflect the distance vanishes.54 In that case the decay curve approaches a between the plates, become smaller. This means that for multi-exponential decay curve. It was also found that the long the extracted dimensions extracted from the q- higher the q-values (or b-values) used in the diffusion space analysis of NMR diffusion data are smaller than experiment, the more exponents are apparent. Using the real ones.48 Callaghan’s formula for diffusion in cylindrical, spheri- Kuchel and Stilbs have used q-space diffusion MRS cal and planar geometries it is possible to simulate the to study the diffusion characteristics of red blood cells.50 signal decay for certain cell diameters.47 Using this They observed the diffraction phenomena for water formula we produced the signal decay for ensembles of molecules in suspensions of red blood cells, which are cylinders having different diameters, as shown in Fig. relatively uniform in shape and size (Fig. 3).50–52 Using 4(A). In such simulations diffraction peaks that represent intra- and extracellular (choline and phosphatidyl cho- the diameter of the cylinder have been observed. Figure line, respectively) markers, it was possible to assign the 4(C) shows the same simulation, but in this case for an maxima and minima of the diffraction peaks to ensemble of cylinders having the diameter distribution Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

520 Y. COHEN AND Y. ASSAF Figure 4. (A) Simulated diffusion signal decay for diffusion in cylinders with different diameters, using the Callaghan formula47 for diffusion perpendicular to the long axis of cylinders. (B) Axon diameter distribution taken from an electron-microscopy analysis of a squirrel's optic nerve.55 (C) Simulated diffusion signal decay for diffusion in cylinders as in (A), but here the decay is calculated for an ensemble of cylinders having the diameter distribution shown in (B) given in Fig. 4(B). It should be noted that the size coefficient, D, and the root mean square displacement distribution shown in Fig. 4(B) represents the axon (DXrms), could be calculated from the full-width at half- diameter distribution as measured from histology of the height of the displacement distribution function (DX0.5) squirrel optic nerve.55 It is clear that the sharp diffraction using eqns (4) and (5), in which td is the diffusion time. peaks, which are clearly apparent when an ensemble of homogenous cylinders is considered, disappear when the X0:5 24D ln td 21=2 4 decay curve is simulated for an ensemble of cylinders of different sizes. In such a case, when no diffraction peaks Xrms 2D td 1=2 0:425 X0:5 5 are observed, it seems that it is much more difficult to extract structural information from the signal decay. Figure 6 shows the q-space analysis, as developed by Cory In 1990, Cory and Garroway utilized the Fourier and Garroway, for an isotropic solution of t-butanol. Figure relation between the decay of the echo intensity, ED(q), 6(A) shows the experimental signal decay of t-butanol at and the displacement probability, Ps(R,D) to extract different diffusion times. Figure 6(B) shows the effect of structural information on an heterogeneous systems from the diffusion time on the displacement distribution profiles such diffusion experiments.41 They used yeast cells to obtained by Fourier transformation of the data shown in show that, by Fourier transformation of the signal decay, Fig. 6(A). The data in Fig. 6 show that, as the diffusion time it is possible to extract the displacement distribution increases, the t-butanol molecules can travel greater profile which give, under the SGP condition, the cells’ distances and the probability for zero displacement diameter as shown in Fig. 5. In that article, Cory and decreases. Using eqn. (4) one can relate the displacement Garroway developed the q-space analysis mathemati- profile to the diffusion coefficient. In order to do that, one cally for displacement distributions having a Gaussian has to acquire the signal decay until it reaches less than 1% shape.41 From the distribution profile, both the diffusion of its original value and to plot it on a logarithmic scale as a Figure 5. Displacement distribution pro®le for water diffusion in yeast cells at diffusion times of 5, 20 and 100 ms. (B) is an expansion of (A) (reproduced by permission of Wiley-Liss, Inc. from Cory and Garroway41) Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 521 Figure 6. (A) Signal decay in PGSE experiment of t-butanol at 25 °C at three diffusion times of 35, 125 and 305 ms. (B) Displacement distribution pro®les calculated by Fourier transformation of the data shown in (A). (C) The rms displacement calculated from the full-width at half-height of the Gaussian displacement distribution pro®les shown in (B) against the square root of the diffusion time. The slope of the straight line in (C) provides the self-diffusion coef®cient of t-butanol (2.7 10 6 cm2 s 1) (reproduced by permission of Wiley-Liss, Inc. from Assaf and Cohen66) function of the q-values [Fig. 6(A)]. The displacement However, as stated previously, the q-space analysis was distribution profiles can be produced by Fourier transfor- developed under the short gradient pulse (SGP) approx- mation of the signal decays in Fig. 6(A) for each diffusion imation, i.e. for cases in which → 0 and D. time [Fig. 6(B)]. The full-width at half-height (DX0.5) is However, in many cases the SGP approximation is then used to calculate the root mean square displacement violated due to insufficient strength of the pulsed (DXrms) from these profiles for each diffusion time. The gradients, which requires the use of long gradient pulses slope of the plot of DXrms as a function of the square root of in order to achieve adequate b or q values. The effect of the diffusion time [Fig. 6(C)] is then proportional to the long gradient pulses was investigated theoretically by diffusion coefficient of the diffusing molecules. Indeed, simulations48,56 which predicted that the measured cell when this approach was performed for a solution of size would be smaller than the real size when the SGP t-butanol it was possible to extract from the slope of the condition is violated. Since, in neurological systems, graph in Fig. 6(C) the actual self-diffusion coefficient of factors other than restriction may influence the signal t-butanol. Therefore, for the case of isotropic diffusion one decay (i.e. compartmentation, exchange) we studied the can relate the displacement profile and the diffusion effect of the pulsed gradient length () on the signal decay coefficient using eqn. (4). In cases of non-Gaussian and the displacement distribution profiles, experimen- diffusion, as is found in biological tissues, one can use tally. Figure 7(A) depicts the normalized signal decay of this relation to calculate the apparent diffusion coefficient water in excised sciatic nerve for a series of diffusion (ADC) if a mono-exponential signal decay is observed. experiments where both and g were varied in a way that The comparison between Figs 5(B) and 6(B), which show kept the q values (and b-values) the same in all the effect of the diffusion time on the displacement profiles experiments. Figure 7(B) shows the displacement dis- for restricted and un-restricted diffusion modes, demon- tribution profiles obtained by FT of the decay curves strates the power of this approach for recognizing restricted shown in Fig. 7(A). In these experiments, we started with diffusion. In the case where restriction is significant, the a combination of and gmax of 4.5 ms and 160 increase in the diffusion time does not result in a wider Gauss cm 1, respectively, for which the mean displace- displacement distribution profile. ments extracted from the q-space analysis were found to be in good agreement with the axons size distribution observed by EM (see below).57 For extreme violation of q-Space analysis and experimental parameters the SGP condition we used the following parameters: gmax = 10 Gauss cm 1 and = 72 ms, which approaches The q-space analysis provides a means of obtaining the experimental parameters used in the clinical set-up of displacement–probability profiles even in complex sys- such diffusion experiments (see below). In these experi- tems by performing a Fourier transformation of the echo ments all other experimental parameters were kept decay with respect to q even when diffraction peaks are constant (TR, TE, q-value, b-value, D). The data in Fig. not observed. This means that q-space analysis of MR 7(A) shows that, as expected for restricted diffusion,48,56 diffusion experiments is suitable for obtaining structural the signal decay becomes smaller when the diffusion information on the investigated sample, non-invasively. gradient duration, , is increased and the apparent relative Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

522 Y. COHEN AND Y. ASSAF Figure 7. The effect of the length of the diffusion gradient pulse () on the water diffusion in sciatic nerve. (A) Effect on the signal decays. (B) Effect on the respective q-space distribution pro®les. The gradients were applied perpen- dicular to the long axis of the nerve. Diffusion experiments were performed on an 8.4 T NMR spectrometer equipped with a micro5 gradient probe driven by a BGU-II system producing pulse gradients of up to 190 Gauss cm 1 in each of the three directions. Diffusion experiments were performed using the PGSE pulse sequence with the following parameters: TR/TE = 3000/206 ms and D = 100 ms. Five sets of experiments were performed with different diffusion gradient duration and amplitude and 24 data points but with the same q- and b-values. The duration of the diffusion gradients was 4.5, 9, 18, 36 and 72 ms with respective gradient amplitudes of 160, 80, 40, 20 and 10 Gauss cm 1 (reproduced by permission of Wiley-Liss, Inc. from Assaf et al.89) population of the slow diffusing component becomes very accurate. Nevertheless, their relative size extracted larger. Consequently, the extracted displacement distri- from such a q-space analysis will reflect the relative bution profiles become narrower and more intense.58 physical sizes of the different compartments. These results show that the displacement extracted from Besides the need to perform the diffusion experiment the q-space analysis of these diffusion data decreased over a long diffusion time scale and under short-pulse- from a value of 3.3 mm at of 4.5 ms to a value of about gradient conditions, in order to extract the real dimen- 1.6 mm at of 72 ms. This implies that, when one is forced sions of the compartment in which the diffusion occurs, to use long , as on clinical MRI scanners, the mean adequate spatial resolution should be used. The spatial displacements obtained from the q-space analysis are resolution of the q-space analysis is determined by the smaller than the physical size of the compartment in maximal magnitude of the pulsed gradients or the which the restricted diffusion takes place. However, the maximal q-value that is used in the measurements (qmax). differences are not very large and, even when is changed In the yeast cells example (Fig. 5) the resolution was by a factor of 16, the extracted mean displacements vary 2.75 mm41 (distance between two adjacent point in the q- only by a factor of two.58 Thus, the utilization of long space profile), while in the mouse brain example the diffusion gradient pulses overemphasizes the restricted resolution was 4.1 mm (see below).59,60 In principle, the component. This experiment suggests that, when extract- spatial resolution can be calculated using eqn. (6): ing an ADC from diffusion experiments in which the diffusion is restricted, the ADC depends not only on the 1 R 6 diffusion time but also on the pulsed gradient strength. In N q this case, it might be possible that for two equivalent b- values, one having short gradient pulse and strong where R is the resolution, N is the number of gradient gradient strength and the other having a long gradient points acquired in the PGSE experiment and Dq is the pulse and small gradient amplitude, two different ADC difference in q-values between two adjacent gradient values will be calculated. These results demonstrate that, points (assuming that the q-values are changed linearly in general, caution should be exercised when comparing and are equally spaced). A gradient system with a ADCs obtained from different experiments performed maximal gradient amplitude of 100 G cm 1 and 2 ms with different experimental parameters. Regarding the gradient pulse duration will give a displacement resolu- structural parameter obtained from a q-space analysis tion of about 11 mm. This resolution is insufficient to when the SGP approximation is violated, it should be measure displacements in biological systems in which the noted that the extracted absolute numbers might be not cell sizes may be of 5 mm or less. Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 523 Figure 8. Increasing q-space resolution using zero ®lling (zf) and data extrapolation. (A) Original data and (D) the Fourier transformation (FT) of the data in (A). (B) Original data and the zero ®lled curve and (E) the corresponding displacement pro®le obtained by FT of the data in (B). Notice that zero ®lling produces wiggles in the displacement distribution pro®le, due to the truncation of the data. (C) Original data and the extrapolated curve and the corresponding displacement pro®le (F) obtained by FT of the data in (C). The resolution is 5.8 mm in (D) and 0.6 mm in (E) and (F) In principle, there are two ways to increase the spatial distribution profile to calculate the fraction of water resolution of the Fourier transformation: zero filling and molecules that exhibits a net displacement of less than data extrapolation to higher q-values, as shown in Fig. 8. 10 mm in normal, ischemic and postmortem brain tissues Enhancement of the resolution by zero filling or using localized diffusion spectroscopy (Fig. 9).59,60 They extrapolation allows extraction of smaller displacement, found that in ischemic or postmortem conditions the even though the hardware limits the resolution. Zero displacement distribution profile becomes narrower, filling is a less subjective method for resolution which is consistent with the ADC reduction observed enhancement and is commonly used when the signal after stroke.59,60 They also found that in ischemic, and decays to less than 2–4% of its original value at qmax. In even more so in postmortem, conditions the fraction of cases where the signal decays to higher values (>4% of water molecules having a mean displacement of less than its original value) zero filling will result in wiggles in the 10 mm increases. In their second article,60 King et al. displacement probability profile, as shown in Fig. 8(E). In discuss the possible causes for the existence of non- these cases, data extrapolation seems to be the better mono-exponential decay curves. They attributed the choice, compared with zero filling. However, it should be non-gaussian behavior of the decay curves to tissue noted that extrapolation requires fitting the data to a heterogeneity and restricted diffusion in the region of model, the accuracy and validity of which may be interest. Despite this heterogeneity and the non-gaussian questioned. decay curves, it is interesting that in these studies a nearly mono-gaussian displacement distribution profile was observed, which in turn was attributed to the large q-Space diffusion MRS in neuronal tissues heterogeneity of the examined ROI and to the effect of exchange. These studies demonstrated that there is a King et al. were the first to use the q-space approach on significant water component that persists even at high neuronal tissues.59,60 They used the displacement diffusion weighting, implying a water population with an Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

524 Y. COHEN AND Y. ASSAF Figure 9. Water displacement pro®les derived from q-space measurements for (A) intact, in vivo mouse brain, (B) ischemic lesion in an in vivo mouse brain and (C) post-mortem, in situ mouse brain (reproduced by permission of Wiley-Liss, Inc. from King et al.60) apparent slow diffusion coefficient. Although their some structural (although limited) information could be studies could not shed light on the origin of the low obtained from the q-space analysis of such diffusion ADC in ischemic brain tissues, they did demonstrate that data. Figure 10. Water signal decay curves in diffusion experiments on excised rat brain at different diffusion times as a function of (A) the b-values, and (B) the q-values. (C) The respective displacement pro®les of the data shown in (B). NAA signal decay in diffusion experiments performed on excised rat brain at different diffusion times as a function of (D) the b-values, and (E) the q- values. (F) The respective displacement pro®les of the data are shown in (E) Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 525 Figure 11. Displacement distribution pro®les from (A) excised bovine optic nerve measured parallel to the long axis of the nerve and (B) excised bovine optic nerve measured perpendicular to the long axis of the nerve. The data was acquired on an 11.7 T spectrometer with a z-gradient probe. The stimulated echo diffusion pulse sequence was used with the following parameters: TR/TE = 3000/70 ms, = 15 ms, gmax = 27 g cm 1 and qmax = 1727 cm 1.66 As of 1996 we were interested in finding a probe that that the diffusion time has a larger effect on the water would allow cell swelling to be measured in neuronal decay curves. However, the interesting point is the effect tissue non-invasively.61 Therefore, we looked for a of the diffusion time on the displacement distribution cellular marker that would report on the size of the profiles. With the increase in the diffusion time the water compartment. Water is found in all compartments and is displacement profiles become wider, indicating that the generally known to exchange across membranes faster water molecules translate larger distances as the diffusion than metabolites.62,63 Indeed, King et al. have noted that, time increased, exhibiting almost free diffusion. In the during the diffusion time of 50 ms that was used in that NAA case, however, the displacement profiles are almost study, the majority of the water molecules have diffused the same at the different diffusion times, suggesting that over a distance greater than three times the cell spacing, the diffusion of this fraction of metabolites is highly probably due to fast exchange between intra- and extra- restricted. These results imply that structural information cellular spaces.60 Therefore, to minimize the effect of about the intracellular compartment could be obtained exchange, we decided to study the diffusion characteris- from diffusion characteristics of metabolites.36,61–64 tics of intracellular metabolites at high b-values and long However, when high b-values water diffusion in optic diffusion time, first using a multi-exponential model and nerve was studied by the q-space approach the existence later using the q-space approach.62–64 Figure 10(A) and of a large water population that exhibits restricted 10(B) shows the water signal decays as a function of the b diffusion was found as shown in Fig. 11.65,66 Interest- and q-values, respectively while Fig. 10(C) shows the ingly, it was found that the effect of diffusion time on the respective displacement distribution profiles for water in water signal decay, as a function of the b-values in the in excised rat brain at different diffusion times. The optic nerve and in brain tissues, shows opposite trends. In displacement distribution profiles shown in Fig. 10(C) nerve, the water signal decay as a function of the were obtained by FT of the data shown in Fig. 10(B). diffusion times behaves similarly to the signal decay of Figures 10(D), 10(E) and 10(F) show the same data set the metabolites in brain tissues (where exchange effects obtained with the same experimental parameters, on the are less important). Figures 10(C) and 11, showing the same samples but for N-acetyl aspartate (NAA). Inter- displacement distribution profiles of water in brain estingly, the effect of diffusion time on the decay curves tissues and in bovine optic nerve in two orientations of water and NAA as a function of the b-values was very [Fig. 11(A) and 11(B)] for different diffusion times, different. For water molecules, it was found that as demonstrate the ability of the q-space analysis to identify (D /3) increased the apparent fraction of the slow restricted diffusion. In the isotropic solution of tert- diffusing component decreased, while the opposite trend butanol, a gaussian distribution profile is obtained [see was observed for the NAA molecules. This is to be Fig. 6(B)] and the mean displacement increases linearly expected, since in the water case, exchange is probably with the square root of td [Fig. 6(C)], as expected from more important than restriction, while in the NAA case it eqn. (4). For water in nerve, however, it is clear that in seems that restriction is the dominant factor.64 When the both orientations there is a component for which the water and the NAA signal decays are plotted against the displacement distribution profile is not affected by the q-values for different diffusion times, one can observe increase in the diffusion time as expected for restricted Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

526 Y. COHEN AND Y. ASSAF Figure 12. Water signal decay in diffusion experiments and the respective q-space pro®les for control and EAN diseased sciatic nerves. Normalized signal intensities for (A) day 9, (B) day 23 post-immunization and (C) for the chronic phase groups along with the signal intensity of the control group. Displacement distribution pro®les for (D) day 9, (E) day 23 post- immunization and (F) for the chronic phase groups along with the q- space pro®le of the control group.67 diffusion [Fig. 11(A) and (B)].65,66 When the diffusion is that a certain fraction of water molecules are restricted to measured perpendicular to the fibers’ direction in the a compartment of about 2 mm. More importantly, nerve, the relative weighting of the restricted component although the compartment size in which the restricted is even larger [Fig. 11(B)]. In the nerve experiments, in diffusion occurs seems to be very similar for the different which diffusion was measured parallel to the long axis of CNS tissues, the relative weighting of this restricted the nerve, mixed behavior was detected; whereas the component is larger in white matter-rich areas and narrow component was not influenced by the diffusion depends on its orientation.66 These observations suggest time, the displacement of the broad (fast) component that this slow and restricted diffusing component is much became larger as the diffusion time increased. A close more prominent in white matter.64–66 Therefore, it was examination of the displacement distribution profile postulated that water diffusion at high b-values and long reveals that, even in brain tissue, there is a small diffusion time (and relatively long TE) may provide a component for which the mean displacement does not useful means for following the pathophysiological state increase linearly with the square root of the diffusion of white matter. time.65,66 Diffusion in the axonal milieu is one possible The results shown in Figs 10(C) and 11 demonstrate explanation for the slow restricted diffusing component. Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 527 Figure 13. The steps for obtaining q-space analyzed MR images for system in which the direction of the ®ber is known a priori and the diffusion needs to be measured only in one direction (perpendicular to the long axis of the ®bers).68 If this assignment is correct, any process of demyelina- space MRS was able to detect the EAN pathology at day tion should have a strong effect on the weighting and on 9 [Fig. 12(A) and 12(D)] before the appearance of any the diffusion characteristics of this slow component clinical signs. In addition, both the deterioration up to day observed at high diffusion weighting. Therefore it seems 23 post-immunization, the time at which maximal logical to study experimental allergic neuritis (EAN) demyelination occurred [Fig. 12(B) and (E)], and the using high-b-value q-space DWI since it is as an animal improvement at the chronic stage [Fig. 12(C) and (F)], model where demyelination occurs.57,67 In this model, could be followed using q-space diffusion MRS. The rats are immunized with myelin basic protein, which structural changes correlated well with electron micro- causes an autoimmune reaction against myelin that scopy.67 results in progressive demyelination on the peripheral All these recent applications of q-space diffusion in nervous system (PNS). In the chronic stages of this biological samples dealt with diffusion-weighted MRS model, remyelination occurs. In the peripheral sciatic and, as such, gave only average values for the entire nerve, a multi-exponential signal decay was also system.41,50–53,57,59,60,64–67 Biological systems are het- observed, as in the cases for CNS tissues [Fig. 12(A)]. erogeneous in nature and it is therefore desirable to obtain In this study the signal of water in the sciatic nerves could the spatial distribution of the parameters extracted from not be fitted accurately even by a bi-exponential function. the q-space analysis for each pixel. We therefore Only a tri-exponential function gave a good fit to the computed displacement and probability MR images or experimental data, motivating us to use q-space analysis maps based on the q-space analysis of high b-value DWI. in this study also.67 Indeed, the results were as expected. These images were used to study the pathophysiological It was found that the weighting of the slow diffusing state of white matter.68 component decreased significantly with the progression of demyelination and increased, although not to control values, when remyelination occurred [Fig. 12(A–C)]. q-Space displacements and probabilities MR The q-space profiles obtained by FT of the data shown in images Fig. 12(A–C), enabled characterization of these changes, as shown in Fig. 12(D–F). These figures show a decrease The procedure for obtaining the q-space MR images was in the population of the component with the narrow described previously and is outlined schematically in Fig. displacement profile and an overall broadening of the 13.68 In principle, a set of diffusion images is arranged in displacement profile following induction of the EAN. It a 3D array in which the x and y coordinates are the image was found that the weighting of the slow diffusing axes and the z direction is that of the q-values. The z component that is restricted decreased by 20% between direction is then zero-filled (in some cases, for compari- control and EAN diseased sciatic nerves at day 9 post- son the z direction was extrapolated using a multi- immunization. Concomitantly, the mean displacement of exponential decay function) in order to increase FT the restricted component increased by 33% from a value resolution. Then the signal decay in each pixel of the of 3.9 0.4 mm for control nerves to a value of image was transformed into displacement distribution 5.2 0.1 mm for diseased nerves.67 The high b-value q- profiles using eqn. (3) by an in-house Matlab1 program. Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

528 Y. COHEN AND Y. ASSAF Figure 14. The effect of spinal cord maturation on (A) the white water signal decay with respect to q, and on (B) the respective displacement distribution pro®les obtained by FT of the data shown in (A). Data obtained when diffusion was measured perpendicular to the long axis on the cord and with a diffusion time of 150 ms. The data were acquired on a Bruker 8.4 T spectrometer equipped with a micro-imaging accessory. The stimulated echo diffusion pulse sequence was used with the following parameters: TR/TE = 1500/30 ms, D/ = 150/2 ms, gmax = 150 g cm 1 and qmax = 1277 cm 1.68 The Fourier transformation of the signal decay with nent.65,66 This conclusion was the driving force for the respect to q produced a non-mono-gaussian displace- attempts to evaluate the diagnostic capacity embedded in ment–distribution profile for each pixel in the image. In this apparent slow-diffusing component. Both white the spinal cord, diffusion was measured perpendicular to matter maturation and degeneration as well as spinal the long axis of the fibers of the spinal cord, resulting in cord trauma were evaluated first by high b-value q-space only one displacement distribution function for each DWI.68,69,72 pixel. Two parameters of the displacement distribution profile, namely the displacement as calculated from the full width at half-height and the probability for zero High b-value q-space DWI of in vitro spinal cord displacement (given by the height of the displacement maturation profile at zero displacement), are then extracted for each pixel. Finally, two sub-images based on these two One of the first models that was used to test the high b- parameters are constructed on a pixel-by-pixel basis. value q-space DWI approach was spinal cord maturation in the rat.68 Figure 14(A) shows the effect of the spinal cord maturation on the water signal decay as a function of APPLICATION OF HIGH b-VALUE q-SPACE q and Fig. 14(B) shows the respective displacement– DWI distribution profiles obtained by FT of the decay curves shown in Fig. 14(A) after extrapolation. This figure As stated previously, the data in Fig. 11 imply that there clearly demonstrates that, as maturation progresses, the is a large fraction of water molecules in optic nerve displacement–distribution profile becomes narrower and whose diffusion is restricted to about 2 m, even when the more intense, indicating an increase in the component of MR diffusion experiments were carried out with a spinal cord water that exhibits restricted diffusion.68 q- diffusion time of 305 ms.65,66 These studies also showed Space analyzed MR images that follow the spinal cord that this slow diffusing population is much larger in white maturation are shown in Fig. 15. This Figure shows the matter than in gray matter and it was found that the displacement and probability MR images of spinal cords displacement of this component hardly changes when the of rats at ages of 3, 7, 17 and 77 days taken with a diffusion time is increased by a factor of about 10.65 It diffusion time of 150 ms. The mean displacement of the was also found that, in optic nerve, the relative fraction of water molecules in the white matter decreased with age, the apparent slow diffusing component depends strongly reaching a value of about 2.2 0.3 mm at 77 days. At 3 on the relative orientation of the diffusion-sensitizing days, the mean displacement in the white matter was gradients with respect to fiber orientation.65,66 Based on similar to that in the gray matter (9.6 0.2 and these findings we suggested that axonal water contributes 9.8 0.2 mm, respectively). Significant changes were significantly to this apparent slow-diffusing compo- also observed in the probability images. Here it was Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 529 Figure 15. (A) q-Space displacement and (B) probability MR images of spinal cords of rats of different ages along with the displacement distribution pro®les of the ROIs depicted on the images. (C) and (D) show the displacement pro®les of gray and white matter of 3-day-old rat spinal cord, respectively, and (E) and (F) show the pro®le in the gray and white matter of a mature rat spinal cord, respectively. For experimental details see Fig. 14 (reproduced by permission of Wiley-Liss, Inc. from Assaf et al.68) found that the probability of zero displacement increased following VaD seem to be related to the progression of with age. Analysis of the pixels in the white and gray demyelination and axonal loss. It is known that stroke- matter of the newborn and mature rat spinal cords prone spontaneous hypertensive rats (SP-SHRs) develop, revealed that the contrast is formed due to the change in under a high-salt diet, chronic hypertension that may lead the diffusion characteristics of the white matter upon to multi-focal stroke lesions in the brain, which in turn maturation.68 As seen in Fig. 15, the mean displacement can lead to neuronal degeneration, even in the spinal in the gray matter barely changed between day 3 [Fig. cord. 15(C)] and day 77 [Fig. 15(E)]. It is the dramatic decrease Indeed, the SP-SHR population developed motor in the mean displacement in the white matter, from 9– impairment under conditions of a high-salt diet.69 It was 10 mm [Fig. 15(D)] to around 2–3 mm [Fig. 15(F)], which suspected that these impairments originated from small brings about the formation of the gray–white matter multiple ischemic lesions in their brains that are also contrast in the mature spinal cord. This data shows that projected into their spinal cords. In order to evaluate the the progression of maturation, which is also accompanied diagnostic capacity of the high b-value q-space DWI by myelination, brings about the restriction observed in approach we computed q-space MR images of spinal cords the mature white matter. of a group of SP-SHRs and a group of age-matched control rats. In this study, diffusion anisotropy at low b-values (bmax = 2500 s mm 2) and high b-value q-space MR High b-value q-space DWI of spinal cord degen- images were acquired for the two groups in vitro. The eration caused by chronic hypertension69 diffusion data collected from ROIs that encompass the entire white matter of the spinal cords of these two groups It is well known that chronic hypertension (high blood are given in Fig. 16. Figure 16(A) shows the natural pressure) is one of the major risk factors for ischemic logarithm of the normalized signal intensity for the low b- white matter lesions (also termed leukoaraiosis) or value range (bmax = 2500 s mm 2) while Fig. 16(B) shows vascular dementia (VaD).70,71 Some of the clinical signs the normalized signal-decay curves, on a logarithmic Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

530 Y. COHEN AND Y. ASSAF Figure 16. Diffusion data for SP-SHR spinal cords (n = 5) and control Wistar rats spinal cords (n = 4). (A) Diffusion anisotropy measurements at low b-values up 2.5 103 s mm 2. The diffusion gradients were applied along the z direction (parallel to the spinal cord long axis, represented by the // symbol) and along the x direction (perpendicular to the spinal cord long axis, represented by the ? symbol). (B) Signal decay as a function of b when diffusion was measured perpendicular to the spinal cord long axis for the high b- value range (up to 1 105 s mm 2). The data were acquired with the same system and with the parameters given in caption of Figure 14.69 scale, of the control and SP-SHR groups for the entire diffusion anisotropy index computed from the low b- range of b-values (up to bmax of 1 105 s mm 2). This values range (up to 2.5 103s mm 2) shows no significant figure shows that the differences between the control and differences between the SP-SHR and the control groups. SP-SHR groups are much more pronounced at the high b- However, the population fraction of the slow-diffusing value range. Figure 16(A) demonstrates also that the component at high b-values was significantly smaller in Figure 17. q-Space displacement image of a spinal cord of a control rat along with (B) EM image (1500) and (C) a magni®cation of a single axon (5000) of the same rat spinal cord shown in (A) showing the nearly intact myelin around the axon. (D) q-Space displacement image of a spinal cord of a SPSHR rat along with (E) EM image (1500) and (F) a magni®cation of a single axon (5000) of the same rat spinal cord shown in (D) showing the progressive demyelination and reduction in number of myelin wraps around the axon. For experimental details see Fig. 14.69 Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

HIGH b-VALUE q-SPACE DWI OF WHITE MATTER PATHOLOGIES 531 Figure 18. q-Space probability MR images (upper panel) and osmium staining (lower panel) of a rat spinal cord 6 weeks after 60 s hemi-crush trauma. The trauma site is the central slice (slice 3 out of the ®ve slices shown).72,73 the SP-SHR group than in the control group (51 2 vs loss associated with spinal cord injury, rely heavily on 67 2% respectively).69 The mean displacement in- experimental models of spinal cord trauma and on creased from a value of 2.2 0.3 mm to a value of methods for evaluating the pathophysiological state of 3.8 0.7 mm (p < 0.03), while the probability for the zero the spinal cord. However, it was recently commented that displacement decreased from 10.6 1.4 (a.u.) to a value one of the main problems in evaluating new spinal cord of 6.4 0.9 (a.u.; p < 0.03).69 These changes in the therapy is the relative difficulty in assessing the extracted q-space parameters, namely the increase in the pathophysiological state of the spine.75 Diffusion in mean displacement and the decrease in the probability for general, and diffusion anisotropy in particular, was zero displacement, are in line with what is expected from identified as a useful means of studying the spinal cord axonal loss and demyelination. Interestingly, it was found in the early days of DWI.23 In recent years, there have that the anisotropy index at the low b-value range showed been significantly more DWI studies of spinal cord only a small reduction from a value of 0.84 0.10 to morphology and pathology.76–79 Here again, in most of 0.76 0.11. This change was statistically non-significant. the spinal cord diffusion studies, the data was analyzed These results suggest the higher sensitivity toward this using a mono-exponential decay, i.e. Stejskal–Tanner pathology of the high b-value q-space DWI, compared equation [eqn. (1)], and in a few the bi-exponential with the conventional diffusion anisotropy index com- function shown in eqn. (2) (where n = 2) was used.40 We puted from low b-value DWI. Figure 17 shows q-space therefore thought that spinal cord trauma, which may displacement MR images of a control and SP-SHR spinal cause axon degeneration, might be more visible when cords, along with electron microscopy (EM) of these using high b-value q-space DWI, especially for areas spinal cords at two different magnifications. As expected distant from the trauma site. Here high b-value q-space for the SP-SHR spinal cord, where the mean displacement DWI was measured perpendicular to the long axis of the extracted from the q-space diffusion data is larger, a clear fibers of the spinal cord as seen in the upper panel of demyelination and neuronal degeneration is observed in Fig. 18.72,73 the EM. The EM images show reduction in the thickness Indeed high b-value q-space DWI was found to be of the myelin in the SP-SHR spinal cords and the sensitive to the hemi-crush model of the spinal cord and formation of large vacuoles between the axon and the provided evidence for damage in areas remote from the disrupted myelin. These vacuoles may explain the increase trauma site even 6 weeks post-trauma for the 60 s hemi- in the average displacement in those spinal cords. crush as seen in Fig. 18. This Figure also demonstrates that at the trauma site and in remote areas where demyelination occurred there was a reduction in the High b-value q-space DWI of spinal cord trau- probability for zero displacement. This decrease in the ma72,73 probability for zero displacement was accompanied by an increase in the mean displacement as expected for a High b-value q-space diffusion MRI was used also to demyelination process. As it was suggested that the study spinal cord trauma.72,73 It is well known that spinal apparent slow restricted diffusing component originates cord injury has devastating consequences and is therefore mainly from intra-axonal water,64,66 for which the myelin studied extensively.74 These studies, which are directed serve as a barrier, the computed q-space images were towards finding ways to reverse or reduce the functional compared with osmium staining for myelin. Indeed, in Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:516–542

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