- Email: [email protected]
Preliminary experimental evaluation of an inverse source imaging procedure using a decoupled coil detector array in magnetic resonance imaging
,Zlrd. I:‘ll&
Ph.
Vol.
17. No. 4. pp. 257-m, 19% Ehier Srience hd Lbr RES Printed in Great Britain I :FJo-433/95 s1o.nn + 0.00
Preliminary experimental evaluation of an inverse source imaging procedure using a decoupled coil detector array in magnetic resonance imaging D. Kwiat and S. Einav Department University,
of Biomedical Engineering, Ramat-Aviv 69978, Israel
Krwivcd
accepted
1992,
October
School
of Engineering,
Tel-Aviv
1993
ABSTRACT Rerently, rue have discussed smeral of the aspects involved in the detector array concept in magnetic monance imaging where an image is obtained by applyi>Lg inverse source procedure,s to the data assembled by an array _ of coil detectors surrouading the object. In this work we describe an ex@rimental setup, where a detector array uas ron\tructvd of 9 coils, that gives a coarse resolution of3 x 3 pixels. By measun’ng the induced current signals over thi.c al-ray of roils, a relationship is established between the set of signals and the structure of the body under investigation. Through matrix inversion. reconstruction of the original source from the detected signals is possible. In this prelimitlary experimental setup, 71~ did not use an actual nuclear magnetic resnnanrc signal. Instead, a miniature .rolunoid was used as a sourre, lo simulate a prece.kng magnetir moment. Keywords Med.
Eng.
Magnetic Phys.,
resonance 199.5,
Vol.
imaging, 17. 257463,
inverse
A new model of MRI modality was described recently’, where MRI procedure is based on the simultaneous recording of signals in an array of closely packed, decoupled, coil detectors instead of zeugmatography. By using an array of N,., mutually decoupled, coils an image with a resolution of & voxels can be obtained, where the whole procedure of image acquisition is performed within the duration of a single FID. The large number of detectors making up this array requires that particular attention be paid to sensitivity, resolution and SNR problems, as well as to the coupling caused by mutual inductance between adjacent coils. Several studies have been published already z-H describing experimental work carried out about SNR improvement and decoupling of coils. In this work we use the partial overlay method for the decoupling of adjacent coils’-“* -8. Based on the previously described analysis’, an array of coil detectors with negligible coupling was constructed.
A discussion is required single coil detector. Two
AND
imaging
procedure,
detector
array
.June
INTRODUCTION
SENSITIVITY RESOLUTION CONSIDERATIONS
source
SNR
of the resolution of a adjacent pixels will have
nearly similar signals seen by the detector. Without noise presence, the resolution of the detector will merely depend on the dynamic range (resolution) of the ADC and the distance r between the detector and the sources (signal intensity is inversely proportional to r7). Let S,,, be the average signal intensity at a detector, obtained from the whole volume activated. Let SNR denote the average signal-to-noise-ratio level at one of the coil detectors. The noise amplitude at a certain instant of measurement may be described by
where t(t) is a random number -1 G 5 G 1. Let q stand for the distance from the nearest source (voxel) to the coil, let x be the distance from an arbitrary source (voxel) to the coil. Let s, (x,,) be the amplitude of the signal at the detector from a single unit source voxel (relative proton density p = 1) at q. Recording this signal repeatedly N times (N acquisitions) and averaging, we showed’ that the difference between two signals (averaged by N samplings) from two adjacent voxels at x and at x+ 6x is given by: (2)
Preliminary
experimental
evaluation
of an inverse
source imaging procedure:
where 0, E @/ax. In order for the system’s resolution to be able to detect it, (as), should be greater than the lowest meaningful signal, which is composed of the lowest detectable clean signal 2-“rS,, plus noise:
(3) where n,+ 1 is the resolution of the ADC used for recording the data. We then obtain:
(4) where R is the distance from a coil’s centre to the array’s centre and gets most of its contribution from the closest lying source and, therefore, from si (q). If all voxels had the same relative proton densities p = 1, and were all located at q, the total signal would have become S,, = N,s, (x,J. However, due to their spatial distribution and different proton densities, and differing coil effective areas (direction in space), the total signal S,, is reduced by a filling factor f < 1 defined by V-’
d3v sr(x) cos 8 p(x) I
j-= =
C
sl(xi)Pi
cos 4
(5) c. Sl(xi)
where, in the integral, cos 8 is the effective area of the coil, as seen by the source at X. Equation (5) can be reformulated in a discrete form to read Stot = sh)Nvf (6) where N, is the total number of voxels into which the imaged volume Vis divided. In the same manner, we shall assume that m can be represented by Sp( 3~0)NJ where lip(q) is the inherent source noise from a unit density source at 3~0.We, thus, obtain the minimum resolution requirement: ln 2 n, = In wJ.fR3 (7) 4 I which allows us to estimate the minimal resolution required of the ADC for the certain system geometry that we have. For a sphere of radius R, uniformly filled, with p - 1, and a small radius coil at a distance 3~0,the following expression for f can be derived:
f = 9 (In 2 + 1) 2 3 In 5 This
258
expression
0 indicates
(8) the correctness
of our
D. Kwiat
and S. Einav
above statement about the decreased sensitivity of n, to variations in 3~0,R and $ By inserting the minimal resolution requirement, eqn (7), into eqn (8) and assuming that the worst resolution is obtained from the innermost voxel where x = R, we obtain the relationship between the SNR value and the minimal number of samplings required: w?f ‘~(‘X/R‘xD,/3 - l/24)’ where SNR’ is given by N>
1 ~- 1 SNR’=SNR+
1 ~ pain SNR’*
(9)
~P(%co)
or
When coming to select the required ADC, two contradicting parameters must be considered, namely, its dynamic range resolution - n, bits, and its throughput sampling rate - in MHz. The the lower is the available higher the resolution, sampling rate of the ADC. the desired performance limits Obviously, could be obtained by increasing Nthe number of samplings. We are, though, limited by the sampling rates of available ADCs. A 12-bit ADC (a DATEL ADC-500BMC with an additional sample and hold circuit SHM-45 for example) is limited to a 1.25 MHz throughput rate, which puts an upper limit on the number N of samples possible within B seconds. We have assumed a 4 ppm spread in proton frequencies which at 1 T field gives Au,,* = 80 Hz and therefore i?J = l/ (~Av,,,) = 6ms. With this estimation of 7?$, the duration of available echo signal for sampling is ^- 2 Q = 12 ms and we are limited to NI 15 000 samplings. In case of a read-gradient, the z value is further reduced as can be seen according to the relation Tg = l/[ (y/27r) GAx]. Thus, for a 1 mm pixel v will be about 5 ms. with Grad = 0.005 T/m, Longer 7$ values may also be obtained by using a thinner slice; however, this will reduce the SNR ratio significantly. As we recall, there is a connection between the closeness of the coils to the centre of the sample, and the number, N,, and radius, a~, of the coils. This close packing estimation can be shown to be given by N rcl 4(R+ c
d*
(10)
at and for a case where M R one obtains the approximation R = !m,,d ( NC). The detected signal is demodulated to the main Larmor frequency and then passed through a all frequency bandpass filter, which removes components of noise outside the practical frequency window. The Nyquist criterion requires the sampling rate to be at least two times the highest frequency. Since we are only allowed for 128 effective (averaged) samplings within the 10 ms duration of the signal, then the effective sampling rate is 12 800 Hz only, and the frequency band is limited to O-6400 Hz. An immediate calculation shows that in this case, with 0.005 T/m gradient,
an upper limit on the slice thickness is 3 cm. For a thicker slice, the number of averages could be reduced. Thus, a 1’Ssamplings average will allow for a 0.3 m slice. Availability of ADC with much higher sampling rates will allow for a full scale slice imaging without losing the averaging capabilities. This averaging procedure reduces the high frequency components of the noise which accompany the original signal but does not reduce the low frequency components. This procedure of averaging is thus equivalent to a seiective low-pass filter, where only high-frequency noise components are filtered out, while true stgnal high-frequency cornpo~len~ are not. Low frequency components of noise are not removed by this method. As we see, this method is limited, with current sampling technologies and SNR values, to quite small samples, and with a coarse resolution. Obviously, the smaller the coil’s radius, the smaller the value of I?, and t.he smaller the required SNR value becomes. We also observe from this relation, that increase in the number of coils increases the value of SNR requirement, not to mention the increase in mutual interactions between coils, which brings us to another important issue to be dealt with in this article, namely, the coupling between acljacent coil detectors.
cables
+ Faraday
RGl58
2 cage
//
-, -- - 5 -
,
Coax
-$300mm
+I
i
EXPERMENTAL A preliminary experimental set-up was tested in our laboratory in order to evaluate some aspects of the proposed method. The unit consisted of a circular array constructed of 9 circular, single-turn coils that are placed symmetrically on a cylindrical perspex of radius R= 5 cm. All coils were similar (40 mm diameter) and directed towards the centre of the cylinder (see Figure 1). Each coil was made of a 2 mm copper wire with an additional RC filter in series (R= 10 R, C= 82 nF) to reduce noise. The coils were partially overlapping to minimize coupling and each coil was connect,ed through a coaxial cable to a separate mechanical switch used to connect/disconnect the detected signal to and from a single amplifier. All mechanical switches, as well as the Faraday cage, were properly grounded to reduce noise. Instead of studying a true nuclear magnetic resonance signal, we studied a much simpler source of RF signal by introducing a miniatLlre multiturns cot1 (a void solenoid) into the cylindrical array of detecting coils. This miniature solenoid was constructed by looping a thin (0.2 mm) copper wire to create three different types of solenoids: (a) a 50 turns, 6 mm diameter solenoid. (b) a 10 turns, I .5 mm diameter solenoid. (c) a 10 turns, 3 mm diameter solenoid.
x 6 mm
length
x 5 mm
length
x 1 mm
length
Each of these solenoids had a permanent fitment to a BNC type connector, which in turn was
Fiie 1 ~onstI~cti(~n of the detector array. The array contains IO coils of which onl\i 9 were used in this experiment (fkr purpose of convenience). The array is made of 10 copper !S mm diameter) coils of 40 mm diameter each and with a 1.7 overlap factor (d/q,) where n is the distance between two adjacent coil centres and (4, is the radius of the coil. This overlap fktor is slightly altered in practice because of the curvature of the array as the coils arc placed in a cylindrical configuration. The actual decoupling is small but no1 completely negligible due to this curvature. F1111 decoupling is achieved by slightly modifying the directions of the edge leads of the coils in space. The array is placed in a Faraday cage to reduce external RF noise. To that Fame purpose. an RC impedance is added in series to each coil detector
fixed to a movable plexiglass plate that made a part of the Faraday cage surrounding the array of detectors. When this top plate was set in place, the solenoid would be at the exact level of the centres of the coils. By moving and rotating the top cover plate, the solenoid could be moved around in the xy plane and placed anywhere in this plane and at any desired direction. A magnetic moment precessing clockwise in the my plane, induces a signal in the detecting coil, which is given by’ & = m+w a- g[sin wtV x (TX P/X-? - cos ot V X 6X r/r31 1 - ii while, on the other hand, an alternating
(11) magnetic
259
Prelimina?
experbnental
evaluation
of an inverse source imaging
procedure:
field in a solenoid directed in the x-direction will create a magnetic moment m(t) = m, 4 cos wt and thus induce a voltage in the coil given by 5AC=m,07T~[sinwtVx(Txr/r5)].;1
(12)
or CR = m, w 7~ d[sin
&A, - cos w BJ
(13)
and [AC=~w~~sinwtA,i where A, and Be are defined respectively, with
(14) by iii * fii and iii * 9
fg = f2(rj - rOJ and fJ = f,(r, - roj) In the above, m,, is the magnetization, o is the precessing (Larmor) frequency, e is the coil’s radius and r is the distance from the coil to the magnetic moment. n is the vector perpendicular to the area enclosed by the coil. It can be shown, that for either point-sources or solenoids, the signal is given by
so that in both cases, the same inverse-source procedure is applicable, though the M matrix is not the same for the two cases. In a detector array designed for recording of a sample of certain size, the array radius R is a fixed and given parameter. The number of detectors in the array, together with the radius a of each detector, are more flexible to design constraints than is the array radius. In order to ensure decoupling of adjacent coils in the array from mutual interactions, the overlap ratio d/a,, = 1.7 must be kept”, where d is the distance between the centres of two adjacent coils (Figure 2). With a total of N, detectors, constructed in groups of exactly n, coils in each group, placed in a concentric structure around the symmetry
D. Kwiat
and S. Einav
axis of the array; the perimeter of each group of detectors is 297R and it can be verified that this perimeter is also equal to n,d, so n, = 2~-R/1.7%. The magnetic field B1, induced by a unit current in a circular coil of radius a, at a distance z from the coil’s centre, along its symmetry axis, is given by 4 B,(z) = kk9 2 (4 t z2)3’2 with p0 = 47r x lo-‘. The magnitude of the reciprocal EMF signal that is induced in the coil due to a precessing magnetic moment located at a distance z from the coil is well known and is given by
Ny7h2% 4 5lcm3 = 16+’ K,T
where N is the number of protons in 1 cm3 of water, y is the proton gyromagnetic ratio, h is the Planck constant, R,, is the constant magnetic field, K, is the Boltzmann constant and T is the sample’s temperature. For a sample at room temperature ( T= 300 K) we obtain 5.41 x lo-‘@ a2 clcrn3 = 0.86 @ B1 = (4 + zy3’* Since the most remote source in the array can be at a distance z,,,~, = 2R, the weakest signal from 1 cm3 of water would be given by 5.41 x lo-’ @ a2 5.41 x lo-’ $ (18) Lnin = ($ + 4R)3’2 = /J&l + 0.29?$)3’2 and for nc + 1 one obtains lcm3 ~ 3.4 x 1C6 G (19) nZ a, We are interested in a large number of coils, the resolution of the n c, since this determines image, and therefore, in order to keep the overlap ratio at 1.7 we have to reduce the radii of the coils. Near the coil centre (z 4 G+,) we have f
BE!%
RC
RC158
Coax
F-m
BNC
RG158
filter
RC
filter
Coax BNC
Figure 2 Due to the spatial behaviour of the magnetic field lines, the mutual induction between two adjacent overlapping coils depends upon the amount of overlap, which we measure by an overlap factor a = d/a,, where d is the distance between the centres of the coils, and a, is their radius. It can be shown”, that when o = 1.7, a full decoupling of the coils from mutual induction is achieved. Each detecting coil is made of a copper (3 mm diameter) wire and has 40 mm diameter. An R= 10 R resistor and a C= 82 nF capacitor are added in series for the purpose of impedance matching
(16)
’
PO
l 2 (L$+zf?)3’2==2n, and therefore the sensitivity of the coil to the signal increases linearly with the decrease in coil’s radius. This is contrary to the situation when the source is located far from the coil (z + a~), where we have a2 ,r”“$ B1 & (21) 2 23 2 (4 + zy* so that the sensitivity of the coil to the signal increases as 4. This phenomenon is well known and by equating aB,(z)/a~, to zero, we can see that for a single surface coil, the optimal working distance from the source is where variations in the source distance from the coil will have a minimal impact on the amplitude of-the induced signal. This happens for z= uJd2 and thus roptimal ~ 0.707~. For a detector array we obtain the result i&3
= 3.4 x 1o-6 @ $ f
c
(22)
and for an array with a radius expected H-NMR signal intensity water is given by (&~
= 7.2 x I&’
B;: 1 72;
of R= 5 cm the from a 1 cm3 of
f23)
The weakest noise signal (voltage, peak-to-peak) that we managed to reduce the system to (including cables) at the amplifier output was approximately 1 mV. Much of this noise can still be removed quite easily, since its spectrum is at frequencies in the range of 10 MHz and higher, while our source and measured signals were in the range of 200 kHz-2 MHz. This noise was measured while a single coil detector was connected to the amplifier, and all the rest were disconnected. connecting several coils sim~lltaneously to the amplifier, did not influence the noise level at all (though there are slight variations (~10%) in noise levels for different coils). Activating or deactivating the RF generator also did not influence the noise level. Let the weakest noise level signal (in volts) be denoted by ‘4. The weakest detectable significant signal level &, from a given pixel, must satisfy then the relation { A, L n,. Since i;, is connected to {,,J through fip, th e number of pixels per 1 cm”, one obtGns (24)
where the noise level n, is expressed in microvolts. The number of pixels per cmJ may be estimated for a 1 cm thick slice by Np =* N,/rl-?! -‘I nz/rrIP, and thus iv n, 42y-j~
(25)
In the case of R = 10 cm and NC = 128 we obtain for a 5 T magnet: A~~67dB+2Olog~dB
(26)
and we can see that a crucial factor for the method to be applicable is a low noise level. If the total noise in the system (including cables and sample) can be kept at levels as low as 10 p,V, an amplification gain of 77 dB will be enough to acquire an image of a 10 cm radius slice, with resolution of 20 cm/128 = 1.6 mm per pixel. The amplifier that was used for our experiment (J&Yu~ 3) gives a 35 dB maximal amplification gain for a 200 kHz sinusoidal signal. The couple of 1 nF capacitors at the output terminals serve as a filter to the high frequency noise, without affecting the amplification gain. This amplifier showed a I mV noise signal at its output when no signal was measured. The true noise level is thus assumed to be at around 20 JLV, a figure which can be further reduced by proper low-pass filtering of the very high frequency components of the noise. We tested the coupling between two adjacent coils by the following procedure. In one coil, the signal induced from the solenoidal source was measured. We then introduced a signal from a
separate source (at the same level as the signal measured in the first coiI), directly into an adjacent coil, and measured the amount of perturbation introduced into the original signal at the first coil. For a 400 mV signal (peak-to-peak) measured originally at coil 1, we measured 2.0-3.0 mV changes when the other coils were activated (one of each of the other coils was activated in turn, in order to record the residual coupling, separately). The coupling between the coils was thus measured to be less than 1% when the array was void. Of course, introduction of the sample into the array, may increase the mutual coupling to some extent, but the main cause for coupling between adjacent coils, namely, mutual induction was successfully removed by the overlapping construction of coils. The decoupling of adjacent coils can be achieved either by the overlappin method or via a special algorithm. Previously’* F;‘, we have calculated the exact coupling that exists between two circular coplanar coils. We have also developed a ~thematical algori~m for removing the coupling between coils in an array. The induced EMF in an adjacent coil was calculated and plotted for values of cy= d/q, (d is the distance between coil centres and q, is their radius) in the range [O2.51, which means that we start from a full overlap configuration (d = 0) through a fully separated tacgential configuration with (Y= 2, and from this point on, the coils do not overlap so the field-lines intensity decreases approximately as I/&. Next, we suggested a mathematical algorithm for reducing the coupling between coils in an array. This algorithm is based on calibration of the initial coupling coefficients between the coils. This iterative procedure is based upon the assumption that the induced currents are coherent, and thus the parameters involved can be assumed time independent. Instead of calculating the matrix M, as needed for the imaging process (see equation (8)), we used the following procedure. We introduced the solenoidal source into the array, and placed it at nine different midpoints which we denoted by letters A to I (see Figure 4). For each location, we recorded the induced signal in each of the nine
261
Pwhinaly Tabie
experimental
n/aluation
of an inverse source imaging
procedure:
D. Kwiak nnd S. Kinav
1
A
B
c
D
E
F
G
H
I
1 2 3 4 5
24.00 10.00 10.40 8.60 8.80
11.60 8.20 10.80 9.20 10.00
7.20 8.00 11.20 10.00 11.60
18.50 17.50 16.00 10.40 9.60
10.40 10.00 12.80 12.80 13.00
7.00 8.40 11.40 12.80 20.00
11.00 19.20 40.00 16.50 10.80
8.00 10.80 20.40 25.60 17.20
7.80 10.00 14.60 19.00 48.00
F 8 9
9.40 9.00 10.40 11.80
12.80 10.80 15.20 12.00
26.50 16.50 20.00 13.40
9.20 9.20 11.20
11.40 11.60 10.80 11.40
16.40 21.20 11.60 12.00
10.00 9.80 9.40 12.00
10.40 11.40 9.80 12.00
21.20 12.80 11.40 14.40
Table
2
0.09 0.02 0.01 0.03 -0.26 0.03 0.00 0.05 0.01
-0.11 0.03 0.00 0.17
0.03 0.01 0.00 -0.05 -0.07 0.01 0.05
-0.07 0.00 -0.03 0.01 0.00
0.00 0.03 0.00 0.05 -0.19 0.03 -0.03 0.12 -0.01
0.00
0.01
detecting coils, constituting the array. In each coil, the peak-to-peak amplitude of the sinusoidal signal was read. The results were put in a matrix form (Tubk I) where each column represents the actual signals (peak-to-peak voltage) at the appropriate location within the q plane. Thus, for instance, with a source placed at the middle of square B, the signal at coil No. 6 was 10.80 mV. The inverse matrix M1 was then calculated ( Tu~~ 2).
Repeating next, for three different types of solenoidal sources, we measured the signals at the detecting coils for three different experiments, where in each, a different type of source was placed at the same location point: ABDE (a point which lies in between squares A, B, D and E). The source was directed at an angle of 270” from the yaxis (0’). Table 3 shows the results obtained. We see that the closest representation to the true image was obtained in case c. This is reasonable, since this source was the smallest in size of all three, and therefore represented more accurately Table
3
coil No.
source signal
1 2 3 4 5 6 7 8 9
262
15.00 10.20 12.00 10.00 10.40 10.40 11.20 11.60 12.00
a
result 0.3 0.0 0.0 -0.1 1.2 -0.1 0.0 -0.2 0.0
source
b
SOurcf! c
signal
result
signal
16.13 11.43 12.50 10.25 10.35 10.15 10.70 11.40 11.60
0.3 0.2 0.0 0.2 0.3 0.0 0.0 0.0 0.0
21.00 13.00 11.40 11.00 10.80 11.20 11.60 12.80 13.20
result 0.4 0.3 0.0 0.4 0.1 0.0 -0.1 0.1 0.0
0.01 0.00 0.02 0.01 -0.06 -0.03 0.00 0.00 0.04
-0.01 0.12 -0.06 0.05 -0.29 0.19 0.00 0.04 -0.03
0.05 -0.22 0.12 -0.03 0.15 -0.07 0.00 -0.02 0.01
-0.06 0.41 -0.02 0.15 -0.72 0.08 -0.02 0.14 0.02
-0.02 -0.38 -0.07 -0.36 1.57 -0.24 0.02 -0.34 -0.04
precise placement. Also its magnetic field is more commensurate with the description of a precessing magnetic moment, since deviations from our theoretical model become significant for a solenoid with increased dimensions. The original source and the reconstructed image for the third are represented in a matrix form (Table 4) where only abstract values are used for the image, and where we have averaged the original unit (normalized) source, equally between the four squares. These results are far from being significant at this stage for the following reasons. First, we have employed a very small number of coils and the resolution is quite bad. Second, we do not have any considerations of the sensitivity of the method to small variations in source power and location. Third, serious work should still be performed concerning the non-singularity of the M matrix as well as its sensitivity and accuracy. Also, one should explain in detail the significance (or insignificance) of the negative signs of some of the results in the output vectors. The experimental work should continue in the following directions: (a) Increase the number of coils. (b) Reduce noise signals. Table
4
Reconstructed source 0.25 0.25 0.00
0.25 0.25 0.00
Image
0.00 0.00 0.00
0.4 0.4 0.1
0.3 0.1 0.1
0.0 0.0 0.0
(c) Employ completely automatic sampling, filtering, and signal processing procedures. (d) Introduce the system into a real NMR environment.
REFERENCES 1.
‘Ih.
3.
CONCLUSIONS
4.
A method of ob~ining an image by solving the inverse source problem, as presented in detail in our earlier work’, meets heavy limitations due to sensitivity, mutual coupling and noise. The receiving coil sensitivity has to be increased by at least an order of magnitude, since in the detector array, each coil has a cross-sectional area that is much smaller than a conventional receiver in MRI. Signals at the coils were recorded and fed into the vector S, from which, the image vector p is calculated. Decoupling of coils from each other was implemented experimentally by overlapping methods as described in recent literature. A simple, small scale, experimental set-up of the method was tested in order to evaluate the feasibility of this imaging modality for imaging an object using the method of multiple coil detectors. SNR problems were considered and included in the discussion. As demonstrated, noise is not a prohibitive factor and can be overcome. Altogether, our experimental set-up was limited, and it managed to verify the anticipated, namely, that a high amplification gain, together with a reduced noise level, are the most important factors in the successful impleInentation of this method. We would however not exclude the possibili~ that in the future this approach would be the preferred choice in cases where ultrafast imaging is required at the expense of image resolution, such as in the case of functional imaging.
5.
6.
?I.
9.
10. 11. 12. 13. 14. 15. 16. I?.
Kwiat D, Einav S, Navon G. A decoupled coil detector array for fast image acquisition in magnetic resonance imaging, Med Phys, 1991; 18(2): 251-65. Roemer PB it al,, Simultaneous multiple surface coil NMR imaging, 7th Annual, Sot Magn Reson Med, August 1988; 875. Hayes CE d al. Volume imaging with MR phased array, 8th Annual, Sot Magn Reson Med, August 1989; 175. Roemer PB et aL, The Z-dimensional NMR phased array, 8th ~~~~nu~l, Sot Magn Reson Med, August 1989: 176. Jesmanowicz A rf al., Parallel acquisition image processing, 8th AnnuaL, Sot Magn Reson Med. August 19893 923. Tropp J, Derby K, Cross-talk and signal LO noise losses in quadrature reception, &h Annual, Sot Magn Reson, August 1989; 940. Prammer MG et al., Optimal decoupling of multipietuned receiver coils, 8th Annual, Sot Magn Reson Med, August 1989; 946. Okamoto K el al., SNR improvement with multiple surface coils and weighted addition, 8th Annual, Sot Magn Reson Mad, August 1989; 953. Lauterbur PC, Image formation by induced local interactions: examples employing nuclear magnetic resonance, Nature Friday (London) 1973; 242: 190-l. Hutchinson M, R&f U, Fast MRI data acquisition using multiple detectors, Map &on Med, 1988; 6: 87-91. Purcell EM, &~tticil)l and Magnetism. Berkeley Physics Course Vol. 2, 2nd edn, McGraw-Hill, 1986. Freeman R, ;2 Handbook of Nuclear Mqnetir Kesonnnce, Longman Scientific and Technical, 1988. Hoult D, Richards R.E, The signal-to-noise ratio of’ the nuclear magnetic resonance experiment. J Magr? !&son, 1976; 24: 51-85. Hoult D, fauterbur PC, The sensitivity of the zeugmatographic experiment involving human samples, J &fagn Reson, 1979; 34Z 425-33. -Jackson .JD, Class&-al ~~~fro~~~arn~~~. Chs. 5-7. 9. John Wiley & Sons, IVew York, 2nd edn, 1975. Plonscy R, Collin RE, P~r~f~~~s und .4~l~r~t~vr~s oJ.~~cfr~ ~~~~t~r fields, Chs. 6-9, McGraw-Hill, New York, 1961. Kwiat D, Saoub S, Einav S, Mutual induction between ii~~ighh(~ring coils, I&YX T~r~ns Riom~rl E:nf, 1992; 39: 5.
263
Ph.
Vol.
17. No. 4. pp. 257-m, 19% Ehier Srience hd Lbr RES Printed in Great Britain I :FJo-433/95 s1o.nn + 0.00
Preliminary experimental evaluation of an inverse source imaging procedure using a decoupled coil detector array in magnetic resonance imaging D. Kwiat and S. Einav Department University,
of Biomedical Engineering, Ramat-Aviv 69978, Israel
Krwivcd
accepted
1992,
October
School
of Engineering,
Tel-Aviv
1993
ABSTRACT Rerently, rue have discussed smeral of the aspects involved in the detector array concept in magnetic monance imaging where an image is obtained by applyi>Lg inverse source procedure,s to the data assembled by an array _ of coil detectors surrouading the object. In this work we describe an ex@rimental setup, where a detector array uas ron\tructvd of 9 coils, that gives a coarse resolution of3 x 3 pixels. By measun’ng the induced current signals over thi.c al-ray of roils, a relationship is established between the set of signals and the structure of the body under investigation. Through matrix inversion. reconstruction of the original source from the detected signals is possible. In this prelimitlary experimental setup, 71~ did not use an actual nuclear magnetic resnnanrc signal. Instead, a miniature .rolunoid was used as a sourre, lo simulate a prece.kng magnetir moment. Keywords Med.
Eng.
Magnetic Phys.,
resonance 199.5,
Vol.
imaging, 17. 257463,
inverse
A new model of MRI modality was described recently’, where MRI procedure is based on the simultaneous recording of signals in an array of closely packed, decoupled, coil detectors instead of zeugmatography. By using an array of N,., mutually decoupled, coils an image with a resolution of & voxels can be obtained, where the whole procedure of image acquisition is performed within the duration of a single FID. The large number of detectors making up this array requires that particular attention be paid to sensitivity, resolution and SNR problems, as well as to the coupling caused by mutual inductance between adjacent coils. Several studies have been published already z-H describing experimental work carried out about SNR improvement and decoupling of coils. In this work we use the partial overlay method for the decoupling of adjacent coils’-“* -8. Based on the previously described analysis’, an array of coil detectors with negligible coupling was constructed.
A discussion is required single coil detector. Two
AND
imaging
procedure,
detector
array
.June
INTRODUCTION
SENSITIVITY RESOLUTION CONSIDERATIONS
source
SNR
of the resolution of a adjacent pixels will have
nearly similar signals seen by the detector. Without noise presence, the resolution of the detector will merely depend on the dynamic range (resolution) of the ADC and the distance r between the detector and the sources (signal intensity is inversely proportional to r7). Let S,,, be the average signal intensity at a detector, obtained from the whole volume activated. Let SNR denote the average signal-to-noise-ratio level at one of the coil detectors. The noise amplitude at a certain instant of measurement may be described by
where t(t) is a random number -1 G 5 G 1. Let q stand for the distance from the nearest source (voxel) to the coil, let x be the distance from an arbitrary source (voxel) to the coil. Let s, (x,,) be the amplitude of the signal at the detector from a single unit source voxel (relative proton density p = 1) at q. Recording this signal repeatedly N times (N acquisitions) and averaging, we showed’ that the difference between two signals (averaged by N samplings) from two adjacent voxels at x and at x+ 6x is given by: (2)
Preliminary
experimental
evaluation
of an inverse
source imaging procedure:
where 0, E @/ax. In order for the system’s resolution to be able to detect it, (as), should be greater than the lowest meaningful signal, which is composed of the lowest detectable clean signal 2-“rS,, plus noise:
(3) where n,+ 1 is the resolution of the ADC used for recording the data. We then obtain:
(4) where R is the distance from a coil’s centre to the array’s centre and gets most of its contribution from the closest lying source and, therefore, from si (q). If all voxels had the same relative proton densities p = 1, and were all located at q, the total signal would have become S,, = N,s, (x,J. However, due to their spatial distribution and different proton densities, and differing coil effective areas (direction in space), the total signal S,, is reduced by a filling factor f < 1 defined by V-’
d3v sr(x) cos 8 p(x) I
j-= =
C
sl(xi)Pi
cos 4
(5) c. Sl(xi)
where, in the integral, cos 8 is the effective area of the coil, as seen by the source at X. Equation (5) can be reformulated in a discrete form to read Stot = sh)Nvf (6) where N, is the total number of voxels into which the imaged volume Vis divided. In the same manner, we shall assume that m can be represented by Sp( 3~0)NJ where lip(q) is the inherent source noise from a unit density source at 3~0.We, thus, obtain the minimum resolution requirement: ln 2 n, = In wJ.fR3 (7) 4 I which allows us to estimate the minimal resolution required of the ADC for the certain system geometry that we have. For a sphere of radius R, uniformly filled, with p - 1, and a small radius coil at a distance 3~0,the following expression for f can be derived:
f = 9 (In 2 + 1) 2 3 In 5 This
258
expression
0 indicates
(8) the correctness
of our
D. Kwiat
and S. Einav
above statement about the decreased sensitivity of n, to variations in 3~0,R and $ By inserting the minimal resolution requirement, eqn (7), into eqn (8) and assuming that the worst resolution is obtained from the innermost voxel where x = R, we obtain the relationship between the SNR value and the minimal number of samplings required: w?f ‘~(‘X/R‘xD,/3 - l/24)’ where SNR’ is given by N>
1 ~- 1 SNR’=SNR+
1 ~ pain SNR’*
(9)
~P(%co)
or
When coming to select the required ADC, two contradicting parameters must be considered, namely, its dynamic range resolution - n, bits, and its throughput sampling rate - in MHz. The the lower is the available higher the resolution, sampling rate of the ADC. the desired performance limits Obviously, could be obtained by increasing Nthe number of samplings. We are, though, limited by the sampling rates of available ADCs. A 12-bit ADC (a DATEL ADC-500BMC with an additional sample and hold circuit SHM-45 for example) is limited to a 1.25 MHz throughput rate, which puts an upper limit on the number N of samples possible within B seconds. We have assumed a 4 ppm spread in proton frequencies which at 1 T field gives Au,,* = 80 Hz and therefore i?J = l/ (~Av,,,) = 6ms. With this estimation of 7?$, the duration of available echo signal for sampling is ^- 2 Q = 12 ms and we are limited to NI 15 000 samplings. In case of a read-gradient, the z value is further reduced as can be seen according to the relation Tg = l/[ (y/27r) GAx]. Thus, for a 1 mm pixel v will be about 5 ms. with Grad = 0.005 T/m, Longer 7$ values may also be obtained by using a thinner slice; however, this will reduce the SNR ratio significantly. As we recall, there is a connection between the closeness of the coils to the centre of the sample, and the number, N,, and radius, a~, of the coils. This close packing estimation can be shown to be given by N rcl 4(R+ c
d*
(10)
at and for a case where M R one obtains the approximation R = !m,,d ( NC). The detected signal is demodulated to the main Larmor frequency and then passed through a all frequency bandpass filter, which removes components of noise outside the practical frequency window. The Nyquist criterion requires the sampling rate to be at least two times the highest frequency. Since we are only allowed for 128 effective (averaged) samplings within the 10 ms duration of the signal, then the effective sampling rate is 12 800 Hz only, and the frequency band is limited to O-6400 Hz. An immediate calculation shows that in this case, with 0.005 T/m gradient,
an upper limit on the slice thickness is 3 cm. For a thicker slice, the number of averages could be reduced. Thus, a 1’Ssamplings average will allow for a 0.3 m slice. Availability of ADC with much higher sampling rates will allow for a full scale slice imaging without losing the averaging capabilities. This averaging procedure reduces the high frequency components of the noise which accompany the original signal but does not reduce the low frequency components. This procedure of averaging is thus equivalent to a seiective low-pass filter, where only high-frequency noise components are filtered out, while true stgnal high-frequency cornpo~len~ are not. Low frequency components of noise are not removed by this method. As we see, this method is limited, with current sampling technologies and SNR values, to quite small samples, and with a coarse resolution. Obviously, the smaller the coil’s radius, the smaller the value of I?, and t.he smaller the required SNR value becomes. We also observe from this relation, that increase in the number of coils increases the value of SNR requirement, not to mention the increase in mutual interactions between coils, which brings us to another important issue to be dealt with in this article, namely, the coupling between acljacent coil detectors.
cables
+ Faraday
RGl58
2 cage
//
-, -- - 5 -
,
Coax
-$300mm
+I
i
EXPERMENTAL A preliminary experimental set-up was tested in our laboratory in order to evaluate some aspects of the proposed method. The unit consisted of a circular array constructed of 9 circular, single-turn coils that are placed symmetrically on a cylindrical perspex of radius R= 5 cm. All coils were similar (40 mm diameter) and directed towards the centre of the cylinder (see Figure 1). Each coil was made of a 2 mm copper wire with an additional RC filter in series (R= 10 R, C= 82 nF) to reduce noise. The coils were partially overlapping to minimize coupling and each coil was connect,ed through a coaxial cable to a separate mechanical switch used to connect/disconnect the detected signal to and from a single amplifier. All mechanical switches, as well as the Faraday cage, were properly grounded to reduce noise. Instead of studying a true nuclear magnetic resonance signal, we studied a much simpler source of RF signal by introducing a miniatLlre multiturns cot1 (a void solenoid) into the cylindrical array of detecting coils. This miniature solenoid was constructed by looping a thin (0.2 mm) copper wire to create three different types of solenoids: (a) a 50 turns, 6 mm diameter solenoid. (b) a 10 turns, I .5 mm diameter solenoid. (c) a 10 turns, 3 mm diameter solenoid.
x 6 mm
length
x 5 mm
length
x 1 mm
length
Each of these solenoids had a permanent fitment to a BNC type connector, which in turn was
Fiie 1 ~onstI~cti(~n of the detector array. The array contains IO coils of which onl\i 9 were used in this experiment (fkr purpose of convenience). The array is made of 10 copper !S mm diameter) coils of 40 mm diameter each and with a 1.7 overlap factor (d/q,) where n is the distance between two adjacent coil centres and (4, is the radius of the coil. This overlap fktor is slightly altered in practice because of the curvature of the array as the coils arc placed in a cylindrical configuration. The actual decoupling is small but no1 completely negligible due to this curvature. F1111 decoupling is achieved by slightly modifying the directions of the edge leads of the coils in space. The array is placed in a Faraday cage to reduce external RF noise. To that Fame purpose. an RC impedance is added in series to each coil detector
fixed to a movable plexiglass plate that made a part of the Faraday cage surrounding the array of detectors. When this top plate was set in place, the solenoid would be at the exact level of the centres of the coils. By moving and rotating the top cover plate, the solenoid could be moved around in the xy plane and placed anywhere in this plane and at any desired direction. A magnetic moment precessing clockwise in the my plane, induces a signal in the detecting coil, which is given by’ & = m+w a- g[sin wtV x (TX P/X-? - cos ot V X 6X r/r31 1 - ii while, on the other hand, an alternating
(11) magnetic
259
Prelimina?
experbnental
evaluation
of an inverse source imaging
procedure:
field in a solenoid directed in the x-direction will create a magnetic moment m(t) = m, 4 cos wt and thus induce a voltage in the coil given by 5AC=m,07T~[sinwtVx(Txr/r5)].;1
(12)
or CR = m, w 7~ d[sin
&A, - cos w BJ
(13)
and [AC=~w~~sinwtA,i where A, and Be are defined respectively, with
(14) by iii * fii and iii * 9
fg = f2(rj - rOJ and fJ = f,(r, - roj) In the above, m,, is the magnetization, o is the precessing (Larmor) frequency, e is the coil’s radius and r is the distance from the coil to the magnetic moment. n is the vector perpendicular to the area enclosed by the coil. It can be shown, that for either point-sources or solenoids, the signal is given by
so that in both cases, the same inverse-source procedure is applicable, though the M matrix is not the same for the two cases. In a detector array designed for recording of a sample of certain size, the array radius R is a fixed and given parameter. The number of detectors in the array, together with the radius a of each detector, are more flexible to design constraints than is the array radius. In order to ensure decoupling of adjacent coils in the array from mutual interactions, the overlap ratio d/a,, = 1.7 must be kept”, where d is the distance between the centres of two adjacent coils (Figure 2). With a total of N, detectors, constructed in groups of exactly n, coils in each group, placed in a concentric structure around the symmetry
D. Kwiat
and S. Einav
axis of the array; the perimeter of each group of detectors is 297R and it can be verified that this perimeter is also equal to n,d, so n, = 2~-R/1.7%. The magnetic field B1, induced by a unit current in a circular coil of radius a, at a distance z from the coil’s centre, along its symmetry axis, is given by 4 B,(z) = kk9 2 (4 t z2)3’2 with p0 = 47r x lo-‘. The magnitude of the reciprocal EMF signal that is induced in the coil due to a precessing magnetic moment located at a distance z from the coil is well known and is given by
Ny7h2% 4 5lcm3 = 16+’ K,T
where N is the number of protons in 1 cm3 of water, y is the proton gyromagnetic ratio, h is the Planck constant, R,, is the constant magnetic field, K, is the Boltzmann constant and T is the sample’s temperature. For a sample at room temperature ( T= 300 K) we obtain 5.41 x lo-‘@ a2 clcrn3 = 0.86 @ B1 = (4 + zy3’* Since the most remote source in the array can be at a distance z,,,~, = 2R, the weakest signal from 1 cm3 of water would be given by 5.41 x lo-’ @ a2 5.41 x lo-’ $ (18) Lnin = ($ + 4R)3’2 = /J&l + 0.29?$)3’2 and for nc + 1 one obtains lcm3 ~ 3.4 x 1C6 G (19) nZ a, We are interested in a large number of coils, the resolution of the n c, since this determines image, and therefore, in order to keep the overlap ratio at 1.7 we have to reduce the radii of the coils. Near the coil centre (z 4 G+,) we have f
BE!%
RC
RC158
Coax
F-m
BNC
RG158
filter
RC
filter
Coax BNC
Figure 2 Due to the spatial behaviour of the magnetic field lines, the mutual induction between two adjacent overlapping coils depends upon the amount of overlap, which we measure by an overlap factor a = d/a,, where d is the distance between the centres of the coils, and a, is their radius. It can be shown”, that when o = 1.7, a full decoupling of the coils from mutual induction is achieved. Each detecting coil is made of a copper (3 mm diameter) wire and has 40 mm diameter. An R= 10 R resistor and a C= 82 nF capacitor are added in series for the purpose of impedance matching
(16)
’
PO
l 2 (L$+zf?)3’2==2n, and therefore the sensitivity of the coil to the signal increases linearly with the decrease in coil’s radius. This is contrary to the situation when the source is located far from the coil (z + a~), where we have a2 ,r”“$ B1 & (21) 2 23 2 (4 + zy* so that the sensitivity of the coil to the signal increases as 4. This phenomenon is well known and by equating aB,(z)/a~, to zero, we can see that for a single surface coil, the optimal working distance from the source is where variations in the source distance from the coil will have a minimal impact on the amplitude of-the induced signal. This happens for z= uJd2 and thus roptimal ~ 0.707~. For a detector array we obtain the result i&3
= 3.4 x 1o-6 @ $ f
c
(22)
and for an array with a radius expected H-NMR signal intensity water is given by (&~
= 7.2 x I&’
B;: 1 72;
of R= 5 cm the from a 1 cm3 of
f23)
The weakest noise signal (voltage, peak-to-peak) that we managed to reduce the system to (including cables) at the amplifier output was approximately 1 mV. Much of this noise can still be removed quite easily, since its spectrum is at frequencies in the range of 10 MHz and higher, while our source and measured signals were in the range of 200 kHz-2 MHz. This noise was measured while a single coil detector was connected to the amplifier, and all the rest were disconnected. connecting several coils sim~lltaneously to the amplifier, did not influence the noise level at all (though there are slight variations (~10%) in noise levels for different coils). Activating or deactivating the RF generator also did not influence the noise level. Let the weakest noise level signal (in volts) be denoted by ‘4. The weakest detectable significant signal level &, from a given pixel, must satisfy then the relation { A, L n,. Since i;, is connected to {,,J through fip, th e number of pixels per 1 cm”, one obtGns (24)
where the noise level n, is expressed in microvolts. The number of pixels per cmJ may be estimated for a 1 cm thick slice by Np =* N,/rl-?! -‘I nz/rrIP, and thus iv n, 42y-j~
(25)
In the case of R = 10 cm and NC = 128 we obtain for a 5 T magnet: A~~67dB+2Olog~dB
(26)
and we can see that a crucial factor for the method to be applicable is a low noise level. If the total noise in the system (including cables and sample) can be kept at levels as low as 10 p,V, an amplification gain of 77 dB will be enough to acquire an image of a 10 cm radius slice, with resolution of 20 cm/128 = 1.6 mm per pixel. The amplifier that was used for our experiment (J&Yu~ 3) gives a 35 dB maximal amplification gain for a 200 kHz sinusoidal signal. The couple of 1 nF capacitors at the output terminals serve as a filter to the high frequency noise, without affecting the amplification gain. This amplifier showed a I mV noise signal at its output when no signal was measured. The true noise level is thus assumed to be at around 20 JLV, a figure which can be further reduced by proper low-pass filtering of the very high frequency components of the noise. We tested the coupling between two adjacent coils by the following procedure. In one coil, the signal induced from the solenoidal source was measured. We then introduced a signal from a
separate source (at the same level as the signal measured in the first coiI), directly into an adjacent coil, and measured the amount of perturbation introduced into the original signal at the first coil. For a 400 mV signal (peak-to-peak) measured originally at coil 1, we measured 2.0-3.0 mV changes when the other coils were activated (one of each of the other coils was activated in turn, in order to record the residual coupling, separately). The coupling between the coils was thus measured to be less than 1% when the array was void. Of course, introduction of the sample into the array, may increase the mutual coupling to some extent, but the main cause for coupling between adjacent coils, namely, mutual induction was successfully removed by the overlapping construction of coils. The decoupling of adjacent coils can be achieved either by the overlappin method or via a special algorithm. Previously’* F;‘, we have calculated the exact coupling that exists between two circular coplanar coils. We have also developed a ~thematical algori~m for removing the coupling between coils in an array. The induced EMF in an adjacent coil was calculated and plotted for values of cy= d/q, (d is the distance between coil centres and q, is their radius) in the range [O2.51, which means that we start from a full overlap configuration (d = 0) through a fully separated tacgential configuration with (Y= 2, and from this point on, the coils do not overlap so the field-lines intensity decreases approximately as I/&. Next, we suggested a mathematical algorithm for reducing the coupling between coils in an array. This algorithm is based on calibration of the initial coupling coefficients between the coils. This iterative procedure is based upon the assumption that the induced currents are coherent, and thus the parameters involved can be assumed time independent. Instead of calculating the matrix M, as needed for the imaging process (see equation (8)), we used the following procedure. We introduced the solenoidal source into the array, and placed it at nine different midpoints which we denoted by letters A to I (see Figure 4). For each location, we recorded the induced signal in each of the nine
261
Pwhinaly Tabie
experimental
n/aluation
of an inverse source imaging
procedure:
D. Kwiak nnd S. Kinav
1
A
B
c
D
E
F
G
H
I
1 2 3 4 5
24.00 10.00 10.40 8.60 8.80
11.60 8.20 10.80 9.20 10.00
7.20 8.00 11.20 10.00 11.60
18.50 17.50 16.00 10.40 9.60
10.40 10.00 12.80 12.80 13.00
7.00 8.40 11.40 12.80 20.00
11.00 19.20 40.00 16.50 10.80
8.00 10.80 20.40 25.60 17.20
7.80 10.00 14.60 19.00 48.00
F 8 9
9.40 9.00 10.40 11.80
12.80 10.80 15.20 12.00
26.50 16.50 20.00 13.40
9.20 9.20 11.20
11.40 11.60 10.80 11.40
16.40 21.20 11.60 12.00
10.00 9.80 9.40 12.00
10.40 11.40 9.80 12.00
21.20 12.80 11.40 14.40
Table
2
0.09 0.02 0.01 0.03 -0.26 0.03 0.00 0.05 0.01
-0.11 0.03 0.00 0.17
0.03 0.01 0.00 -0.05 -0.07 0.01 0.05
-0.07 0.00 -0.03 0.01 0.00
0.00 0.03 0.00 0.05 -0.19 0.03 -0.03 0.12 -0.01
0.00
0.01
detecting coils, constituting the array. In each coil, the peak-to-peak amplitude of the sinusoidal signal was read. The results were put in a matrix form (Tubk I) where each column represents the actual signals (peak-to-peak voltage) at the appropriate location within the q plane. Thus, for instance, with a source placed at the middle of square B, the signal at coil No. 6 was 10.80 mV. The inverse matrix M1 was then calculated ( Tu~~ 2).
Repeating next, for three different types of solenoidal sources, we measured the signals at the detecting coils for three different experiments, where in each, a different type of source was placed at the same location point: ABDE (a point which lies in between squares A, B, D and E). The source was directed at an angle of 270” from the yaxis (0’). Table 3 shows the results obtained. We see that the closest representation to the true image was obtained in case c. This is reasonable, since this source was the smallest in size of all three, and therefore represented more accurately Table
3
coil No.
source signal
1 2 3 4 5 6 7 8 9
262
15.00 10.20 12.00 10.00 10.40 10.40 11.20 11.60 12.00
a
result 0.3 0.0 0.0 -0.1 1.2 -0.1 0.0 -0.2 0.0
source
b
SOurcf! c
signal
result
signal
16.13 11.43 12.50 10.25 10.35 10.15 10.70 11.40 11.60
0.3 0.2 0.0 0.2 0.3 0.0 0.0 0.0 0.0
21.00 13.00 11.40 11.00 10.80 11.20 11.60 12.80 13.20
result 0.4 0.3 0.0 0.4 0.1 0.0 -0.1 0.1 0.0
0.01 0.00 0.02 0.01 -0.06 -0.03 0.00 0.00 0.04
-0.01 0.12 -0.06 0.05 -0.29 0.19 0.00 0.04 -0.03
0.05 -0.22 0.12 -0.03 0.15 -0.07 0.00 -0.02 0.01
-0.06 0.41 -0.02 0.15 -0.72 0.08 -0.02 0.14 0.02
-0.02 -0.38 -0.07 -0.36 1.57 -0.24 0.02 -0.34 -0.04
precise placement. Also its magnetic field is more commensurate with the description of a precessing magnetic moment, since deviations from our theoretical model become significant for a solenoid with increased dimensions. The original source and the reconstructed image for the third are represented in a matrix form (Table 4) where only abstract values are used for the image, and where we have averaged the original unit (normalized) source, equally between the four squares. These results are far from being significant at this stage for the following reasons. First, we have employed a very small number of coils and the resolution is quite bad. Second, we do not have any considerations of the sensitivity of the method to small variations in source power and location. Third, serious work should still be performed concerning the non-singularity of the M matrix as well as its sensitivity and accuracy. Also, one should explain in detail the significance (or insignificance) of the negative signs of some of the results in the output vectors. The experimental work should continue in the following directions: (a) Increase the number of coils. (b) Reduce noise signals. Table
4
Reconstructed source 0.25 0.25 0.00
0.25 0.25 0.00
Image
0.00 0.00 0.00
0.4 0.4 0.1
0.3 0.1 0.1
0.0 0.0 0.0
(c) Employ completely automatic sampling, filtering, and signal processing procedures. (d) Introduce the system into a real NMR environment.
REFERENCES 1.
‘Ih.
3.
CONCLUSIONS
4.
A method of ob~ining an image by solving the inverse source problem, as presented in detail in our earlier work’, meets heavy limitations due to sensitivity, mutual coupling and noise. The receiving coil sensitivity has to be increased by at least an order of magnitude, since in the detector array, each coil has a cross-sectional area that is much smaller than a conventional receiver in MRI. Signals at the coils were recorded and fed into the vector S, from which, the image vector p is calculated. Decoupling of coils from each other was implemented experimentally by overlapping methods as described in recent literature. A simple, small scale, experimental set-up of the method was tested in order to evaluate the feasibility of this imaging modality for imaging an object using the method of multiple coil detectors. SNR problems were considered and included in the discussion. As demonstrated, noise is not a prohibitive factor and can be overcome. Altogether, our experimental set-up was limited, and it managed to verify the anticipated, namely, that a high amplification gain, together with a reduced noise level, are the most important factors in the successful impleInentation of this method. We would however not exclude the possibili~ that in the future this approach would be the preferred choice in cases where ultrafast imaging is required at the expense of image resolution, such as in the case of functional imaging.
5.
6.
?I.
9.
10. 11. 12. 13. 14. 15. 16. I?.
Kwiat D, Einav S, Navon G. A decoupled coil detector array for fast image acquisition in magnetic resonance imaging, Med Phys, 1991; 18(2): 251-65. Roemer PB it al,, Simultaneous multiple surface coil NMR imaging, 7th Annual, Sot Magn Reson Med, August 1988; 875. Hayes CE d al. Volume imaging with MR phased array, 8th Annual, Sot Magn Reson Med, August 1989; 175. Roemer PB et aL, The Z-dimensional NMR phased array, 8th ~~~~nu~l, Sot Magn Reson Med, August 1989: 176. Jesmanowicz A rf al., Parallel acquisition image processing, 8th AnnuaL, Sot Magn Reson Med. August 19893 923. Tropp J, Derby K, Cross-talk and signal LO noise losses in quadrature reception, &h Annual, Sot Magn Reson, August 1989; 940. Prammer MG et al., Optimal decoupling of multipietuned receiver coils, 8th Annual, Sot Magn Reson Med, August 1989; 946. Okamoto K el al., SNR improvement with multiple surface coils and weighted addition, 8th Annual, Sot Magn Reson Mad, August 1989; 953. Lauterbur PC, Image formation by induced local interactions: examples employing nuclear magnetic resonance, Nature Friday (London) 1973; 242: 190-l. Hutchinson M, R&f U, Fast MRI data acquisition using multiple detectors, Map &on Med, 1988; 6: 87-91. Purcell EM, &~tticil)l and Magnetism. Berkeley Physics Course Vol. 2, 2nd edn, McGraw-Hill, 1986. Freeman R, ;2 Handbook of Nuclear Mqnetir Kesonnnce, Longman Scientific and Technical, 1988. Hoult D, Richards R.E, The signal-to-noise ratio of’ the nuclear magnetic resonance experiment. J Magr? !&son, 1976; 24: 51-85. Hoult D, fauterbur PC, The sensitivity of the zeugmatographic experiment involving human samples, J &fagn Reson, 1979; 34Z 425-33. -Jackson .JD, Class&-al ~~~fro~~~arn~~~. Chs. 5-7. 9. John Wiley & Sons, IVew York, 2nd edn, 1975. Plonscy R, Collin RE, P~r~f~~~s und .4~l~r~t~vr~s oJ.~~cfr~ ~~~~t~r fields, Chs. 6-9, McGraw-Hill, New York, 1961. Kwiat D, Saoub S, Einav S, Mutual induction between ii~~ighh(~ring coils, I&YX T~r~ns Riom~rl E:nf, 1992; 39: 5.
263