Determination of proton magnetization transfer rate constants in heterogeneous biological systems

Determination of Proton Magnetization Transfer Rate Constants in Heterogeneous Biological Systems Douglas Brooks, Kazuo Kuwata, Thomas Schleich

Two procedures are currently. in use for the determination of proton magnetization transfer rate constants between macromolecular tissue components and water. The first method assumes that there are only two spin baths (macromolecular plus solvent) and that during off-resonance irradiation complete saturation of the “immobile” proton spin bath occurs (S. H. Koenig, R. D. Brown, 111, R. Ugolini, Magn. Reson. Med. 29, 311 (1993)). This approach neglects the possibility of incomplete saturation and polydispersity, and yields an apparent magnetization transfer rate constant, Kap,,. The second approach utilizes a formalism which can account for polydispersity and incomplete saturation of the immobile spin bath (K. Kuwata, D. Brooks, H. Yang, T. Schleich, J. Magn. Reson., in press). In this work magnetization transfer rate constants derived by the use of both methods for two systems, ocular lens tissue and cross-linked bovine serum albumin (BSA) were compared. For both samples Kapp was dependent on B, off- resonance irradiation frequency and power when the first method was used. The second method provided values of the magnetization transfer rate constant that were similar to the values obtained by the first method, as the limit of complete saturation was approached. Key words: magnetization transfer; cross relaxation; bovine serum albumin (BSA); ocular lens.

INTRODUCTION

The transfer of proton magnetization between the macromolecular and solvent (water) components of tissue has generally been modelled phenomenologically in terms of two spin baths representing solid-like or immobile macromolecular protons and mobile solvent protons (1-17). To accommodate the complexities of the ocular lens, a tissue that includes both moderately mobile (7, - 100 ns) and solid-like protein macromolecular components (18, 19), Kuwata et al. (16) extended the two spin bath formalism to include two macromolecular proton spin baths plus a solvent spin bath. Moreover, this model of magnetization transfer was cast in the form of a matrix formalism enabling the generalization to N-proton spin baths. Parameters characteristic of both two and three spin bath models include the intrinsic spin-lattice rate constants and the spin-spin relaxation times for each of the spin baths, the rate constant for magnetization trans-

MRM 31:331-336 (1994) From the Department of Chemistry and Biochemistry, Sinsheimer Laborato- ries, University of California, Santa Cruz, California. Address correspondence to: Thomas Schleich, Ph.D., Department of Chem- istry and Biochemistry, university of California, Santa Cruz, CA 95064. Received July 6, 1993; revised October 20, 1993; accepted November 1, 1993 This research was supported by National Institutes of Health Grant EY 04033.

Copyright 0 1994 by Williams & Wilkins All rights of reproduction in any form reserved.

0740-3194/94 $3.00

fer between the spin baths, and the ratio of the number of macromolecular protons in each spin bath to the solvent proton spins.

For a two spin bath system when complete saturation of the restricted population of spins is assumed, the equations described by the generalized method (16), as we show below, simplify to the equations for a two spin bath system obtained by others (2, 10, 14). Thus, the apparent magnetization transfer rate constant (Kapp) obtained by assuming a two spin bath system and saturation of the motionally restricted spin bath is not identical to the magnetization transfer rate constant (RAC or RT) obtained from the generalized formalism unless complete saturation of the restricted population of spins is attained. Magnetization transfer rate constants representing the transfer of magnetization from water to the macromolecular phases have been reported in the literature using the following representations: k, (2), RT (3,4, 6), 4

We wish to draw attention to a fundamental difference in the two approaches currently used to determine the proton magnetization transfer rate constant, namely the formalism that assumes complete saturation of the solid- like phase (2, 10, 14), and the other, which does not require this assumption (3, 6, 15, 16), and demonstrate that the value of the magnetization transfer rate constant will not be the same unless saturation of the solid-like spin bath is achieved. The results of this study demonstrate that Kapp is dependent on the field strength and frequency of the off-resonance irradiation applied during the preparation period of the off-resonance rotating frame spin-lattice relaxation experiment. Furthermore, the value of Kapp tends to approach that obtained from an analysis of intensity ratio dispersion data as more complete saturation is attained at smaller off-resonance irradiation frequencies and stronger Bz fields. Results for the apparent magnetization transfer rate constant of a model tissue system presented recently in the literature (14) are dependent, as we show experimentally in this study, on the unwarranted assumption of complete or total saturation of the solid-like macromolecular spin bath. The relevant magnetization transfer parameters were derived from off-resonance rotating frame proton spin-lattice relaxation data using simulated annealing combined with the method of steepest descents employing the formalism of Kuwata et al. (16).

To make a direct comparison of the magnetization transfer rate constant obtained by assuming total saturation of the solid-like spin bath and by a formalism that does not require this assumption (16), we have chosen two systems: a model system of cross-linked bovine serum albumin, which has been assumed to consist of two proton spin baths (macromolecular and water) (3,14) and

(101, K (141, RAE (161, and RAC (16).

331

332 Brooks et al.

the nuclear region of the ocular lens, which represents a tissue system containing three spin baths (two macromolecular + water) (16, 18, 19).

THEORY

The phenomenologically based formalism for the behav- ior of proton off-resonance rotating frame spin-lattice relaxation involving N macromolecular spin baths has been presented elsewhere (16). The relevant steady state expression relating the intensity of the water proton reso- nances in the presence (M;) and absence ( M t ) of off- resonance radiation for a three-spin bath system (water (A) + 2 macromolecular ( B [moderately mobile], C [motionally restricted]) is:

f(gRiR,R*,RA + gR*CRBRAB + g R i R C R A C

R A - M: + fRBRBCRAC + gRCRABRBC - f R A R i C l ~~1 M: - D

where RA is the intensity ratio of the A spins, f is the ratio of B-spins to A-spins, and g is the ratio of C-spins to A-spins. RAE, RBC, and RAC are the magnetization transfer rate constants between the different spin populations (A -+ B, B + C, and A -j C, respectively), and Rj is the intrinsic longitudinal relaxation rate constant (i.e., in the absence of cross-relaxation) for each of the spin baths. D is given by:

D = fgRlRiR: - gR',RiB - fRiRic

- f ' R P i c - 2fTIABRBcRAc

where

RL = R, + RAE + RAc + 2WA

RAE f R i = R, + - + RBc + ZWB

RAC f R*,= R, + - + -RE, + 2Wc g g

and lineshape functions (2Wi) for each component [i = A, B, C) are given by

2 " .

where Aj is the offset frequency in Hz, wli is the field strength (rad/s) of the B2 RF irradiation field (= yHBz, where yH is the gyromagnetic ratio) and TZi is the trans- verse relaxation time for components with a Lorentzian line-shape. TZc for the gaussian component is defined from its lim-width.

1 uc = -

2VTZC For two magnetically coupled spin baths, one with a

Lorentzian lineshape representing bulk water (A) and a

motionally restricted species represented by a gaussian line shape (C), the equations simplify to

by setting RBc and RAB = 0, where

RA + RAc + 2WA €3; = R i = R c + - + 2 W c RAC

g and using the lineshape functions previously defined. This equation is analogous to Eq. [Za] of ref. 6 with the motionally restricted spin bath represented by a gaussian line shape in place of a Lorentzian.

When complete saturation of the restricted mobility spin bath is assumed Eq. [5a] of ref. 16 reduces to:

[31

Neglecting off-resonance effects on the water (which have been shown to be negligible under most commonly used experimental conditions) the resulting equation becomes:

where the experimentally determined parameter, TtsAT is given b y

Thus,

1 - RA R A C = 7

I S A T

and

RA TISAT

RA = -

Eqs. [5a] and [5b] are identical to Eqs. [8a] and [8b] derived by Koenig et al. (14) and thus Kapp, representing the apparent magnetization transfer rate constant between the two spin populations, becomes equivalent to RAc only when total saturation of the restricted mobility spin bath is achieved. The usual practice, however, is to assume that Kapp = RAC without ensuring that saturation of the solid spin bath is achieved. An additional conse- quence of incomplete saturation is that the value of RA obtained from inversion recovery experiments in the presence of an off-resonance irradiation field by way of Eq. [5b] is over estimated. In this study we equate RA obtained from inversion recovery with saturation experiments to RA/SAT/, and discuss below the utility of relaxation rate measurements made in the presence of an off- resonance irradiation field for the estimation of RA and RAC .

Analysis of Proton Magnetization Transfer 333

EXPERIMENTAL

Sample Preparation

Bovine serum albumin (BSA), essentially fatty acid free, was obtained from Sigma Chemical Co., St. Louis, MO, and used without further purification. Cross-linked solutions of BSA were prepared by placing 10% wlw BSA and 25% glutaraldehyde (Sigma) in an ice bath for 1 h (5). The two solutions were then quickly mixed before being transferred to a 5 mm NMR tube (50 pl 25% glutaraldehyde was used per milliliter of BSA solution to give a molar ratio of 67:l glutaraldehyde to BSA). The BSA concentration was determined by measuring the absor- bance at 280 nm using an extinction coefficient, F” = 6.7 ml cm-l g-’ (20). Calf lens nuclear homogenate was prepared as previously described (16).

Instrumental

NMR experiments were performed on a General Electric GN 300 spectrometer operating at 7.05 T using a variable temperature 5 mm lH probe as previously described (16). The proton decoupler was used to provide continuous wave off-resonance rf irradiation ( B , field). B2 field cali- bration was achieved by determining the nutational frequency of the decoupler at a given power output by ac- curately measuring the time required for a d 2 RF pulse. Calibrations were performed for each sample, and off- resonance field strengths used ranged from 0.06 to 0.6 gauss. For each sample 4 or 5 B, field strengths with 30 off-resonance irradiation frequencies between 0 to 50 kHz from the water resonance were used to obtain intensity ratio dispersion curves. Spectra obtained in the presence of off-resonance irradiation were normalized to a control spectrum acquired in the absence of RF irradiation. The off-resonance preparation pulse was applied for 9 s. Spectra were collected in quadrature with 4096 data points and at a sweep width of 10 kHz using CYCLOPS phase cycling. Typically eight acquisitions were collected per spectral data point.

The longitudinal relaxation time of water, designated TISAT, of water was measured by inversion recovery in the presence of an off-resonance saturation field (4) applied at 10, 30, and 45 kHz off-resonance at the field strengths stated above. TISAT values were evaluated by the method of Levy and Peat (21). The water proton T2 was determined by use of the Carr-Purcell spin echo pulse sequence ( d 2 - T - 7r - T - acquire) (22) with relaxation delays less than 100 ms.

Intensity Ratio Dispersion Curve Analysis

Optimization of the relevant NMR parameters was ac- complished by a combination of simulated annealing and the method of steepest descents to minimize the RMSD between the observed and theoretically calculated intensity ratio dispersion curves as previously described (16). The curve fitting routines were written using Borland Turbo Pascal (version 6.0) (Source code may be obtained by sending an E-mail request to: [email protected]). An estimation of the errors associated with each of the fitted parameters was obtained by the parabolic method (23).

The standard deviation in the RMSD was obtained from individual fits of four or five intensity ratio dispersion curves obtained at the different B, field strength values used. Because of the possibility of over estimating the value of RA obtained from TISAT measurements, we also used the average value of RA(SAn as a constraint for the maximum value of RA in the simulated annealing procedure. The value of RA obtained by the constrained fitting procedure (RA 5 RACSAn) was denoted RAIFfn.

RESULTS AND DISCUSSION

Two different procedures were used to obtain the magnetization transfer rate constant by analysis of experimental proton off-resonance spin-lattice relaxation experiment intensity ratio data. The results obtained using the first method were based upon Eqs. [4] and [5] (or equivalently Eq. [8] of ref. 14) which assumed complete saturation of the immobile spin bath. Apparent values of the magnetization transfer constant, Kapp, obtained for nuclear homogenate (35.4% v/v) and cross-linked BSA (10% w/w) are shown in Figs. 1A and 1B, respectively,

20.0 I I I

15.0 1 A

T

I v)

El 0.0 0

-O Y i a

5.0 1 . . 0.10 0.20 0.30 0.40

B2 (Gauss)

5.0 I , I I I , , , , I , , ,

t 1 4.0 -

A 7 ’ V ) 3.0 - v

Q a

yo 2.0 -

1.0 .-

em 0

0.10 0.20 0.30 0.40 0.50 0.60 0.70 b B, (Gauss)

FIG. 1. Apparent magnetization transfer rate constant, Kapp ( s - ’ ) ~ as a function of off-resonance frequency (.,I0 kHz; ., 30 kHz; V, 45 kHz) and off-resonance irradiation field strength for (a) calf lens nuclear homogenate (35.4% v/v, temperature = 18°C) and (b) cross-linked BSA (10% w/w, temperature = 25°C). Kapp was obtained using Eqs. [4] and [5] (or equivalently Eq. [8] of ref. 14), which assumed complete saturation of the immobile spin bath.

334 Brooks et al.

and were derived from experiments employing differing off-resonance irradiation frequencies and B, RF power. Three off-resonance frequencies 10, 30, and 45 kHz and nine B, field strengths, ranging from 0.063 to 0.35 gauss, were used for nuclear lens homogenate, whereas the same three off-resonance frequencies, and 10 B, field strengths, ranging from 0.098 to 0.646 gauss were used for cross-linked BSA. For nuclear lens homogenate an average value for RA(SAn of 0.97 2 0.01 s-l (27 combinations of vOff and B,), whereas for cross-linked BSA it was 0.50 ? 0.04 s-' (30 combinations of vOff and B,). Off- resonance rotating frame spin-lattice relaxation effects on the water present in the protein preparations were observed to be minimal (2WA << RA, RAc). For both samples, the apparent magnetization transfer rate constant, Kapp, assuming Eq. [5a], was found to decrease monotonically with increasing off-resonance irradiation frequency, and to increase with off-resonance field strength. The apparent magnetization transfer rate constant, Kapp, for nuclear lens homogenate was found to increase by a factor of 450 when observed at a high B2 field strength (0.35 gauss) with irradiation 10 kHz off- resonance relative to Kapp evaluated at a low B2 field strength (0.06 gauss) and 45 kHz off-resonance. For cross- linked BSA, Kapp increased by a factor of 275 when measured at a B, field strength of 0.646 gauss (10 kHz off- resonance) relative to 0.098 gauss (45 kHz off-resonance).

It is evident from the Figs. 1A and 1B that when incomplete saturation of the restricted mobility spin bath occurs Kapp is dependent upon B, RF field power and off-resonance irradiation frequency. Thus a potential problem exists in those instances where it is necessary to compare apparent magnetization transfer rate constants obtained under different experimental conditions (off- resonance field strength and frequency) since complete saturation of the restricted mobility spin bath may not be achieved. Furthermore, polydispersity, which in turn results in differing macromolecular mobilities, and hence, degrees of saturation, may also be a contributing factor.

Experimental water 'H intensity ratio dispersion curves obtained using the off-resonance rotating frame spin-lattice relaxation experiment for nuclear lens homogenate (35.4% v/v) and cross-linked BSA (10% w / ~ ) are shown with the theoretically fit intensity ratio dispersion curves in Figs. 2 and 3, respectively. The theoretical fits and the derived magnetization transfer parameters (tabulated in Table 1) were based on Eq. [I] and [2], respectively, with RA equated to the average value of RA(sAT). Table 1 also includes the derived magnetization parameters for cross-linked BSA using a constrained value for RA (RA 5 RA(SAT1). These two approaches yielded similar, though not identical magnetization transfer parameters for cross linked BSA. The minimum number of spin baths needed to achieve adequate fits for intensity ratio dispersion curves obtained from eye lens tissue was three (16). This is consistent with other experimental results (18,191, which demonstrated the presence of both relatively mobile and solid-like macromolecular components (spin baths) in eye lens tissue. By contrast, the minimum number of spin baths required to adequately fit the cross-linked BSA magnetization trans-

1 .o

.o 0.8 0 tY -+

0.6 x u .-

0.4 a,

K cr - 0.2

0.0

FIG. 2. Experimental calf lens nuclear homogenate (35.4% v/v) 'H intensity ratio curves of water (RA = MzA/MoA) versus RF irradiation offset frequency (An) at field strengths of 0.0385 gauss (0), 0.0559 gauss (O), 0.0934 gauss (a), and 0.151 gauss (W). The solid line indicates the best fit theoretical curve using the three- proton spin bath formalism (Eq. [l]). The derived magnetization transfer parameters are tabulated in Table 1. Temperature = 18°C. Data taken from Kuwata et a/. (16).

1 .o

2 0.8 4 2 0 tY x (I)

a, K - 0.2

0.6 cr .- K 0.4

cr

0.0 0 10 20 30 40 50

Voff (kHz1

FIG. 3. Experimental cross-linked BSA (10% w/w) 'H intensity ratio curves of water (RA = MzA/MoA) versus RF irradiation offset frequency (A,) at field strengths of 0.0827 gauss (V), 0.0925 gauss (O), 0.1022 gauss (O), 0.1175 gauss (W), and 0.1416 gauss (+). The solid line indicates the calculated curve using the two- proton spin bath formalism (Eq. [2]). The derived magnetization transfer parameters are tabulated in Table 1. Temperature = 25°C.

fer data was found to be two, a value consistent with the assumption made by others (3-5, 13, 14).

At relatively high B, irradiation power and small off- resonance frequencies (e.g., 0.65 gauss and 10 kHz, respectively) the value of Kapp, as shown in Fig. 1B, approaches that obtained using the formalism described by Eq. [2], and suggests that the magnetization transfer rate constant determined for cross-linked BSA using this formalism is similar, if not identical, to that which would be obtained if complete saturation was achieved. The value of Kapp obtained for cross-linked BSA at a B, irradiation power of 0.1 G and 10 kHz off-resonance is in excellent agreement with a previous determination (14) made under the same irradiation conditions at 4.7 T and a slightly

Arialysis of Proton Magnetization Transfer 335

Table 1 Magnetization Transfer Parameters for Calf Lens Nuclear Homogenate and Cross-linked BSA Obtained using Simulated Annealing

Calf lens nuclear Cross-linked Cross-linked

(3-spin bath)a (2-spin bath)b (2-spin bath)c Parameter homogenate BSA BSA

9 RMSD

0.97 2 0.01 0.50 2 0.04 0.40 % 0.01 0.97 -t 0.10 0.63 2 0.20 0.05 2 0.80 0.05 t 0.02 9.69 2 0.03 69.4 2 0.2 69.4 ? 0.2

50 2 20 12.0 -+ 0.2 12.9 2 0.3 12.88 5 0.01 1.7 2 1.0 15 2 6 5.8 ? 1.7 4.22 t 0.02 8 2 7

0.11 5 0.04 0.61 -t 0.10 0.038 2 0.003 0.029 t 0.001

0.01 7 0.01 3 0.01 3

a Data from Kuwata eta/. (16). Temperature = 18°C. Eq. [l] was employed for the fit. RA was set equal to the average value of R A ~ s A ~ .

Temperature = 25°C. Eq. [2] was employed for the fit. RA was set equal to the average value of RACSAQ.

Temperature = 25°C. Eq. [2] was employed for the fit. RA was obtained from a simulated annealing constrained fit (see text for details).

lower temperature (14). A similar comparison for nuclear homogenate (three spin bath model) can not be made directly because more than one magnetization transfer rate constant characterizes the magnetization transfer between water and macromolecular species. However, it appears that the observed variation in Kapp with B, field strength and off-resonance irradiation frequency is pre- dominately due to incomplete saturation of the macromolecular spin baths.

As previously noted by Kuwata et al. (16), the diffi- culty in obtaining unique and physically significant values for magnetization transfer parameters increases as the number of parameters included in the fitting model increases. Therefore, it is important to obtain as many independent values of the parameters as possible through auxiliary experiments. In this work we used experimentally determined values of TZA, and through the use of Eq. [5b] obtained an average value for RA(sAT, for both BSA and lens nuclear homogenate. Although Eqs. [5al and [5b] are used throughout the literature for the determination of Kapp and RA(sAT), these equations are often used incorrectly, because the condition of complete saturation is rarely attained. Our rationale for using an average value for RA(sATI was its observed insensitivity to changes in off-resonance field strength and frequency. Although saturation was not achieved, the intrinsic longitudinal relaxation rate, RAISAT), was found to be essentially independent of off-resonance irradiation frequency and field strength when we assumed Eq. [5b] for saturat- ing FW power. By obtaining R A from an auxiliary NMR experiment the number of parameters which are vari- ables in the curve fitting procedure is decreased. A draw- back of determining R A from inversion recovery with saturation experiments is that R A is over estimated, i.e., the T 1 A used in curve fitting will be shorter than the intrinsic longitudinal relaxation time constant. For cross- linked BSA it was possible to evaluate the value of the

intrinsic longitudinal relaxation rate constant, &(FIT], by constraining the value of R A in the fitting procedure to a value less or equal to than that of the experimentally determined average value of RA(SAT1. The difference obtained for R A by inversion recovery (&(SAT] = 0.50 t 0.04 s-l) and that obtained from fitting (RA(FIT1 = 0.40 t 0.01 s-l) was small, and the differences, as noted above, obtained by curve fitting for the other parameters with these two different values of R A were also not greatly different [see Table 1). However, in the analysis of off- resonance nuclear lens homogenate data using the three- spin bath model, without an independent assessment of RA from RA(SAT1, we were unable to arrive at unique values for the values of the parameters which describe the magnetization transfer phenomenon. Although the actual value of R A cannot be determined from inversion recovery experiments, unless complete saturation is achieved, it does provide a constraint for the maximum value that R A may assume in the fitting procedure. A similar constraint can also be determined from inversion recovery experiments for the minimum value of Kapp in two component spin systems. In polydisperse systems the same constraints can not be reliably made because of the oc- currence of multiple magnetization transfer rate constants.

REFERENCES 1. S. H. Koenig, R. G. Bryant, K. Hallenga, G. S. Jacob, Magnetic

cross relaxation among protons in protein solution, Bio-

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

chemistry 17, 43484358-(1978). S. D. Wolff, R. S. Balaban, Magnetization transfer contrast (MTC) and tissue water proton relaxation in vivo, Magn. Res. Med. 10, 135-144 (1989). 1. Grad, R. G. Bryant, Nuclear magnetic cross-relaxation spectroscopy, ]. Magn. Reson. 90, 1-8 (1990). J. Grad, D. Mendelson, F. Hyder, R. G. Bryant, Direct mea- surement of longitudinal relaxation and magnetization transfer in heterogeneous systems, ]. Magn. Reson. 86,416- 419 (1990). R. G. Bryant, D. A. Mendelson, C. C. Lester, The magnetic field dependence of proton spin relaxation in tissues, Magn. Reson. Med. 21, 117-126 (1991). G. H. Caines, T. Schleich, J. M. Rydzewski, Incorporation of magnetization transfer into the formalism for rotating frame spin-lattice relaxation in the presence of an off-resonance irradiation field, ]. Magn. Reson. 95, 558-566 (1991). J. Grad, D. Mendelson, F. Hyder, R. G. Bryant, Applications of nuclear magnetic cross relaxation to tissues, Magn. Re- son. Med. 17, 452459 (1991). C. C. Lester, R. G. Bryant, Water proton studies nuclear magnetic relaxation in heterogeneous systems: hydrated ly- sozyme results, Magn. Reson. Med. 22, 143-153 (1991). X. Wu, Lineshape of magnetization transfer via cross relaxation, ]. Magn. Reson. 94, 186-190 (1991). R. S. Balaban, T. L. Ceckler, Magnetization transfer contrast in magnetic resonance imaging, Magn. Reson. Q. 8,116-137 (1992). T. L. Ceckler, S. D. Wolf, V. Yip, S. A. Simon, R. S . Balaban, Dynamic and chemical factors affects water proton relaxation by macromolecules, ], Magn. Reson. 98, 637-645 (1992). H. N. Yeung, S. D. Swanson, Transient decay of longitudinal magnetization in heterogeneous spin systems under selec- tive saturation, ]. Magn. Reson. 99, 466479 (1992).

336 Brooks et al.

13. S. H. Koenig, R. D. Brown, 111, R. Ugolini, A unified view of relaxation in protein solutions and tissue, including hydra- tion and magnetization transfer, Magn. Res. Med. 29, 77-83 (1993).

14. S. H. Koenig, R. D. Brown, 111, R. Ugolini, Magnetization transfer in cross linked bovine serum albumin solutions at 200 MHz: a model for tissue, Magn. Res. Med. 29, 311-316 (1993).

15. R. M. Henkelman, X. Huang, Q. S. Xiang, G. J. Stanisz, S. D. Swanson, M. J. Bronskill, Quantitative interpretation of magnetization transfer, Magn. Reson. Med. 29, 759-766 (1993).

16. K. Kuwata, D. Brooks, H. Yang, T. Schleich, Relaxation matrix formalism for rotating frame spin-lattice proton NMR relaxation in the presence of an off-resonance irradiation field, J. Magn. Reson., in press.

17. T. Schleich, G. H. Caines, J. M. Rydzewski, Off-resonance rotating frame spin-lattice relaxation: theory, and in vivo MRS and MRI applications, in “Biological Magnetic Reso- nance” L. J. Berliner, J. Reuben (Eds.), vol. 11, p. 55-134, Plenum Press, New York, NY, 1992.

18. C. F. Morgan, T. Schleich, G. H. Caines, P. N. Farnsworth, Elucidation of intermediate (mobile) and slow (solid-like) protein motions in bovine lens homogenates by carbon-13 NMR spectroscopy, Biochemistry 28, 5065-5074 (1989).

19. J. M. Rydzewski, S. X. Wang, A. Stevens, C. Serdahl, T. Schleich, C-13 NMR studies of protein motional dynamics in bovine, human, rat, and chicken ocular lenses, Exp. Eye Res. 56, 305-315 (1993).

20. H. A. Sober, R. A. Harte (Eds.), in “Handbook of Biochem- istry’’ 2nd ed., p. C71, Chemical Rubber Co., Cleveland, OH, 1970.

21. G. C. Levy, I. R. Peat, The experimental approaches to ac- curate carbon-13 spin-lattice relaxation measurements, J. Magn. Reson. 18, 500-521 (1975).

22. J. A. Pople, W. G. Schneider, H. J. Bernstein, in “High-Reso- lution Nuclear Magnetic Resonance,” p. 85, McGraw-Hill, New York, NY, 1959.

23. Y. Watanabe, T. Kubozoe, T. Mukoyama, Error estimation in fitting x-ray spectrum by non linear least squares method, Nucl. Instr. Meth. Phys. Res. B44, 35-39 (1989).

Documents

Determination of proton magnetization transfer rate constants in heterogeneous biological systems