high-resolution nmr chemical-shift imaging with reconstruction by the chirp z-transform

I90 IEEE TRANSACTIONS ON MEDICAL IMAGING. VOL 9. NO 2 . J U N t 1990

High-Resolution NMR Chemical-Shift Imaging with Reconstruction by the Chirp z-Transform

TIE NAN MA A N D KUNIO TAKAYA

Absrracr-A study of a new NMR chemical-shift image reconstruction method with a high chemical-shift resolution achieved by the chirp :-transform (CZT) is presented. The approach uses phase encoding for the spatial coordinates .Y and x, and reserves the frequency coordinate especially for the chemical-shift. The Fourier transform (FT) image reconstruction algorithm, which forms the basis of the new CZT image reconstruction method, is first introduced. The new method, which employs the CZT instead of the FT to evaluate the chemical-shift spectrum at a much higher resolution, is then studied. The chemical-shift resolutions, achieved by the FT and the CZT, are theoretically studied from the aspect of the peak height and the peak width of chemical-shift spectra. The chemical-shift spectra calculated at a selected point in the image plane, and the chemical-shift images reconstructed hy the new method, are shown for a simple phantom containing ethanol and nieth- a n d at different locations. The results obtained by the new method and by the existing FT method are compared and discussed. The experimental results have shown that a chemical-shift as small as 39 Hz, relative to the proton resonance frequent) of 21.34 hlHz, can be success- fully resolved by the new method without the need of any improvements in magnetic field homogeneity.

I . INTRODUCTION NMR (Nuclear Magnetic Resonance) imaging,

rhemical-shif t imaging is increasingly attracting more attention because of the inherent nature of NMR. Histor- ically, NMR spectroscopy has been used in the areas where quantitative chemical analysis is needed to differ- entiate substances included in a sample. With the advent of the MRI (Magnetic Resonance Imaging), the scope of the quantitative chemical analysis has been tremendously expanded from analysis merely in a test tube to the analysis of an object which has a spread over either two- or three-dimensional space. Using the fundamental capabil- ity of MRI, that is, the ability to map the spatial distribution of an NMR-nucleus, researchers have attempted to perform chemical analysis right at a specified site within an enclosed body or a surface. This concept naturally leads the method to an ultimate goal to produce an image for every detectable chemical component by NMR, which is contained in the subject under study. Chemical-shift imaging is a technique used to produce images corresponding to different resonance frequencies, which in turn correspond to different chemical species or to different

Manuscript received February 3, 1989: revised December 12, 1989. This work was supported by the Natural Sciences and Engineering Research Council Canada (NSERC) under Grant A-4460. The work of T. N . Ma was supported by the University of Saskatchewan under a graduate scholarship.

The authors are with the University of Saskatchewan. Saskatoon, Saak.. Canada S7N OW0.

IEEE Log Number 9034447.

molecular structures separately observable by chemical- shift.

In the case of chemical-shift imaging, a constraint needs to be imposed on the spatial encoding necessary to iden- tify where an NMR resonance signal originates. Since a resonance signal, more specifically, the FID (Free Induc- tion Decay) signal, conveys the information about chemical-shift in terms of frequency, frequency encoding is not permissible for the purpose of tagging the resonance signal with positional information, as usually done in MRI. A gradient magnetic field superimposed to the static magnetic field shifts the resonant frequency. so that the location of the source can be immediately identified by ex- amining the frequency of the resonance. Spectral analysis by means of the Fourier transform usually translates the frequency contents of an NMR signal into its spatial distribution. However, different frequencies in chemical-shift imaging mean different chemical structures of an NMR sensitive nuclear species or even different species. In order to leave the frequency axis for chemical-shift. the spatial coordinates must be attached to FID signals by some other means.

Among several chemical-shift imaging methods proposed in [1]-[9], the method which uses phase encoding for both the x-coordinate and y-coordinate, and reserves the frequency coordinate especially for chemical-shift, has been experimentally implemented and its feasibility has been tested. This method is a very straightforward exten- sion of the widely used Fourier zeugmatography. Al- though this method is more time consuming compared to other methods, such as the planar echo method [ I O ] , its high resolution in the chemical-shift axis warrants the true chemical-shift analysis at the image level, almost of the same degree as in NMR spectroscopy. Also, the adoption of another image reconstruction technique based on the chirp z-transform [ 111 is possible to further improve the spectral resolution. This method subsequently discussed in this paper is capable of separating spectra which are normally merged and appear as a single peak.

In MRI, the FID signal is the fundamental unit of measurement. A large number of FID signals need to be measured to reconstruct a cross-sectional image. The FID signal is basically an exponentially decaying sinusoid. The profile, of course, depends on the source distribution if they are subjected to the gradient fields, or on the substances affecting the frequency contents in the case of chemical-shift imaging. The rate of decaying and the fre-

0278-0062/90/0600-0190$01 .OO 0 1990 IEEE

MA AND TAKAYA HIGH-KESOI.IITION NMR CHkMICAL-SHIFT IMAGING 191

quency of every decaying sinusoid involved in an FID signal are represented by a pole in the complex plane of the Laplace transform. The Fourier spectral analysis of FID signals is equivalent to evaluating the inverse of the distance to a pole from a chosen point on the imaginary axis of the s-plane, and totaling for all the poles of the FID signal. The closer the distance of a pole to the imaginary axis, the sharper is the peak of resonance. The chemical-shift is usually as small as tens of ppm, and the inhomogeneity of the static magnetic field makes the spin- spin relaxation time T2 become a substantially smaller TF. As a consequence, poles are located far from the imaginary axis, and smaller and broadened peaks result in the spectrum. Therefore, chemical-shifts are often dif- ficult to be distinctly resolved.

Improving the homogeneity of the static magnetic field is a key issue in MRI. What this means mathematically is to move the poles of FID signals closer to the imaginary axis of the s-plane. Even with advanced techniques, inhomogeneity is inevitable. As a result, the poles of FID signals are never close to the imaginary axis. Viewing the problem from another angle, moving the path used to evaluate the distances closer to the poles has the same effect as moving the poles closer to the imaginary axis. If a path other than the imaginary axis but closer to the poles is chosen, the peaks in the spectrum should be distinctly resolved as separate peaks. The chirp z-transform pro- vides a means of performing the spectral analysis from an arbitrarily chosen path, in another word, a contour in the z-plane. The method to implement the chirp z-transform in NMR chemical-shift imaging is proposed in this paper.

11. MEASUREMENT OF NMR SIGNALS Measurement techniques of NMR signals for both spec-

troscopy and imaging and related instrumentation have been well established. When a paramagnetic nuclear species, such as proton H', is placed in a uniform static magnetic field, and is irradiated by electromagnetic waves of a radio frequency (RF) specific to the nuclear species, it gives rise to a resonance at the same frequency. The RF electromagnetic wave associated with the resonance is detected by a set of pick-up coils. The detected RF signal is then frequency shifted to its baseband by a quadrature detector. The demodulated signal is the FID signal, which represents precessional motion of bulk magnetization, or spin in another word, about the axis parallel to the static magnetic field, when observed from the rotating coordinate system [ 121.

All MRI systems are furnished with three sets of gradient coils, one for each x-, y-, and z-axis, in order to attach the positional information to resonance signals in the form of either a frequency shift or a phase shift. Gra- dient coils are arranged to add or subtract a certain amount of magnetic field to or from the static magnetic field depending on the position in the static magnetic field. The nonuniform magnetic fields, linearly varying in intensity as a function of position, encode FID signals in frequency, if applied while FID signals are being acquired.

RF-pulse

data acquisition

Fig. I . A control sequence of double phase encoding for chemical-shift imaging by means of the Fourier transform and :he chirp z-transform.

If they are applied prior to data acquisition only for a finite time interval, FlD signals are phase encoded. The gradient field is also used along with RF pulses to excite only the NMR spins within a specified slice thickness.

NMR resonance signals are detected by a pair of sole- noid coils, of which the axis is usually placed in the x-y plane. Spins are disturbed by RF irradiation and start pre- cessing about the z-axis at an angular velocity corresponding to the resonance frequency U". The x- and y-vectorial components of the precession induce emf (electro-motive- force) in the detector coils. An RF pulse referred to as a 90" RF pulse nutates spins by 90" when observed from the rotating coordinate system and maximizes FID signals. Those spins nutated by a 90" RF pulse start dis- persing within the x-y plane due to the inhomogeneity of the static magnetic field. This dispersion of spins makes the vectorial sum of spin vectors gradually smaller as time elapses. However, another RF pulse called 180" pulse, applied at a time T after the 90" pulse, brings the dis- persed spins back into a bundle, generating the second FID signal not interfered by RF pulses at 2T. This ma- neuver, known as spin echo technique, is adopted in the imaging method dealt with in this paper.

Measurement of NMR signals requires precise timing for different events which takes place in sequence. A 90" and a 180" RF pulse must be applied with an accurate interval of T s. Furthermore, setting proper timing and duration for gradient pulses and data acquisition is important to obtain a set of data from which images can be reconstructed. A control sequence describes the detailed measurement procedure which depends on the imaging reconstruction method to be employed. The control sequence for phdse encoding is depicted in Fig. 1. The gradient fields for x-axis and y-axis of varied intensities are applied twice per sequence. Two gradient pulses opposite in polarity are applied before and after the 180" pulse to cancel unnecessary transient effects associated with the gradient pulses [SI. The net effect of two gradient pulses

192 IEEE TRANSACTIONS ON MEDICAL IMAGING. VOL. 9. NO. 2. J U N E 1990

is determined by the difference between the duration of the two pulses and the amplitude common to both. Spins precess at an increased or a decreased angular velocity compared with wo while the gradient fields are applied. At the termination of the gradient fields, a phase shift of either lagging or leading phase is gained as a result of the frequency varied by the gradient fields, and the normal angular velocity wo resumes thereafter.

111. MATHEMATICAL EXPRESSION OF FID SIGNALS FID signals are the fundamental elements necessary to

reconstruct an image out of measurements. An FID signal here is more specifically referred to a portion of the pulse echo covering the time after 2T s . The signal from a pair of detector coils is an exponentially decaying sinusoid having the spin-spin relaxation time T: as a decay time constant, if a point source is assumed. If only the sinu- soidal part is considered for simplicity, the signal is expressed as

w ( t ) = sin [ (ao + A w ) t + e ] . The quadrature demodulator multiplies this signal w ( t ) and sin (wet), also w ( t ) and cos (wet), to produce a set of two orthogonal signals, which can be represented by a complex function. Multiplications yield

O.S(sin [(2wo + A w ) t + e] + sin ( A w r + e ) } 0.51 -COS [(2w0 + A w ) t + e ] + cos ( A u t + O)}.

The first terms are filtered out by low-pass filters leaving the second terms to form an FID signal,

( 1 ) S ( t ) = eJAwr+fle-( l /T:) l

The resonance frequency wo related to the static magnetic field Ho by Larmor's equation,

WO = YH",

has been removed by the demodulation process. Assum- ing e = 0, the frequency shift Aw can be considered as a chemical-shift. Although inhomogeneity in the static magnetic field also contributes to A w in practice, this can be assumed zero without losing the generality. In addition to the frequency shift representing the chemical-shift, x- and y-gradient fields defined by

aH G, = -

ax aH

G,. = - ax introduce another frequency shift y ( G,x + G , y ) which depends on the location of a point source (x, y ) . When the duration of the gradient fields is finite, i.e., A T , the signal gains a phase shift of y ( Grx + G, y ) A T. The FID signal of the point source located at (x , y ) having a source density p(x , y ) is now expressed as

Since the FID signal obtained from the detector coil is a collection of all FID signals of point sources distributed over a 2-dimensional plane, and of all chemical-shifts involved,

s ( r ) = icl s(x , y , A @ , t ) d a w dx dy ( 3 )

yields the overall representation of the FID signal.

IV. IMAGING BY THE 3-D FOURIER TRANSFORM

Chemical-shift imaging is mathematically interpreted as a problem of finding p (x , y ) of chemical-shift A w from the collection of the FID signals s ( t ) given in (3). The 3-D Fourier transform is used as a means to solve the integral equation (3) based on the fact that s ( x , y, A o , t ) is separable in terms of x, .y, and Aw. In order to show that a point source having a specific chemical-shift Aw appears exactly at the location corresponding to its position in the image representing the chemical-shift frequency, the exponential decay in (2) is intentionally ignored here. The decay in FID signals merely produces a spread in the spectrum.

First, discretize independent variables G,, G,, and t for the Fourier transform as follows:

G, = AGr ( I - F) I = o , 1, . . . , M - 1

r = A m n = 0 , 1, - * e , N - I

Since the FID signal from a point source with a chemical-shift angular frequency w L , located at (x, y ) , is under the influence of the Ith x-gradient field and the mth y-gradient field, the signal can be written in terms of 1, m, and n as

/ Y A G , ( / - M / 2 ) rAT s ( l , m, n ) = ~ ( x , y ) e

. (4) . e j ~ ? r C , ( t n - M/Z )\ATe ~ w r A i i i

Applying the 3-dimensional DFT (Discrete Fourier Transform) to s ( I , m , n ) ,

M - I M - 1 N - I

S [ S ( I , m , n ) ] = C C C $ ( I , m , n ) / = o ,n=o n = o

Now, discretize x and y according to x = A x i and y = A y j , and introduce the following relationships:

2a yAG.,AxAT = -

M

2 a M

y A G , A y A T = -.

MA A N D TAKAYA: HIGH-RESOLUTION NMR CHEMICAL-SHIFT IMAGING I93

These two conditions set the size of the field of view, 2n

x,,, = AxM = ~

y A G , A T

2n y A G,. A T '

ymax = A y M =

= M 2 N p ( i , j ) ( - l ) l + ' G ( i - p )

This result indicates that the DFT S ( i , j , r ) represents the point source p ( i , j ) and has nonzero spectrum if r = ( f k / f o ) . Wyere fk is the chemical-shift frequency and fO = ( N A r ) - . In other words, only such rth image out of N generated images contains the chemical-shift of h.

V. THE CHIRP Z-TRANSFORM In the reconstruction of a chemical-shift image, the

3-dimensional Fourier transform has been applied to a set of M X M FID signals of N samples. The Fourier transform with respect to the time variable n represents the spectrum due to a chemical-shift. r is a discrete spectrum number. Although the exponential decay was omitted in the discussion to show the validity of the image reconstruction method, this decay inevitably contributes to the spread of each peak in the spectrum. The chirp z-transform treats this decay more positively to sharpen the broadened spectral peaks than the Fourier transform does. The last Fourier transform in (6) with respect to n can be considered separately from the other two Fourier transforms, which play a role only to reconstruct an image by the double phase-encoding scheme, since all three Fourier transforms are completely separable and independent.

Let x, be a finite sequence of FID samples. x,, is complex here, due to the quadrature demodulation employed. The z-transform of this finite sequence is

N - 1

X ( z ) = c x,,z-ff f1 = 0

( 7 )

The z-transform X(z) can be evaluated at any arbitrary point in the z-plane, because the z-transforms of finite se- quences exist everywhere in the z-plane. When the z-transform is evaluated at

z = 2,

= AW-' r = 0, 1, * , L - 1 (8)

for arbitrary complex numbers,

A = AOeJon, W = (9)

X ( z , . ) is referred to as the chirp z-transform. L is an integer. The case of A = 1, L = N , and W = places N discrete points on the unit circle in the z-plane. Hence, it corresponds to the Fourier transform. The z-plane contour of the chirp z-transform usually begins with point z = A and, depending on the value of W , spirals in or out with respect to the origin. If WO = 1, the contour is an arc of a circle. The angular spacing between samples is &. Using the sampling interval A r , the equivalent s-plane contour begins with the point

1 so = - In A

A T

= - ( l n ~ , 1 + j O O ) . A T

The points corresponding to z,. in the s-plane are

1 A T

s,. = - (In A - r In W )

, L - 1. r = 0, 1, - 1 -

s,. lies on a straight line segment determined by A and W. FID signals are, in general, exponentially decaying si-

nusoids. More precisely speaking, an FID signal is a linear combination of damped sinusoids having different rates of decaying and different frequencies. This means that FID signals have poles in the left half of the s-plane. By properly choosing AO, &, WO, and &, in (9), a contour can be made to pass through the neighborhood of the poles of the FID signal, especially those dominating in terms of magnitude. Letting A w = wk in ( 2 ) reveals that one of such poles is at ( - 1 /T:, w k ) in the s-plane. The contour in the s-plane passing through its origin and this particular pole is expressed as a function of w ,

0 s = -- w k T ; + j w .

w is discretized as U, with an increment of &,. Therefore,

Since z = esA7, the contour in the z-plane is

, r = 0 , 1 , - . . , L - l j ( l a r / ~ ) zr = e - ( ~ s r / ~ w i

( 10)

This is one of many possible contours that could enhance the chemical-shift spectrum at w k . In this particular case, OO = 0, A , = 0, and & = 2 n / L . The chirp z-transform of the FID sequence x, can now be calculated by (7) and (8) with the z , given in (10).

194 IEEE TRANSACTIONS ON MEDICAL IMAGING. VOL. 9. NO. 2 . J U N E 1990

VI. COMPARISONS ON PEAK HEIGHT AND WIDTH A . Spectrum Based on DFT

As described in the previous two sections, two image reconstruction methods, the Fourier method and the chirp z-transform method, differ only in the contour chosen to transform the original FID data into a spectrum. The imaging by two-dimensional phase encoding is not affected by this choice. In order to clarify the merits of using the ctirp z-transform, comparative studies have been con- ducted. The height and the width of a peaking spectrum resulting from a single pole were studied for both methods. In the discussion of image reconstruction methods, the decay of FID signals was ignored for the sake of simplicity in explanations. Since the decay, more specifically, an exponential decay in the case of a point source, is a determining factor for the spread in peaking spectra, it must be taken into consideration in the discussion of peak height and width. Now, ignoring the phase shift caused by gradient fields, an FID sampled signal is written as follows. It is assumed that the resonant frequency f k is an integer multiple of the fundamental frequency considered in DFT, i.e., fo = 1 / N A r .

, n = 0 , 1, . . . , N - 1. s ( n ) = e ~ ~ i A r r ~ e - ( A ~ n / T ; i

(11)

Let the DFT of s ( n ) be

N - I

~ ( r ) = S(n)e-J('rI')rn , r = O , 1, e . . , N - 1.

n = O

When r = &/so, S( r ) gives its maximum value

It can be proved [13] that

and, in general, I S ( r ) 1 < N . When T; << 1 , then

J S ( r ) J << N .

Therefore, the peak height (meaning the maximum value of a peak) of the FID signal is bound by N and is usually much smaller than N . Since the magnitude of I S ( r ) I drops to 1 /& of the peak height at

and

the band-width A f defined at half-power points is

A f = ( r ? - . I ) f O

1 TT; '

-~ -

As expected, the resonance peak will spread wider as T ; becomes smaller.

B. Spectrum Based on CZT The chemical-shift spectrum obtained by applying CZT

is discussed in what follows. The CZT basically means the evaluation of the z-transform on a circular or a spiral contour which begins and ends at any arbitrary points in the z-plane, instead of the ordinary contour of the unit circle in the case of DFT. Substituting the FID signal s ( n ) defined by (1 1) for the discrete sequence nn used in (7),

N - 1

qr) = e ~ u " i A ~ n e - ( A r n / T : ) Z,", n = O

r = 0 , 1, * * , L - 1, (16)

where L is an arbitrary integer number. Without losing generality, consider a contour which begins at the unity of the real axis and completes one revolution around the origin in the z-plane with the total of L points. Also, make the contour pass through the pole of the FID signal as specifically given by (10). Presumably, this contour is the best possible one that most sharpens the spectral peak of the FID signal given by (11). Referring to (9), A. = 1 and Bo = 0 and c $ ~ = 2 a / L are used. With this particular contour, the rth spectrum calculated by CZT now becomes

r = 0, 1, . , L - 1, (17)

where r f = r - fk/fo andfo = 1 / L A r . Making the same assumption that&/fo is an integer as was made in the previous section for the DFT, choose r such that r = fk/fo. Then, it is mathematically possible to prove [13] that

The result is significant to the application of CZT for FID signals, because the CZT spectrum measured exactly at the pole representing an FID signal is finite and constant. The peak height is neither a function of Tf nor a function of f k . Although the peak height of the FID spectrrm by CZT is easily found by analytic means, the peak width defined by half-power points requires the solving of a transcendental equation given by (17). Instead of attempt- ing exhaustive studies about the analytical solution for the peak width, peak widths were numerically calculated by varying the spin-spin relaxization time T;. Illustrated in Fig. 2 are graphs showing the results of the computations. Changes in peak height and width are shown as a function

MA A N D TAKAYA. HIGH-RESOLUTION NMR CHEMICAL-SHIFT IMAGING

300 -

4 6 200-

3 !2 100-

.e

X

I95

100

O o L BY DFT Legend 0 a = 1500 Hz A 5 = 3000 Hz

c

BY CZT L = 256

0.005 0.01 0.05 0.1 0.005 0 01 0.05 0.1

Legend I BY DFT "'1 BY CZT L = 256

0 1 0-

T; T;

0.005 001 0 0 5 0 1 0.005 0 0 1 0.05 0 1

(b) Fig. 2. Comparison between the DFT and CZT methods in peak height

and peak width using spin-spin relaxization time T-? as a variable. The total of 256 data samples were used for calculations for each of FID frequencies of 1500 Hz. 3000 H z , and 4500 Hz.

6000

PEAK WIDTH = 13.93 Hz

0 (-A, 3000) c 3000 (- & ,3000)

E

0 3000 5000 -200 -150 -100 -50 0

-1/T; Requency (Hz) (a) (b)

'"1 PEAK HEIGHT = 246.24 PEAK WIDTH = 13.60 Hz

200 BY CZT

g 100

3000 5000

Requency (Hz) (C? "'1 PEAK PEAK HEIGH; WIDTH = 49.46 33.36 Hz , ~ '"k, PEAK WIDTH = 13.59 Hz

' 200 BY FFT -o 200 BY CZT .z % M 3 100 2 100

0 0

0 3000 5000 0 3000 5000

Requency (Hz) Frequency (Hz)

(d) (e)

Fig. 3 . The influence of the CZT contours on N M R chemical-shift spectra. (a) Two different contours passing through i) wI = 2 a x 3000 Hz, TT = 0.01 s, and ii) w/ = 2a X 3000 Hz, TT = 0.1 s; (b) DFT and (c) CZT for TT = 0.01 s: and (d) DFT and (e) CZT for T: = 0.1 s . Param- eters used are N = M = 256, A T = 0.0002 s.

.-


of T,* for both DFT and CZT. It is apparent from the graph that the ratio of peak height to peak width, of CZT spectrum, is always higher than that of DFT spectrum and stays almost constant regardless of changes in T:. In con- trast with the spectrum by CZT, peaks in the spectrum by DFT usually become spread, meaning lower and wider, as T: decreases.

C. Effects of Contours on Chemical-Shift Spectrum As discussed in the previous sections, CZT as opposed

to DFT has the freedom to choose a contour depending on the location of the poles of a given signal. The CZT contour is usually chosen such that it passes through a dominant pole or the neighborhood of the dominant pole. The effects of properly chosen contours are best seen by comparing two chemical-shift spectra, one obtained by CZT and the other by DFT, for different pole locations. Fig. 3 illustrates such comparisons. Fig. 3(a) shows two poles corresponding to two separate FID signals having the same frequency of 3000 Hz but different values of time constant T;, 0.1 and 0.01 s. Two CZT contours, passing through each of the respective poles, are also shown. Fig. 3(b) shows the DFT of the signal with a frequency of 3000 Hz and T: = 0.1. Fig. 3(c) is the CZT of the same signal obtained by using the contour passing through its own pole. Fig. 3(d) shows the result of the DFT of the signal with a frequency of 3000 Hz and T: = 0.01, and Fig. 3(e) is the result of the CZT of this signal.

From the results shown in Fig. 3, it is apparent that a smaller T: will result in a wider and lower spectral peak for the DFT, whereas the spectral peak achieved by the CZT is always narrower than that by the DFT, and is little affected by the value of T;. The shorter the T:, the more prominently the effect of the CZT will be seen. This is important because T; in practice is a substantially small value.

VII. EXPERIMENTAL RESULTS The new NMR chemical-shift imaging method pre-

sented in this paper has been verified by using a set of synthetic data generated by the MRI system simulator de- veloped earlier [ 141, [ 151 and a set of real FID data obtained from a 0.5 Tesla MRI research system using a su- perconducting magnet (Central Research Laboratory, Hitachi Ltd. 181). The MRI system has a bore size of 20 cm and the gradient fields can be changed upto the maximum of 5 X Gauss/cm for all x , y , and z axes. Two orthogonal FID signals are made available by a built- in quadrature detector. A local microcomputer controls timing and magnitudes of different events necessary for a control sequence specified by a main computer. The experiments for double phase encoding chemical-shift imaging employed the control sequence shown in Fig. 1. This control sequence required approximately 4 s of data collection time per FID signal.

A . Experiments with Synthetic Data The synthetic FID data are generated by a simulator for

a phantom consisting of three test tubes placed at different

sites. See the cross-sectional diagram shown in Fig. 4(a). The ( x , y ) image plane consists of 16 x 16 pixels. All the test tubes are assumed to contain the same nuclear species, such as H'. Each of them, however, experiences a different chemical environment from the others. They are named A , B , and C. For instance, they could be H+ in the form of CH3-, or C2H5-, or OH- as will be seen in the case of real data. The frequencies of demodulated FID signals produced from A , B , and C are set to be 2850 Hz, 3000 Hz, and 3100 Hz, respectively. In other words, the chemical-shift is 150 Hz between A and B , and 100 Hz between B and C. The values of T: are assigned to be 0.0125, 0.011, and 0.01 for A , B , and C , respectively. The sampling frequency for FID signals is set to be 5000 Hz, or 0.0002 s, in sampling interval. The total number of FID signals obtained for varied x- and y-gradient fields is 16 X 16. The contour for CZT was chosen so that it begins at the origin and passes through the pole of the component B as shown in Fig. 4(b). The number of sampling points N in time and the number of sampling points L in frequency are made both equal to 256. The frequency spacing for CZT and FFT is therefore 19.53 Hz.

After completing the entire image reconstruction process, the chemical-shift spectra at the sites of the sources A , B , and Clocated respectively at ( 3 , 14) , ( 12, lo) , and (5 , 5 ) are extracted and shown in Fig. 4(c) for DFT and in (d) for CZT. In the figures, the spectrum of A , B , and C was drawn by a solid line, a dashed line, and a broken line, respectively. Some of NMR chemical-shift images reconstructed from the same synthetic data set are shown in Fig. 5(a) for DFT and in (b) for CZT. Each of the figures contains nine pictures at selected chemical-shift frequencies of 40-60 Hz apart, arranged in increasing order from left to right, then from top to bottom.

Comparison between the two sets of chemical-shift spectra, shown in Fig. 4(c) and (d), reveals clearly that the heights of peaks have been markedly increased and the widths have been decreased by the CZT method. The improvements can be noticed even more clearly by comparing two sets of pictures shown in Fig. 5(a) and (b). Different chemical-shift components are completely separated from others and appear only in a picture corresponding to each of the chemical-shift frequency. In the case of the DFT method, the image of the source A , B , or C is observed in pictures other than its own frequency. However, in the CZT method, the leakage of the source image is hardly noticed and the sources A , B , and C are perfectly separated. The improvements achieved by the CZT consequently mean the improvements in the sense of spectral resolution and sensitivity.

The CZT method is then applied to another set of synthetic FID data in which chemical-shift frequencies are much closer to each other compared to the first experiment. Furthermore, the magnitude of the chemical-shift component B is made smaller to an extent that the spectrum is merged and hidden in the spectra of A and B by the DFT method. The differences in chemical-shift frequency between A and B , and between B and C , are set

M A A N D T A K A Y A - HIGH-RESOLUTION N l l K CHEMICAL-SHIFT IMAGIUG

16

h

1

5500

S - PLAN E

4000

CONTOUR

500

16 -200 -160 -120 -80 -40

-1,'T; ( b )

25000 25000 1 BY FFT

20000

15000

20000 250 HZ -

8 g 10000 15000;

Chemical Shif t

(c)

Fig. 4. (a) An illustration of spatial locatlons o f test tubes in the artificial phantom described in text: and (b) the locations of poles of synthetic FlD data generated by a simulator for the phantom. and a contour for the CZT: (c) the NMR chemical-shift spectra computed by the FFT. and (d) by the CZT.

I07

250 HZ 3

Chemical Shif t

(d)

(a) (b)

Fig. 5. (a) NMR chemical-shift images of the phantom, obtained by using the FFT. and ( b ) by using the CZT. Nine selected pictures are 5hown in increasing order of chemical-shift frequency from left to right then from top to bottom

to be 39 and 78 Hz, respectively. The poles representing A , B , and C and the contour used for the CZT which passes through the pole B of the FID signals are shown in Fig. 6(b). The chemical-shift spectra at the source sites of A , B , and C are computed in the similar manner as the first experiment by the FFT and the CZT. and are shown in Fig. 6(c) and (d), respectively. As shown in Fig. 6(d), the spectrum for the source B is significantly enhanced by the CZT and is outstanding distinctly by itself without being immersed in the large spectra of A and C. The chemical-shift images generated by the conventional DFT, and by the CZT, are shown in Fig. 7(a) and (b), respectively. The frequency difference between two adjacent picture is approximately 20 H z . The results obtained by the CZT are satisfactory as well as in the case of the first experiment.

B. Experinleiits Leith Real Durci The proposed CZT imaging method is applied to the

real FID data to prove that the method is truly effective

even for such data as affected by magnetic field inhomogeneity, instrumentational noise, timing misalignments in a control sequence, and other artifacts. A phantom consisting of five glass tubes, in which three are filled with a chemical substance methanol ( CH,-OH ) and the remain- ing two tubes are filled with ethanol ( CH3-CH2-OH ), was scanned by the MRI system described earlier. The config- uration of the phantom is shown in Fig. 8(a). To cover the 2 X 2 cm2 cross-sectional area of the phantom, an image plane of 16 x 16 pixels is used to keep the measurement time reasonable ( 17 min for this experiment). The sampling frequency used to sample FID signals is 1600 Hz.

When FID signals are acquired from a real MRI system, their pole locations. that is. their frequencies and exponential decay constants ( T T ) , are unknown. The spectrum by the DFT obtained from the largest among all the FID signals, which is usually the one measured with no gradient fields applied, can be used to estimate the locations of poles involved in the set of acquired FID sig-

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Fig. 6. (a) An illustration of spatial locations of test tubes in the artificial phantom described in text; and (b) the locations of poles of synthetic FID data generated by a simulator for the phantom described in the second experiment, and a contour for the CZT; (c) the NMR chemical-shift spectra computed by the FFT, and (d) by the CZT.

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(a) (b) Fig. 7 . (a) NMR chemical-shift images of the phantom, obtained by using

the FFT, and (b) by using the CZT. Nine selected pictures are shown in increasing order of chemical-shift frequency from left to right then from top to bottom.

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Fig. 8. (a) An illustration of spatial locations of test tubes in the phantom described in the text; and (b) a estimeted optimum contour for the CZT. (c) The chemical-shift spectra of real FID data computed by the FFT, and (d) by the CZT.

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199 MA AND TAKAYA: HIGH-RESOLUTION NMR CHEMICAL-SHIFT IMAGING

(a) (b) Fig. 9. (a) NMR chemical-shift images of real experimental FID data gen-

erated by the conventional DFT method. and (b) by the new CZT method. Nine pictures are shown, which are selected out of 256 images, with the picture at the upper-left from the low end of the frequency spectrum. and successive pictures occur from left to right, then from top to bottom.

nals. The degree of spreading in peaks or simply the width is indicative of the exponential decay constants. An in- teractive search algorithm for pole locations has been de- veloped based on this notion and the details are described in [ 131. The optimum contour for CZT determined by the pole search method is shown in Fig. 8(b).

The chemical-shift spectra for methanol ( CH,-OH ) located at (9, 9 ) in the image plane, and ethanol ( CH,-CH,-OH) at ( 14, 4 ) , are plotted in Fig. 8(c) obtained by the FFT and in (d) by the CZT. The corresponding NMR chemical-shift images achieved by the FFT method and by the CZT method are shown in Fig. 9(a) and (b), respectively. The spacing between adjacent pictures is approximately 20 Hz. From the specta shown in Fig. 8(c), it can be seen that 1) CH,- in ethanol and CH,- in methanol have their peaks at different frequencies; 2) CH,- in methanol and -CH,- in ethanol have almost the same chemical shift frequency, but the magnitude of CH,- in methanol is far greater than -CH,- in ethanol; 3) the chemical-shift frequency of -OH is common for both methanol and ethanol. The three major peaks are of CH3- of ethanol, CH3- of methanol, and OH- of both. It is therefore expected that the three test tubes of methanol and the two test tubes of ethanol can be separated at the two different chemical-shift frequencies of CH,-, and all five test tubes are seen at the frequency of -OH-. In the chemical-shift images shown in Fig. 9(a), the frame in the center shows three test tubes of methanol, the middle frame at the bottom shows two test tubes of ethanol, and all five test tubes appear in the right frame of the top row, as expected. However, due to the broadened foot of peaks, undesirable leakage of spectrum has occurred. For example, the frame at the center shows a somewhat weaker pattern of two test tubes of ethanol. On the contrary, the spectral peaks shown in Fig. 8(d) by the CZT appear more distinct and well separated. The images shown in Fig. 9(b) have much less leakage of undesired patterns. The pictures corresponding to the major spectral peaks show only

the pattern which belongs to the particular chemical-shift frequency.

VIII. DISCUSSION Although the CZT method improves the resolution in

chemical-shift frequency remarkably, there are some con- cerns particularly with respect to noise. Practically speaking, the CZT changes a damped sinusoid into a less damped sinusoid, which can almost be an unattenuating sinusoid if the contour is passing very close to the pole representing the damped sinusoid. The same effect will occur for noise. Since noise such as white Gaussian noise is characterized only in the sense of the Fourier transform, the locations of noise poles are uncertain. How- ever, the frequency components of noise at the neighborhood of the spectral lines are usually enhanced by the CZT. The spectrum of an FID signal under the influence of noise is studied by using a simulated FID signal and computer generated noise.

Fig. 10(a) shows the contours used to evaluate the spectrum of a simulated FID signal, which is a damped sinusoid having a magnitude of 100 and a decay time constant equivalent to T; = 0.0167 s. A white Gaussian noise with statistic characteristics of mean equal to 0 and standard deviation equal to 10 is added to the FID signal. The contour-0 corresponds to the Fourier transform, and the contours -1, -2, and -3 are used by the CZT. The S I N (signal-to-noise ratio) defined for spectral lines under noise [16] is used to quantify the influence of noise. The graph of S I N versus contour number is shown in Fig. 10(b), in which the cases for three different sampling frequencies are depicted. From the graph, it can be seen that the S I N stays constant until the contour comes very close to the pole of the FID signal. This means that the CZT equally enhances signal and noise. The decreased S I N at the contour-3 is explained from the fact that the CZT of the Gaussian white noise exhibits a linearly increasing spectrum in the log frequency scale. When the contour


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Fig. 10. (a) Contours used for FT and CZT to evaluate signal-to-noise ratios, and (b) the relationship between signal-to-noise ratios and CZT contours plotted for 3 sampling frequencies. 5000 Hz. 7500 Hz. and I O 000 Hz.

largely deviates from the imaginary axis of the s-plane, the slope significantly becomes greater [ 111. In other words, this increase in noise spectrum exceeds the degree of the magnitude enhancement by the CZT for the FID signal, in the case of the contour-3. Fortunately, this lower S I N always occurs at the higher end of the spectrum where the spectral lines involved in an FID signal are less likely to exist and no imaging is required. When T; becomes smaller, the FID signal rapidly decays. At a higher sampling frequency, the time duration for acquiring a fixed number of data samples will become shorter, thus eliminating the tail end of the decaying FID signal. As this avoids capturing a long train of data samples domi- nated by noise, the S I N is higher for a higher sampling frequency. In practice, it is desirable to use as high a sampling frequency as possible and terminate data acquisition when the signal diminishes.

The two axes associated with the CZT-amplitude and frequency-often invite confusion. The frequency axis used in the CZT is exactly identical to that of the FT, whereas the amplitude means something different from that of the FT. Since Bo + C#Ior in (9) is used as the frequencies of the CZT, equally spaced with C#IO, there is no difference in the definition. Obviously, the spectral lines resulting from the CZT are enlarged and sharpened giving a larger amplitude to each spectral line. The primary purpose of using the CZT is to separate closely located spectral lines of chemical-shift and to make individual images representing those elements of chemical-shift clearly dis- tinguishable. The relative intensities among pixels must be maintained within a picture representing a chemical- shift element. The quantitative comparison among different images is, however, less important. In spectroscopy for which the FT plays a key role, the quantity of an element corresponding to a chemical-shift spectral line is determined by the area under the spectral peak. This rule does not hold in the case of the CZT, although its mathematical relationship could possibly be derived.

IX. CONCLUSIONS The new NMR chemical-shift image reconstruction

method, using the CZT approach, has shown very good performance and a significant advantage over the conventional FT image reconstruction method in providing higher chemical-shift resolution. This is largely due to the flex- ibility that the CZT permits to evaluate the z-transform on any general contour outside or inside the unit circle in the z-plane, so that it is possible to choose an optimum contour just passing through or coming near the dominant (or the most interesting) poles of the FID signal. As a result, the spectral peaks corresponding to these poles can be sharpened. Thus, the chemical-shift resolution is increased. On the contrary, the FT image reconstruction method evaluates the spectrum on the fixed contour of the imaginary axis in the s-plane, in spite of the fact that the poles of the FID signals are always far away from the imaginary axis because of the small spin-spin relaxation time T;. The spectral peaks are often too wide to be resolved when they are mutually close.

The new CZT NMR chemical-shift image reconstruction method can directly replace the FT method without the need of revising the schemes of the control sequence regarding RF pulses and gradient fields or modifying the hardware on the existing MRI system. Although the CZT method will require approximately three times as much computation time for processing FID data and recon- structing resultant images, as compared to the FT method, this increase is fractional in the total time required for imaging. A large portion of time to be spent in acquiring the FID data will not change in both the methods. The increase in data processing time does not make an appre- ciable difference for the whole imaging procedure. There- fore, the new image reconstruction method appears to be practically feasible and could be implemented.

The contour chosen close to the poles corresponding to the chemical components of particular interest can selec- tively enhance the images corresponding to these com-

MA A N D TAKAYA: HIGH-RESOLUTION NMR CHEMICAL-SHIFT IMAGING

ponents. The contours used in the experiments described in this paper are only straight lines in the s-plane. There is, however, no reason to doubt that a more general contour, for example, a piecewise linear contour passing through most poles of a given set of FID signals, can be adopted for the CZT. The images corresponding to all chemical components involved can be thus improved. Chemical-shifts are very small as usually measured in ppm referenced to the NMR resonance frequency. All chemical components causing chemical shifts occupy only a small frequency range of spectrum. Therefore, a shorter contour segment which covers all the chemical-shift components is sufficient to obtain all the images related to the chemical-shift. Computation time can be somewhat saved in this way [ 171.

ACKNOWLEDGMENT A portion of the experiments herein described was con-

ducted at the Central Research Laboratory, Hitachi Lim- ited. The cooperation given to the authors is gratefully acknowledged.

REFERENCES A. Volk, B. Tiffon, J . Mispelter, and J . Lhoste, “Chemical shift- specific slice selection: A new method for chemical shift imaging at high magnetic field.” J. Magnetic Resonance. vol. 71, pp. 168-174. 1987. W . Dixon. “Simple proton spectroscopic imaging.” R n d i o l o ~ ~ . vol. 153, pp. 189-194, 1984. H. Yeung and D. Kormos, “Separation of true fiit and water iinagcs by correcting magnetic field inhomogeneity in situ,” Radiology, vol. 159, pp. 783-786, 1985.

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P. Bottomley, T . Foster, and W . Leue, “ / / I viiw nuclear magnetic resonance chemical shift imaging by selective irradiation.” Proc. Natl. Accid. Sri. USA, vol. 81, pp. 6856-6860, Nov. 1984. A. Haase and J . Frahm, “Multiple chemical-shift-selective NMR imaging using stimulated echoes,” J. Magneric Resonace, vol. 64. pp.

A. Maudsley, S. Hilal, W . Perman, and H . Simon, “Spatially resolved high resolution spectroscopy by ‘four-dimensional’ NMR.” f . Magnetic Resonance, vol. 51, pp. 147-152, 1983. I. Pykett and B. Rosen, “Nuclear magnetic resonance: I n vivo proton chemical shift imaging,’’ Radiology, vol. 149, pp. 197-201, Oct. 1983. K . Takaya, “Chemical-shift imaging by means of 3-D Fourier transform,” Res. Rep., Central Research Lab., Hitachi Ltd., 1984. T . R. Brown, B. M. Kincaid, and K. Ugurbil, “NMR chemical shift imaging in three dimensions.” Proc. Na f . Acad. Sci. USA, vol. 78, pp. 3523-3526, June 1982. M. Tropper, “Image reconstruction for the NMR echo-planar technique, and for a proposed adaptation to allow continuous data acquisition,” J. Magnetic Resonance, vol. 42, pp. 193-202, 1981. L. Rabiner, R. Schafer, and C . Rader. “The chirp z-transform algorithm and its application,” BellSyst. Tech. J., pp. 1249-1292, May- June 1969.

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[I21 T . Farrar and E. Becker, Pulse and Fourier Transform NMR: Inrro- duction to theory arid Methods. New York: Academic Press, Inc., 1971.

[I31 T . N . Ma, “High resolution and sensitivity NMR chemical-shift imaging method achieved by chirp ;-transform,” Master’s thesis, Univ. Saskatchewan, 1988.

[ 141 K . Takaya and K. Chu. “Digital simulation for NMR-CT imaging,” in Proc. CMBEC-13, June 1987, pp. 63-64.

[I51 K . Chu, “Development of a simulator for NMR imaging,” Master’s thesis. Univ. Saskatchewan. 1988.

[ 161 D. Gadian, Nuclear Magnetic Resonunce und Its Application to Liv- ing Systems. London, England: Clarendon and Oxford University Press, 1982.

[I71 T . N. Ma and K . Takaya, “Application of the short contour of the chirp z-transform on MR chemical-shift imaging,” in Proc. CMBEC-15, July 1989, pp. 17-18.

high-resolution nmr chemical-shift imaging with reconstruction by the chirp z-transform

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