Journal of Magnetic Resonance 216 (2012) 28–36

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Journal of Magnetic Resonance

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The c parameter of the stretched-exponential model is influenced by internalgradients: Validation in phantoms

Marco Palombo a,b, Andrea Gabrielli a,c, Silvia De Santis a,d, Silvia Capuani a,b,⇑a Physics Department, Sapienza University of Rome, P.le Aldo Moro, 5, 00185 Rome, Italyb CNR-IPCF UOS Roma Sapienza, Physics Department, Sapienza University of Rome, P.le Aldo Moro, 5, 00185 Rome, Italyc ISC-CNR, Via dei Taurini, 19, 00185 Rome, Italyd CUBRIC, School of Psychology, Cardiff, South Glamorgan, United Kingdom

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 August 2011Revised 12 December 2011Available online 9 January 2012

Keywords:Stretched exponential modelAnomalous Diffusion ImagingDiffusionInternal gradientsMagnetic susceptibility differencesNMR PGSTEPolystyrene micro-beadsTiO2 micro- and nano-beads

1090-7807/$ - see front matter � 2012 Elsevier Inc. Adoi:10.1016/j.jmr.2011.12.023

⇑ Corresponding author. Address: CNR-IPCF UOSSapienza University of Rome, P.le Aldo Moro, 5, 00649913928.

E-mail address: silvia.capuani@roma1.infn.it (S. Ca

In this paper, we investigate the image contrast that characterizes anomalous and non-Gaussian diffusionimages obtained using the stretched exponential model. This model is based on the introduction of the cstretched parameter, which quantifies deviation from the mono-exponential decay of diffusion signal as afunction of the b-value. To date, the biophysical substrate underpinning the contrast observed in c maps,in other words, the biophysical interpretation of the c parameter (or the fractional order derivative inspace, b parameter) is still not fully understood, although it has already been applied to investigate bothanimal models and human brain. Due to the ability of c maps to reflect additional microstructural infor-mation which cannot be obtained using diffusion procedures based on Gaussian diffusion, some authorspropose this parameter as a measure of diffusion heterogeneity or water compartmentalization in biolog-ical tissues. Based on our recent work we suggest here that the coupling between internal and diffusiongradients provide pseudo-superdiffusion effects which are quantified by the stretching exponentialparameter c. This means that the image contrast of Mc maps reflects local magnetic susceptibility differ-ences (Dvm), thus highlighting better than T�2 contrast the interface between compartments character-ized by Dvm. Thanks to this characteristic, Mc imaging may represent an interesting tool to developcontrast-enhanced MRI for molecular imaging.

The spectroscopic and imaging experiments (performed in controlled micro-beads dispersion) that arereported here, strongly suggest internal gradients, and as a consequence Dvm, to be an important factorin fully understanding the source of contrast in anomalous diffusion methods that are based on astretched exponential model analysis of diffusion data obtained at varying gradient strengths g.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Pulse field gradient NMR (PFG NMR) methods, used to measuremolecular diffusion in liquid systems, provide relevant microstruc-tural information from the media in which molecules diffuse [1–5].The attenuation of PFG signal depends on the b-value, which inturn depends on both gradient strength, g, and diffusion time, D.If the short pulse approximation holds, PFG signal decay is relatedto the molecular motion propagator (MP) by a simple mathemati-cal relationship: the shape of signal attenuation is the Fouriertransform (FT) of the MP. As a consequence, the mono-exponentialStejskal–Tanner decay [6] of the PFG signal as a function of b, re-flects a Gaussian MP. On the other hand, a deviation of the PFG sig-nal decay from the mono-exponential decay inherently assumes a

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Roma, Physics Department0185 Rome, Italy. Fax: +39

puani).

non-Gaussian propagator to describe the molecular motion. In thelatter case, it refers to anomalous diffusion. A deviation of the PFGexperimental data from the mono-exponential decay has been ob-served in many experiments performed in heterogeneous systems[7–9]. This deviation suggests a non-Gaussian or anomalous diffu-sion of water in the media under investigation.

We have recently proposed a new MR methodology [10] that ob-tains microstructural information from heterogeneous media basedon the anomalous diffusion theory as described in a generalizedrandom walk framework using the fractional calculus introducedby Metzler and Klafter [11]. Temporal and spatial fractionalexponents, a and l, introduced within the framework of continuoustime random walk (CTRW) [11], were simultaneously measured byPFG NMR technique in samples of micro-beads dispersed in aque-ous solution. As water behavior in this kind of micro-pore systemis known to be Brownian or subdiffusive, we suggested [10] thatthe anomalous diffusion parameter l does not quantify real super-diffusion processes, rather pseudo-superdiffusion phenomena due

M. Palombo et al. / Journal of Magnetic Resonance 216 (2012) 28–36 29

to artifactual effects generated by magnetic susceptibility differ-ences (Dvm) at the interface between beads and diffusing water[10]. We also found that l depends on both bead sizes and localinternal gradient (Gint) within porous samples.

In recent years, several approaches have been proposed toquantify non-Gaussian [12–16] and anomalous diffusion processesin heterogeneous systems [17–22]. Among these, the stretchedexponential model [17,20,21], which is based on the introductionof the c stretching parameter, quantifies deviations from themono-exponential decay of PFG signal as a function of b. Althoughthis is an empirical approach, the stretched exponential methodhas already been used to investigate both animal models [23]and human brain [19,20,24,25], showing the ability of c maps todiscriminate between different brain structures on the basis oftheir microstructural complexity. However, the biophysical sub-strate underpinning contrast obtained using the stretched expo-nential model required further clarification, especially withrespect to the ability of c maps to discriminate between differenttissues for which no significant differences in mean diffusivity(MD) can be detected [20,22]. An alternative analytical derivationfor the stretched exponential model, using fractional order spacederivatives, has been recently proposed by Magin et al. [19]. Theseauthors emphasized that the potential utility of fractional order bparameter (which is, in practice, equal to c parameter) to charac-terize the environment for molecular diffusion (as a complementto apparent diffusion coefficient) might lead to a new way to inves-tigate tissue structural features [24,26]. However, the biophysicalbasis of fractional order derivative in space b (i.e. c parameter)[22] remains elusive.

To the best of our knowledge, there are no previous studiesavailable that have investigated the relationship between c con-trast and Gint.

In this report we demonstrate that the c parameter of thestretching exponential model is equal to l/2, thus suggesting thatc likely quantifies pseudo-superdiffusion of water due to Dvm. Wefirst performed spectroscopic and imaging experiments to measurel and c in controlled phantoms characterized by restricted Gauss-ian or sub-diffusion processes only. We then quantified Dvm in allsamples by measuring R�2 ¼ 1=T�2 and background or internal gradi-ents (Gint) [27]. Since l < 2 quantifies pseudo-superdiffusion pro-cesses due to Dvm when it is measured using variable g PGSTE[10], a source of contrast in the stretched exponential model isthe Dvm. In other words, we demonstrate here that the contrastin c maps in heterogeneous systems is not only due to restricteddiffusion processes but it is also due to local Dvm at the interfacesbetween different compartments.

2. Theory

It is well known that the NMR signal observed in a PFG [3,5]experiment has a Fourier relationship with the probability of spinmotion, the so-called displacement propagator. The signal, E, ob-tained in a PFG experiment can be written as [3]:

Eðq; tÞ /Wðq; tÞ ¼Z 1

�1PsðR; tÞ � exp½i2pq � R�dR ð1Þ

where PsðR;tÞ is the ensemble average displacement propagator, orthe probability of a spin to undergo a displacement R = r0 � r, pro-ceeding from an initial position r to a final position r0 during a cer-tain diffusion time t = D. The quantity q ¼ 1

2p cgd is the wave vector,where c is the spin gyromagnetic ratio, |g| = g is the diffusion gradi-ent strength, and D is its pulse duration.

When PsðR;tÞ is Gaussian, PFG signal attenuation is a mono-exponential decay as a function of the single variable b ¼ ðcjgjdÞ2

D ¼ j2pqj2D. When PsðR;tÞ is non-Gaussian, under certain hypoth-

eses about the spatio-temporal scaling of the elementary steps ofthe motion, it is possible to fit the average MP with the solutionof fractional diffusion equations introduced within the frameworkof the CTRW model [11]. Briefly, to investigate superdiffusion pro-cesses characterized by a divergence of the jump length variance ofthe particles, it is possible to use the following function [10,11]:

Wðq;DÞ � exp½�Klj2pqjlD� ð2Þ

where Kl is a generalized diffusion constant, whose units are ml/s,and 0 < l < 2.

In order to evaluate l experimentally, expression (2) has to befitted to data obtained by PFG signal decay as a function of b, col-lected by changing g at a constant value of D.According to the Ma-gin derivation of the spatial fractional exponent procedure [19],since D is a constant, the stretched exponential function used toquantify c can easily be derived from expression (2). Substitutingj2pqjl ¼ bl=2

Dl=2 in Eq. (2), the following relations can be derived:

Wðq; tÞ � exp � Kl

Dðl=2�1Þ bl=2� �

¼ exp �Dq2ðl=2�1Þ

Dðl=2�1Þ bl=2� �

¼ exp �Dl=2eff bl=2

h i¼ exp �Dc

eff bc

h ið3Þ

where D is the diffusion coefficient, q2(l/2�1) and D(l/2�1) are frac-tional order space and time constants that preserve units, and Deff

is an effective diffusion constant. Expression (3) clearly shows thatwhen PFG signal is collected by changing g at a constant value ofD, measured c values are equal to l/2 ones. In particular, by mea-suring values of l and c along x, y and z axes, (ðliÞi¼x;y;z andðciÞi¼x;y;z), it is possible to compute values of l and c averaged acrossdirections: Ml ¼ 1

3

Pi¼x;y;zli and Mc ¼ 1

3

Pi¼x;y;zci. As a consequence,

from expression (3) it is possible to derive the following relation:

Mc ¼ 12

Ml ð4Þ

3. Materials and methods

All measurements were performed on a Bruker 9.4 T Avancesystem, operating with a microimaging probe (10 mm internaldiameter bore) and equipped with a gradient unit characterizedby a maximum gradient strength of 1200 mT/m, and a rise timeof 100 ls.

3.1. Spectroscopic measurements

Six samples in total were carried out using polystyrene micro-beads (microbeads AS, Norway) with nominal average diameterof 40, 30, 20, 15, 10 and 6 lm and characterized by a|Dvm| = |vm

H2O � vmPolystyrene| = 1.59 � 10�6 in SI units [28]. Six

8 mm NMR tubes were filled up to a volume of approximately1.6 cm3 with beads mono-dispersed in a solution of polyoxyethy-le-sorbitan-mono-laurat (Tween 20) 10�6 M and deionized water(conductivity � 10�6 O�1/cm). Samples were sonicated and inves-tigated 1 month after their preparation. Finally, one tube filledwith free water was also used as control.

Spectroscopic pulsed field gradient stimulated echo (PGSTE)and a BPP-LED PGSTE (bipolar pulse field gradient with eddy cur-rent delay) [29] pulse sequences using: TE/TR = 6.2/5000 ms, diffu-sion gradient pulses delay D = 80 ms, diffusion gradient pulsesduration D = 4.4 ms and diffusion gradient strength g applied alongthe x, y and z axes using 48 gradient amplitude steps from 2.6 to1000 mT/m, eddy current delay equal to 5 ms, were used to collectspectroscopic diffusion data.

Values of l and c, along x, y and z axes (ðliÞi¼x;y;z and ðciÞi¼x;y;z),were obtained by fitting expressions (2) and (3) to data, respec-

30 M. Palombo et al. / Journal of Magnetic Resonance 216 (2012) 28–36

tively. The mean values of l (Ml) and c (Mc) were obtained byaveraging across directions: Ml ¼ 1

3

Pi¼x;y;zli and Mc ¼ 1

3

Pi¼x;y;zci.

Gint was measured by using a spectroscopic SE sequence(TR = 1500 ms, NS = 8) with N = 64 data points (corresponding to64 echoes refocusing every 2 ms from 1 to 125 ms), as previouslydescribed [30].

Fig. 1. Mc vs Ml graph for samples of packed (mean effective porosity, p = 0.28± 0.03) polystyrene micro-beads of differing sizes (listed in the insert pannel) mono-dispersed in a solution of polyoxyethyle-sorbitan-mono-laurat (Tween 20) 10�6 Mand de-ionized water (conductivity � 10�6 X�1/cm). The solid line representsequation Mc = C Ml fitted to data. The coefficient C = 0.4998 ± 0.0006 extractedfrom the fit and the Pearson’s R coefficient close to 1 experimentally verify therelationship in (4). FWHM of samples ranged from 15 to 24 Hz for 6 lm and 40 lmbeads, respectively.

3.2. Imaging measurements

A phantom comprising of six capillaries placed in a 0.8 cm NMRtube filled with 10 lm packed polystyrene beads in aqueous solu-tion was prepared. The first capillary was filled with free water, thesecond with 10 and 6 lm polystyrene beads poly-dispersed inaqueous solution, the third and fourth with 6 lm polystyrenebeads mono-dispersed in aqueous solution and, finally, theremaining two with 5 lm and 50 nm TiO2 beads mono-dispersedin aqueous solution, respectively.

An imaging version of PGSTE sequence using TE/TR = 21/3000 ms, D = 80 ms, d = 4 ms, 10 b-values (0, 730, 1070, 1530,2050, 2520, 3050, 3500, 4030, 4550, 5200 s/mm2), gradient direc-tion along x, y and z axis, matrix 128 � 128, field of viewFOV = 0.8 cm, slice thickness STH = 1 mm, number of averagedscans NSA = 16, was used to obtain both conventional mean diffu-sivity maps (MD-maps), and Mc-maps, as previously described[20]. Since the phantoms used in our experiments are character-ized by a cylindrical geometry, we measured MD, Mc and Ml alongx, y and z axis only. However, in more complex geometries, a tensoranalysis should be used [20].

A MSME (multi slice multi echo) sequence, with TR = 3000 ms,50 values of TE from 3.1 ms to 155 ms with step of 3.1 ms, matrix128 � 128, FOV = 0.8 cm, STH = 1 mm, NS = 8, was used to extractGint map from the spin-echo (SE) decay, as previously described[30].

Finally, a T�2 map was obtained by a GEFI (gradient echo fastimaging) sequence with TR = 3000 ms, 10 values of TE (1.7, 2.5, 4,6, 8, 10, 12, 15, 20,30 ms), matrix 128 � 128, FOV = 0.8 cm,STH = 1 mm, NSA = 8.

All fitting procedures were performed by means of Levenberg–Marquardt algorithm using homemade scripts in MATLAB R 2009b.

Fig. 2. Mean c values (Mc), measured using PGSTE (black data points) and BPP-LEDPGSTE (gray data points), as a function of the internal magnetic field gradientstrength (Gint) for samples of different suspensions of mono-dispersed beads,characterized by the sizes displayed in the left side insert.

4. Results and discussions

The plot of Mc as a function of Ml shown in Fig. 1 verifies Eq. (4)experimentally, for which Mc = 1/2 Ml. In Fig. 2, plot of Mc as func-tion of Gint for all mono-dispersed phantoms (including an aqueoussolution sample) is displayed. This graph clearly shows that Mc ob-tained by using a PGSTE sequence strongly depends on the effec-tive Gint extracted from SE decay. Gint is an effective gradientbecause it depends on both water diffusion regime and Dvm [31].Specifically, the higher the Gint, the lower the Mc value. In otherwords, since Gint quantifies Dvm, and c is equal to 1 in case ofPGSTE mono-exponential decay (or, equivalently in the case ofGaussian diffusion), the higher the Dvm, the higher the deviationfrom the mono-exponential decay of the PGSTE signal.

The graph displayed in Fig. 2 shows that each phantom charac-terized by mono-dispersed micro-beads with a well defined diam-eter can be identified by a well defined Mc value. In Fig. 2, dataillustrated in gray color are obtained using bipolar diffusion gradi-ents that, in principle, eliminate the static (in space and time)dependence of the signal on background gradients [32,33]. How-ever, the initial and final time intervals of the PGSTE sequencemust also be considered which contribute to the non-completeelimination of internal gradients. As expected, all gray colored dataare close to Mc = 1, showing a poor dependence of Mc on Gint whencompared to data obtained by using monopolar diffusion gradients

(illustrated in black color in Fig. 2) due to residual background gra-dients. Values of c lower than 1 have been traditionally associatedwith anomalous diffusion of compartmentalized water in complextissue structures [20,21,23], intravoxel water diffusion heterogene-ity [25] or to increasing cellular heterogeneity [24]. Results dis-played in Fig. 2 strongly suggest that c values lower than 1 arealso due to Dvm in all samples investigated using variable g PGSTE.In order to give more insight into the meaning of c parameter andits interplay between coupling diffusion processes and backgroundgradients, Mc, Gint, MD, R�2 maps, T�2 and T2-weighted image, wereobtained from an NMR tube comprised of six capillaries filled withpacked polystyrene beads in aqueous solution (Fig. 3). The Mc mapclearly shows a different image contrast when compared to thecorrespondent MD map (see Fig. 3a–c). In the Mc map there is abetter discrimination of capillaries filled with mono-dispersedTiO2 beads in water than those filled with polystyrene micro-beadsin water. This is due to the high sensitivity of Mc to Dvm which isabout one order of magnitude higher in TiO2 beads in water sample[34] than in polystyrene beads in water sample. According to ourprevious results [10], we speculate that the Mc contrast depends

Fig. 3. Mc map (a), Gint map obtained by using TE = (1 125 ms) (b), MD map obtained using b = 1000 s/mm2 (c), R�2 map obtained by using TE = (1.7 20 ms) (d), T2 mapobtained using TE = 4 ms with depicted ROIs (e), and T�2 image obtained using TE = 18 ms (f) are shown. Maps were obtained from a phantom comprised of six capillariesplaced in a 0.8 cm NMR tube filled with 10 lm packed polystyrene beads in aqueous solution. The displacement of each capillary is shown in the SE image reported in (e). Thecapillary defined by ROI 1 is filled with free water, capillary defined by ROI 2 is filled with poly-dispersed 10 and 6 lm polystyrene beads, capillaries defined by ROIs 3 and 4are filled with mono-dispersed 5 lm and 50 nm TiO2 beads, respectively; capillaries defined by ROIs 5 and 6 are filled with mono-dispersed 6 lm polystyrene beads and ROI 7defines a region in NMR tube filled with mono-dispersed 10 lm polystyrene beads. All maps displayed are characterized by an in plane resolution 63 lm � 63 lm andSTH = 1 mm. SNR of Mc, Gint, MD and T�2 maps, evaluated considering ROI 7, is approximately equal to: 8, 4, 10 and 9 respectively.

Table 1MD, Mc, Gint, R�2 ¼ 1=T�2 mean ± SD values obtained from each map (a–d of Fig. 3) inROIs depicted in the SE image displayed in Fig. 3e.

ROI MD (10�4 mm2/s) Mc Gint (mT/m) R�2 ¼ 1=T�2 (Hz)

1 7.07 ± 0.87 0.965 ± 0.029 0.00160 ± 0.00023 3.58 ± 0.292 2.44 ± 0.54 0.899 ± 0.070 0.642 ± 0.014 14.7 ± 3.53 2.62 ± 0.57 0.683 ± 0.094 1.50 ± 0.19 28.6 ± 1.84 3.33 ± 0.96 0.68 ± 0.13 2.10 ± 0.14 45.5 ± 2.75 2.94 ± 0.55 0.896 ± 0.062 0.548 ± 0.077 4.7 ± 3.36 2.59 ± 0.57 0.897 ± 0.058 0.1574 ± 0.0089 7.3 ± 1.17 3.26 ± 0.27 0.915 ± 0.056 0.4904 ± 0.0080 7.41 ± 0.52

M. Palombo et al. / Journal of Magnetic Resonance 216 (2012) 28–36 31

on both the water dynamics and the Dvm. In particular, by chang-ing the g strength in a PGSTE sequence it is possible to probe sev-eral Gint of differing strengths (namely it is possible to probe themicroscopic distribution of Gint) in order to collect more informa-tion about what affects the water diffusion than the conventionalMD method based on the acquisition of diffusion signal at a fixedor a short range of g values. All these bits of information are sum-marized in the c parameter. In other words, the c parameter in themesoscale region (namely the averaged c measured in each voxelof the Mc image) summarizes the contribution of the distributionof Dvm microscopic variations in the sample to the mesoscopicscale. When internal and diffusion PGSTE gradients are of the sameorder of magnitude, their coupling influences the PGSTE signalattenuation in each image voxel. Since phase shift is proportionalto coupling gradients strength, some spins increase the degree ofPGSTE signal attenuation (those most affected by the Gint, i.e. thosecloser to the water-bead interface), while others (which can be dis-tant from the former) will acquire a phase causing a signal in-crease. Due to the impossibility of distinguishing water spins, theabove illustrated mechanism mimics a superdiffusion regime ofwater molecules whose signal disappears in one spot and appearsin another, generating a modulated signal of the image, closelylinked to magnetic susceptibility micro-distribution [10]. Otherauthors have observed super-diffusive phenomena by using non-Gaussian NMR approaches. As an example, in the Tabesh et al. pa-

per [35] the authors discuss the ‘‘empirical evidences in the brainthat suggest a super-Gaussian displacement distribution’’.

In summary, the Mc map displayed in Fig. 3a is characterized byan image contrast that embodies information deriving from bothGint and MD map (shown in Fig. 3b and c, respectively). However,unlike the Gint map, based on a CPMG sequence which assumes aconstant g in each voxel, Mc (obtained using a large range of g) ischaracterized by a better ability to resolve details. In particularwe would like to underline that the c parameter embodies all themicroscopic (i.e. in the subvoxel scale) details related to themicroscopic Gint which influences the mesoscopic scale (i.e. in thevoxel scale).

Fig. 4. Plots of Mc vs Gint (a), Mc vs 1=T�2 ¼ R�2 (b), MD obtained at b = 1000 s/mm2 vs Gint (c), MD obtained at b = 1000 s/mm2 vs R�2 (d), MD obtained at b = 3000 s/mm2 vs Gint (e)and R�2 vs Gint (f) are shown. Each data point depicted with a number from 1 to 7 represents the mean values obtained from the corresponding ROI (1–7), displayed in Fig. 3e.Solid lines in every plot show the linear correlation (Pearson’s coefficient, R, is also reported).

32 M. Palombo et al. / Journal of Magnetic Resonance 216 (2012) 28–36

Usually, in order to highlight Dvm at the interface between dif-ferent tissues, T�2-weighted images are obtained by using gradientecho (GE) type sequences. By comparing the Mc map with the T�2weighted image displayed in Fig. 3a and f, respectively, it is evidentthat the former is characterized by a contrast that defines bound-aries and contours of the heterogeneous phantom better than theT�2 image. The contrast-to-noise ratio calculated in pixels close tothe boundary and contained within the capillaries ranges from0.21 to 1.9 and from 1.7 to 4.0 in T�2 and Mc maps, respectively.

Indeed, signal coming from GE type sequences is mainly af-fected by the so-called static dephasing regime (SDR) [36]. SDR oc-curs when the magnetic moment dephasing (resulting from thelocal differences in the nuclear frequencies) is faster than the aver-aging of difference nuclei phases caused by diffusion phenomenamanage. In this case, all magnetic field inhomogeneities (bothinternal, due to susceptibility differences, and external, inherentto the static magnetic field) [31] contribute to the image contrast.Conversely, signal generated by a PFG type sequence is a lot lesssensitive to the fast dephasing of spins due to external magneticfield inhomogeneities. This is because the 180� radio-frequency

pulse (or a right combination of 90� pulses) usually used in PGSEor PGSTE sequences refocuses all static magnetic field inhomoge-neities. As a consequence, the interplay between diffusion andinternal gradients becomes one of the main mechanisms for irre-versible signal loss due to Dvm in diffusion maps. By increasing dif-fusion g, the coupling between diffusion gradients and the highestGint (that is strictly close to interfaces) provides an irreversible sig-nal loss that can be modeled as a pseudo-superdiffusion process[10]. Compared to conventional MD maps, the interface contrastin Mc maps is emphasized due to a signal loss close to thewater–glass capillaries boundaries and a signal gain far from theinterface.

To better understand the mechanism underlying Mc contrast,and the interplay between Mc, MD, R�2 and Gint, all parameters wereevaluated in defined regions of interest (ROIs) as illustrated inFig. 3e and summarized in Table 1. Plots expressing the linear cor-relation between the measured parameters are reported in Fig. 4together with their Pearsons’s coefficients (R). As expected, wefound a strong positive correlation between the two parametersthat quantify Dvm: R�2 ¼ 1=T�2 and Gint (R = 0.962) as along with a

Fig. 5. Suface plots obtained from normalized c and ADC maps of the phantoms depicted in (a). The DL–DT surface plot (reported in (c)) is obtained from the differencebetween ADC measured along then z direction of the static magnetic field, H0, (DL), and ADC measured along the directions x and y orthogonal to the direction of H0, (DT), usinga PGSTE imaging sequence. Similarly, cL � cT histogram reported in (b) is obtained from the difference between c measured along the direction of H0 (cL) and c measuredalong the direction orthogonal to H0 (cT), using a g variable PGSTE imaging sequence. Surface plots, shown in this figure, strongly demonstrate the use of c contrast inhighlighting boundaries and barriers between substances characterized by different magnetic susceptibilities. Moreover the results reported here, demonstrate the centralrole of the coupling between internal and diffusion gradients in providing the c contrast.

M. Palombo et al. / Journal of Magnetic Resonance 216 (2012) 28–36 33

strong negative correlation between these parameters and Mc(R = 0.945 and R = 0.927, see Fig. 4a and b, respectively). A poor cor-relation was found between MD, and R�2, and between MD and Gint

(R = 0.470 and R = 0.433, respectively). Finally, a moderately posi-tive correlation (R = 0.743) was found between MD obtained athigh b-value (b = 3000 s/mm2) and Gint (see Fig. 4e). Graphs illus-trated in Fig. 4 unequivocally indicate that Mc is more correlatedto Dvm than MD.

The local nature of the image contrast provided by c allows usto better highlight the capillary boundaries in Mc map comparedwith MD, R�2 and Gint maps. To confirm this we also derived the sur-face plots reported in Fig. 5, where cL � cT and DL-DT surface plots

are displayed in Fig. 5b and c, respectively. They were obtained bysubtracting the c and the apparent diffusion coefficient (ADC)measured along the directions x and y, orthogonal to the direc-tion of H0 (transversal c, cT and transversal ADC, DT) from c andADC measured along the z direction parallel to the direction ofthe static magnetic field H0, (longitudinal c, cL and longitudinalADC, DL).

Results shown in Figs. 3 and 5 strongly demonstratet the abilityof c contrast to detect barriers and boundaries located betweencompartments characterized by different magnetic susceptibilities.To give more quantitative information about surface plots dis-played in Fig. 5, the behavior of cL, cT, DL and DT as a function of

Fig. 6. Values of DL, DT, cL, and cT, (measured using PGSTE and a g variable PGSTE imaging sequence, respectively) as a function of increasing voxel number from the capillaryboundary to the capillary interior, quantified in the ROIs: ROI 1, ROI 2, ROI 4 and ROI 6, which are highlighted in Fig. 3e. Moreover, the histograms of the percentage difference(|DL � DT|)/DL and (|cL � cT|)/cL in each voxel are reported for each graph (from a) to (d), and from (e) to (h), respectively). A significant difference between cL and cT is observedin voxels close to the boundary. Conversely, DL is approximately higher than DT in all voxels. Note that, as expected, c values in ROI 4 (which identifies TiO2 beads in aqueoussolution) are lower than those in the other ROIs.

34 M. Palombo et al. / Journal of Magnetic Resonance 216 (2012) 28–36

increasing voxel number from the voxel located close to theboundary to the inner of the capillary, are reported in Fig. 6. Dueto the higher Gint located at the interface between capillaries andmicro-bead dispersion along the transverse direction, when com-pared to local Gint along the longitudinal direction, a significant dif-ference between cL and cT was found in voxels close to theboundary (Fig. 6a–d). Conversely, DL is approximately higher than

DT in all voxels (Fig. 6e–h). According to results displayed in Fig. 4,the percentage difference between cL and cT evaluated in voxelsclose to the boundary, were higher than those between DL and DT

(Fig. 6a–h). Moreover, as expected, both cL and cT values in ROI 4(which identifies TiO2 beads in aqueous solution, Fig. 6c) are lowerthan those in the other ROIs. These findings are due to a gradientinterference effect close to the wall of capillaries which provides

M. Palombo et al. / Journal of Magnetic Resonance 216 (2012) 28–36 35

a stronger anisotropy in c than in ADC parameter. Results and con-siderations reported here are in agreement with observationsunderlined in Cho et al. paper [37], where internal magnetic fieldgradients arising from susceptibility contrast between an array ofcylindrical glass tubes and surrounding water in a uniform appliedmagnetic field are reported.

In conclusion, this work based on controlled polystyrene micro-beads dispersed in water, provides robust experimental evidencethat Dvm represents a source of contrast in anomalous diffusionmethods based on c stretched exponential model. In particular,the coupling between internal and diffusion gradients results inpseudo-superdiffusion effects [10] which can be quantified bythe stretching exponential parameter c. The resulting image con-trast of Mc maps strictly reflects local Dvm, and offers a betterdetection than that achievable by T�2 images at the interface be-tween two substances characterized by a different magnetic sus-ceptibility. Thanks to these characteristics of Mc contrast, wesuggest the use of Mc maps to perform experiments based onsuperparamagnetic nanoparticles for contrast-enhanced MRI andmolecular imaging [38,39]. Further work is needed to define thefeatures and the specificity of Mc imaging by investigating ex-tracted and in vivo tissues. However, as we have used a phantomthat mimics human tissue (namely characterized by pore size from4 to 10 lm and Dvm � 10�6 in SI), we would suggest considerationof the dependence of v on Dvm in ‘‘in vivo’’ investigations, partic-ularly when they are performed at high magnetic fields. Moreover,a better correlation between Mc and T�2 compared to MD and T�2 hasalready been observed in vivo in different brain regions [40].

Finally, the coupling between internal and diffusion (external)gradients, quantified by the c exponent, may be useful in clarifyingthe intermolecular double quantum coherences (iDQCs) contrastmechanism in porous systems. Indeed, the possibility of measuringmicrostructures in porous system using iDQCs contrast, which de-pends on local Dvm [41–43], has previously been proposed.

In light of the experimental results reported here, which havebeen obtained in a controlled phantom comprised of packed mi-cro-beads in water, we suggest that Dvm should be taken as asource of contrast rather than multicompartimental restricted dif-fusion when non-Gaussian methods are used. We hope that thephysical interpretation of the c parameter proposed here will beuseful as a reference for investigators who are about to embarkon studies related to non-Gaussian or anomalous diffusion basedon variable g PGSTE (or PGSE) sequence.

Acknowledgments

Thanks go to Peter Basser and Evren Özarslan for fruitfuldiscussions.

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