High-Speed Spiral-Scan Echo Planar

Transcription

1 2 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. MI-5, NO. 1, MARCH 1986 High-Speed Spiral-Scan Echo Planar NMR Imaging-I C. B. AHN, STUDENT MEMBER. IEEE, J. H. KIM, AND Z. H. CHO, FELLOW, IEEE Abstract-An improved echo planar high-speed imaging technique using spiral scan is presented and experimental advantages are discussed. This proposed spiral-scan echo planar imaging (SEPI) technique employs two linearly increasing sinusoidal gradient fields, which results in a spiral trajectory in the spatial frequency domain (k-domain) that covers the entire frequency domain uniformly. The advantages of the method are: 1) circularly symmetric T2 weighting, resulting in a circularly symmetric point spread function in the image domain; 2) elimination of discontinuities in gradient waveforms which in turn will reduce initial transient as well as steady-state distortions; and 3) effective rapid spiral-scan from dc to high frequency in a continuous fashion, which ensures multiple pulsing with interlacing for further resolution improvement without T, decay image degradation. Some preliminary experimental results will be presented and further possible improvements suggested. I. INTRODUCTION IN the development of the NMR imaging technique [l]- [3], one of the inherent drawbacks is in the imaging speed or time necessary for the imaging data collection, for instance, compared to X-ray CT. This speed limitation is largely due to the spin-lattice relaxation time (T1) which requires a certain period of relaxation before one spin system can be reexcited. This drawback usually limits the NMR imaging to a time frame of a few minutes. More recently, very high-speed imaging in the time frame of a few tens of milliseconds using echo planar technique was developed by Mansfield and later a number of its variations were suggested [4]-[91. All of these techniques basically utilize multiple echos by fast gradient alternation. Although the methods appear attractive, they suffer two main technical drawbacks, namely, the difficulty of realizing large high-speed gradient fields and the resolution limit imposed by finite TX decay. In addition, Mansfield's original echo planar technique suffers nonisotropic resolution in the image domain due to the different weighting of T, decay in directions x and y. In this paper, a modified echo planar technique using the spiral-scan technique known as sprial-scan echo planar imaging (SEPI) is proposed, and its experimental results are presented [8]. It utilizes a pair of linearly increasing sinusoidal gradient fields so that in the frequency space, Manuscript received January 2 1, 1985; revised January 13, C. B. Ahn and J. H. Kim are with the Department of Electrical Science, Korea Advanced Institute of Science, Seoul, Korea. Z. H. Cho is with the Department of Radiological Sciences, University of California, Irvine, CA 92717, on leave from the Department of Electrical Science, Korea Advanced Institute of Science, Seoul, Korea. IEEE Log Number the entire domain is uniformly covered by a spiral shape after completion of the scan. This ensures both rapid scanning as well as uniform coverage of the entire frequency space of interest. As will be detailed, this spiral scan also simplifies the sampling of the data in radial directions once the scan is completed. These practical advantages further facilitate accurate sampling of data in both radial and angular directions. The latter is related to the number of views required in conventional projection reconstruction. Although the data collected by the SEPI technique can be processed entirely by the Fourier method with interpolation, projection reconstruction has been found to be more convenient due to the fact that the radial data are readily available directly from the scanned frequency domain data through l-d FFT. II. BASkC TECHNIQUE OF THE SPIRAL SCAN ECHO PLANAR IMAGING (SEPI) The well-known 3-D Fourier NMR imaging algorithm with three time-varying gradient fields is given by S(t) = p(x, y, z) exp iy i {xg1(t') + yg,(t') + zg_(t')} dt' exp (-t) dx dy dz where S(t) is FID signal, p(x, y, z) is spin density distribution, and G (t), G,.(t), and G (t) are the time-varying gradient fields of the x, y, and z coordinates, respectively. The exponential term e-' 1' is the T2 decay term that often appears as a limiting factor in high-resolution imaging as will be discussed later. General 3-D imaging can easily be reduced to 2-D forms for analysis and the T2 decay term can also be neglected for convenience. The FID signal obtained for a selected 2-D slice at z = z, is given by S(t) = p(x, y, zo) exp [i {kr(t) x + ky.(t) y}i] dx dy - where S(kr!, kv) /86/ $ IEEE.k-C(t) = ey G,G(t') dt', 0 (1) (2)

2 AHN et al.: HIGH-SPEED SPIRAL-SCAN ECHO PLANAR IMAGES and t = k, (t) 'y 0 Gl. (t') dt'. From (2) and (3), it can easily be shown that p(x, y) and S(k_, k-.) are the Fourier transform pair. In original echo planar imaging, a small but constant gradient is applied in the y direction while an alternating rectangular gradient field is concurrently applied along the x direction. The equal high-frequency cutoffs for both x and y directions, provided sampling conditions are properly met, are given by k1(tc) kj.(t,.). (4) From (4) and Mansfield's original paper [41, the following relation should also be satisfied: (3) T, = nt, (5) where n corresponds to the number of echos to be generated by the alternating gradient pulses in the x direction. One of the main drawbacks of this technique is that the alternating gradient to be applied is a series of rectangular pulses which are often difficult to generate when the required gradient power and frequency are high. Another drawback of the method is that the y directional sampling in the k-domain is determined by the number of echos that can be generated by the number of alternating gradient pulses, which in turn is limited by the intrinsic NMR property, i.e., 712 decay. In short, the original echo planar technique has several drawbacks such as practical realizability and poor spatial resolution, especially in y directional resolution (see the point spread functions and discussions on the original echo planar technique versus spiral-scan echo planar technique). To alleviate these drawbacks, a new approach in gradient pulsing using the spiral formula is implemented. In this scheme, the entire frequency domain is covered uniformly by a form of a sprial scan and thereby a circularly symmetric response function is obtained. Let us consider a continuous but circularly symmetric frequency or k-domain scan and the necessary conditions to achieve these goals. Clearly, the simultaneous k-domain sinusoidal pulse pair will result in a set of circularly symmetric concentric circles with different radii in the k-domain, i.e., kt (t) = an ji(t) cos it k,,(t) -yq1q(t) sin it (6) where qj (t) is the discrete amplitude as a function of time and t is a constant yet to be determined. From (6), by slight modification, one obtains continuous, rather than discrete, circles, i.e., spiral pattern which would be easy to implement in practice kx(t) = ayt cos it k, (t) = -yrt sin Vt. (7) Simultaneous application of these two result in continuously increasing circles, i.e., a spiral scan in the k-do- FID Iv fcan- --. l - t I I~~ ~ Fig. 1. The pulse sequence for the SEPI technique. Fig. 2. An illustration of the sampling interval of the SEPI in radial and angular directions and the conjugate symimietry filling. main. From (3) and (7), the desired time domain gradient pulse forms which will generate spiral scan in the k-domain can be obtained as suggested by Ljunggren [81, d G,(t) -- [k,(t)] = - cos it - nit sin it 'y dt I d G,(t) = -- [kjt)] = -q sin it + rqt cos (t. (8) -y dt These equations represent sinusoidal signals with linearly increasing amplitude as a function of time (see Fig. 1). As will be seen in the following, the resulting spiral scan will represent close approximation of the concentric circles that cover the entire frequency range of interest, thereby providing the circularly symmetric resolution (see Figs. 2 and 5). To define several necessary conditions, let us derive the t *0 TD 3

3 4 polar coordinate k-domain functions, i.e., from (7), kr(t) = k7(t) + k,(t) kr(t) and k0(t) IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. MI-5, NO. 1, MARCH 1986 ko (t) = tan - = t. (9) Note here that both k, (t) and k0 (t) are explicit functions of time. As is shown, both radial and angular increments, i.e., Ak(t) and Ak0(t) are related to the sampling interval AT in FID domain as Akr = yqnnat Ak/ = tat (10) where No is the number of samples in one complete rotation. The required Nyquist sampling rate can also be defined as 27r Ak = 2NrAr 2wr Ak6= - (11) No where Nr and Ar denote the number of rotations in the k- domain and image resolution in the spatial domain, respectively. From (10) and (11), the constants -q and t can be determined, i.e., 7r yn6n,oa TAr =No AT (12) The total required image data acquisition time TD, which is related to T2 relaxation time, can be defined as - TD = NrN6AT T2. (13) The maximum gradient amplitude required can also be determined by Gmax =?0TD. (14) From (13), maximum radial frequency kr,l, derived and is given by can also be krmiiax = -y1jtd = NrAkr. (15) The kr,u,ax is also the determining factor for the resolution that can be attained with the method. As is seen, resolution is explicitly related to the number of circles or rotations of the sprial scan with a given excitation, i.e., dependent on TD or Nr. Even with limited TD due to T2 at each scan, Nr and krmax can be increased by repeated scanning so that the unfilled radial frequency space can be filled. This can easily be satisfied with what is called the "interlacing technique." An additional advantage noted in conjunction with Nr is the conjugate symmetric property of the spiral scan as shown in Fig. 2, e.g., point 2' in the figure is a complex conjugate of point 2, while 3' t=o Fdternqg Bockpojcton Im e (p -Filter) tti Fig. 3. A schematic illustration of the reconstruction procedure for the spiral-scan sampled data (solid line), while the dotted line denotes the ideal reconstruction process. can be obtained from point 3, and so on. Corresponding FID data are shown in Fig. 3 together with the overall image reconstruction process. A typical example of spiral scan EPI in an attempt to reconstruct the image of an object of diameter 30 cm in a matrix size of 128 x 128 would require N, = 32 (considering the conjugate filling discussed above), t/2 = 300 Hz with sampling time interval of AT = 8 Its, and the total image data acquisition time of 100 ms. Image reconstruction of SEPI involves formation of the projection data via Fourier transform of the k-domain data in the radial direction as shown in Fig. 2. The k-domain data Q corresponding to the lth view can be written as Q= S[1AT + (Nr 1) N0AT], S[IAT + (Nr -2) NoAT], *** S[lA T], SL la T + AT j S IA T + (N. - 1) N0A T + -o A T {SN,- + P/N,, SN, -2 + I/No' S/INo' S I/2 + I'No, SN, - 1/2 + IlNo } (16) By use of the projection theorem, it can be shown that the Fourier transform of (16) will be the form of projection data, Pl(k), which is given by N- I (U S- ~ m+inh W2Nr In =0 P(U) = Z-(,n + I/No)u Nrv ~~~~~(vin'#+ I1/2 + IINO)U + mn' =0 Sm''+ 1/2 + INo W2NI N, - I + 1/NE S -(m + l/no)u + n + IINO W2N ( 2 -Nr- + 0Sm'+ -tn '= 0 W 1/2 + lno W2N (17)

4 AHN t (a/.: HIGH-SPEED SPIRAL-SCAN ECHO PLANAR IMAGES Note that the center point of FID (dc point), which will be filtered out by the convolution process, is disregarded in (17). Since the projection data are real, (17) can also be written as P'(u) = Re fp'(u)] N, - I = Re E S,,, + IN, W' 4 t" + I/NO)u N, I v* ~~~~~i- ( In ' + I1/2 + 11NO)u + Ei S ' IlNo W2N, / /I' = 0 (18) Equation (18) can be further simplified for a more convenient form of FFT operation as where -2N,- - I Pl(u) = Re (I/No)u) E =RjW2N' 4N, -- 2N, - I = Re W-(I/No)ti) E ewnn =O U 0, = 1, q(2m) = Sin + I/No q(m) WN, q(m) WzNS 1 I (19) q(2m + 1) = Si*n+1/2+ 1NO' for m = 0, 1, 2, Nr- 1. (20) Equation (19) can be easily computed by 4 Nr point FFT with 2N,. point zero appending. The computational flow schematic diagram is given in Fig. 3. The final image is reconstructed using convolution backprojection for the projection data set obtained according to (16)-(20). III. EXPERIMENTAL RESULTS AND DiscuSSIONS To test the viability of the proposed spiral-scan echo planar imaging technique, and to make comparisons to other similiar techniques, a number of experiments were performed and results are shown in the following. For the comparative study, modified echo planar imaging (MEPI) whose increments in the k,. direction as a stepwise rather than a zigzag trajectory of original EPI is considered. The pulse sequence and corresponding k-domain trajectory of modified EPI are shown in Fig. 4 and, respectively. In the experiment, the following two aspects have been studied to confirm the expected advantages of the spiralscan echo planar technique in comparison to the others, e.g., modified EPI. A. Circularly Symmetric k-domain Sampling of SEPI in Comparison to the Modified EPI Under the Presence of T2 Decay As has been mentioned, the proposed method has several unique advantages over other methods, among others, circularly symmetric sampling property under T2 decay in k-domain which will result in circularly symmetric image domain resolution regardless of the number of ro- RF G1 90, 80 Gy n n n n ;. FID t o I,I I-~~- 1 1 rzizizi ri I z, m 4...-i I _ Ḵ I 4.0 fz~~~~~~~~~~~~~~~~2 Ky ZI~~~~~~~~~~~~~~~ _IZ., * * * *.,... Fig. 4. The pulse sequence and corresponding k-plane representation of the sampled data obtained by the modified EPI technique. The pulse sequence for modified EPI. The ideal trajectory for modified EPI in the k-domain. tations of the spiral scans. This condition is not usually met in conventional EPI techniques [for both original and modified EPI techniques, noticeable asymmetric point spread functions are common, for example, see Fig. 5]. This is mainly due to the fact that the conventional EPI techniques are unable to scan in both the x and y directions with equal T2 decay. In spiral scan EPI, however, a perfectly symmetric point spread function was obtained as it is seen from Fig. 5. It should be also worth noting that the number of spiral rotations N0 is the key parameter which ultimately affects the overall system resolution in SEPI technique. Increase of N, by "interlacing" scanning is, therefore, suggested for the high-resolution SEPI technique and in fact SEPI has been found to be most optimal for the adoption of the interlacing technique. B. Limitations Arising from the Frequency Response of Gradient Coils Unlike the spectroscopic NMR system, whole body NMR imaging system requires large size gradient coils to TD Lz -A 5

5 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. MI-5, NO. 1, MARCH Modeling of Gradient System L RD: Dummy Lood RL: Resistance of Grodient Coil L: Inductance of Gradient Coil Fig. 6. A simplified equivalent circuit for the gradient generating system. Fig. 5. Point spread functions of SEPI and modified EPI are shown in log scale, where the top is 0 db and the bottom is -48 db. The data acquisition time is assumed to be three times as long as its T2 relaxation time. The PSF of SEPI and the PSF of modified EPI. accommodate the entire human body. Together with the large current and power requirements for large gradient fields, overall frequency response of the coil severely limits the performance of the EPI imagings. This was indeed the main obstacle of the original EPI, especially in generating and applying such a high field strength (- I G/ cm) rectangular alternating gradient pulse. Let us investigate the electrical properties of gradient coils such as the frequency response or the transfer function. In Fig. 6, the equivalent circuit of the gradient coil system, which has system transfer function of Y(iw) = (RD + RL + icol)'- is shown. This simple circuit has lowpass filter characteristics and severely degrades some input gradient pulses such as the rectangular alternating gradient shown in Fig. 4. These low-pass filter characteristics affect the imaging performance of both the conventional EPI as well as the SEPI technique. In the latter, however, low-pass filtering affects only the phase delay and can easily be corrected. In the conventional EPI imaging, however, effects are detrimental and seriously degrade high-frequency characteristics, creating severe aliasing artifacts and resolution degradation. Computer simulated images of this low-pass filtering effect are shown in Fig. 7. In Fig 7, an image obtained by the SEPI is shown, while (c) is the one obtained by the conventional modified EPI. Both images are obtained with a coil which has relatively high-frequency response, i.e., 1.6 khz bandwidth. As is seen, in the conventional EPI, (c) (e) (d) Fig. 7. Finite bandwidth effects on the reconstructed images. The frequency of the oscillating gradients is 625 Hz. Original images. Reconstructed image using the SEPI technique with a 1.6 khz bandwidth system. (c) Reconstructed image using the modified EPI technique with a 1.6 khz bandwidth system. (d) Reconstructed image using the SEPI technique with a 160 Hz bandwidth system. (e) Reconstructed image using the modified EPI technique with a 160 Hz bandwidth system. the aliasing effect severely degrades images. In the spiral however, only slight image rotation is observed, which can eventually be corrected later. In computer simulation, further bandwidth reduction (160 Hz) resulted in severe image degradation for the modified EPI', while image degradation in SEPI is hardly noticeable [see Fig. 7(d) and (e)]. In Fig. 8 an experimentally obtained image by the proposed SEPI technique is shown. For this particular exscan, 'rhese artifacts can be reduced if one inserts some idle time interval after changing steps in the ki direction, or if one uses sinusoidally oscillating gradients.

6 AHN et al.: HIGH-SPEED SPIRAL-SCAN ECHO PLANAR IMAGES 7 Phantom Diagram 50- o0ni: mm Through experiments and computer simulations, one finds that there are three important advantages in the SEPI technique as compared to other existing fast imaging algorithms, for example, the original EPI or modified EPI. These advantages are identified as follows. 1) The inherent T2 effect appears as a circularly symmetric blurring rather than as the one-dimensional blur observed in original or modified EPI scan imaging. 2) Negligible transient effect, i.e., transient effect appears as an equiangular delay in the FID domain, which results in angular rotation in the reconstructed image by the same amount without image distortion. 3) Finally, high resolution imaging is possible by use of the interlacing technique which can easily be implemented in the case of SEPI due to its circularly symmetric property in the data acquisition. An experimentally obtained image of 64 x 64 pixel size shown in Fig. 8 strongly supports the above notions. Fig. 8. An experimentally obtained image with KAIS 1.5 kg resistive magnet system. Configuration of the used phantom. Experimentally obtained image of 64 x 64 pixels. The total image data acquisition time was 64 ms. periment, a small object (4 cm diameter phantom) is chosen to satisfy the requirements of the high-gradient fields and homogeneity of the main magnet. As one can see, in comparison to our previuos experimental results, there has been substantial improvements in both image resolution and uniformity [7]. IV. CONCLUSIONS The proposed SEPI method has been experimentally tested as a fast NMR imaging modality. Performance of the SEPI technique is examined from the viewpoint of physical implementation. REFERENCES [I] P. C. Lauterbur, "Image formation by induced local interactions: Examples employing nuclear magnetic resonance," Nature, vol. 242, pp , [2] P. Mansfield and Grannell, "NMR 'diffraction' in solids," J. Phys. C., vol. 6, pp. L422-L426, [3] Z. H. Cho, H. S. Kim, H. B. Song, and J. Cumming, "Fourier transform nuclear magnetic resonance tomographic imaging," Proc. IEEE, vol. 70, no. 10, pp , [4] P. Mansfield, "Multi-planar image formation using NMR spin echoes," J. Phys. C., vol. 10, pp , [5] M. M. Tropper, "Image reconstruction for the NMR echo-planar technique, and for a proposed adaptation to allow continuous data acquisition," J. Magn. Reson., vol. 42, pp , [6] P. Mansfield, R. Rzedzian, M. Doyle, B. Chapman, D. Guilfoyle, R. Coupland, A. Chrispin, and P. Small, Abstracts of papers presented at the Third Annual Meeting of the Society of Magnetic Resonance in Medicine, Magn. Reson. Med., vol. 1, pp. 197, [7] H. Y. Kwon, "A study on fast NMR imaging modality EPIKWE," M.S. thesis, Korean Advanced Inst. Sci., Seoul, Korea, [8] S. Ljunggren, "A simple graphical representation of Fourier-based imaging method," J. Magn. Reson., vol. 54, pp , [9] C. B. Ahn, C. Y. Rew, J. H. Kim, 0. Nalcioglu, and Z. H. Cho, Abstracts of papers presented at the Fourth Annual Meeting of the Society of Magnetic Resonance in Medicine, Magn. Reson. Med., vol. 2, pp , 1985.