Suppression of Static Magnetic Field in Diffusion Measurements of Heterogeneous Materials

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1 PIERS ONLINE, VOL. 5, NO. 1, Suppression of Static Magnetic Field in Diffusion Measurements of Heterogeneous Materials Eva Gescheidtova 1 and Karel Bartusek 2 1 Faculty of Electrical Engineering and Communication, Brno University of Technology Kolejni 2906/4, Brno , Czech Republic 2 Institute of Scientific Instruments, Academy of Sciences of the Czech Republic v.v.i Kralovopolska 147, Brno , Czech Republic Abstract The paper describes a magnetic resonance MR) method for establishing the diffusion coefficients in heterogeneous materials. The pulsed field gradient stimulated-echo methods have a reduced coupling between the applied magnetic field gradient and a constant internal magnetic field gradient caused by different susceptibilities throughout the sample. When studying systems where it is necessary to keep the duration of the pulse sequence at a minimum or to study diffusion as a function of observation time, the spin-echo method should be chosen. The basic idea is to acquire the spin echo amplitude with pulsed field gradient of opposite signs and to subtract in a suitable way the NMR signals measured. The measuring method and the digital signal processing enable eliminating the effect of static magnetic field on the accuracy of measuring. The method proposed can be used to measure diffusion-weighted images of liquids found in porous materials, and in the development of new MR tomography measuring methods in the Institute of Scientific Instruments of the Academy of Sciences of the Czech Republic, v.v.i. 1. INTRODUCTION Imaging based on nuclear magnetic resonance is one of important methods for the study of tissues and molecules. The knowledge of the diffusion and other motions of the nuclei of different substances in the porous material under examination can bring new diagnostic results. The MR method enables measuring the slow motion of nuclei and molecules. In heterogeneous systems, where relaxation time T 2 is shorter than T 1 and where diffusion coefficient D is small, the use of the Pulsed Field Gradient Spin Echo PFG-SE) sequence may be advantageous [1]. More sophisticated methods that eliminate the effect of magnetic susceptibility of biological materials have been introduced [2 7]. They are based on producing a stimulated echo, for which the diffusion time can be extended even in materials with a short relaxation time T 2. These techniques eliminate the effect of the so-called cross terms related to the space-dependent gradient of the basic field of the MR system. The proposed method employs the PFG-SE pulsed sequence for three magnitudes of diffusion gradient and enables calculating the diffusion constant of the material being measured. The above procedure for calculating the b-factor makes it possible to eliminate the effect of the heterogeneity of static magnetic field, which is due to the magnetic susceptibility of the material being measured. 2. METHOD The principle of measuring in the current spin-echo pulse sequence consists in applying two diffusion gradients of length [8]. The first of them is located between two RF pulses and serves the spins being brought out of phase in a defined way while the other gradient is applied after the 180 pulse and serves to bring the spins into phase again, Fig. 1. For the whole period of measuring, a static gradient magnetic field G 0 is acting on the spins, which is due to the magnetic susceptibility of the material being measured. The effect of this field on the precision of measuring the diffusion coefficient should be minimized. If due to the diffusion the spins move randomly, the MR signal attenuation M can be described by a simple exponential equation M = M 0 e bd, 1) where M 0 is the signal intensity without diffusion e.g., measured by a sequence without both the diffusion gradients and the static gradient magnetic field G 0 ). The constant b the so-called

2 PIERS ONLINE, VOL. 5, NO. 1, B 1 90 o 180 o Echo G D 1 τ +1 2τ t G 0 2 t Figure 1: PFG-SE sequence. b-factor) gives the pulse sequence sensitivity to diffusion, and is given by the integral b = γ 2 2τ 0 t 0 Gt )dt 2 dt. 2) Equation 2) is used to calculate the b-factor of pulse sequences of any effective gradient waveform. For the proposed technique [1] it can be derived that the drop in spin echo magnitude as expressed by the b-factor will be b = γ 2 [ a 1 G 2 D a 2 G D G 0 + a 3 G 2 0], 3) where a 1 = γ 2 2 ) [ ], a 2 = γ ) ) τ 2, a 3 = γ τ 3. The effect of term 3) can be eliminated by measuring in the presence of gradient G 0 and in the presence of both gradients, G 0 and G D. After mathematical re-arrangement we obtain b = γ 2 { 2 3 ) G 2 D [ ) ) τ 2 ] G D G 0 }. 4) For the ratio of the magnitudes of spin echoes measured with and without the diffusion gradient, G D = 0) M GD and M 0 ), it holds ) MGD ln = γ 2 [ a 1 G 2 ] D a 2 G D G 0 D. 5) M 0 By measuring the spin echo amplitudes M GD, M GD, and M 0 and calculating according to relation 5) it is possible to calculate from three experiments the diffusion coefficients, according to the relation ) M GD M GD ln M 2 G D=0 D = 2γ 2 2 ). 6) G 2 D 3 The accuracy of the measurement of diffusion coefficients depends on the inaccuracy of the diffusion gradient magnitudes, timing and determination of the spin echo magnitude. The timing error can be neglected in current tomography systems. The accuracy of determining the spin-echo magnitude greatly depends on the signal-to-noise ratio and on the drop in echo magnitude for the diffusion gradient used. The diffusion coefficient in heterogeneous materials calculated by relation 4) carries an error that is due to the cross term. In this case, the relative error due to the error in measuring the amplitude of NMR signal is given by the relation D = ln 4 M M GD M GD M 2 G D=0 ) 7) The error M depends on the magnitude of signal-to-noise ratio in MR signal or in MR image.

3 PIERS ONLINE, VOL. 5, NO. 1, EXPERIMENTAL VERIFICATION Some experimental tests were made by measuring the diffusion coefficient of water both inside and outside of selected samples of porous materials of different properties. The change in the diffusion of water in porous materials was studied. The measuring method was experimentally tested on the MR tomograph 200 MHz/120 mm 4.7 T) in the ISI ASCR in Brno. The 6-interval sequence PFG- SE), shown in Fig. 1, was used in the measurement. The error measured for the determination of spin echo magnitude for G D = 0 and G 0 = 0 is M = 1.8%. When greater amplitudes of the two gradients are applied, the magnitude of spin echoes decreases to as little as one third, with the magnitude of noise remaining the same and with the error M increasing. The non-suppressed effect of background field gradient will lead to a greater error of diffusion coefficient measurement than the error of spin echo determination is. For standard diffusion measurements with the spin or the stimulated echo one usually acquires several echo amplitudes as a function of the b-factor. This enables performing a fit or even a deconvolution of the data, and thus increasing the certainty of the results and gaining the distribution of diffusion coefficients. We believe that the method of three measurements provides for this kind of processing. The samples were immersed in a beaker with deionized water and placed, together with the beaker, in the working space of tomograph. The diffusion coefficients were measured for the diffusion gradients G D = 0 and ±161 mt/m by the method of three measurement. Transverse slices, 3 mm thick, were measured in all samples. The images detected were pixels. For each sample, the diffusion was determined in several different areas and the resultant value was determined as the arithmetic mean of these values. The first sample to be measured was a mm cylinder of a material used in industry for mechanical filters with diameter 0.5 mm pores. Fig. 2 gives the MR image of the sample, weighted by spin density and diffusion, with areas of diffusion evaluation marked out. Figure 2: Measurement results for a sample of porous material, with 0.5 mm pores. a) photograph of sample, b) NMR image weighted by spin density, and c) MR image weighted by diffusion. temperature of 20.5 C is D = m 2 s 1. The diffusion inside of the sample is D = m 2 s 1 [9]. The difference between these diffusions amounts to m 2 s 1 and is proportional to the size of pores in the sample being measured. The measurement error, determined from the magnitude of standard noise deviation in the area of diffusion evaluation is m 2 s 1. The second sample to be measured was a mm cylinder of a material used in industry for mechanical filters with diameter 3.5 mm pores. Fig. 3 gives the photograph of the sample, its image weighted by spin density and its diffusion image with areas of measurement marked out. temperature of 20.5 C is D = m 1 s 1. The diffusion inside of the sample is D = m 2 s 1. The change in the diffusion of water due to porous material is m 2 s 1. The measurement error is m 2 s 1. The third sample to be measured was a white porous material used in catalysts of mm in dimensions. Fig. 4 gives the photograph of the sample, the MR image of water weighted by spin density, and the diffusion MR image with areas of measurement marked out.

4 PIERS ONLINE, VOL. 5, NO. 1, Figure 3: Measurement results for a sample of porous material with 3.5 mm pores. a) photograph of sample, b) MR image weighted by spin density, and c) MR image weighted by diffusion. Figure 4: Measurement results for a sample of porous material with 1.5 mm pores. a) photograph of the sample, b) MR image weighted by spin density, and c) MR image weighted by diffusion. temperature of 20.5 C is D = m 2 s 1. The diffusion inside of the sample is D = m 2 s 1. The change in the diffusion of water in porous material is m 2 s 1 and the measurement error is m 2 s 1. When measuring the diffusion map of thin slices, the noise in the image is pronounced and higher than in the measurement of the whole volume on an NMR spectrometer. Time averaging is often used to increase the signal-to-noise ratio. Gradient pulses in imaging sequences in all directions readout, phase and slice selection) can affect the accuracy of diffusion measurement. The methods proposed reduce this effect. These methods can be applied in measurements on current MR tomography systems, which are in most cases furnished with the standard spin-echo method for diffusion measurement. 4. CONCLUSIONS In the paper, the measurement of the diffusion coefficients of water in heterogeneous systems is described. It is characterized by a special method of measuring, digital image processing, and calculation of the diffusion coefficients. An advantage of the three measurement arrangement is the elimination of both the cross terms G D G 0 and the term with G 2 0. The diffusion constant being measured depends on the time parameters of measurement, stability of the RF channel for nucleus excitation and MR signal reception, accuracy of the determination of spin echo magnitude, and on the magnitude of the diffusion gradient. The technique will be made use of in the measurement of diffusion-weighted images of solids or gas found in porous materials, and in the development of new MR tomography measuring methods in the Institute of Scientific Instruments of the Academy of Sciences of the Czech Republic, v.v.i.

5 PIERS ONLINE, VOL. 5, NO. 1, ACKNOWLEDGMENT This work was supported within the framework of project No. 102/07/0389 of the Grant Agency of the Czech Republic. REFERENCES 1. Stejskal, E. O. and J. E. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, J. Chem. Phys., No. 42, 288, Tanner, J. E., Use of the stimulated echo in NMR diffusion studies, J. Chem. Phys., Vol. 52, , Cotts, R. M., M. J. R. Hoch, T. Sun, and J. T. Markert, Pulsed field gradient stimulated echo methods for improved NMR diffusion measurements in heterogeneous systems, J. Magn. Reson., Vol. 83, , Sorland, H. G., B. Hafskjold, and O. Herstad, A stimulated-echo method for diffusion measurements in heterogeneous media using pulsed field gradients, J. Magn. Reson., Vol. 124, , Sun, P. Z., J. G. Seland, and D. Coryb, Background gradient suppression in pulsed gradient stimulated echo measurements, J. Magn. Reson., Vol. 161, , Galvosas, P., F. Stallmach, and J. Kärger, Background gradient suppression in stimulated echo NMR diffusion studies using magnetic pulsed field gradient ratio, J. Magn. Reson., Vol. 166, , Sorland, H. G., D. Aksnes, and L. Gjerdaker, A pulsed field gradient spin-echo method for diffusion measurements in the presence of internal gradients, J. Magn. Reson., Vol. 137, , Bartusek, K. and E. Gescheidtova, MRI method of diffusion measurement in heterogeneous materials, Measurement Science and Technology, Vol. 19, 1 8, Holz, M., S. R. Heil, and A. Sacco, Temperature-dependent self-diffusion coefficients of water and six selected molecular liquids for calibration in accurate 1 H NMR PFG measurements, Phys. Chem. Chem. Phys., Vol. 2, , 2000.