INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

Pulsed field gradient spin echo (PGSE) nuclear magnetic resonance technique has been used to probe the structures of porous materials and the diffusion of the confined spin-bearing molecules (Tanner and Stejskal, 1968; Callaghan, 1991). It is well known that, in q space experiments there are diffraction-like patterns shown in the PGSE attenuation curves for various cases of restricted diffusion (Callaghan et al., 1991; Balinov et al., 1994; Kuchel et al., 1997; Callaghan, 1995; Codd and Callaghan, 1999). For restricted diffusion in single pores, the characteristic diffraction-like pattern may reflect the size of the pore, whereas, for restricted diffusion among well-separated multipores, the patterns reflect the mean distance between the pores.

Kärger (1985) developed an analytical approach to examining PGSE attenuation through two-region exchange approximation. Because there is exchange between two freely diffusing phases, the calculated double exponential decay profile does not display the diffraction-like pattern as observed in the PGSE experiment on the erythrocyte suspension system. Stanisz et al. (1998) modified Kärger's interpretation for the PGSE attenuation of the erythrocyte suspension, in which they considered the diffusion within an erythrocyte as one-dimensional restricted diffusion. They also estimated an apparent diffusion coefficient from the PGSE attenuation feature by considering one-dimensional restricted diffusion and then calculated the profile of PGSE attenuation using the two-region exchange model. Similarly, Price et al. (1998) derived the apparent diffusion coefficient from the PGSE attenuation of the restricted diffusion in a spherical pore. Also, Peled et al. (1999) applied the same strategy to studying water diffusion in the frog sciatic nerve and derived the apparent diffusion coefficient from the PGSE attenuation in a cylindrical pore to mimic the restricted diffusion in the nerve cells. All those results involve the usage of the apparent diffusion coefficient to illustrate the effects of the restricted diffusion in the cells and, in turn, to explain the diffraction-like pattern of the PGSE attenuation curve.

Kuchel et al. (1997) explored the PGSE experiments for the erythrocytes suspension system and estimated two physically significant lengths from the two diffraction-like patterns of the attenuation profile. They assigned the two lengths obtained to the size of the erythrocyte and the average distance of extracellular pore spacing. Such a treatment is similar to that adopted by Callaghan et al. (1991) for the pore-like space between the polystyrene spheres. They have used this method to derive the PGSE attenuation for the restricted diffusion among multipores of the same size. Also, they introduced an effective diffusion coefficient to describe self-diffusion for long-range migration between pores. The formulation used by Stanisz et al., (1998), Peled et al., (1999), and Price et al., (1998) can be used to interpret the diffraction-like pattern caused by the restricted diffusion in the erythrocytes, but it cannot be used to explain the first diffusion-like peak observed by Kuchel et al. (1997). It implies that there exists some constraint in extracellular diffusion. Therefore, a new model is needed to include the effects of the size and the arrangement of the erythrocytes and the external pores between the erythrocytes. Because such systems are too complicated to be solved exactly by the general diffusion equation, the modified pore-to-pore hopping model may be used as an approximation for studying the diffusions in the erythrocyte suspension system.

In the present work, we develope a simplified diffusion model for a two-phase system, represented by the coupled master equations (Haus and Kehr, 1987) with pore-to-pore hopping exchange between two different pores. We take into account the effects of the pore size, the spatial arrangement of the pores, and the variation of water-residing times in each phase on the PGSE attenuation. Then we calculate the PGSE attenuation of diffusion among multipores of the same size and compare the results with those derived by Callaghan (1995). Also, we applied the proposed model to analyzing the results of the PGSE experiments for the erythrocyte suspension system.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

The general formulation

The formulation of the PGSE attenuation for the molecular diffusion among multipores can be easily derived based on the "pore equilibrium" condition (Callaghan et al. 1991). This assumption is suitable for a porous medium with well-defined pore-channel structure, i.e., the size of the channels is much smaller than that of the pores. With the pore equilibrium condition and the short gradient pulse approximation (Tanner and Stejskal, 1968; Linse and Söderman, 1995), where the waveform of the gradient pulse is considered to be a -function in time domain, the PGSE attenuation of diffusion among multi-identical pores can be expressed in q space by (Callaghan, 1991)

(1)

where q = G/2,  is the gyromagnetic ratio,  is the duration of the magnetic field gradient pulses, G is the strength of the gradient; and (r₀) is the density of the spins within the 0th pore initially, and  is the interval between the two pulsed gradients. The subscript 0 represents the 0th pore, e.g., the pore where the spin is situated at the initial time, and the subscript n indicates the nth pore where the spin is situated at time . V₀ and V_n are the internal space of the 0th and the nth pores, respectively. r₀ and r_n are the position vectors of the pore centers at the 0th and the nth pores, respectively. In Eq. 1, S(q)  _V dr(1/V)exp(i2q · r) is called the "structure factor" of the pore, where 1/V is the density of the spins, which normalizes the amount of the spins in a pore, and V is the volume of a single pore. P(n, ) is the probability of a spin existing in the nth pore at time , R_n is the position vector of the center of the nth pore relative to that of the 0th pore, and

(2)

Thus P(q, ) describes the arrangement of the pores as expressed in q space, and one may derive it in terms of the hopping exchange model described in the master equation in the next section.

Diffusion among identical pores

For restricted diffusion in a single pore with a permeable wall, one may treat the boundary as a semi-adsorptive wall (Barzykin et al. 1995),

(3)

where D is the diffusion coefficient, H is a constant to represent the transport ability of the boundary; r₀ and r are the position vectors; P(r₀|r, t) is the probability of finding a particle initially at r₀ and at r after a time t, and a represents the position where the boundary exists. By solving the diffusion equation with the boundary condition described by Eq. 3, the total probability within the pore, e.g., _V drP(r₀|r, t), gives an exponential decay form with a characteristic lifetime. By analogy, we adopted the same idea to the case of the exchange diffusion among multipores and take the lifetime as the time needed to travel from one pore to another. Then, the master equation, which is analogous to that of the multisite jump diffusion (Haus and Kehr 1987), can be constructed. The master equation of hopping diffusion among pores of the same size can be written as

(4)

where P_i(t) is the probability of a spin existing at the ith pore at time t, P_ij(t) is the joint probability of a spin existing at the jth pore neighboring to the ith pore, N is the number of the first shell of pores, and W = 1/N is the pore-to-pore exchange rate, where  is the spin residence lifetime in a pore.

In Eq. 2, P(q, t = ) may be obtained readily by the Fourier-Laplace transform of Eq. 4 and then solved by inverse Laplace transform, which yields

(5)

where
is called the configuration integral of this multipore system and R_j is the position vector of the center of the jth pore in the first shell relative to that of the central pore.

Equation 5 was in the same form as that derived previously by Callaghan et al. (1991), except that the term /_j is replaced by D_eff/6l² in their formulation, where D_eff is the long-time limit (effective) diffusion coefficient among the pores, and l is the distance between two pores.

Application to cell suspension systems

The formulation may be extended with the help of Eq. 1 to condensed cell suspension systems. As shown in Fig. 1, the condensed cell suspension system can be approximated as a system consisting of two kinds of pores. We denote the cells as pores C. The space enclosed by the cells can be considered as the external pores, which are denoted as pores S. One may suppose that the diffusion of spin-bearing molecules starting from a pore C (or S) initially can be found either in a pore C or in a pore S at time , and thus four cases must be considered. Particularly, for those spins existing in pores C initially, and at a later time  in pores S, the contribution to the magnetization is given by

(6)

where

(7)

and where S_C(n, ) is the probability for a spin diffusing from a pore C initially to the nth pore S at time . R_Sn is the position vector at the center of the nth pore S relative to that of the initial pore C. V_C and V_S are the volumes of a single pore C and pore S, respectively. r_C and r_S are the position vectors within a pore C and a pore S relative to their pore centers, respectively. The expressions for the other cases, CC (pore C to pore C), SC and SS are similar. Here, the subscript C means the starting point is a pore C, and the subscript S means the starting point is a pore S. There are only three hopping rate constants considered in this system because the cells do not connect to each other directly. Pores S are all connected to their neighboring pores S with a hopping rate of W_SS. Pores C are isolated from the other pores C, but are connected to their neighboring pores S with hopping rate constants W_CS = 1/(N_C) and W_SC = 1/(N_S), where _C and _S are the spin residence lifetimes of pores C and pores S, respectively.

View larger version (22K): [in this window] [in a new window]   FIGURE 1   The pictorial representation of the simplified two-phase model consisting of the cells (solid circle, pores C, solid line represents the membrane) and the pores between the cells (dotted circle, pores S). RC and RS are the radius of pores C and pores S, respectively. The mean distance between two pores S is d. The mean distance between a pore C and a pore S is b. The hopping rates between pores are also marked by W with appropriate subscripts to indicated the exchanging species.

Furthermore, in the PGSE experiment, the time interval between the two gradient pulses is set shorter than the transverse spin relaxation, and the observed magnitudes of PGSE attenuation for various q is normalized by the observed value at q = 0. Thus, for this simplified system, the effect of the transverse spin relaxation processes is cancelled. The master equation may be written without the spin relaxation term as

(8a)

(8b)

where C_n(t), S_n(t), C_ni(t), and S_ni(t) are the corresponding probabilities at time t, N is the coordination number of a pore C surrounded by S pores, and M is that of a pore S surrounded by other S pores. For simplicity, we set the same value N as the coordination number of a pore S surrounded by pores C. Then we follow the hopping exchange model and obtain C_C(q, ), S_C(q, ), C_S(q, ), and S_S(q, ) by solving the coupled master equation after the Fourier-Laplace transform and the inverse Laplace transform as derived in the Appendix. M_CS may be calculated in accordance with Eq. 6. By analogy, M_CC, M_SC, and M_SS can be obtained accordingly. Consequently, for random pore arrangement, we obtain

(9a)

(9b)

(9c)

(9d)

where the parameters x_i and y_i (i = CC, SC, CS, and SS) are defined in the Appendix. They are related to the hopping rate and the pore-to-pore distance. The exponent parameters, ₁, and ₂, in Eqs. 9 are given by

(10)

where  is defined by

(11)

On the basis of the completely random arrangement of pore C and pore S, the derivation procedures are readily presented in the Appendix. By summing up the four parts as given in Eq. 9, we obtain the PGSE attenuation

(12)

where the structure integral for a single pore C and a single pore S are given by
and
respectively. 1/(V_C + V_S) is the density of the spins, which normalizes the amount of the spins in one pair of pore C and pore S.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

Preparation of erythrocyte suspension

Blood was obtained from a healthy human volunteer. The erythrocytes were centrifugally washed (3000 × g, 10 min) two times in cold glucose-enriched saline solution (154 mM NaCl, 10 mM glucose, 4°C). The plasma and the buffy coat were discarded. The appropriate amount of cold glucose-enriched saline solution was then added to form the erythrocyte suspension. All erythrocyte suspensions were gently bubbled with carbon monoxide for 5 min to transform the hemoglobin into a stable low-spin diamagnetic state. For the experiments on inhibiting transmembrane water exchange, p-Chloromercuribenzenesulfonate (p-CMBS) (Sigma, St. Louis, MO) was added (1.9 mg to 1 ml of suspension) and the suspension was kept at 37°C for 1 hr before doing the PGSE experiments.

PGSE experiment

PGSE experiments were performed on a (Bruker Analytik GmbH, Rheinsteuen, Germany) MSL-500 spectrometer, operating at a 11.4-T magnetic field, equipped with a Bruker DIFF-25 gradient probe capable of a maximum gradient of 10 T m¹. The use of the actively shielded gradient coil in the probe and the precompensation function of preemphasis unit greatly reduce the effect of eddy current on diffusion measurements. A blanking unit is open 200 µs before the gradient pulse and stays on during the gradient pulse and the ring-down period to allow the preemphasis to work. The eddy current generated after the gradient pulse is less than 2% of the static value of the gradient pulse within 150 µs. It rings down to less than 1% after 250 µs. In all the experiments, the standard PGSE pulse sequence and phase cycles were used (Kuchel et al., 1997). The duration of the 90° pulse was 25 µs; that of the two gradient pulses, , was 1.2 or 2 ms. Because the proton transverse relaxation times inside and outside the erythrocyte (Stanisz et al., 1998) yield 160 and 400 ms, respectively, to achieve significant signal in our experiments, one may set the time interval between the two gradient pulses to be shorter than 160 ms. Here the time interval between the gradient pulses was set to 15 or 40 ms. The relaxation delay between transients was 8 s; and the number of transients per spectrum was 80. The probe temperature was maintained at 298 ± 0.3 K to minimize the convection. The S/N for full magnetization was higher than 2000 in all the experiments.

Orientation observation

Gelatin solution was prepared by adding an appropriate amount of gelatin (Sigma, St. Louis, MO) into a saline solution. The erythrocyte suspensions (with or without p-CMBS treatment) were added to the gelatin solution and kept at 37°C for at least 3 h within the 11.4-T magnetic field to ensure the formation of gelatin network so that the erythrocytes could become readily oriented. For the controlled experiments, the samples were kept outside the magnetic field. The temperature was then cooled to 20°C to make the sample gel. The orientation of the erythrocytes fixed in the gelatin gel was observed under a light microscope (original magnification ×400).

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

Permeability effect

Considering the condensed spherical cell (pores C) suspension systems, the pores (pores S) between pores C may also be approximated to be a sphere, as shown in Fig. 1. For the case with the volume ratio of pore C to pore S to be V_C:V_S = 0.7, we obtain the radius ratio of pore C to pore S, R_C:R_S=. The mean distance b between pore C and pore S may be approximated by the sum of their radii because of the compact stacking of the two kinds of pores (see Fig. 1). Moreover, if one considers the 3-dimensional cubic packing of two kinds of pores, the mean distance d between two S pores may be set to ~b. Here the same density of spins is assigned to each pore, i.e., _C/_S = 1.0, and thus the population ratio of the spins in pore C to pore S gives P_C/P_S = V_CC/V_SS = 0.7, which also implies the ratio of the two rate constants W_SC/W_CS = P_C/P_S = 0.7 in accordance with the principle of detailed balance. Then, we set  = 1.5/MW_SS, which is 1.5 times the lifetime of a spin in pore S. In the present work, we first considered a situation in which the spin-bearing molecules may not penetrate the cell membrane, e.g., W_CS = W_SC = 0, but W_SS  0 because there must be connections between pores S. In this case, we obtain exp(₁) = 1 and ₂ = MW_SS[1  sinc(qd)]. Consequently, the attenuation reduces to

(13)

where the first part, F/[V_C(V_C + V_S)], results solely from the restricted molecular diffusion in pores C, and the second part, F exp(₂)/[V_S(V_C + V_S)], from the molecular diffusion in the external pores. Furthermore, one may enhance W_SC and W_CS to the same magnitude of W_SS to investigate the effect of the permeability. However, when W_SC and W_CS are significant, the PGSE attenuation is no longer dominated by the spins in pores C and pores S only. Instead, the effect of spin diffusion from pores C(S) into pores S(C) is considered (see Eq. 9). In those cases, the exp(₁) versus qb plot is presented in Fig. 2 A, and the exp(₂) versus qb plot is presented in Fig. 2 B. We can clearly see that exp(₁) oscillates with the same periods as exp(₂) does. The position of the first peak of exp(₁) and exp(₂) is situated at qb = 0.72, which is close to qb = 0.57 (qd = q ·  b  1). As compared with the PGSE results for the diffusion among pores of the same size, the position is characterized by the length between two pores S, or two pores C. The increasing oscillation magnitude of the exp(₁) term with increasing W_SC and W_CS shows the effect of the pore arrangement with the exchange process.

View larger version (18K): [in this window] [in a new window]   FIGURE 2   The two exponential terms as expressed in Eq. 9 versus qb plots. (A) exp(1) versus qb plot; (B) exp(2) versus qb plot. The solid curves a, b, c, d, and e correspond to NWSC/MWSS = 0, 0.25, 0.50, 0.75, and 1.00, respectively.

The E(q, ) versus qb curves with different NW_CS values are plotted in Fig. 3. The individual contribution to the magnetization, e.g., M_CC, M_CS, M_SC, and M_SS for three cases, NW_SC/MW_SS = 0, 0.5, and 1.0, are shown in Fig. 4 for comparison. Apparently, in Fig. 3, there are characteristic peaks at qR_C or qR_S  q · (b/2)  1 for all of the five curves with various magnitudes of NW_SC. The intensity of the characteristic peak decreases as the permeability increases. As shown in Fig. 4, the PGSE attenuation may be analyzed in detail as follows. The characteristic peaks result mainly from M_CC, but not from M_SS, because the high molecular mobility of the spins in pores S causes more attenuation of M_SS in q space. Analogously, the intensity of the characteristic peak in M_CC decreases with increasing W_SC (W_CS), which reduces the intensity of the characteristic peak in E(q, ). The increasing decay rate of E(q, ) at small q with increasing W_SC (W_CS) comes mainly from the change of the relative proportion of each component. As shown in Fig. 4 A, M_CC decreases with increasing W_SC (W_CS), whereas the fast decaying M_CS, M_SC, and M_CC dominate the shape of the curve at small q. In addition, when W_SC (W_CS) is about the same magnitude of W_SS, i.e., NW_SC/MW_SS = 1.0, represented by the dotted line in Fig. 4 A, there is also a fast decay in M_CC at small qb. When NW_SC/MW_SS < 0.5, because M_CC is large, the first diffraction-like pattern of M_SS, reflecting the distance between two pores S, is under the shadow of M_CC, and therefore it is invisible on the E(q, )-qb plot.

View larger version (17K): [in this window] [in a new window]   FIGURE 3   The PGSE attenuation E(q, ) versus qb plot. The solid curves a, b, c, d, and e correspond to NWSC/MWSS = 0, 0.25, 0.50, 0.75, and 1.00, respectively.

View larger version (15K): [in this window] [in a new window]   FIGURE 4   The magnetization contributed from each part of the spins with varying NWSC. (A) MCC, (B) MSS, (C) MCS and MSC. Solid lines, NWSC = 0; dashed lines, NWSC/MWSS = 0.5; dotted lines, NWSC/MWSS = 1.0. There are two features in (C): 1) there is no contribution of MCS and MSC for the case of NWSC = 0; and 2) the values in the arch with x on it are the absolute values of the negative calculated results.

Orientation effect

For a nonspherical cell suspension system, the orientations of the cells may affect the PGSE attenuation. As shown in Fig. 5, we may consider the disk-shape cells (pores C) suspension system with disk thickness t₁, radius R₁ and the spacing between the cells x. The cells are randomly packed. We may then simply take the averaged shape of the medium (pores S) separating the cells as a disk with radius R_S and thickness t_S estimated from the lattice model shown in Fig. 5 B. The PGSE attenuation for a specific cell orientation  (see Fig. 5 A) may be derived as

(14)

where the cell orientation  is defined as the angle between the magnetic field and the central axis of the pore C. Other parameters have been defined previously for Eq. 9. The structure integral, F_C(), for a pore C with the cell orientation  is expressed by

(15)

where J₁(2q sin  · R₁) is the first-order Bessel function of the first kind. Similarly, the structure integral, F_S(), for a pore S with the specific cell orientation  may be expressed as

(16)

To demonstrate the effect of the cells orientation, we may consider the case of R₁ = 1.5l, t₁ = l and x = l/4 where the length, l, is arbitrary. We then set  = 1.5/MW_SS and NW_SC = MW_SS. The PGSE attenuation of different cell orientations versus ql plot is presented in Fig. 6. We can find that the characteristic peak shifts from ql = 0.56 to a higher ql value while the cell orientation changes from 90° to 45°. The characteristic peak shifts to an even higher ql value while  = 0° and cannot be seen in the low ql region. Moreover, the PGSE signal attenuates slower in the low ql region while  = 0° as compared to the cases of  = 90° and  = 45°. The effects of cell orientation merit attention.

View larger version (21K): [in this window] [in a new window]   FIGURE 5   (a) The cell orientation  is defined as the angle between the magnetic field direction (q direction) and the central axis of the cell. (b) The lattice model of the equally separated disk-shaped cells with disk radius R1 and thickness t1. The averaged shape of the medium (pores S) can be approximated as a disk with radius RS and thickness tS. There is one pair of pore C and pore S in a unit cell. The distance between a pore C and a pore S is denoted by b. The distance between two pores S is denoted by d. The distance between cells is denoted by x. RS is the radius of a pore S. b = , d = , RS = (l1 + l2)/2, l1 =  b1  R1, l2 = b1.

View larger version (12K): [in this window] [in a new window]   FIGURE 6   The PGSE attenuation E(q, ) versus q plot. The curves a, b, c represent the simulation data of a 51% hematocrit erythrocyte suspension at  = 90°,  = 45° and  = 0°, respectively. The simulation parameters for curve a, a = 3.06 µm, b = 4.12 µm, and d = 8.24 µm; curve b, a = 3.06 µm, b = 3.1 µm, and d = 6.2 µm; and curve c, a = 3.06 µm, b = 1.51 µm, and d = 3.01 µm. The parameters NWCS = 100 s1 and MWSS = 120 s1 are for all three curves.

Observation of erythrocyte orientation

As shown in Fig. 7 A, pure erythrocytes are oriented with their disk plane parallel to the magnetic field direction. For the controlled experiments in the absence of magnetic field, the erythrocyte orientations were random. These results were the same as those obtained by Higashi et al. (1993). Besides, Kuchel et al. (2000) used the diffusion tensor method to analyze the PGSE attenuation of the erythrocyte suspension system, and they found that the diffusion tensor component at z-direction (the direction parallel to the magnetic field of the NMR spectrometer) is larger, which also verifies the anisotropic orientations of the erythrocytes. The erythrocytes with p-CMBS added, as shown in Fig. 7 B, are oriented with their disk plane perpendicular to the magnetic field direction.

View larger version (112K): [in this window] [in a new window]   FIGURE 7   (a) Pure erythrocytes and (b) erythrocytes with p-CMBS added inside a 11.4-T magnetic field. The magnetic field direction is normal to the test plane. Pure erythrocytes were photographed on their edge so that they were orientated with their disk plane parallel to the magnetic field direction. Erythrocytes with p-CMBS added were photographed on their edge so that they were orientated with their disk plane normal to the magnetic field direction. For the controlled experiments in the absence of magnetic field, the erythrocyte orientations were random. The controlled results were the same as those obtained by Higashi et al. (1993).

Applications to the erythrocyte suspension system

Experimental PGSE results

View larger version (14K): [in this window] [in a new window]   FIGURE 8   The experimental PGSE attenuation E(q, ) versus q plot and the simulated results. Solid circles and dashed line stand for the experimental and fitted results, respectively, for the 48% hematocrit erythrocyte suspension with p-CMBS added (NMR parameters:  = 1.2 ms and  = 40 ms). Open circles and solid squares stand for the experimental result of  = 2 ms and  = 1.2 ms, respectively, for the 51% hematocrit pure erythrocyte suspension. Solid line is the fitted result for 51% hematocrit pure erythrocyte suspension. The time interval of the two gradient pulses was set to 15 ms. Open inverted triangles and dotted line stand for the experimental and fitted results, respectively, for the 40% hematocrit pure erythrocyte suspension (NMR parameters:  = 2 ms and  = 40 ms).

Analysis of the experimental results

(17)

(18)

(19)

(20)

View larger version (30K): [in this window] [in a new window]   FIGURE 9   The side view and the front view of the erythrocyte and the approximated shape of the erythrocyte.

CC

SS

1

View larger version (15K): [in this window] [in a new window]   FIGURE 10   The magnetization contributed from each part of the spins in the 51% hematocrit erythrocyte suspension, and the experimental PGSE attenuation E(q, ) versus q plot (solid circles) and the simulated results. (a) E(q,