Constant gradient FEXSY: A time-efficient method for measuring exchange

Journal of Magnetic Resonance 311 (2020) 106667

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Constant gradient FEXSY: A time-efficient method for measuring exchange Yuval Scher a,⇑, Shlomi Reuveni a, Yoram Cohen b a School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel b School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, The Sagol School of Neuroscience, Tel Aviv University, Tel Aviv 6997801, Israel

a r t i c l e

i n f o

Article history: Received 29 October 2019 Revised 1 December 2019 Accepted 6 December 2019 Available online 16 December 2019 Keywords: CG-FEXSY Pulsed-field gradient PFG Filter-exchange PFG FEXSY FEXI Constant gradient PFG CG-PFG Transmembranal water exchange rate Intra-cellular mean residence time

a b s t r a c t Filter-Exchange NMR Spectroscopy (FEXSY) is a method for measurement of apparent transmembranal water exchange rates. The experiment is comprised of two co-linear sequential pulsed-field gradient (PFG) blocks, separated by a mixing period in which exchange takes place. The first block remains constant and serves as a diffusion-based filter that removes signal coming from fast-diffusing water. The mixing time and the gradient area (q-value) of the second block are varied on repeated iterations to produce a 2D data set that is analyzed using a bi-compartmental model which assumes that intra- and extracellular water are slow and fast diffusing, respectively. Here we suggest a variant of the FEXSY method in which measurements for different mixing times are taken at a constant gradient. This Constant Gradient FEXSY (CG-FEXSY) allows for the determination of the exchange rate by using a smaller 1D data set, so that the same information can be gathered during a considerably shorter scan time. Furthermore, in the limit of high diffusion weighting, such that the extra-cellular water signal is removed while the intra-cellular signal is retained, CG-FEXSY also allows for determination of the intra-cellular mean residence time (MRT). The theoretical results are validated on a living yeast cells sample and on a fixed porcine optic nerve, where the values obtained from the two methods are shown to be in agreement. Ó 2019 Elsevier Inc. All rights reserved.

1. Introduction Determining the structure and dynamic properties of microscale porous systems is of great interest in many fields [1,2]. Diffusion-magnetic resonance (diffusion-MR) techniques were shown to allow for such determination completely noninvasively. Thus, these methods are applicable, inter alia, to in vivo study of biological tissues, and particularly to the study of neuronal tissues, which exhibit distinct structural characteristics. Indeed, diffusion-MR is essential in characterization of normal tissues and in diagnosis of pathological tissues and diseases, in both biomedical research and the clinics [3–12]. Conventional diffusion-MR studies of biological systems utilize, in general, pulsed-field gradient (PFG) MR experiments [13,14], in which the signal is sensitized to the translation of water molecules. These experiments are usually conducted with diffusion times, D, long enough for water molecules to probe the boundaries of their compartments, and gradient pulse durations, d, short enough for ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Scher), [email protected] (Y. Cohen). https://doi.org/10.1016/j.jmr.2019.106667 1090-7807/Ó 2019 Elsevier Inc. All rights reserved.

diffusion during the gradient pulse to be ignored (the short gradient pulse [SGP] approximation) [15]. Under such conditions, information can be inferred from the dependence of the signal attenuation on D and on the ‘reciprocal spatial vector’ q, whose magnitude is proportional to the area of the gradient pulse. The gradient area (q-value) can be varied either by controlling the strength, g, or the duration, d, of the gradient pulse [1,2,13–15]. For a single freely-diffusing water population the signal attenu2

ation is of the form e
Dq D , where D is the diffusion coefficient of water in the medium [13–15]. Biological tissues, however, are complex multicompartmental systems with semi-permeable membranal boundaries. The signature of the attenuation in such systems is affected by a large number of parameters which include: the number of distinct water populations, their intrinsic diffusion coefficients, the structural features of the compartments within which they reside, and the exchange between them. The diffusional exchange of water molecules across axonal membranes and myelin sheaths is generally assumed to be in the slow-exchange regime, hence it is often neglected when modeling diffusion in neuronal tissues [16,17]. Nonetheless, transmembranal water exchange is affected by a myriad of processes that take place in living cells, and could thus potentially provide indication on

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tissue state. Indeed, in the last few decades, MR measurement of exchange rates were shown to be sensitive to a diverse set of cellular conditions and pathologies [18–22]. Furthermore, recent studies suggest that diagnosis of central nervous system (CNS) pathologies based on exchange measurements is possible [23]. A series of PFG experiments with constant gradient strength (CG-PFG) and varied diffusion times was shown, by Pfeuffer et al. [24,25] and Meier et al. [26,27], to allow for the determination of transmembranal exchange rates in bi-compartmental systems, assuming the bi-exponential Kärger model [28–30]. This model is of two exchanging spin populations, which are usually associated with intra- and extra-cellular water. While the Kärger model assumes free-diffusing populations, later works by Price et al. [30] and Meier et al. [26] revised it to fit cases where one of the populations is restricted to a sphere or a cylinder, respectively. Callaghan and Komlosh introduced a two-dimensional (2D) diffusion-NMR method for measuring exchange, called diffusion exchange spectroscopy (DEXSY) [31,32]. Briefly, the DEXSY pulse sequence is comprised of two sequential PFG blocks, separated by a mixing time t m , in which the system evolves unperturbed. Each block records the displacement at a different time. Spins that exchange diffusion sites during the mixing time appear as cross peaks in a computed 2D-correlation map. The strength of the method is that producing the 2D-correlation map does not require any model, thus allowing to easily identify multi-sites exchange processes. However, quantitative exchange measurement with DEXSY requires a 3D acquisition scheme where different mixing times are probed as well. 3D-DEXSY is extremely demanding in terms of scan time and is thus scarcely used. Recently, a paper by Benjamini et al. suggested a sampling and processing strategy that greatly reduces 3D-DEXSY scan time [33]. Cai et al. used a reparameterizaion approach that utilizes the symmetry of the 2D-correlation map, originally proposed by Song et al. for analysis of 2D relaxation-MR experiments [34], to futher reduce the scan time of the DEXSY experiment [35]. However, these novel methods are yet to be tested on a biological tissue. Åslund et al. [36] proposed setting the first block of a DEXSY experiment to be a co-linear constant preparation module. The preparation module serves as a displacement-based filter that precedes the conventional PFG experiment. In bi-compartmental systems, the filter selectively removes the signal of the fast diffusing water population. By changing the mixing time, t m , in repeated iterations, it is possible to measure the exchange rate between the filtered population and the slowly diffusing population - an experiment termed filter-exchange spectroscopy (FEXSY, Fig. 1). The imaging version of FEXSY, known as filter-exchange imaging (FEXI), was shown to allow for in vivo diagnosis of pathological tissues on clinical scanners [20,23].

FEXSY results are analyzed using a Kärger-like model, where both intra- and extra-cellular water are assumed to be freelydiffusing, albeit with apparent diffusion coefficients (ADCs) which depend on hindrance and restriction. In this description, the slowly diffusing population is associated with the intra-cellular water. Thus, a strong enough filter can remove the extra-cellular signal while retaining most of the intra-cellular signal. During the mixing period, some of the intra- and extra-cellular water diffuse across the membrane, changing their affiliation accordingly. As the mixing time is increased, the filtered signal is redistributed and the measured fractional populations approach equilibrium again. The FEXSY data set is acquired by repeated iterations of the sequence in Fig. 1, varying the mixing time and the q-value of the second PFG block to produce a 2D data set. The collected data is then globally fitted to a bi-compartmental model [36]. Alternatively, ADCs are calculated for multiple 1D slices in which the mixing time is constant, and then fitted to a phenomenological model [23]. In both methods the entire 2D data set is used. In this paper, we show that in order to determine the exchange rate and the intra-cellular mean residence time (MRT) it is sufficient to acquire a partial FEXSY data set, such that a single q-value is kept fixed and only the mixing time is varied. The outcome is a variant of the original method which we call: constant gradient FEXSY (CGFEXSY). The size of a FEXSY data set varies with respect to scan time limitations, but generally 10–15 different q-values are used when fitting the data according to the procedures described above. Thus, reduction in scan time by up to an order of magnitude is possible by collecting data for a single q-value (in addition to a normalization slice where the q-value is set to zero). We corroborate our theoretical predictions with experiments on a bi-compartmental system (yeast cells) and a multicompartmental system (fixed porcine optic nerve), where the values obtained from FEXSY and CG-FEXSY are shown to be in agreement. This is true even when using clinically feasible gradient strengths, suggesting that CG-FEXSY can safely replace FEXSY as an efficient method for measuring apparent exchange rates. 2. Theory 2.1. FEXSY Consider a simple model of a cell (Fig. 2) composed of two distinct water populations, intra- and extra-cellular, which differ in their ADCs (Di and De , respectively). Depending on the dilution of the sample, the extra-cellular water can be freely-diffusing or hindered. However, the diffusion times in the PFG blocks are taken to be long enough such that intra-cellular water is restricted by the cell walls, thus leading to a situation where the intra-cellular ADC is considerably lower than the extra-cellular one (i.e.,

Fig. 1. The PGSTE version of the FEXSY pulse sequence. Two co-linear sequential PGSTE blocks are separated by a mixing time t m . The first block is kept constant and serves as a diffusion-based filter. To collect the data, the mixing time and the area of the gradients in the second block are varied. In the present work, a p2-pulse is used to place the magnetization on the z-direction prior to the application of the second PGSTE block. During part of D1 ; D2 and tm , the magnetization is in the longitudinal direction, bypassing T 2 -relaxation (but at the cost of losing half of the signal intensity per each interval). Spoiler gradients are used to remove any residual transverse magnetization that might be left during these intervals.

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Fig. 2. The bi-compartmental exchange model. An illustration of a cell system where intra- and extra-cellular water populations, with apparent diffusion coefficients Di and De , exchange reversibly with rates ki and ke .

Di  De ). The two populations undergo exchange across the cell membrane [36]: ki

Intra-cellular water Extra-cellular water:

ð1Þ

ke

Denoting the fractional populations of intra- and extra-cellular water f i and f e respectively, we have f i þ f e ¼ 1. The fractional population of the extra-cellular water at a given mixing time t m follows from Eq. (1) [37],

f e ðt m Þ ¼

eq fe







eq fe




ktm


f e ð0Þ e

;

ð2Þ

where k ¼ ke þ ki and the superscript ‘eq’ denotes the respective equilibrium fraction. The echo intensity attenuation is then simply the superposition of two attenuations, weighted by their fractional populations:

Eðq; t m Þ ¼ f e ðt m Þe
De q

2D

þ ½1
f e ðt m Þe
Di q D ; 2

ð3Þ

where we have used the relation f i ðt m Þ ¼ 1
f e ðt m Þ. Eqs. (2) and (3) eq are used to analyze the FEXSY data set and to extract k and f e . Note that in the limit t m ! 1 FEXSY is equivalent to a PFG experiment which is not preceded by a filter block (neglecting relaxation effects). eq eq For a system in equilibrium we have ki f i ¼ ke f e , which in turn gives eq

ki ¼ f e k;

ð4Þ
1

from which ki and its inverse, the intra-cellular MRT, si ¼ ki , can both be calculated. Finally, note that the treatment here neglects T2 -relaxation effects, as was done in most FEXSY related works [20,23,36]. In a recent work, Eriksson et al. were able to evaluate the effect of T2 differences on the exchange rates extracted from FEXSY in yeast cells [21]. The authors did this with the aid of an additional 2D diffusion-relaxation experiment and found that FEXSY underestimated the exchange rate by about 20% when the total echo time was minimized to 17 ms. Here the total echo time is similar. 2.2. CG-FEXSY To develop CG-FEXSY, we rearrange Eq. (3) to obtain

Eðq; t m Þ ¼ aðqÞ þ bðqÞf e ðtm Þ;

ð5Þ

where aðqÞ ¼ e
Di q D and bðqÞ ¼
aðqÞ þ e
De q D . Note that in the limit of high q-values bðqÞ !
aðqÞ, since it is assumed that Di  De . Substituting Eq. (2) into Eq. (5), we get 2

2


eq  eq Eðq; t m Þ ¼ aðqÞ þ bðqÞf e
bðqÞ f e
f e ð0Þ e
ktm : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð6Þ

independent of t m

Thus, for a data set with a fixed q-value (constant gradient), we can simply fit the echo intensity using
kt m

Eðt m Þ ¼ a þ be

;

ð7Þ


eq  where aðqÞ ¼ aðqÞ þ and bðqÞ ¼
bðqÞ f e
f e ð0Þ . Assuming that the signal from the extra-cellular water is completely removed by the filter (f e ð0Þ ¼ 0) and taking the high q-value limit we obtain eq bðqÞf e

eq

fe ¼

b ; aþb

ð8Þ

where we have used bðqÞ !
aðqÞ. The intra-cellular MRT can be q!1

estimated from data by using Eqs. (4), (7) and (8). 3. Experimental 3.1. Sample preparation 3.1.1. Yeast A colony of the BY4741 strain of S.Cerevisiae was grown on a petri dish with agar. The dish was kept at 4 °C. Each NMR sample (N = 6) was prepared by taking a small amount of yeast from the colony and suspending it in 45 mL of Synthetic Defined Medium. The suspension was then left in an incubator shaker overnight (30 °C, 140RPM). During the night, an amount befitting a 5 mm NMR tube had grown. The yeast were harvested from the suspension by centrifugation and removal of the supernatant, then washed (resuspended and harvested) three times with autoclaved double-distilled water. Finally, the yeast cells were transferred to a 5 mm NMR tube. The NMR tube was centrifuged and the supernatant was removed. Note that yeast cells, previously used to challenge the accuracy of NMR methods for compartment size

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measurement, were reported to have average cell diameter of 4– 5 lm [38,39]. 3.1.2. Porcine optic nerve Optic nerves (N = 3, diameter of 4 mm) were dissected from 5.5 month-old porcines at the Institute of Animal Research Lahav CRO (Kibbutz Lahav, Israel). Each nerve was immediately immersed in Formalin solution (4% Formaldehyde) and sent to Tel-Aviv University (2hrs), where it was left overnight (4 °C) in order to achieve full fixation, followed by 24hrs wash in saline. The nerve from the saline solution was carefully dried, inserted into a 5 mm sleeve (to ensure correct alignment) and transferred to a 10 mm NMR tube filled with Fluorinert (Sigma Ltd., Saint Louis, MO, USA) [40–43]. The tube was placed in the magnet such that the nerve fibers were parallel to the direction of the B0 field and perpendicular to the direction of the diffusion sensitizing gradients. Note that the average axon diameter of optic nerves is 2 lm [40–42]. 3.2. NMR hardware All NMR experiments were performed on a Bruker 9.4T WB NMR spectrometer, equipped with a Bruker Micro5 probe, capable of producing pulsed field gradients of up to 300 g/cm. The probe temperature was kept constant at 25 °C. Sine-shaped pulses were used. In all diffusion experiments the relation deff ¼ 2d p was used to determine the effective duration of the gradient pulses [44]. The effective pulse ramp up/down time was 104 ls. A stabilization time of at least 1 ms was always allowed after the application of a gradient. The p2 RF pulse was calibrated for each sample and found to be approximately 15 ls. 3.3. FEXSY experiments 3.3.1. Yeast The FEXSY experiment on the yeast cells sample was performed with TR of 2000 ms and TE1 ¼ TE2 of 8 ms, D ¼ 13 ms and deff ¼ 2 ms for each block. The gradient strength (g) for the filter block was 150 g/cm and for the measurement block g was incremented from 5 to 150 g/cm in 30 equally spaced steps. The experiment was performed with the following mixing times: tm ¼ 8, 25, 30, 50, 100, 150, 200, 250, 300, 400, 500, 750, 1000, 1250 ms. The

‘‘no filter” acquisition was a PGSTE with the same parameters as the measurement block. There the SNR for the zero gradient spectrum was approximately 20,000. This acquisition was not included in the global fit. 3.3.2. Porcine optic nerve The FEXSY experiment on the fixed porcine optic nerve was performed with TR of 2000 ms and TE1 ¼ TE2 of 8 ms, D ¼ 35 ms and deff ¼ 2 ms for each block. The gradient strength (g) for the filter block was 285 g/cm and for the measurement block g was incremented from 5 to 275 g/cm in 55 equally spaced steps. The experiment was performed with the following mixing times: t m ¼ 8, 25, 30, 50, 100, 150, 200, 250, 300, 400, 500, 750, 1000, 1250 ms. The ‘‘no filter” acquisition, was again, a PGSTE with the same parameters as the measurement block, where the SNR for the zero gradient spectrum was found to be approximately 4,000. This acquisition was not included in the global fit. Note that the ADCs in the optic nerve and the difference between them are smaller than in yeast cells, therefore here we have used stronger gradient pulses and longer diffusion time in our FEXSY measurements. 4. Results 4.1. Yeast The complete FEXSY data set taken for a yeast cells sample is shown in Fig. 3. A global fit was performed using Eq. (3) to give
1

eq

k ¼ 3:95 s
1 ; s ¼ k ¼ 253 ms and f e ¼ 0:71. Using the relation in Eq. (4) we get ki ¼ 2:8 s
1 , which corresponds to an intra
1

cellular MRT of si ¼ ki ¼ 357 ms. To check how the use of partial temporal data affects these numbers, we repeated the fit with t max m values of 500 and 250 ms, instead of the tmax ¼ 1250 ms that was m used for the full data set. The calculated s values in the analysis were found to be 228 and 205 ms, respectively. Comparing these values to the values obtained from the full data set, we conclude that partial temporal data leads to a slight decrease in the calculated value. We then looked at constant-gradient (CG) slices of the data shown in Fig. 3 and fitted each slice according to Eq. (7) (Fig. 4). We repeated the fit for the same three values of t max as before, m and plotted the CG-FEXSY results vs. the gradient strength (the

Fig. 3. FEXSY data set of a yeast cells sample. The normalized natural logarithm of the signal attenuation is plotted vs. the diffusion weighting for different mixing times. Mixing times were varied to span the range of 8–1250 ms. The gradient strength of the filter block was set to 150 g/cm and that of the measurement block was varied between 5–150 g/cm. The gradient duration and the diffusion times were kept constant at deff ¼ 2 ms and D ¼ 13 ms. The ‘‘no filter” acquisition (black line) is a PGSTE experiment with the same parameters as the measurement block. As expected, the attenuation curves approach the ‘‘no filter” curve as the mixing time is increased. In this work we show that instead of globally fitting the entire data set, a normalized CG slice as shown in the figure is sufficient to determine the exchange rate. Note that full circles represent the experimental data while the lines are just to guide the eye.

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Fig. 4. CG-slices of the yeast FEXSY data set. Normalized signal attenuation for five CG slices of the FEXSY data set presented in Fig. 3 vs. tm . In each slice, the two PGSTE blocks are constant and the mixing time is varied between 8–1250 ms. Note that full circles represent the experimental data while the lines are fits according to Eq. (7).

gradient duration was kept constant), as shown in Fig. 5. The corresponding values from the FEXSY global fits were also plotted as horizontal lines for comparison. It can be appreciated that the CG-FEXSY results are in good agreement with FEXSY results for all gradient strengths, and especially in the case of t max = 1250 ms where full temporal information is used for the m analysis. eq To obtain the intra-cellular MRT, we calculated f e for each CG eq slice using Eq. (8). In Fig. 6, the calculated f e values are plotted vs. gradient strength and compared to the corresponding value from the FEXSY global fit (dashed horizontal line). As predicted, CG-FEXSY values approach the FEXSY value as the gradient strength is increased. At g-values higher than 100 g/cm (with deff eq of 2 ms) the same values for f e are obtained from the two NMR experiments. The FEXSY global fit, that is more demanding in term of acquisition time and uses high q-values in the analysis, gives additional

eq

eq

Fig. 6. FEXSY vs. CG-FEXSY - f e in yeast. The f e values obtained from CG-FEXSY eq and FEXSY experiments performed on a yeast cells sample. The f e value obtained by globally fitting the FEXSY data set in Fig. 3 is plotted as a horizontal dashed line eq while the f e values obtained for each CG slice using Eq. (8) are plotted as full circles. As predicted, the two methods are in agreement in the high diffusion weighting regime.

valuable information such as the ADCs and f e ð0Þ. A full FEXSY data set analysis of the 6 different yeast samples gave: s ¼ 240 eq 17 ms; f e ¼ 0:66  0:03; f e ð0Þ ¼ 0:03  0:01; Di ¼ 3:95  0:67 10
11 m2 =s; De ¼ 1:32  0:08  10
9 m2 =s. 4.2. Fixed porcine optic nerve

Fig. 5. FEXSY vs. CG-FEXSY - exchange times in yeast. A comparison between the
1 mean exchange times s ¼ k obtained from FEXSY and CG-FEXSY experiments performed on a yeast cells sample. Analysis was performed for tmax = 250, 500 and m 1250 ms. The s values obtained by globally fitting the FEXSY data set presented in Fig. 3 are plotted as horizontal dashed lines while those extracted for each CG slice using Eq. (7) are plotted as full circles. The results of the two methods are in agreement, especially when all temporal information is used for the analysis (i.e., tmax = 1250 ms). m

To asses whether the current CG-FEXSY method can be extended to treat complex white matter (WM), we repeated the procedure described above for a porcine optic nerve. The raw data is shown in Fig. 7. Five CG-slices are shown in Fig. 8 as an example. eq The global fit for tmax ¼ 750 ms gave k ¼ 4:39 s
1 and f e ¼ 0:31 for m eq
1 g max ¼ 150 g/cm, and k ¼ 4:76 s and f e ¼ 0:38 for g max ¼ 275 g/ cm. Using the relation in Eq. (4) we get ki ¼ 1:36 and 1:80 s
1 respectively, which corresponds to intra-cellular MRTs of

si ¼ k
1 ¼ 735 and 552 ms. i Note that all attenuation curves in Fig. 7 are situated well above the ‘‘no filter” curve. This is in contrast to the yeast experiment, where attenuation curves at high tm coincide with the ‘‘no filter” curve, as expected for a bi-compartmental system. A possible

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Fig. 7. FEXSY data set of the optic nerve. The normalized natural logarithm of the signal attenuation in a FEXSY experiment performed on a fixed porcine optic nerve is plotted vs. the diffusion weighting for different mixing times. Mixing times were varied to span the range 8–1250 ms. The gradient strength of the filter block was set to 285 g/cm while that of the measurement block was varied between 5–275 g/cm. The gradient duration and the diffusion times were kept constant at deff ¼ 2 ms and D ¼ 35 ms. The ‘‘no filter” acquisition is a PGSTE with the same parameters as the measurement block. Contrary to the yeast sample, the attenuation curves stay well above the ‘‘no filter” curve even for t m ¼ 1250 ms. The quality of the data for tm higher than 750 ms was poor, hence was not used for the quantitative analysis, and is shown here only for the sake of completeness. Note that full circles represent the experimental data while the lines are just to guide the eye.

Fig. 8. CG-slices of the optic nerve FEXSY data set. Normalized signal attenuation for five CG slices of the FEXSY data set presented in Fig. 7 vs. tm . In each slice, the two PGSTE blocks are constant and the mixing time is varied between 8–1250 ms. Note that full circles represent the experimental data while the lines are fits according to Eq. (7).

explanation would be the existence of additional non-exchanging water populations in the nerve. As the filter would remove the signal coming from these populations indefinitely the system would not return to its steady state even at very long times. Alternatively, it is possible that the exchange process occurs on a time-scale of several seconds, which is detectable only with extremely long mixing times that are difficult to obtain because of T1 -relaxation. Figs. 9 and 10 were produced using the exact same methodology used to produce Figs. 5 and 6 but for the optic nerve. For low q-values (g-values lower than 35 g/cm), the CG-FEXSY results greatly diverge from the corresponding FEXSY values (and are hence not shown). Fig. 9 shows that the s values extracted from CG-FEXSY decrease moderately as q-values are increased, but nonetheless seem to roughly approximate the corresponding FEXSY values. Contrary to the yeast experiment, FEXSY values are strongly dependent on tmax m , and are also dependent on the maximum gradient strength of the measurement block, g max . For low tmax m , however, this latter dependence is weak, and the methods eq are in a better agreement. Fig. 10 shows that the f e values extracted from CG-FEXSY increase as the q-value is increased, but do not reach the corresponding FEXSY values which by themselves are strongly dependent on g max . However, at high q-values, they do roughly approximate the corresponding FEXSY values.

Fig. 9. FEXSY vs. CG-FEXSY - exchange times in the optic nerve. A comparison
1 between the mean exchange time s ¼ k obtained from FEXSY and CG-FEXSY experiments performed on porcine optic nerve. Values extracted for s are plotted vs. g and for tmax = 250, 500 and 750 ms. The s values obtained by globally fitting the m FEXSY data set presented in Fig. 7 are plotted as horizontal dashed and dasheddotted lines while those extracted for each CG slice using Eq. (7) are plotted as full circles. Note that the FEXSY values change considerably as tmax is varied. m

Y. Scher et al. / Journal of Magnetic Resonance 311 (2020) 106667

7

exchange rates and MRTs. As mentioned in the introduction, CGPGSTE [24–30] is also used for this purpose and the resemblance between the two is clear: in both methods gradients are kept constant and signal attenuation is measured as a function of a temporal variable. However, in CG-FEXSY the temporal variable is the mixing time while in CG-PGSTE it is the diffusion time. Moreover, the measurement block in CG-FEXSY is preceded by a filter which perturbs the existing equilibrium by removing signal coming from fast-diffusing water. Thus, in contrast to CG-PGSTE where the system is initially at equilibrium, in CG-FEXSY one extracts information by observing how the system returns to equilibrium following an initial perturbation (filter). A systematic comparison between FEXSY/CG-FEXSY and CG-PGSTE is beyond the scope of this communication and will be presented elsewhere. A first step in this direction has already been reported by Tian et al. [46]. eq

eq

Fig. 10. FEXSY vs. CG-FEXSY - f e values in optic nerve. The f e values obtained by globally fitting the FEXSY data set presented in Fig. 7, for g max of 150 and 275 g/cm (measurement block), are plotted as horizontal dotted and dashed-dotted lines. The eq full circles represent f e values obtained for each CG slice using Eq. (8). For gradient eq strengths lower than 30 g/cm, the calculated f e values fluctuated greatly and eq hence are not shown. For higher g-values, the calculated f e values increase with g until they roughly approximate the corresponding FEXSY value, but do not reach it eq as in the case of the yeast sample. Note that here the FEXSY f e values differ significantly, whereas halving g max from 150 g/cm to 75 g/cm in the yeast sample eq did not change the FEXSY f e values at all.

As for the yeast cells, full FEXSY data set analysis for 3 different eq optic nerves was performed to give: s ¼ 197  17 ms; f e ¼

0:41  0:03; f e ð0Þ ¼ 0:14  0:01; Di ¼ 0:94  0:05  10
11 m2 =s; De ¼ 0:21  0:01  10
9 m2 =s. 5. Discussion

Reducing scan time is paramount when studying biological systems and even more so in clinical studies, where it is costly and limited. Here, we showed that the scan time required to measure exchange rates can be greatly reduced by adopting CG-FEXSY – a constant gradient variant of FEXSY [36]. The two methods were compared on a yeast cells sample, were we have shown that measured exchange rates are in agreement. Moreover, the exchange rates measured using CG-FEXSY were found to be virtually independent, above a certain threshold, of the constant gradient (q-value) used. This in fact testifies to the robustness of the method. To further obtain the intra-cellular MRT we implemented CG-FEXSY using q-values higher than  9  104 m
1 . Here, the high q-value regime was reached by increasing the gradient strength, such that the SGP approximation was always valid, and indeed good agreement with FEXSY was found. In clinical MRI systems, where high gradient strengths are not available, the gradient duration can be increased instead, at some accuracy cost [45]. FEXSY and CG-FEXSY were also compared on a fixed porcine optic nerve where agreement between the methods was maintained. Note, however, that in this case results were more sensitive to variations in experimental parameters, e.g., the maximal mixing time t max and maximal gradient g max . This lack of robustness, m together with the fact that even at high mixing times signal attenuation does not agree with that observed in the absence of a filter, suggests that further development of the methods and analysis procedures is required to better handle a system as complex as white matter. Further studies will determine whether these methods are adequate for depicting grey matter, which is closer in its morphology to the yeast cells sample studied here. Concluding, the work presented here asserts that acquiring a CG-FEXSY data set is sufficient when coming to determine

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We thank Natalya Weber for the graphical design of the visual abstract and Fig. 2. We thank Lihi Gershon and Shay Bramson from the Kupiec group at Tel-Aviv University for help in producing the yeast samples. S.R. acknowledges support from the Azrieli Foundation and from the Raymond and Beverly Sackler Center for Computational Molecular and Materials Science at Tel Aviv University. Y. C. acknowledges the Israel Science Foundation (ISF, Jerusalem) for a grant that partially supported the purchase of the 9.4T MR system used in this study. References [1] P.T. Callaghan, Translational Dynamics and Magnetic Resonance: Principles of Pulsed Gradient Spin echo NMR, Oxford University Press, Oxford, 2011. [2] D.K. Jones, Diffusion MRI, Oxford University Press, Oxford, 2010. [3] M.E. Moseley, Y. Cohen, J. Mintorovitch, L. Chileuitt, H. Shimizu, J. Kucharczyk, M. Wendland, P.R. Weinstein, Early detection of regional cerebral ischemia in cats: comparison of diffusion and T2-weighted MRI and spectroscopy, Magn. Reson. Med. 14 (2) (1990) 330–346. [4] M.E. Moseley, Y. Cohen, J. Kucharczyk, J. Mintorovitch, H.S. Asgari, M.F. Wendland, J. Tsuruda, D. Norman, Diffusion-weighted MR imaging of anisotropic water diffusion in cat central nervous system, Radiology 176 (2) (1990) 439–445. [5] P.J. Basser, J. Mattiello, D. LeBihan, MR diffusion tensor spectroscopy and imaging, Biophys. J. 66 (1) (1994) 259–267. [6] P.W. Kuchel, A. Coy, P. Stilbs, NMR ‘‘diffusion-diffraction” of water revealing alignment of erythrocytes in a magnetic field and their dimensions and membrane transport characteristics, Magn. Reson. Med. 37 (5) (1997) 637– 643. [7] P.J. Basser, S. Pajevic, C. Pierpaoli, J. Duda, A. Aldroubi, In vivo fiber tractography using DT-MRI data, Magn. Reson. Med. 44 (4) (2000) 625–632. [8] A.C. Guo, J.R. MacFall, J.M. Provenzale, Multiple sclerosis: diffusion tensor MR imaging for evaluation of normal-appearing white matter, Radiology 222 (3) (2002) 729–736. [9] Y. Cohen, Y. Assaf, High b-value q-space analyzed diffusion-weighted MRS and MRI in neuronal tissues-a technical review, NMR Biomed. 15 (7–8) (2002) 516–542. [10] Y. Cohen, D. Anaby, D. Morozov, Diffusion MRI of the spinal cord: from structural studies to pathology, NMR Biomed. 30 (3) (2017) e3592. [11] D.S. Novikov, E. Fieremans, S.N. Jespersen, V.G. Kiselev, Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation, NMR Biomed. 32 (4) (2019) e3998. [12] D.C. Alexander, T.B. Dyrby, M. Nilsson, H. Zhang, Imaging brain microstructure with diffusion MRI: practicality and applications, NMR Biomed. 32 (4) (2019) e3841. [13] E.O. Stejskal, J.E. Tanner, Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient, J. Chem. Phys. 42 (1) (1965) 288– 292. [14] J.E. Tanner, Use of the stimulated echo in NMR diffusion studies, J. Chem. Phys. 52 (5) (1970) 2523–2526.

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