Quantification of water compartmentation in cell suspensions by diffusion-weighted and T 2 -weighted MRI

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1 Available online at Magnetic Resonance Imaging 26 (2008) Quantification of water compartmentation in cell suspensions by diffusion-weighted and T 2 -weighted MRI Yiftach Roth a, Aharon Ocherashvilli a, Dianne Daniels a, Jesus Ruiz-Cabello b, Stephan E. Maier c, Arie Orenstein a, Yael Mardor a, 4 a Advanced Technology Center, Sheba Medical Center, Tel-Hashomer 52621, Israel b Universidad Complutense, Madrid 28040, Spain c Department of Radiology, Brigham and Women s Hospital, Harvard Medical School, Boston, MA 02115, USA Received 5 October 2006; revised 22 April 2007; accepted 24 April 2007 Abstract When studying water diffusion in biological systems, any specific signal attenuation curve may be reproduced by a broad range of mathematical functions. Our goals were to quantify the diffusion and T 2 relaxation properties of water in a simple biological system and to study the changes that occur in osmotically stressed cells. Human breast cancer cells were incubated in isotonic or hypotonic osmotic buffers. Diffusion-weighted and T 2 -weighted magnetic resonance images were acquired during sedimentation over 12 h. Diffusion-weighted imaging (DWI) data were analyzed with a biexponential fit, the Kärger model for exchange between two freely diffusing populations and the Price-modified Kärger model accounting for restricted diffusion in spherical geometry. We found that only the Price model provided an accurate quantitative description for water diffusion in both cell systems, independent of acquisition parameters, over the entire density range. Model-derived cell radii, intracellular volume fractions and transmembrane water exchange times were in good agreement with results calculated from light microscopy and with model-free exchange times. T 2 data indicated two populations in fast exchange, with volume fractions clearly different from DWI populations. Hypotonic stress led to higher slow apparent diffusion coefficient, longer T 2 and lower membrane permeability. The tortuosity in a hypotonic cell suspension complied with the Wang model for spherical geometry. Quantitative characterization of biological systems is obtainable by DWI, using appropriate modeling, accounting for water restriction and exchange between compartments. D 2008 Elsevier Inc. All rights reserved. Keywords: Water diffusion; Compartmentalization; Diffusion-weighted MRI; Hypotonic stress; Tortuosity 1. Introduction Water molecules in biological tissues partition among various compartments and are characterized by diversity in magnetic resonance (MR) properties such as relaxation times and diffusion coefficients. Hence, the signal in an MR image of biological tissue generally contains contributions from several water populations. Diffusion in biological tissues is affected by the characteristics of the compartment in which they reside and by exchange between compartments. The presence of barriers with various biophysical 4 Corresponding author. Tel.: , (mobile); fax: address: yael.mardor@sheba.health.gov.il (Y. Mardor). properties may affect water diffusivity by hindering Brownian motion. Diffusion-weighted imaging (DWI) enables the noninvasive measurement of apparent diffusion coefficients (ADCs). ADCs may be affected by cell swelling or cell density; thus, DWI can detect pathologies such as cerebral ischemia [1,2]. Heavily diffusion-weighted (high b value) magnetic resonance imaging (MRI) can separate the signal of intracellular water, which is restricted by macromolecules and the cell membrane, from the signal of extracellular water, where apparent diffusion is an order of magnitude faster [3,4]. A common finding of high b DWI studies is nonmonoexponential signal attenuation indicating nonsimple diffusion of water molecules [4 11]. Nevertheless, the X/$ see front matter D 2008 Elsevier Inc. All rights reserved. doi: /j.mri

2 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) volume fractions of slow-diffusing and fast-diffusing water populations extracted from simplistic biexponential analysis deviate significantly from known ratios of intracellular/ extracellular water in brain tissue [4,5,8] and in perfused brain slices [7]. This difference may originate partly from T 2 relaxation differences between compartments. It was shown that, in general, T 2 relaxation decay in biological tissues is nonmonoexponential [12 14]. Yet, the assignment of T 2 populations to tissue compartments and their relations with populations extracted from DWI are nontrivial [15,16]. Additional mechanisms that may be involved are the presence of more than two compartments, exchange between compartments, anisotropy and complete or partial restriction of intracellular water by cell membranes [6,17 22]. Moreover, several studies demonstrated that nonmonoexponential water diffusion behavior may be observed within the intracellular space of large cells, such as Xenopus oocyte [23] and Aplysia neurons [24]. Phenomenological characterization of experimental diffusion decay curves was achieved by models assuming a continuous distribution of tissue diffusion coefficients, rather than two distinct diffusion coefficients as in the biexponential function [25,26]. Yet, the extracted apparent model parameters are sensitive to acquisition details such as diffusion time. In general, a broad range of mathematical functions with a sufficient number of free parameters may enable the reproduction of any specific signal attenuation curve. The accuracy and applicability of the model to a specific biological system may be tested in two ways: (a) a comparison of biophysical parameters extracted from the model to independently known parameters; and (b) the degree of robustness of the model, which is determined by the sensitivity of model parameters to experimental acquisition parameters. The goal of the present study was to obtain a quantitative detailed understanding of the diffusion and T 2 relaxation properties of water in a relatively simple biological system. Another goal was to study the origins of changes in MR properties observed in osmotically stressed cells. For these purposes, we used suspensions of breast cancer cells incubated either in regular or in hypotonic osmotic conditions. Both cell suspensions were monitored over time during cell sedimentation, and the effect of cell density on diffusion-weighted and T 2 -weighted images was assessed. The data were fitted to the widely used biexponential function and, in addition, by two more complex mathematical models. The dependencies of each model parameter on biophysical properties such as cell size and density, on one hand, and on experimental parameters, on the other hand, were studied. In addition, we tested the compliance of each model with non-mri parameters obtained from light microscope images. We believe that these results may enable a better understanding of mechanisms determining the multiexponential diffusion behavior observed in biological tissues and of the relations between biophysical parameters and apparent parameters calculated using recurrent biexponential fit. 2. Methods 2.1. Cell suspensions The MDA-MB231 human breast cancer cell line was maintained at 378C with 5% CO 2 at 95% humidity in Dulbecco s modified Eagle s medium enriched with 10% fetal calf serum, 1% penicillin streptomycin and 1% glutamine, and was subcultured twice a week. The cells were incubated in isotonic (0.9% NaCl saline) or hypotonic (0.18% NaCl saline and 20% of standard concentration) osmotic solutions and then centrifuged at 1100 rpm for 5 min at 48C. Cells were counted by trypan blue. A suspension of 5 ml of each sample containing cells from the appropriate kind was prepared, and the two vials (7 ml in size) were placed in the MRI system, at a temperature of 17F18C, as controlled by the temperature control system of the MR system. The cells were continuously monitored by MRI during sedimentation over 12 h. Three samples were extracted from each cell suspension prior to MR acquisition, and another three samples were extracted 12 h later to detect possible changes over time. All samples were photographed using an inverted light microscope with a magnification of 200. The percentages of cell killing were calculated by trypan blue, and the means and standard deviations of cell size and eccentricity were calculated from each photograph. The whole experiment was repeated three times, where, in each experiment, the cell suspensions were monitored over 12 h and the reproducibility of the results was tested. All results are given as the means and standard deviations of the three experiments Data acquisition Data were acquired at the Haim Sheba Medical Center using a General Electric 0.5-T interventional MRI system (Signa SP/i) with a gradient intensity of up to 1 G/cm and with line scan diffusion imaging (LSDI) [27] pulse sequence. A specially designed animal volume coil 5 cm in diameter was used for data acquisition. Axial T 2 -weighted images and diffusion-weighted images were acquired during cell sedimentation over 12 h. Axial T 2 -weighted fast spin echo (FSE) MR images were acquired with the following: matrix = ; field of view (FOV)=129 cm 2 ; repetition time (T R )=3000 ms; slice thickness=4 mm. T 2 curves at a diffusion-weighting factor of b = 0 were obtained at different sedimentation temporal points by a series of T 2 -weighted FSE images with an incremented echo time (T E ) between 38 and 589 ms. Line scan diffusion-weighted images were acquired using pulsed gradient spin echo sequence [28] with the following: matrix=128128; FOV=129 cm 2 ; T R =3000 ms; slice thickness = 4 mm. Several diffusion curves were derived from constant diffusion time experiments, where, in each experiment, 19 values of b ranging from 225 to 4000 s/mm 2 were used, which were obtained by incrementing diffusion gradient intensity up to 0.9 G/cm, with all three gradients

3 90 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) turned on at the same time. The diffusion gradient duration d was 73 ms, and the separation times between gradients D used were 77.3, 86.3, 101.3, 116.3, 131.3, and ms, which resulted in T E values of 152, 170, 200, 230, 260, 280 and 340 ms, respectively. Such experiments were repeated at different sedimentation temporal points. Since T E and D changed simultaneously, we obtained a twodimensional data set in b value and T E, from which T 2 curves at b values of 1063, 1902 and 2741 s/mm 2 were obtained. The resulting acquisition times of the diffusion experiments were between 7 min 25 s and 10 min 25 s Data analysis Image analysis was performed using the Interactive Data Language software package (Version 3.6.1; Research Systems, Inc.). Two regions of interest (ROI) were defined in a central slice of a hypotonic cell vial (bottom and top), and one ROI was defined in an isotonic cell vial (Fig. 1). Since the vials were not moved during the experiment, it was easy to guarantee that exactly the same ROI were analyzed at all time points. The two ROI in hypotonic cell suspension differed naturally in their cell densities, and data points from both ROI were used in the analysis. The means and standard deviations of signal intensity in each ROI were computed for each T 2 -weighted and diffusionweighted image, at each time point. These average data were used to create diffusion curves and T 2 curves. Curve fitting was performed with MATLAB (Version 6.5; The MathWorks, Inc.) using the Gauss Newton nonlinear least squares method Diffusion-weighted curve analysis In biological systems, there are, in general, at least two water compartments: intracellular and extracellular. The diffusion curves reflecting signal attenuation with increasing b values were fitted by three models A biexponential fit In case of two freely diffusing noninteracting water populations, the normalized intensity of water signal is given by: I I 0 ¼ A f expð bd f ÞþA s expð bd s Þ ð1þ where I and I 0 denote signal intensities in the presence and in the absence of diffusion-sensitizing gradients, respectively; A f and A s are volume fractions (A f +A s =1); and D f and D s are diffusion coefficients of the fast and slow water populations, respectively. The diffusion-weighting factor b is given by: b ¼ q 2 t D ¼ ðcdgþ 2 ðd d=3þ ð2þ where c is the gyromagnetic ratio of nuclei, and g and d are diffusion gradient amplitude and duration, respectively. The effective diffusion time traversed by spins during measurement is t D =D d/3, where D is the separation time between diffusion gradients. By varying b, a diffusion curve for each ROI is obtained, in which ln(i/i 0 ) is plotted as a function of b. The b (b =q 2 t D ) value can be varied by incrementing q (through g or d) while keeping the diffusion time constant (c t, diffusion curve) or by varying t D (through D) while keeping the gradient strength q constant (c g, diffusion curve). The biexponential model ignores the effects of exchange between water compartments and the (partial) restriction of slow/intracellular water molecules by cell membranes Two-site Kärger model This model [29] describes two freely diffusing populations with exchange. Signal attenuation is now given by: I ¼ A 1 expð bd 1 ÞþA 2 expð bd 2 Þ ð3þ I 0 where D 1,2 and A 1,2 are the apparent self-diffusion coefficients and population fractions, respectively, which are related to real biophysical parameters by: D 2;1 ¼ 0:5fD f þ D s þ ð1 þ n bs s þ 4nt exp 2 ðbs s Þ 2 Þt exp # 1=2 ) " F D s D f þ ð1 nþt 2 exp bs s ð4þ ð A 2 ¼ 1 A sþd f þ A s D s D 1 D 2 D 1 ð5þ Fig. 1. Typical LSDI MR image of vials containing suspensions of cells that were incubated under isotonic (left) or hypotonic (right) osmotic conditions. The two ROI in the hypotonic vial and the ROI in the isotonic vial, which were used in the analysis at all sedimentation time points, are shown. The image was acquired with a b value of 644 s/mm 2, a diffusion gradients separation time of 77.3 ms and an FOV of 129 cm 2. A 1 ¼ 1 A 2 ð6þ where n =A s /(1 A s ), t exp is experimental diffusion time (as explained below) and the free biophysical parameters of the Kärger model are the diffusion coefficients D f and D s, the

4 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) slow volume fraction A s and the mean residence lifetime in the intracellular compartment s s. Note that the residence lifetime in the extracellular compartment s f is given by s f =(A f /A s )s s. In Eq. (4), the term 1/q 2, which usually appears in the presentation of Kärger equations [29], was replaced by t exp /b. The two terms are identical in the case of short gradient pulse (SGP) approximation, where the gradient duration d is very small compared to the separation between gradients D. In such a case, t exp =t D =D d/3~d. In this study, SGP approximation is not fulfilled, since d and D are in the same order of magnitude. Lori et al. [30] suggested replacing D d/3 with D+d in order to better account for the finite d effect. In our analysis, we studied both cases and performed curve fitting for each experimental curve with t exp =D d/3 and t exp =D+d. The obtained v 2 and the correspondence between model-derived and microscope-derived cell radii were studied in each case The modification of Price et al. to the Kärger model This model [18] accounts for two exchanging water populations, while one population is restricted by cell membranes with spherical geometry and the other population experiences free diffusion. The attenuation of intracellular signal due to restriction in a sphere is given by (Eq. (7) in Price et al. [18]): ð I int ¼ 3j 1ðÞ a Þ 2 a þ 6a 2X l n¼0 jn VðÞ a X 2 m¼0 l ð2n þ 1Þk 2 nm exp k2 nm D sd=r 2 k 2 nm n2 n k 2 2 a2 where a =qr, R is the cell radius, j n (x) is the spherical Bessel function of the first kind and k nm is the mth nonzero root of the equation j n V(x)=0. I int expresses the attenuation of the signal originating from intracellular spins due to membrane restriction and diffusion within the intracellular space. The Price-modified Kärger model was used in three versions First-order Price model. With a long time limit, the condition: D s D H1 ð8þ R2 holds. In this case, the first-order approximation in the dependence of signal attenuation on qr is applicable, and Eq. (7) reduces to: 2 3j1 ðaþ I int ¼ : a nm ð7þ ð9þ The physical meaning is that, at long-enough diffusion time, all spins feel the effect of the cell membrane, and the attenuation of the signal arising from intracellular water depends mainly on cell size and on the exchange between the compartments and does not originate from diffusional decay within the intracellular space. In our study, the condition given by Eq. (8) was partially fulfilled; hence, the data were analyzed with both first-order and higherorder models. In the first-order case, the intracellular diffusion coefficient D s may be completely eliminated from exchange equations, and attenuation is given by Eq. (3) but with modified Kärger equations (Eqs. (21) (23) in Price et al. [18]): D 2;1 ¼ 0:5fD f þ ð1 þ nþt exp F D f þ ð1 n bs s bs s 1=2 þ 4nt2 exp ðbs s Þ 2 Þt 2 exp ð A 2 ¼ 1 A sþd f D 1 ð1 A s þ A s I int Þ ðd 2 D 1 Þð1 A s þ A s I int Þ ð10þ ð11þ A 1 ¼ 1 A 2 : ð12þ Thus, now, the free biophysical parameters now D f, A s, s s and R. As for the Kärger model, each experimental curve was analyzed with t exp =D d/3 and t exp =D+d Higher-order Price model, Version 1. When the condition given by Eq. (8) does not hold, higher orders have to be considered in Eq. (7), and D s becomes an additional free parameter, through the dependence of I int on D s. Nevertheless, where there is a significant restriction effect, the system should not be treated as two exchanging populations with two distinct diffusion coefficients D f and D s, as in the Kärger model (Eqs. (4) (6)). A more realistic approach is to describe intracellular molecules as having an effective (relatively high) D s far from cell boundaries, which decreases considerably when approaching the membrane. Thus, intracellular molecules exchanging with the extracellular domain are almost not affected by D s. Hence, in the first version of the higher-order Price model, we used Eq. (7) in conjunction with Eqs. (10) (12), where D s is eliminated Higher-order Price model, Version 2. Where the effect of restriction is small, it can be accounted for as a small correction to the original Kärger model. This may be done by using Eq. (7) in conjunction with Kärger equations (Eqs. (4) (6)), where A s is replaced by A s I int. In both versions of the higher-order Price model, we used up to n =3 in the sum of spherical Bessel functions (Eq. (7)), since higher orders led to a change of b1% in all fit results T 2 -weighted curve analysis In cases where populations have significant differences in relaxation times, data analysis should include appropriate

5 92 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) corrections. For the case of two compartments (intracellular and extracellular) with two diffusion coefficients D f and D s, there are, in general, two spin spin relaxation times T 2f and T 2s of the fast and slow populations, respectively, which may have exchanges between them. Yet, for strong diffusion weighting (high b value), the contribution of the fast population may be neglected, and normalized signal intensity is given by: I ¼ A s expð bd s Þexpð T E =T 2s Þ: ð13þ I 0 where A s is the volume fraction of slow/intracellular molecules. T 2 -weighted images were taken at different T E values at strong diffusion-weighting conditions at various b values, and the T 2 relaxation time of the slow population was calculated. The validity of the approximation of neglecting the fast population s contribution was assessed by studying the dependence of the calculated T 2 on the b value and on sedimentation time (cell density). T 2 curves were also obtained at no diffusion-weighting conditions (b =0). The data were fitted to monoexponential and biexponential functions and to the Kärger model, using residence lifetimes s s extracted from diffusion curves data. In the Kärger model analysis of T 2 curves, D f and D s were replaced by 1/T 2f and 1/T 2s in Eqs. (4) (6), and q 2 s s was replaced by s s. In this way, T 2 relaxation times and volume fractions of different T 2 populations were obtained at several time points Extraction of intracellular residence lifetime In systems with two or more compartments, there is, in general, water exchange between compartments. The rate of water exchange can be described as the reciprocal of the mean residence time s s in the intracellular compartment. Residence time can be extracted from the slope of the decay of signal intensity in a c g diffusion curve as a function of separation time D [19]. When q is high enough so that the contribution of the extracellular signal is dephased and when D is long enough so that most intracellular spins have spanned the cell interior, signal attenuation is governed by intracellular contribution and is given by: I I 0 ¼ expð T E =T 2 D=s s Þ: ð14þ In the LSDI sequence installed in our open MRI system, T E and D were related by: T E ¼ 2D 2:6 ms: ð15þ After the substitution of Eq. (15), we obtain the dependence of signal attenuation on D: ln I ¼ 2:6 2 þ 1 D: ð16þ I 0 T 2 T 2 s s When the relaxation time T 2 is known, s s can be extracted from the slope of signal attenuation versus D, using Eq. (16). The residence times s s were also assessed independently from the Price model by estimating the quality of the fit as a function of s s. Model quality was determined by a combination of two criteria: (a) the compliance of model parameters with non-mri parameters obtained from light microscope images, and (b) the dependencies of model parameters on the experimental separation time D. The detailed method of model quality estimation is described in Appendix A Tortuosity of extracellular water The diffusion of water molecules in the extracellular compartment is hindered by collisions with cell membranes. This effect leads to reduction in extracellular ADC with increasing cell density, due to increased tortuosity in the extracellular medium. Tortuosity factor can be defined as D f /D saline and reflects the reduction in fast ADC relative to ADC in cell-free saline. The effect of cell density on the extracellular diffusion coefficients of hypotonic and isotonic cell suspensions was studied. The exact functional relation between D f and A s depends on cell geometry. A commonly used equation describing monodisperse spherical particles was derived by Jonsson et al. [31]: D f ¼ 1 D saline 1 þ A : ð17þ s 2 Another equation accounting for tortuosity effect was presented by Wang [32]: D f ¼ 1 A s D saline 1 þ A : ð18þ s 2 An approximate equation analogous to Eq. (18) for cylindrical geometry, applicable for low and medium A s [33], is obtained using the equation introduced by Fricke [34] with the correction of the (1 A s ) factor: D f ¼ 1 A s : ð19þ D saline 1 þ A s In this study, we compared the experimental data of the two cell suspensions with the above functions. 3. Results and discussion 3.1. Light microscopy The cells were placed on an MR scanner at room temperature for 12 h. Under such conditions, they may suffer from hypoxia or ischemia. Yet, a viability of N90% was detected in cell counting by trypan blue assay, following incubation in both isotonic and hypotonic conditions. The percentage of cell killing after 12 h was negligible. Cell size and shape were studied from light microscope images acquired before MR acquisition and after 12 h. The means and standard deviations of cell radii and eccentricity are listed in Table 1. No significant change

6 Table 1 Results of the biexponential fit, the Karger model, the Price-modified Karger model and light microscope images a Biexponential model Karger model Price model Microscope images D b [ms] D c s [10 4 mm 2 /s] R c [Am] D c s [10 4 mm 2 /s] R c [Am] D c s [10 4 mm 2 /s] R c [Am] R d [Am] E d Hypotonic cells Isotonic cells Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) F F F F0.3 66F F F F F F F F F F F F F F F F F F F F F F F F0.17 a Data are presented as meanfs.d. b Diffusion gradients separation time. c Slow ADC D s and cell radius R as obtained from the analysis of DWI data. d Cell radius R and cell eccentricity q (defined as the ratio of the major axis to the minor axis) as obtained from light microscope images. in cell radius R with time was found for both cell suspensions (isotonic cells: R = 4.4F2.8 Am before acquisition and R =4.3F2.3 Am after 12 h; hypotonic cells: R = 5.8F3.4 Am before acquisition and R = 6.1F3.2 Am after 12 h; meanfs.d.). Moreover, the slow diffusion coefficients, intracellular residence times, membrane permeabilities and intracellular T 2 were found to be independent of time for both cell suspensions (see below). Hence, it seems that the effect of subtle possible ischemic changes on diffusion measurements was small. Hypotonic cells were significantly larger (n = 159 for isotonic cells and n =162 for hypotonic cells; P b.0001, twotailed t test), as expected, and were less eccentric (n =123 and 142; P b.0001). Both cell systems exhibited a wide distribution in cell size T 2 -weighted curve analysis Interestingly, all T 2 curves, with no exception, showed a clear monoexponential behavior. Relaxation times calculated at b values of 0, 1063, 1902 and 2741 s/mm 2 are plotted in Fig. 2 as a function of sedimentation time for both cell systems. For b =0, there was a clear decrease in T 2 with this time (cell density), apparently due to the increased volume fraction of the slow population having shorter T 2. For b values of s/mm 2, T 2 was independent of time and b values for both systems, apparently reflecting a single T 2 population of slow/intracellular water molecules (T 2s ). In the hypotonic cell solution, T 2s =514F28 ms, while for isotonic cells, T 2s =270F12 ms. The longer T 2s in hypotonic cells may result from a smaller interaction with macromolecules and less surface relaxation effects [35] in hypotonic/swollen cells. The measured T 2 relaxation time in cell-free buffer was 2020F40 ms. The estimated T 2 of the fast/extracellular population in both cell suspensions (T 2f ) varied between this value and the minimal possible values, which are the measured T 2 for b =0 at the minimal time (700 ms for isotonic cells and 1000 ms for hypotonic cells; Fig. 2). The values of T 2s and T 2f, together with intracellular residence lifetimes extracted as explained in the Methods section, were used to fit the T 2 curves data obtained at b =0 by the Kärger model function. This analysis showed that the monoexponential behavior of T 2 curves could be well explained by the fast exchange between two T 2 populations and yielded apparent volume fractions of the populations Biexponential analysis of diffusion-weighted curves Several diffusion curves from ROI in hypotonic and isotonic cells tubes are shown in Fig. 3, along with noise level, as obtained from image background. At each time Fig. 2. T 2 relaxation times obtained from T 2 curves acquired at b values of 0 (diamond), 1063 (square), 1902 (triangle) and 2741 () s/mm 2 are plotted as a function of time for isotonic (left) and hypotonic (right) cell suspensions. Note that, unlike at b =0, T 2 is independent of time at high b values. Fig. 3. Plots of signal attenuation as a function of b value from isotonic (left) and hypotonic (right) cell suspensions, for various time points. Shown are couples of curves acquired at close time points with diffusion gradients separation times of 77.3 ms (open symbols) and ms (closed symbols). The noise level is shown for reference. Note the different dependence on diffusion time for the two cell systems.

7 94 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) point, diffusion curves obtained using two values of the separation time are shown: D=77.3 ms and D=141.3 ms. As time goes by, there is more cell sedimentation and hence increased cell density. It can be clearly seen that for both cell systems, signal attenuation as a function of b value became smaller for longer sedimentation times, reflecting increased slow volume fraction. A comparison of diffusion curves obtained with different D values reveals different behaviors in the two systems: In isotonic cells, increasing D from 77.3 to ms led to increased signal attenuation as a function of b value, although the difference decreased with sedimentation time (density). On the other hand, in hypotonic cells, one can see the same trend, but at high-enough cell density (sedimentation times longer than 280 min), signal attenuation became smaller for longer D (Fig. 3). The results of fitting the diffusion curves obtained with two separation times (D=77.3 ms and D=141.3 ms) by a biexponential function are shown in Fig. 4, illustrating the change in A s and D s with the sedimentation time for both cell suspensions. It can be seen in both cell cases that A s increased with time since the sedimentation increased cell density. On the other hand, D f decreased with this time due to increased tortuosity in the extracellular medium (not shown). D s was independent of time, as expected, since sedimentation should not affect intracellular conditions. The effect of using a D longer than ms was manifested as a consistently smaller A s at each time point in both cell suspensions (isotonic cells: P b.0005; hypotonic cells: P b.0001, paired two-tailed t test) and significantly smaller D s for hypotonic cells ( P b.0001), while for isotonic cells, D s did not change with D. Slow ADC can be used for a rough estimation of the molecular displacement of intracellular water, using the Einstein relation (in its three-dimensional form): p R ¼ ffiffiffiffiffiffiffiffiffiffiffi 6D s t D : ð20þ This apparent R represents average displacement from a Gaussian distribution of freely diffusing molecules. Where the diffusion of intracellular molecules is (partially) restricted by the cell membrane, the apparent D s depends on t D (or D), and R incorporates information about cell dimensions. The results of D s and R obtained by the three models used in this study for hypotonic and isotonic cells, using D=77.3 ms and D=141.3 ms, are shown in Table 1. The increase with D in apparent R implies that the restriction of intracellular water is partial Water exchange between compartments The decrease in A s with D, as observed in biexponential analyses (Fig. 4A and C) of both cell suspensions, indicates a significant transmembrane exchange between the compartments. Where D is longer, more spins cross the cell membrane into the other compartment, and the apparent result of this exchange is a decrease in A s as calculated in the biexponential model. Hence, it is clear that a proper description should account explicitly for this process. Fig. 4. The results of fitting the data with a biexponential equation are shown for the two cell systems as a function of time. Shown are A s (A) and D s (B) for isotonic cells, and A s (C) and D s (D) for hypotonic cells. In each graph, results obtained at D=77.3 ms (diamonds) and D=144.3 ms (squares) are presented.

8 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) Fig. 5. The quality factor Q, reflecting the quality of the Price model fit, is plotted as a function of the residence time s s for hypotonic and isotonic cells. The means, upper limits and lower limits of standard deviations of s s as obtained from c g diffusion curves for the two cell suspensions are shown as well. It can be seen that, for both cell suspensions, the minimum of Q occurs well within the range ofs s as obtained from c g diffusion curves. Plots of signal attenuation as a function of D were obtained for three q values of 164, 179 and 184 mm 1, and residence lifetimes s s were extracted from the slopes of these c g curves at four sedimentation time points, for both cell systems. Since T 2 data exhibited monoexponential behavior, T 2 contribution could be easily separated (Eq. (16)), and s s could be extracted. The extracted s s values were independent of sedimentation time and q value, as expected for a real biophysical parameter. The average s s values obtained were 160F25 and 110F20 ms (meanfs.d.) for hypotonic and isotonic cells, respectively. The standard deviations of s s between different c g diffusion curves were large due to the low signal-to-noise ratio (SNR) at the magnetic field of our open-system MRI. Nevertheless, when accounting for the number of c g curves obtained (n =22 for hypotonic cells and n =11 for isotonic cells), the difference in s s between the two cell suspensions was statistically significant ( P b.0001, two-tailed t test). An independent assessment of s s was obtained from the Price model by estimating the quality of the fit as a function of s s, as explained in detail in Appendix A. The quality factor Q is plotted in Fig. 5 as a function of s s for hypotonic and isotonic cells. It can be seen that for both cell suspensions, the minimum of Q occurs well within the range of s s as obtained from c g diffusion curves. These values of s s were used in models accounting for transmembrane water exchange, as described in Sections and The finding of T 2 populations in the fast-exchange regime, while diffusion populations are in the slowexchange regime, may seem puzzling at first sight. Yet, when accounting for the relations of s s with relaxation times, on one hand, and with diffusion coefficients, on the other hand, the puzzle is resolved. In both cell suspensions, it is found that s s bt 2s,f (hence, T 2 populations experience fast exchange) and, simultaneously, s s N( q 2 D s,f ) 1 (hence, DWI populations are in the slow-exchange regime) Two-site Kärger model The results of fitting the diffusion curves obtained with two separation times (D=77.3 ms and D=141.3 ms) by a Kärger model [29] function are shown in Fig. 6, illustrating Fig. 6. The results of fitting the data with the K7rger model are shown for the two cell systems as a function of time. Shown are A s (A) and D s (B) for isotonic cells, and A s (C) and D s (D) for hypotonic cells. In each graph, results obtained at D=77.3 ms (diamonds) and D=44.3 ms (squares) are presented.

9 96 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) the change in A s and D s with sedimentation time for both cell suspensions. As in the biexponential fit (Fig. 4), A s increased with sedimentation time and cell density, while D f decreased with time (not shown) and D s was independent of this time, as expected. Nevertheless, the effects of D on fit parameters were different for the two models. The slow volume fraction A s was independent of D in the Kärger model, while the use of longer D yielded significantly smaller D s not only for hypotonic cells (as in the biexponential fit) but also for isotonic cells (Table 1; P b.0001 for hypotonic cells and P b.0002 for isotonic cells; two-tailed t test). On the other hand, the apparent R (as obtained from Eq. (20)) was independent of D. D f was independent of D in both fits. The Kärger model accounts explicitly for exchange between water populations, but not for the fact that intracellular diffusion is partially restricted. Hence, membrane effect was manifested by a sharp decrease in D s with D, which was sharper in hypotonic cells Price-modified Karger model: exchange with restriction The modification of Price to the Kärger model [18] accounts for a partial restriction of intracellular water molecules in cells with spherical geometry. As explained in the Methods section, we performed three kinds of analyses using a first-order model and two versions of higher-order approximations First-order Price model. Fig. 7A F shows the results of fitting diffusion data by the first-order Price model. The free model parameters are A s, D f and cell radius R, shown as a function of sedimentation time. The trends of A s and D f with sedimentation time were similar to the previous models and to the trends detected by Thelwall et al. [10] in erythrocyte ghosts with various densities. Yet, a striking result is that, unlike in other models, changing the separation time from D=77.3 ms to D=141.3 ms did not considerably change any of the free Fig. 7. The results of fitting the data with the first-order Price-modified K7rger model are shown for the two cell systems as a function of time. In each graph, results obtained at D=77.3 ms (diamonds) and D=144.3 ms (squares) are presented. (A) A s for isotonic cells. (B) D f for isotonic cells. (C) A s for hypotonic cells. (D) D f for hypotonic cells. (E) Cell radius R for isotonic cells. The means, upper limits and lower limits of standard deviations of radius distribution as calculated from light microscope images are shown. (F) Cell radius R for hypotonic cells. The means, upper limits and lower limits of standard deviations of radius distribution as calculated from light microscope images are presented.

10 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) parameters. Fig. 7E and F shows the means and standard deviations of cell radii as calculated from light microscope images for isotonic and hypotonic cells, respectively (see also Table 1). It can be seen that cell radii calculated from MRI using the Price model were well within the radii distribution range, as calculated from light microscope images, although on the upper part of the range. The model seems to be somewhat less accurate when using longer separation times, as the cell radius tended to increase with D, especially for isotonic cells (Fig. 7E and F). As explained in the Methods section, the experimental diffusion time t exp was allowed to vary between t D =D d/3 and D+d. It was found that, in most cases, curve fitting with the lowest v 2 was obtained when using t exp =D d/3; hence, these values of t exp were used in the data analysis presented in Fig. 7. When we used the exact means of cell radii as calculated from light microscope images (R a =0.006 Am and R a = Am for hypotonic and isotonic cells, respectively), a good curve fitting was obtained in most cases with longer t exp times, which were closer to D+d (data not shown), although with higher v 2 values. Lori et al. [30] found that using an effective t exp of D+d instead of D d/3 may account correctly for the finite d effect in case of heterogeneous diffusion in several microscopic Gaussian domains, with permeable boundaries having free exchange between compartments. In this study, like in many biological systems, cell membranes were semipermeable; hence, both effects of exchange and restriction were important. Avram et al. [36] found that acquisition with large d values yielded a compartment size smaller than the real ones in ensembles of long impenetrable cylinders, using q- space analysis. In this study, increasing t exp towards D+d yielded cell radii values closer to radii means obtained from microscopy. A detailed inquiry into finite d effects on the results of the Price model analysis requires the use of samples with distinct compartment dimensions with semipermeable and nonpermeable boundaries. In any case, the results demonstrate that, even in cases where the SGP approximation condition is not fulfilled, structural information may be obtained using the Kärger model (for heterogeneous diffusion in compartments with permeable boundaries) or the Price-modified Kärger model (for heterogeneous diffusion in compartments with semipermeable boundaries). These results may indicate that, in the Price model, there was no underparameterization; hence, it enabled a quantitatively accurate description of the two cell suspensions, even when experimental separation time was changed. Yet the slow diffusion coefficient D s was not one of the free parameters in the first-order model. The values of D s calculated using the biexponential model or the Kärger model were largely underestimated due to cell membrane effects. Hence, in order to try to obtain more realistic estimates for D s, we used the higher-order Price model Higher-order Price model, Version 1. The intracellular diffusion time D s was added as an additional free parameter. The other free parameters were not significantly different from the results of the first-order fit (not shown). D s was independent of D (indicating the robustness of the model) and of sedimentation time (indicating the stability of intracellular diffusion conditions) for both cell suspensions (not shown). D s was higher in hypotonic cells, probably indicating a freer diffusion inside swollen cells. The means and standard deviations of D s values for both cell systems obtained using D=77.3 ms and D=141.3 ms are shown in Table 1. The values of D s obtained from the Price model were significantly higher than those in the biexponential model and the Kärger model, where D s values were considerably reduced due to boundary effects. The results obtained by the higher-order Price model, Version 1, enable us to also estimate cell membrane permeability in the two cell systems. The restrictive effect of the membrane is best described by the reduced permeability p. In our study, we can estimate p by [37]: p ¼ 3D s s s =R 2 1: A s D s =A f D f 0:2 ð21þ After the substitution of the parameters extracted from the Price model, we obtain p = 0.18F0.04 and p = 0.29F0.08 for hypotonic and isotonic cells, respectively, resulting in a significant difference between the two cell suspensions ( P b.0001, two-tailed t test). The rate of transfer of the membrane P (measured in mm/s) is defined by P=pD s /R and can be roughly estimated as and mm/s for hypotonic and isotonic cells, respectively. Membrane permeability depends on various factors, including the aquaporin water channels concentration and type, membrane lipid composition, temperature and ph, and may vary considerably [38,39]. Specific examples at a temperature of about 208C are human red blood cells (0.05 mm/s) [40], V-79W hamster fibroblast cells (0.041 mm/s) [38] and human bone marrow stem cells (0.006 mm/s) [39]. This study demonstrates that DWI, even at a clinical scanner, may detect small permeability variations. Membrane water permeability was found to be related to the metastatic potential of tumor cells [41]; thus, DWI may be applicable for tumor characterization and grading Effects of hypotonic stress This study implies that hypotonic stress led to cell swelling and lowered membrane permeability. These processes cause an increase in slow volume fraction and a decrease in the global ADC of water molecules in the sample, as is well known in cerebral ischemia. On the other hand, our results indicate an increase in intracellular ADC in hypotonic cells, in contrast to cerebral ischemia, where several studies observed a decrease in intracellular ADC [42]. It is possible that the reduction in global water ADC associated with cerebral ischemia is caused by a combination of mechanisms

11 98 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) in the intracellular level and of mechanisms with characteristics common to osmotic stress Higher-order Price model, Version 2 This version of the model can be accounted for as a small correction to the Kärger model and may describe the data reliably in cases of high cell membrane permeability. The results of data analysis with the higher-order Price model, Version 2, were similar to those with the Kärger model in the sense that D s decreased significantly with D for both cell suspensions. The effect of accounting explicitly on restriction (through Eq. (7)) was manifested as values of A s smaller than those obtained with the Kärger model and not significantly different from those of the first-order Price model (data not shown). The dependence of D s on D in this version suggests that, in the experimental conditions of this study, Version 1 of the higher-order Price model is preferred to Version 2. The difference between the two versions is related to the strength of the restriction effect exerted by cell membranes, which is reflected in the residence time s s. For large-enough diffusion times (or D), the signal attenuation resulting from transmembrane exchange behaves like exp( D/s s ) (i.e., Eq. (14)). From Eq. (7), we can see that diffusional decay within the intracellular space is governed by exp( k 2 11 D s D/R 2 ). Hence, the relative importance of the two processes may be expressed in the condition: k 2 11 D ss s R 2 H1: ð22þ When this condition holds, exponential decay due to exchange is much slower than intracellular diffusional decay, and Version 1 of the higher-order Price model should be preferred to Version 2. This means that intracellular spins have an effective mean diffusivity far from cell boundaries, while close to cell boundary, diffusivity is negligible and signal attenuation is governed by exchange. Substituting k 11 = and the parameters obtained in this study, we find that this condition holds for both cell systems Comparison of slow volume fractions Fig. 8. Plots of slow volume fraction A s as a function of time for isotonic (A) and hypotonic (B) cell suspensions. Comparisons of A s as obtained from fitting diffusion data by the biexponential fit (triangles) and the Pricemodified K7rger model ( ) and as obtained from T 2 data analysis by the K7rger model (diamonds) are presented. The lower error bars of T 2 data indicate the variation of T 2f value from the value measured in cell-free buffer to the minimal possible value. The circle in each graph represents the results of A s calculation according to light microscope images. Note that this result is in good agreement with the results of the Price model for both cell systems. Fig. 8 shows a comparison of the slow volume fraction A s as calculated from the biexponential fit, the Price model and the fitting of T 2 data by the Kärger model. The lower error bars of T 2 data indicate the variation of T 2f values from the value measured in cell-free buffer (2020F40 ms) to the minimal possible values, which are the measured T 2 for b =0 at the minimal time (700 and 1000 ms for isotonic and hypotonic cells, respectively; Fig. 2). A clear discrepancy is seen between the values of A s obtained from T 2 data and the values of A s obtained from DWI data, which abides for any possible T 2f value. A similar finding of volume fractions of short T 2 population that were considerably larger than the volume fractions of the slow diffusion component was demonstrated in a recent study of normal and edematous rat muscles [16]. This phenomenon may be explained by an extracellular water layer interacting with cell membranes. In any case, it seems that the assignment of volume fractions extracted from T 2 decay curves analysis, with actual water compartments, is far from being straightforward. An independent estimation of A s was performed according to light microscope images. The total volume of sedimented cells was obtained from T 2 -weighted MRI after 12 h. Intracellular volume was computed by multiplying cell volume as calculated from the light microscope image by the number of cells in the tube, assuming that, by that time, almost all cells have sedimented. The value of A s obtained in this method is represented by the circle in Fig. 8A and B. It can be seen that this result agrees well with the values obtained from the Price model, for both cell systems, while biexponential analysis overestimates A s. Several clinical [2,5,8,11] and ex vivo [7,16] studies observed an opposite finding, namely, underestimation of intracellular volume fraction in the biexponential model. One suggested explanation for this discrepancy was appreciable exchange between compartments [11]. Another explanation was a T 2 effect [5,7]: T 2 relaxation time is generally shorter in the intracellular compartment due to interactions with macromolecules and surfaces. However, Clark and Le Bihan [11] and Niendorf et al. [2] found that biexponential model parameters were independent of echo time, suggesting that either the relaxation times in the different compartments were similar or they were in the fast-exchange regime. In this study, we observed a difference between intracellular and extracellular T 2, but T 2 populations (unlike diffusion populations) were in the fast-exchange regime and exhibited monoexponential behavior. Hence, there was no T 2 effect on

12 Y. Roth et al. / Magnetic Resonance Imaging 26 (2008) the volume fractions of diffusion populations. In general, in cases where the populations are not in the fast-exchange regime and especially in realistic cellular tissues, the problem of coupling between relaxation and restricted diffusion is much more complicated and may require numerical methods. The overestimation in A s in the biexponential model in this study resulted from the restricting effect of cell membranes. The biexponential and Kärger models, unlike the Price model, do not explicitly account for the partial restriction of intracellular molecules, and this effect is implicitly manifested as an increased A s Tortuosity of extracellular water The tortuosity effect was clearly manifested in all fit methods as a consistent reduction in fast ADC (D f ) with time and cell density. The tortuosity factor was defined as D f /D saline. Fig. 9 depicts the relation between the tortuosity factor and the slow volume fraction A s, as obtained from the Price model for the two cell systems. D saline was found to be F mm 2 /s and was constant over time, as expected. It can be seen that the tortuosity of hypotonic cells is in good agreement with the Wang equation [32] (Eq. (18)). In contrast, the reduction in D f with A s in isotonic cells was larger not only than that predicted by the Wang equation but also relative to the prediction of Eq. (19). This may imply that other factors, such as interactions between cells, may contribute to the increased tortuosity effect in this system. The eccentricity of hypotonic cells as obtained from light microscope images was significantly smaller than that of isotonic cells (Table 1); hence, they may be better described by the spherical geometry assumed in the Price model. Fig. 9 clearly shows that the equation derived by Jonsson et al. [31] (Eq. (17)) is not adequate to describe the tortuosity effect in this study. The two equations deviate by the factor (1 A s ). We can deduce that the Jonsson equation is applicable for the description of self-diffusion in a colloidal system containing particles impenetrable to water, as demonstrated by the authors [31], Fig. 9. Tortuosity factor, defined as the ratio between fast ADC, D f and the ADC in free saline D saline, is plotted against the slow volume fraction A s for hypotonic (diamonds) and isotonic ( ) cells. Also plotted are the Jonsson equation (Eq. (17), dotted line), the Wang equation (Eq. (18), solid line) and Eq. (19) for cylindrical geometry (dashed dotted line). Note that the data of hypotonic cells are in good agreement with the Wang equation. while in cases more relevant to biological systems, with exchange between the compartments, the Wang equation should be used. These results demonstrate that valuable morphological information may be extracted by an analysis of DWI data. Additional studies, and probably more complicated models and numerical methods, are required to obtain an adequate analysis of more complicated and less isotropic systems, such as neuronal tissues Comparison between models The analysis of water diffusion in biological systems is complex and requires consideration of several processes. An indication for an accurate quantitative modeling of a system is the independence of free model parameters from experimental acquisition parameters such as diffusion time and q. In this study, we used three mathematical models to describe hypotonic and isotonic cell suspensions at a broad range of cell densities and with variable experimental separation times. All models enabled the characterization of each decay curve separately and enabled changes in the biophysical properties of the biological systems to be tracked. Yet, only the modification of Price et al. [18] to the Kärger model succeeded to describe both biological systems with biophysical parameters that were independent of experimental conditions over the entire range of measurement time and cell density. In addition, the model results were in good agreement with non-mri-independent measures. (a) Cell radii were within the radius distribution range as calculated from light microscope images. This agreement was somewhat better for hypotonic cells, which were less eccentric spheroids as revealed with this technique. (b) Slow volume fraction A s after 12 h was in good agreement with A s deduced from microscopy, for both cell suspensions. In addition, model-derived transmembrane water exchange times were in good agreement with model-free exchange times Practical application While the clinical scanner and gradient strength used in this study are suboptimal, it was demonstrated that such methodological examinations can be performed in clinical scanners. Besides the obvious shortcomings of the smaller magnetic field and gradient strength, they have the advantage of larger space, enabling the examination of several samples simultaneously. The main limiting factors are the size and homogeneity of the radiofrequency coil. When considering the clinical application of advanced analysis methods such as the Price model, one has to consider the additional information versus the cost of additional acquisition time. The residence lifetime may be extracted phenomenologically from c g diffusion curves, where diffusion time is incremented while the diffusion