True threedimensional nuclear magnetic resonance imaging by Fourier reconstructionzeugmatographyChingMing Lai Citation: Journal of Applied Physics 52, 1141 (1981); doi: 10.1063/1.329728 View online: http://dx.doi.org/10.1063/1.329728 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/52/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ultrafast-based projection-reconstruction three-dimensional nuclear magnetic resonance spectroscopy J. Chem. Phys. 127, 034507 (2007); 10.1063/1.2748768 Magnetic resonance imaging based digitally reconstructed radiographs, virtual simulation, and three-dimensionaltreatment planning for brain neoplasms Med. Phys. 25, 1928 (1998); 10.1118/1.598382 Threedimensional imaging with a nuclear magnetic resonance force microscope J. Appl. Phys. 79, 1881 (1996); 10.1063/1.361089 Surface plasmon resonance microscopy: Reconstructing a threedimensional image Appl. Phys. Lett. 64, 1330 (1994); 10.1063/1.111924 Threedimensional audio image reconstruction J. Acoust. Soc. Am. 78, S60 (1985); 10.1121/1.2022911

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True three-dimensional nuclear magnetic resonance imaging by Fourier reconstruction zeugmatography

Ching-Ming Lai a)

Department o/Chemistry, Slate University 0/ New York at Stony Brook, Stony Brook, New York 11794

(Received 24 March 1980; accepted for publication 2 December 1980)

A new method of obtaining three-dimensional nuclear magnetic resonance images is introduced. This imaging method, Fourier reconstruction zeugmatography, reconstructs true three­dimensional images directly from transient NMR signals by a Fourier transform technique. A theoretical analysis and experimental results are presented. The method is simple in operation and provides a safe, efficient, and versatile alternative to other methods for medical diagnostic imaging.

PACS numbers: 07.55. + x, 76.90. + d, 87.60. - f

I. INTRODUCTION

The possibility of reconstructing two-dimensional (2D) images from ID projections has been realized for decades. 1-3

In the past decade it became a common technique in medical imaging, particularly in x-ray computerized tomography (CT). Nuclear magnetic resonance (NMR) images can be re­constructed from projections as well. 4 The NMR zeugmato­graphic projection, a one-dimensional spatially-resolved NMR spectrum, is generated by imposing a small linear field gradient on the main magnetic field. It is a plane integral over a 3D region defined by the active volume of the radio frequency transmitter-receiver coil system. Complete iso­tropic 3D images reconstructed from these ID projections have been demonstrated by a two-stage convolution tech­nique. 5 Such true 3D imaging is more efficient than ordinary tomographic imaging from projections which are line inte­grals over a 2D region, from which only an image of a single slice is reconstructed.

The NMR zeugmatographic projection is actually ob­tained from the Fourier transform (Ff) of a transient NMR signal, the free induction decay (FID) following a single rf excitation pulse. The availability of FID signals suggests a simpler way of obtaining 3D images directly from the time domain FID information by an Ff technique.6 This new approach to reconstructing true 3D NMR images will be called Fourier reconstruction zeugmatography (FRZ). The principle of FRZ will be described in Sec. II, and two exam­ples of FRZ imaging will be given. Comparison with similar techniques will also be discussed.

II. THE PRINCIPLE OF FOURIER RECONSTRUCTION ZEUGMATOGRAPHY

It is well known that given a 2D function the line inte­grals normal to a line and the values along this line in its 2D Fourier transform space are 1 D FT pair. 2 Extension of this concept to 3D function is not obvious. Thus, in the first step, the relationship between the plane integrals of a 3D function and the 3D Fourier transform of this function will be ana­lyzed. 7 Once this relation is established, the back-projection

"'Present address: Analogic Corporation, Audubon Road, Wakefield, Mas­sachusetts 01880.

equation for 3D FRZ imaging will be derived.

A. Plane integrals and 3D Fourier transform of nuclear density function

LetF(x, y,z) be the nuclear density function andP8~(R ) be its NMR zeugmatographic projection in a field gradient G oriented along the direction (O,¢ ). Also, let SB¢> (x, y,z) be the family of planes perpendicular to Gas shown in Fig. I(a). That is,

S6</>(X, y,z) = x sinO coS¢ + y sinO sin¢ + z cosO. (I)

The projection value at R is then the plane integral of F(x,y,z) on S8</>(X,y,z) = R,

P8</>(R) = III F(x,y,z)dxdydz

s •• ~ R

= I: 00 I: 00 I: 00 F(x, y,z)8(S8</> - R )dx dy dz. (2)

Denote the 3D FTof F(x,y,z) by/(u,v,w). We have

F(x,y,z)

x (a)

I~ 00 J."" oc I: oc /(u,v,w)e - i21Tjxu + yv +Zw1du dv dw

(3)

U (b)

w

FIG. 1. (a) The coordinate system in xyz space, the object space, and one of the planes normal to the field gradient vector G. (b) The coordinate system in the uvw domain, which is the 3D Fourier transform of the xyz domain.

1141 J. Appl. Phys. 52(3), March 1981 0021-8979/81/1141-05$01.10 © 1 981 American I nstitute of PhYSics 1141

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and its inverse FT relation

f(u,v,w) = f: "" f: "" f: "" F(x,y,z)ei211jux+vY+WZidxdydz.

(4)

In terms of spherical coordinates in uvw space, as shown in Fig. lIb), let

Io( p,a,/3 ) f(u,v,w)

= flp sina cos/3,p sina sin{3,p cosa). (5)

Eq. (3) becomes

F(x,y,z) = 1"" fIT IT fo(p,a,/3)

X exp[ - i21T(Xp sina cos/3 + y P sina sin{3 + zp cosa)] p2 sina da d{3 dp

= L"" fIT lIT fo(p,a,/3) exp( - i21TpSa(3)p2 sina da d{3 dp.

(6)

Now take the inverse FT of Poq,(R). Using Eq. (2) and integrating with respect to R, we have

f: 00 P (J.p (R )ei2ITpRdR

= ["" f: 00 f: 00 f: "" F(x,y,z)O(S(J.p - R )ei2ITpR

xdR dxdydz

= f: oc f: 00 f: "" F(x, y,z)e,"2ITps··dx dy dz. (7)

On the other hand, Eqs. (4) and (5) give

fo(p,a,/3) = f: 00 f: "" f: 00 F(x,y,z)

X exp[i21T(Xp sina cos/3 + yp sina sin{3 + zp cosa)]dx dy dz

= S: oc S: oc S: oc F(x,y,z) e,21TpS"ffdx dy dz.

Equation (7) and (8) show that

folp,a,/3) = f: "" Pa{3(R )ei2ITpR

dR.

(8)

(9)

Here we obtain the important relation thatfo( p,a,/3) and Pa {3(R ) are a ID FT pair.

B. Application to FRZ Imaging

In NMR imaging, the recorded signal is a time-domain FID. Let h(J.p(t) be the FID signal with the field gradient oriented at (O,t/J). The FT ofthe FID gives an NMR spectrum.

V(J.p(w) = 1"" h(J.p(t)e - iW'dt.

1142 J. Appl. Phys., Vol. 52, No.3, March 1981

----

.. .. .. ...... ... ........

, .. ..... ... .. ..

... .. .........

. . .

FIG. 2. Plots of the function p2 (dots). an example of the filtering function \(1 + cosp)(dashed line). and their producqp2(1 + cosp). the FID modifi· ~r (solid line).

The spectral frequency w is related to the object projection coordinate by yw = RG, where y is the gyromagnetic ratio and G is the strength of the linear field gradient. We can relate the ID NMR zeugmatographic projection to the FID by

Poq,(R) = Voq,(RGly) = Loo h(J.p(t)e-iRG,IYdt.

Using the above relation, Eq. (9) becomes

fo( p,a,/3) = L"" f: "" haP(t)e - iR (G' Iy - 2ITPidR dt

(OC hap (t)c5 (~ _ p) dt Jo 21TY 21TY

= G haP (21TpyIG ). (10)

Substituting Eq. (10) into Eq. (6), we can express the density function in

__ i._._

a 10 mm 'TT ._L

:n 13 mm 46 mm -T-

_____ L

FIG. 3. Sketch of a phantom consisting of seven vials supported by glass plates. These tubes were filled with an aqueous solution ofNiCl, with a spin­lattice relaxation time of about 50 ms. The overall appearance and individ­ual views of the three layers are shown.

Ching-Ming Lai 1142

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(a)

(e)

FIG. 4.(a) One of the FlO signals of the phantom, at a gradient orientation oft&,,p) = (6',0'). (b) The same FID modified by \ p'( I + cosp). (c) The Four-ier transform of the modified FlO. -

F(x,y,z) = 11'y dfJ 2 i 2lT itT Goo

X sinada fa'" ha(J (2~Y)e - '2"pS",1 p2dp

= (~)2 t" dfJ f" sina da 211'Y Jo Jo

xlooh ( ) - 'pGS,,'l/y 2d a(J pep p. o

In practice, the FID, haP (p), is multiplied by a filter

1143 J. Appl. Phys., Vol. 52, No.3, March 1981

function g( p) to smooth the projection. The truncation of the FID can be considered as a gating by g(p). That is,

g(p) = 0 for p <0 andp>M.

For \k;,p<M, g(p) is usually a damping function, such as an exponential decay function or a Gaussian function. When no filtering is applied, g( p) is treated as a constant. Including this filter function, we have

where

Ha(J(R)== 1M p2g( p)hap ( pIe - '2"pRdp (12)

is the FT of p2g( p)hap ( pI, the modified FID.

C_ Further consideration

Suppose there is a total of N FID's, corresponding to N field gradients oriented at (a, ,fJ,), with angular intervals of (.1a, ,.1fJ,). The integration in Eq. (11) may then be replaced by the summation

F(x,y,z) = ~ i sina,·.1fJ, . .1a,-Hu (J (GSa(J). (13) 211'Y ,~ 1 211'Y

The above equations show that the 3D density function is the back projection of H a(J'

The choice for g( p), including its cutoff limit M, is not unique. Excessive filtering, however, would sacrifice resolu­tion for an increase in signal-to-noise ratio of the image, and vice versa. An example of a g( p) function, !(1 + cosp), is

FIG. 5. Fourier reconstruction zeugmatograms of the phantom displayed consecutively along the z axis as xy plane slices.

Ching-Ming Lai 1143

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FIG. 6. Fourier reconstruction zeugmatograms of a pig heart with slices 6 to 17 shown in (ai, and slices 18 to 29 shown in (bl. Some of the details are lost in the reproduction.

plotted in Fig. 2, along with the correspondingpz and modi­fied pZg( pl.

The factor sinai . ..1,8; ·L1a; represents the solid angle span of the gradient vector associated with the ith FlD. Strictly speaking, all of the solid angle spans should have isotropic geometry and equal magnitude in order to obtain a 3D image with isotropic resolution. This requirement can be approximated by choosing

L1a; = L1a = constant,

..1,8, = L1a/lsinaJ

Eq. (13) then becomes

( GL1a)o N (GSa(3) F(x,y,z)= -- - L H a (3 --. 21TY ;~I 21TY

III. EXPERIMENTAL RESULTS

(14)

Two examples of proton imaging are given here to dem­onstrate the feasibility and quality of FRZ imaging. The ob­jects were placed in the center of a 42-cm bore magnet, oper­ating at a field strength of93.8 mT. The NMR signals, at a resonance frequency of 4.0 MHz, were single-phase detect-

1144 J. Appl. Phys., Vol. 52, No.3, March 1981

ed. The spectrometer employed a 3D gradient control device to reorient the field gradient,S and a software program to correct for the phase distortion. 'I With such an arrangement, the experimental data were automatically acquired and pro­cessed. The filter function !( 1 + cosp) was selected for these experiments, and the images were reconstructed on a 33 X 33 X 33 matrix.

To achieve nearly isotropic and equal solid angle spans for the gradient orientation, the 3D gradient controller var­ied the azimuth angle increment, ..1,8, in proportion to 1!lsinal while maintaining a constant increment L1a in the polar angle. Since this device can only generate angles whose values in degrees are integers, ..1,8 was programmed to the nearest integer smaller than L1a/lsinal, the worst case.

An FRZ scan is characterized by the reorientation se­quence of its gradient vector, defined by the initial angle (a, ,,8;), the increment (L1a; ,..1,8;), and the final angle (af',8f)· These parameters had the values of

a; = 6°, L1a = 12°, af = 354°,

,8; = 0°, ..1,8 = n~, ,8f = 359°,

wherena is an integer with 12/Isinal-l<n,,<12/Isinal. During the scan, the controller incremented,8 by n~ until it exceeded ,8f. At that time,,8 returned to,8; and a jumped by 12°. This sequence continued until the final cycle, for which a = af' was completed. There was a total of 632 steps. The back projection, Eq. (13), has

( GL1a)z 63Z .

F(x,y,z) = -- L sllla;.n".Ha (3· 21TY ;~I

(15)

The dimensions and geometry of a phantom are shown in Fig. 3. It was constructed from seven vials of aqueous NiCl2 solution with a spin-lattice relaxation time of about 50 ms. These tubes were arranged to form the characters "TL1 V" stacked along the z axis, and separated by about 5 mm. A typical FlD signal, at the gradient direction (a,,8) = W,OO), is shown in Fig. 4(a) before modification, and in Fig. 4(b) after it had been modified by !p2(1 + cosp). Its FT, the back-projection function H6,O(cU), is also shown in Fig. 4(c). The reconstructed zeugmatograms are displayed in Fig. 5 as xy plane slices. The number on the corner of each picture marks it location on the z axis. A total of 33 slices were generated, but only slices from the central region con­taining the object are shown. The tubes are well resolved in the pictures, and bear a close resemblance to the object.

Another example, a preserved pig heart with empty chambers, was imaged by the same procedure. The results are shown in Fig. 6. The structure of the heart is clear. Some fine details and low-intensity regions, such as thin walls and small vessels, are visible when viewed directly on the video display, but are lost in this reproduction. The image intensity was dig­itized into 16 gray levels, although the actual dynamic range of the images is at least twice that value. The linear resolu­tion is slightly better than 3 mm, and is limited only by the total number of independent gradient orientations.

In these examples, both a and,8 vary through four quadrants. Note, however, that the orientation of (360° - a, 180° +,8) is identical to that of (a ,,8 ), and that the vectors for (180° - a, 180° +,8) and (180° + a,,8) are the in-

Ching-Ming Lai 1144

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verse of that for (a,/3). Thus there are only 158 independent orientations, and the back projection, Eq. (15), is averaged four times with two directly opposed to the other two. The purpose of this averaging scheme is to minimize instrumen­tal errors such as the drift and inhomogeneity of the main magnetic field and the phase instability in the spectrometer. At the time when these scans were taken, these problems were significant. The expected gain in signal-to-noise ratio, by a factor of 2, from the four-fold averaging was actually lost in compensating for these instrumental errors. If these defects did not exist, the redundancy would not be necessary and the images could be reconstructed on 65 X 65 X 65 ma­trix from 632 independent orientations, to give a resolution of 1.5 mm for the above example.

The total scan time was 8 min, corresponding to 15 s per slice. The speed with which data could be transferred to the magnetic tape was the limiting factor. Given a fast data stor­age system, the scan time could be reduced considerably.

IV. DISCUSSION AND CONCLUSION

FRZ is a true 3D imaging method. It reconstructs a complete 3D image directly from FID signals, generated from a large volume of the object. It is different from those techniques which employ some kinds of selective excitation or selective detection to isolate the NMR signals of a small region from a greater volume. 10-12 Complete 3D imaging by these other techniques would require scanning section by section. Because potential NMR signals are not fully used, they are less efficient than true 3D imaging. However, FRZ is not the only method that can be used to achieve true 3D imaging; other approaches are possible.

One possibility is 3D reconstruction by two-stage con­volution. 5 This technique requires equal spacing in a and/3. The gradient orientations become redundant at small polar angles. Consequently, a scan time more than 40% greater is needed to achieve the same resolution as an FRZ scan. Com­parison between Fig. 5 and the results of similar phantoms imaged by the two-stage convolution technique confirms that this is indeed true. Alternatively, we can, instead ofmul­tiplying the FID by a filter function, convolute the FT of the FlD with an appropriate function. It is then a single-stage convolution reconstruction with the FT of the FID modify­ing function, p2g( pI, as its convolving function. In principle, this approach is equivalent to FRZ, but the multiplication of the FID in FRZ is faster and more straightforward than the convolution operation.

Another NMR imaging method resembling FRZ is Fourier transform zeugmatography.'3 This method uses a transient x gradient for a period of t", followed by ay gradi­ent for another period t

", before switching on a z gradient

and recordingh",/, (t",), the FID as a function of the time t",. This FlD signal is actually the functionf(u,v,w) described in Eq. (4). The necessary information about thex andy compo­nents of the density function is obtained by varying the times t" and t" respectively. The complete density function

1145 J, Appl. Phys., Vol. 52, No.3, March 1981

F (x, y,z) is then obtained from the 3D FT off(u,v,w) in Carte­sian coordinate, as expressed in Eq. (3), in contrast to FRZ which reconstructs F(x, y,z) by computing the FT of fo( p,a,(3) in spherical coordinate. Except on the three axes, the digitized dataf(u,v,w) on a Cartesian grid are not the FT of a projection, and Fourier zeugmatographic imaging is not a reconstruction from projections in the usual sense. Al­though Fourier zeugmatography has been demonstrated in a preliminary 2D imaging experiment, there is no 3D imaging result available for comparison. Because of the fast gradient switching required, before the loss of phase memory, Fourier zeugmatography imposes stringent requirements on the field gradient instrumentation. Also, unlike FRZ, real-time construction of the image may not be possible.

The instruments used for the above experiments were not optimized in performance. Improvements are possible in a number of areas, such as operating at higher field, using quadrature phase detection, devising a more sensitive rf probe, etc. If the instrumentation were upgraded to the state of the art, it should be possible, in a head or whole-body system, to resolve the FRZ images into a 129 X 129 X 129 matrix, with 64 or more intensity levels, within a scan time of about 10 min. The resolution and the speed, in terms of the scan time per slice, would be comparable to those of the latest x-ray CT scanners.

It is clear that the 3D FRZ is a practical and efficient method of obtaining NMR images. It offers many potential advantages for application in medical diagnosis, biomedical research, and other nondestructive investigations.

ACKNOWLEDGMENTS

The author is indebted to Professor P. C. Lauterbur for his careful reading of the manuscipt. This investigation was partly supported by Grant No. CA-153000, awarded by the National Cancer Institute, DHEW, and by Grant No. HL1985101Al, awarded by the National Heart, Lung and Blood Institute, DHEW.

IJ, Radon, BeL Verh, Saechs, Akad, Wiss, Leipzig, Math. Phys, KI. 69, 262 (1917),

'R, N, Bracewell and A, e. Riddle, Astrophys, J, 150,427(1967), 'A. M, Cormack, Phys. Med, BioI. 18, 195(1973). 4p, e. Lauterbur, Nature 242,190 (1973), 'p, e. Lauterbur and e. -M, Lai, IEEE Trans, NucL Sci, 27, 1227(1980), 'Filtering of spin echoes in the time domain has been used as a step in 20 image reconstruction, See1, M, S, Hutchison, Proceedings of the 7th L H, Gray Conf., Medical Images, Leeds (Wiley/Ins!. Physics, Bristol, 1976), p, 135,

7 A similar derivation is also given by L A, Shepp independently in his recent paper in J, Computer Assisted Tomography 4, 94(1980),

xe. -M, Lai and P. e. Lauterbur, J, Phys, E 13, 747(1980), 'Ie. -M, Lai and p, e. Lauterbur, J. Phys, E (in press), lOW. S, Hinshaw, J, Appl. Phys, 47, 370911976), lip. Mansfield and A. A, Maudsley, J. Mag, Reson, 27, 101(1977). lOR, Damadian, M, Goldsmith, and L. Minkoff, Physiol Chern, Phys. 9, 97

(1977). "A. Kumar, 0, Welti, and R, R. Ernst, J. Mag, Reson. 18,6911975).

Ching-Ming Lai 1145

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