Selective imaging among three or more chemical species

The harmonic relationship of certain chemical species is exploited to produce an MRI image of a single chemical species in the presence of at least two other chemical species with the acquisition of as few as two NMR images. If the chemical shift frequencies at a particular polarizing field strength can be approximated as the ratio of two odd integers, an evolution time can be chosen for the images acquired to cancel the contributions of two of the species in one image with a corresponding contribution in the other image. A image of the uncanceled species or of the cancelled species alone may be generated. A third image may be used to correct for inhomogeneities in the polarizing B.sub.0 field.

FIELD OF THE INVENTION 
This invention relates to nuclear magnetic resonance (NMR) imaging methods 
and apparatus and more particularly to a method of providing an image of a 
particular chemical species in the environment of two other chemical 
species. 
BACKGROUND OF THE INVENTION 
Breast augmentation or reconstructive surgery may employ implants 
containing silicone. The silicone used in breast prostheses is composed of 
poly-dimethylsiloxane with varying degrees of polymerization. Dow 
Corning's implants are approximately 40% polymerized. 
Rupture and leakage of the membrane containing the silicone is a known 
complication of these procedures. The prevalence of complications is not 
known because patients may be asymptomatic, however, in light of anecdotal 
reports of a possible link between silicone leakage and systemic 
autoimmune disease, is important to develop a sensitive noninvasive method 
to detect leaks. 
The leak or rupture may occur anywhere over the surface of an implant and 
therefore the use of three-dimensional medical imaging techniques is 
desirable. Such imaging would, in theory, allow careful scrutiny of the 
entire surface of the implant and the detection of even small pockets of 
migrating silicone near that surface. 
NMR imaging ("MRI") is one technique capable of the necessary three 
dimensional imaging. A uniform magnetic field B.sub.0 is applied to an 
imaged object along the z-axis of a Cartesian coordinate system, the 
origin of which is within the imaged object. The effect of the magnetic 
field B.sub.0 is to align the object's nuclear spins along the z-axis. In 
response to radio frequency (RF) pulses of the proper frequency oriented 
within the x-y plane, the nuclei resonate at their Larmor frequencies 
according to the following equation: 
EQU .omega.=.gamma.B.sub.0 ( 1) 
where .omega. is the Larmor frequency, and .gamma. is the gyromagnetic 
ratio which is a property of the particular nucleus. Water, because of its 
relevance abundance in biological tissue and the properties of its proton 
nuclei, is of principle concern in most imaging. The value of the 
gyromagnetic ratio .gamma. for protons in water is 4.26 kHz/Gauss and 
therefore in a 1.5 Tesla polarizing magnetic field B.sub.0, the resonant 
or Larmor frequency of water protons is approximately 63.9 MHz. The other 
primary constituent of biological tissue is fat. Larmor frequency of 
protons in fat is approximately 203 Hertz higher than that of the protons 
in water in a 1.5 Tesla polarizing magnetic field B.sub.0. 
In the well known slice selective MRI sequence, a z-axis magnetic field 
gradient, G.sub.z is applied at the time of an RF pulse so that only the 
nuclei in a slice through an object in the x-y plane are excited into 
resonance. The coherence between the nuclei decays as characterized by two 
relaxation times T.sub.1 and T.sub.2. After excitation of the nuclei, 
magnetic field gradients are applied along the x and y axes and an NMR 
signal is acquired. The gradient field along the x-axis, G.sub.x, causes 
the nuclei to precess at different resonant frequencies depending on their 
position along the x-axis; that is, G.sub.x spatially encodes the 
precessing nuclei by frequency. Similarly, the y-axis gradient, G.sub.y, 
encodes y position into the change of magnetization or NMR signal phase as 
a function of G.sub.y gradient amplitude. This process is typically 
referred to as phase encoding. 
From this data set, an image may be derived according to well known 
reconstruction techniques. The image comprises an array of complex pixel 
values having magnitude and phase. Typically the magnitudes of the pixels 
are mapped to a gray scale to form the visual image. 
In a 1.5 Telsa B.sub.0 field the Larmor frequency of the silicone protons 
is approximately 102 Hertz higher than the protons of fat and 305 Hertz 
higher than the protons of water. The difference between the Larmor 
frequencies of different isotopes or species of the same nucleus, viz., 
protons, is termed chemical shift, reflecting the different atomic 
environments of the species. 
As noted above, the silicone used in such breast implants is composed of 
poly-dimethylsiloxane with varying degrees of polymerization. The primary 
NMR signal is from magnetically equivalent protons on the methyl groups 
which rapidly rotate about the Si-C bond axis. The single resonance has 
fairly long T.sub.1 and T.sub.2 relaxation times. Other protons in the 
silicone gel are present in very low concentrations (e.g., residual D4 
monomers) or have very short T.sub.2 relaxation times (e.g., cross-links) 
and are not detectable by MR imaging. 
Critical to imaging a breast prosthesis is the ability to isolate the 
silicone signal from the water and fat signals comprising the majority of 
the breast tissue. In theory, because the silicone protons have a discrete 
and separate resonance from fat or water protons, the signal from the 
silicone should be capable of isolation from that of fat and water. 
Nevertheless, the small difference between the frequency of resonance of 
the fat and silicone protons at even high field strengths of 1.5 Tesla 
restricts the use of selective excitation techniques, or saturation of the 
silicone resonance, to cases of extremely good B.sub.0 field homogeneity. 
Three Point Dixon 
As described above, an NMR signal vector is composed of a magnitude and an 
angle. Only the magnitude, which represents the density of detected spins, 
is usually displayed in an image. The angle provides the relative phase of 
the detected spins. 
An NMR image can be decomposed into several chemical shift components by 
combining two NMR images S.sub.0 and S.sub..pi. in which the spins of the 
two species are in phase and out of phase by .pi. radians respectively 
("Dixon technique"). For example, images of fat or water alone may be 
constructed by adding or subtracting the complex numbers representing each 
pixel of these two images S.sub.0 and S.sub..pi., on a pixel by pixel 
basis, to cancel the unwanted species. The phase shift between fat and 
water components of the images may be controlled by timing the RF pulse of 
the NMR sequence so that the signal from the fat image evolves in its 
phase with respect to the water signal by the proper angle of exactly .pi. 
before the NMR signal is acquired. 
The phase evolution is caused by the chemical shift between the two species 
and is strongly dependent on the strength of the polarizing magnetic field 
B.sub.0. Variations or inhomogeneities of B.sub.0 caused both by imperfect 
shimming of the polarizing magnet or the effect of the imaged object on 
the magnetic field can change the degree of phase evolution causing the 
decomposed images to contain admixtures of the two species. The accuracy 
of such chemical shift "Dixon" techniques is therefore often unreliable. 
The reliability of the Dixon technique can be improved by adding a third 
image, S.sub.2.pi., to the three point Dixon technique. The third image is 
selected to have both chemical species in-phase after an additional 
evolution time from the original in-phase image. The variation in phase 
between the first and last image is used to deduce the effects caused by 
magnetic field inhomogeneity. Thus, this third image can be used to detect 
and correct for the effect of B.sub.0 magnetic field inhomogeneities. 
This process of deducing the effects of inhomogeneities in the magnetic 
field from the third image requires determination of a "switch function" 
which can be either plus or minus one. This switch is a function of the 
"wrapping" around of trigonometric functions at large angles. The three 
point Dixon technique and a method for determining the switch function is 
described in detail in U.S. Pat. No. 5,144,235 to Glover et al. assigned 
to the same assignee of the present application and hereby incorporated by 
reference. 
SUMMARY OF THE INVENTION 
The present invention recognizes that the three point Dixon technique can 
be extended, in certain situations, to distinguish one chemical species 
from not only a second species, as has been done with previous three point 
Dixon techniques, but from a second and third chemical species. Because 
most body tissue include both water and fat, the invention provides an 
important ability to image a foreign, non-fat, non-water, chemical 
species, such as silicone, introduced into the body. 
The use of phase differences to separate one material from two (as opposed 
to one) other materials is problematic. With two materials, it is a simple 
matter to adjust the relative phase of the materials in each image to the 
proper relationship, by waiting an evolution time based on their chemical 
shift difference. But with three species, a single evolution time will 
generally only produce the desired phase shift between two of the species, 
the phase shift between the third and the other two will be an undesired 
arbitrary amount, the function of a different chemical shift. Thus, one 
might expect it to be extremely difficult if not impossible to produce the 
desired phase shifts between three chemical species with the three point 
Dixon technique. 
To the contrary, the present invention recognizes that an important class 
of chemical species can be separated from water and fat by three point 
Dixon techniques and discloses the technique for determining the proper 
evolution time for such separation. 
Specifically, to produce an image of a first chemical species in the 
presence of a second and third chemical species, the second species having 
a chemical shift frequency difference of .DELTA..omega..sub.1,2 with 
respect to the first chemical species and the third chemical species 
having a chemical shift frequency difference of .DELTA..omega..sub.1,3 
with respect to the first chemical species, both in the presence of a 
polarizing magnetic field B.sub.0, the following steps are taken. First, 
frequencies .omega..sub.a and .omega..sub.b are identified approximating 
frequencies .DELTA..omega..sub.1,2 and .DELTA..omega..sub.1,3 and so that 
the ratio .omega..sub.a :.omega..sub.b equals a ratio of two odd integers, 
i.sub.a and i.sub.b. From these frequencies, an evolution time .tau. is 
selected to equal 
##EQU1## 
. As few as two NMR images are then acquired, the first having an 
evolution time of k.tau., where k is an even integer including zero, in 
which the relative phase of the three species is equal, and is second 
image having an evolution time of l.tau. where l is an odd integer. These 
images are combined to produce a chemical species image with reduced 
contribution from the second and third species. 
For example, in the case of isolating an image of silicone in the presence 
of fat and water in a polarizing field B.sub.0 of 1.5 Tesla, 
.DELTA..omega..sub.1,2 =102 H.sub.3 and .DELTA..omega..sub.1,3 =305 
H.sub.3. Values .omega..sub.a =100 and .omega..sub.b =300 may be selected 
having a ratio of 1 to 3. The evolution time is i.sub.a =1 divided by (4 
times 100), i.e. 1/400. 
It is thus one object of the invention to provide a simple technique for 
isolating one chemical species from an environment containing two other 
chemical species. The use of as few as two images eliminates the need for 
additional NMR acquisitions thus shortening the examination time, and in 
some instances reducing the noise of the acquired signals. A third image 
may be employed for the reduction of the effects of inhomogeneities in the 
polarizing magnetic field B.sub.0. 
For the isolation of chemical species with short T.sub.2 relaxation times, 
the cancellation of the unwanted species is accomplished by combination of 
the second and third acquired images. For the isolation of chemical 
species with long T.sub.2 relaxation times, the cancellation of the 
unwanted species is accomplished by combination of the first, second and 
third acquired images. 
It is thus another object of the invention to improve the cancellation of 
the unwanted species from an NMR image by selection among the three 
acquired images depending on the characteristics of the species being 
isolated. As will be explained in detail below, the use of three images 
can introduce errors in the cancellation process when the T.sub.2 
relaxation time is short, such errors being lessened by the use of only 
the later two images. 
Other objects and advantages besides those discussed above shall be 
apparent to those experienced in the art from the description of a 
preferred embodiment of the invention which follows. In this description, 
reference is made to the accompanying drawings, which form a part hereof, 
and which illustrate one example of the invention. Such example, however, 
is not exhaustive of the various alternative forms of the invention, and 
therefore reference is made to the claims which follow the description for 
determining the scope of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
Referring to FIG. 1, an NMR imaging system of a type suitable for the 
practice of the invention includes a computer 10 which controls gradient 
coil power amplifiers 14 through a pulse control module 12. The pulse 
control module 12 and the gradient amplifiers 14 together produce the 
proper gradient waveforms G.sub.x, G.sub.y, and G.sub.z, as will be 
described below, for a gradient echo pulse sequence. The gradient 
waveforms are connected to gradient coils 16 which are positioned around 
the bore of the polarizing magnet 34 so that gradients G.sub.x, G.sub.y, 
and G.sub.z are impressed along their respective axes on the polarizing 
magnetic field B.sub.0 from magnet 34. The magnet 34 homogeneity can be 
adjusted by means of shimming coils 40 and a power supply 38. 
The pulse control module 12 also controls a radio frequency synthesizer 18 
which is part of an RF transceiver system, portions of which are enclosed 
by dashed line block 36. The pulse control module 12 also controls a RF 
modulator 20 which modulates the output of the radio frequency synthesizer 
18. The resultant RF signals, amplified by power amplifier 22 and applied 
to RF coil 26 through transmit/receive switch 24, are used to excite the 
nuclear spins of the imaged object (not shown). 
The NMR signals from the excited nuclei of the imaged object are picked up 
by the RF coil 26 and presented to preamplifier 28 through 
transmit/receive switch 24, to be amplified and then demodulated by a 
quadrature phase detector 30. The detected signals are digitized by an 
high speed A/D converter 32 and applied to computer 10 for processing to 
produce NMR images of the object. 
Referring to FIG. 2(a), a gradient recalled echo sequence begins with a 
transmission of a narrow bandwidth radio frequency (RF) pulse 50. The 
energy and phase of the initial RF pulse 50 may be controlled such that at 
its termination, the magnetic moments of the individual nuclei of the 
imaged object are precessing about the z-axis within the x-y plane shown 
generally in FIG. 3. A pulse of such energy and duration is termed a 
90.degree. RF pulse. 
The result of the combination of RF pulse 50 and a z-axis gradient pulse 
G.sub.z is that the nuclear spins of a narrow slice in the imaged object 
along an x-y plane are excited into resonance. Only those spins with a 
Larmor frequency, under the combined field G.sub.z and B.sub.0, equal to 
the frequencies contained within the bandwidth of the RF pulse 50 will be 
excited. Hence, the position of the slice may be controlled by the 
gradient G.sub.z offset or the RF frequency. 
After the RF pulse 50, the precessing spins begin to dephase according to 
their chemical shifts which cause the spins of certain chemical species to 
precess faster than others. At a time T after the RF pulse 50, an NMR 
signal 59 is acquired during the application of a x-axis gradient G.sub.x. 
The G.sub.x gradient signal produces a gradient recalled echo as is 
understood in the art. 
The shape of the G.sub.x pulse is such that the water, fat and silicone 
proton spins are aligned at time T having been brought into alignment by 
the portion of the G.sub.x gradient prior to time T. After time T, the 
silicone, fat and water spins begin to dephase. 
A second acquisition of signal may occur at an evolution time .tau. after 
the first acquisition 59' where the silicone, fat and water spins are no 
longer in alignment. The degree of phase difference between the spins will 
be a function of the time .tau. and the magnetic field strength. 
Similarly, with different evolution times, other signals may be acquired. 
The above sequences are repeated with different gradients G.sub.y 57 as is 
understood in the art to acquire multiple NMR signals 59 which may be 
reconstructed according to conventional reconstruction techniques. 
Referring to FIG. 2(b), a spin echo pulse sequence also begins with the 
transmission of a narrow bandwidth radio frequency (RF) pulse 50. Again, 
energy and the phase of this initial RF pulse 50 are controlled such that 
at its termination, the magnetic moments of the individual nuclei are 
precessing around the z axis within the x-y plane. 
After the 90 RF pulse 50, the precessing spins begin to dephase according 
to their chemical shifts which cause the spins of certain chemical species 
to precess faster than others. At time TE/2 after the application of 90 RF 
pulse 50, a 180 RF pulse 54 may be applied which has the effect of 
rephasing the spins to produce a spin echo 56 at time TE after the 90 RF 
50. This spin echo signal 56 is acquired during a read out gradient 53. 
As is understood in the art, a dephaser pulse 52 is applied after the 90 RF 
pulse but before the read out gradient to center spin echo within the read 
out gradient. 
With the 180 pulse 54 centered at time TE/2 the fat, water and silicone 
proton spins will be completely rephased and hence have no phase shift 
with respect to each other at the time of the spin echo 56. The time of 
the 180 pulse 54, however, may be shifted forward or back by time .tau. 
from the time TE/2. In this case, the fat, water and silicone proton spins 
will not be in phase but will be shifted with respect to each other. 
The above sequences are repeated with different G.sub.y gradient pulses 57, 
as is understood in the art, to acquire an NMR data set from which a 
tomographic images S.sub.0, S.sub..pi. and S.sub.2.pi. of the imaged 
object may be reconstructed according to conventional reconstruction 
techniques using the Fourier transform. 
Three tomographic images: S.sub.0, S.sub..pi. and S.sub.2.pi. of the 
imaged object are acquired as will now be described. Referring to FIG. 4, 
in the prior art two-point Dixon technique, an evolution time .tau. was 
chosen so that the two images obtained have the phase differences between 
the fat and water spins (58 and 55 in FIG. 3) of 0 and .pi. in images 
S.sub.0 and S.sub.x.pi. respectively. Adding and subtracting these images 
S.sub.0 and S.sub..pi. provides separate fat and water images. 
In the ideal case, the frequency of the RF modulator 20 and phase detector 
30 are adjusted to match the Larmor frequency of the water. If the 
polarizing magnetic B.sub.0 is uniform, this resonance condition is 
achieved throughout the entire subject. Similarly, the out-of-phase 
condition (.pi. radians) for the fat component is achieved for all 
locations in the subject under homogeneous field conditions as shown in 
FIG. 4. In this case, the two point decomposition into the separate images 
is ideal in that fat is completely suppressed in the water image, and vice 
versa. When the polarizing field is inhomogeneous as shown in FIG. 5, 
however, there are locations in the subject for which the water is not on 
resonance. In this case, the accuracy of the decomposition breaks down and 
the water and fat images contain admixtures of the two species. This 
derives from additional phase shifts of the NMR signal caused by the 
B.sub.o inhomogeneities. The degree to which the off resonance condition 
holds is, in general, not known. The accuracy of such chemical shift 
"Dixon" techniques is therefore often unreliable. 
Field inhomogeneities may result from improper adjustment or shimming of 
the polarizing magnetic field B.sub.0, but are more typically the result 
of "demagnetization" effects caused by the variations in magnetic 
susceptibility of the imaged tissue, such as between soft tissue and air, 
or bone and soft tissue, which locally distort the polarizing magnetic 
field B.sub.0. These demagnetization effects may be of short spatial 
extent but of high magnitude, and therefore may not be removed by 
conventional linear or higher order shimming techniques. 
The influence of demagnetization may be accommodated, however, by an 
imaging technique that uses three images S.sub.0, S.sub..pi., and 
S.sub.2.pi., with the phase evolution times adjusted so that the fat and 
water components of the images are in phase, out of phase by .pi., and 
in-phase by 2.pi. respectively. The complex pixels in each of the three 
images after conventional reconstruction may be represented as follows: 
EQU S.sub.0 =(.rho.1+.rho.2)e.sup.i.phi.0 (2) 
EQU S.sub..pi. =(.rho.1-.rho.2)e.sup.i(.phi.+.phi.0) (3) 
EQU S.sub.2.pi. =(.rho.1+.rho.2)e.sup.i(2.phi.+.phi.0) (4) 
where .rho..sub.1 is the (real) relaxation weighted spin density and hence 
the amplitude of the pixel contributed by the water component, .rho..sub.2 
is the (real) relaxation weighted spin density or amplitude of the pixel 
contributed by the fat component, and .phi..sub.0 is the phase shift 
common to all acquisitions that is caused by RF heterogeneity due to 
penetration effects, phase shifts between the RF transmitter and receiver, 
and other systematic components. These effects are independent of chemical 
shift but dependent on spatial location. In image S.sub..pi., the 
amplitudes .rho.1 and .rho.2 are subtracted because of the .pi. phase 
shift between the fat and water components as previously described The 
phase shift .phi. is caused by the unknown resonance offset that results 
from B.sub.0 heterogeneity. 
The phase offset .phi..sub.0 may be eliminated from equations (2)-(4) from 
S.sub.0, since the .rho..sub.i values are real quantities, by determining 
its argument .phi..sub.0. The argument .phi..sub.0 may then be eliminated 
from the equations (2)-(4) yielding: 
EQU S'.sub.0 =S.sub.0 e.sup.-i.phi.0 =(.rho.1+.rho.2) (2') 
EQU S'.sub..pi. =S.sub..pi. e.sup.-i.phi.0 =(.rho.1+.rho.2)e.sup.i(.phi.)(3') 
EQU S'.sub.2.pi. =S.sub..pi. e.sup.-i.phi.0 =(.rho.1+.rho.2)e.sup.i(2.phi.)(4') 
The values of p.sub.1 and p.sub.2 may be determined from the measured 
values of S'.sub.0, S'.sub..pi. and S'.sub.2.pi. according to equations 
##EQU2## 
where s is a "switch function" which may be either +1 or -1 thus 
determining the sign of the average. The latter equations (5') and (6') 
provide arithmetic rather than geometric averaging of S'.sub.0 and 
S'.sub.2.pi.. 
The choice of the sign of the averages is difficult because the 
demagnetization effects may cause abrupt changes in the local polarizing 
magnetic field B.sub.0 which cause the switch function to change in value 
from pixel to pixel. A method of determining the value of the switch 
function is provided in U.S. Pat. No. 5,144,235 entitled: Method 
Decomposing NMR Images by Chemical Species, assigned to the same assignee 
as the present invention and hereby incorporated by reference. 
Referring to FIG. 5, the three point Dixon method may be compared to the 
two point Dixon method of FIG. 4. Here three images S.sub.0, S.sub..pi. 
and S.sub.2.pi. are obtained with relative phase shifts between the fat 
and water of 0, .pi. and 2.pi.. The phase shift caused by B.sub.0 
inhomogeneities is shown for each of the evolution times and simply adds 
to the phase shift caused by the chemical shift of the fat and water 
components. The third image S.sub.2.pi. can be used to deduce this 
B.sub.0 effect which may then be subtracted out of the S.sub.0 or 
S.sub..pi. images in principle to produce a decomposition as desired. 
Referring now to FIG. 6, a spectrum encountered with a breast prosthesis 
having voxels of water, fat and silicone exhibits three resonant peaks. 
The water and fat peaks common to most in vivo imaging are separated by 
203 Hertz in a 1.5 Tesla field. The fat resonance contains several protons 
species: methyl CH.sub.3, methylene CH.sub.2, and methyne CH, with 
slightly different resonant frequencies. In addition, adjacent protons are 
coupled via indirect dipolar couplings (J-couplings) to each other. These 
two effects cause the fat resonance to have a natural line width of about 
40 Hertz. Only one resonance is observed for silicone, the methyl protons 
resonate at 305 Hertz above water. 
Three point Dixon technique of FIG. 5 will not work for arbitrary three 
species systems. For Dixon three point decomposition, each of the species 
must be substantially either in-phase or out of phase with the species 
being isolated. This allows the images to be added or subtracted to 
eliminate contributions from the undesired species. Although any two of 
the species may be placed in an arbitrary phase relationship by the 
appropriate evolution time .tau., in general, the third species will have 
a phase relationship unsuitable for decomposition. 
Nevertheless, as illustrated in FIG. 6, the ratio of the chemical shift 
frequencies of silicone, fat and water are such that three point 
decomposition may be undertaken. Referring also to FIG. 7, and assuming 
that the B.sub.0 field is perfectly homogenous, three images S.sub.0, 
S.sub..pi. and S.sub.2.pi. may be obtained by allowing the fat component 
to have a phase shift of .pi. and 2.pi. with respect to the silicone, as 
shown in FIG. 7 with the water component having a phase shift of 3.pi. and 
6.pi. with respect to the silicone, which is essentially equivalent to 
.pi. and 2.pi. phase difference of fat. Generally then, these images may 
be added to produce a silicone-only image (.rho..sub.3). 
More specifically, the acquisitions taken produce the following images: 
EQU S.sub.0 =(.rho..sub.1 +.rho..sub.2 +.rho..sub.3) (7) 
EQU S.sub..pi. =(.rho..sub.1 e.sup.i(.theta.=3.pi.) +.rho..sub.2 
e.sup.i(.theta.=.pi.) +.rho..sub.3)e.sup.i(.phi.+.phi.) (8) 
EQU S.sub.2.pi. =(.rho..sub.1 e.sup.i(.theta.=6.pi.) +.rho..sub.2 
e.sup.i(.theta.=2.pi.) +.rho..sub.3)e.sup.i(2.phi.+.phi..sub.0)(9) 
Where .pi.1, .pi.2 and .pi.3 are the amplitude of the water, fat and 
silicone components respectively and .phi..sub.0 is a phase offset 
resulting from RF penetration and other systematic phase shifts. .phi. is 
the phase shift change caused by magnetic field homogeneity and magnetic 
susceptibility during the Dixon delay time .tau.. 
Subtracting the offset phase .phi..sub.0 from each of the images produces: 
EQU S'.sub.0 =(.rho..sub.1 +.rho..sub.2+.rho.3) (10) 
EQU S'.sub.90 =(-.rho..sub.1 -.rho..sub.2 +.rho.3)e.sup.i.phi. (11) 
EQU S'.sub.2.pi. =(.rho..sub.1 +.rho..sub.2 +.rho.3)e.sup.2i.phi.(12) 
The phase terms of S'.sub.2.pi. and S'.sub..pi. are subtracted, fit and 
then unwrapped producing a combination magnetic susceptibility and B.sub.0 
field inhomogeneity map .phi.'. The corrected phase difference map is used 
to determine a switch function s as will be described and subsequently to 
correct the magnitude images. 
EQU S".sub.0 =S'.sub.0 =(.rho..sub.1 +.rho..sub.2 +.rho..sub.3)(13) 
EQU S".sub.90 =S'.sub.90 '.pi.e.sup.-i.phi. =(-.rho..sub.1 -.rho..sub.2 
+.rho..sub.3) (14) 
EQU S".sub.2.pi. =S'.sub.2.pi. e.sup.-2i.phi. =(.rho..sub.1 +.rho..sub.2 
+.rho..sub.3) (15) 
and then to compute the pure silicone (.rho.3) and complimentary water-fat 
images (.rho..sub.1 +.rho..sub.2). 
##EQU3## 
EXAMPLE I 
Referring to FIG. 8, an in vivo measurement of two volunteers was made and 
the percent total water, fat and silicone signal in the images processed 
per equation (16) was plotted against the frequency difference between 
silicone and fat assumed in the calculation of .tau.. The RF transmitter 
was centered on the silicone methyl proton resonance and scan parameters 
were TR=100 milliseconds, TE=25 milliseconds, FOV 16 cm, slice thickness 4 
millimeters, matrix 128.times.256, one excitation. The optimum silicone 
fat frequency difference was found at the minimum percentage fat in the 
silicone only image. This appears at about 102 Hertz. 
Referring now to FIG. 9(a) and (b), the relative phase of the fat spins 58 
and water spins 55 with respect to the silicone spins 51 and the absolute 
phase differences are plotted against evolution time .tau.. Although a 
given .tau. affects the phase of both fat and water simultaneously, it 
will be noted that because of the particular ratio of chemical shift 
frequencies of fat and water with respect to silicone, that the phases of 
both fat and water converge at .vertline..pi..vertline. at given evolution 
time .tau.. While this will not be true for all possible three species 
systems, a large number of systems will meet the requirements necessary 
for successful decomposition. 
The requirements for such decomposition may be established as follows. 
Referring still to FIG. 9, for an arbitrary three species system, the 
relative chemical shift of the second species with respect to the first 
species is .DELTA..omega..sub.1,2 and the relative chemical shift 
frequency of the third species with respect to the first species is 
.DELTA..omega..sub.1,3. The first species is the species of which an 
isolated image will be constructed. A phase shift between the first and 
second chemical species of .pi. (or any odd multiple of .pi.) will occur 
at evolution times: 
##EQU4## 
where n is a nonnegative integer. Likewise for the third chemical species, 
a phase difference of magnitude .pi. will occur at evolution times 
##EQU5## 
For both the second and third chemical species to have phase shift with 
respect to the first chemical species of .pi., it is required only for 
some m and n: 
##EQU6## 
By inspection then, the requirement is simply that the rate of the chemical 
shift of the second and third species be that of a ratio of odd integers. 
Thus ratios of 1/1, 1/3, 3/5 etc. will allow the present three point 
technique to separate one distinct chemical species from two others. 
Clearly, in situations where the chemical shift is not exactly a ratio of 
odd integers but may be approximated as such, chemical species isolation 
may still be performed with some minor degradation in the separation. For 
example, the chemical shift frequencies of water and fat with respect to 
silicone are in fact a fraction 
##EQU7## 
and only approximately 1/3. Nevertheless, adequate images may be obtained 
assuming a 1/3 ratio. Thus frequencies .DELTA..omega..sub.1,2 and 
.DELTA..omega..sub.1,3 are approximated by frequencies .omega..sub.a and 
.omega..sub.b having the desired odd integer ratio. 
Then, .tau. in general, from equation (18), will be 
##EQU8## 
where i.sub.a =(2n+1) where n satisfies equation (21). Alternatively, from 
equation (19) .tau. may be where i.sub.b =(2m+1) where m satisfies 
equation (21). Preferably, i.sub.a and/or i.sub.b will be the smallest 
possible integers satisfying equation (21). Likewise, the image S.sub.0 
may be taken at even multiples of .tau. and not just at zero evolution 
time. 
Referring again to FIG. 7 and equations (5) and (6), a switch function s 
must be determined. In general, the switch function depends on the 
presence of variations in the polarizing magnetic field B.sub.0 and may be 
determined by the information contained in the three acquired images. The 
process of determining the switch function s is described in detail in 
U.S. Pat. No. 5,144,235 to Glover et al. issued September 1, 1992 assigned 
to the assignee of the present invention hereby incorporated by reference. 
Generally, as indicated in the discussion associated with FIG. 5, the 
phase offset of the third image S.sub.2.pi. is used to deduce 
inhomogeneity effect. 
Referring to FIG. 10(a), depending on the chemical species being imaged, 
the NMR signal, as characterized by the relaxation times T.sub.1 and 
T.sub.2 of the second and third species, will become progressively weaker 
for increasingly long evolution times .tau.. For chemical species where 
the T.sub.2 decay is quite rapid, the calculations of equations 5 and 6, 
and 16 and 17 will produce considerable error owing to the effective 
linear interpolation accomplished by the term 
##EQU9## 
which averages the S".sub.0, S".sub.2.pi. images. For any two pixels 
having magnitudes 100 and 102 of the S".sub.0 and S".sub.2.pi. images 
respectively, the linear interpolation for a rapidly decaying signal will 
produce a magnitude 104 greater than the value 106 of the image 
S".sub.2.pi. which must be subtracted from the value 104 to cancel the 
contributions of the unwanted species. This difference between 104 and 106 
will cause incomplete cancellation of the unwanted species in the image. 
Preferably, therefore, in this situation, the images are combined 
according to a new set of equations as follows: 
EQU .rho..sub.3 =(S".sub.2.pi. +sS".sub.90 )/2 (22) 
EQU .rho..sub.1 +.rho..sub.2 =(S".sub.2.pi. -sS".sub..pi.)/2 (23) 
Equations (22) and (23) take advantage of the fact that point 102 is a 
better approximation of point 106 than is interpolated points 104 for 
species with short T.sub.2 relaxation times. The disadvantage to this 
approach is a loss of signal-to-noise ratio as a result of the use of 
weaker signals in generating the selective image. Equations (22) and (23) 
may also be used for the separation of two species under similar 
circumstances. 
Referring to FIG. 10(b), in situations where the decay of the NMR signal is 
relatively long compared to the expected evolution time, the equations 
(16) and (17) which provide a linear interpolation, will provide both an 
good estimate of the zero phase image S.sub.0 at the evolution time of the 
S.sub..pi. images and will produce better signal-to-noise ratio in the 
resulting selective image owing to the combination of a greater number of 
images and the use of the S.sub.0 image having the greatest 
signal-to-noise ratio. 
While this invention has been described with reference to particular 
embodiments and examples, other modifications and variations will occur to 
those skilled in the art in view of the above teachings. For example, the 
technique is not limited to the isolation of one chemical species from two 
others but may be employed to isolate one chemical species from any group 
of other species provided the chemical shift frequency differences are 
such that with some evolution time .tau. the other chemical species may be 
made to evolve to a phase difference of .pi. with respect to the species 
to be isolated. Of course, it will be understood that the invention is 
useful for the imaging of materials other than silicone and that the 
frequencies provided for the imaging of silicone are a function of the 
B.sub.0 polarizing field. Other field strengths B.sub.0 may be used by 
scaling these frequencies proportionally upward (and the evolution times 
proportionally downward) for increases in B.sub.0 away from the value of 
1.5 Tesla considered herein. 
Accordingly, the present invention is not limited to the preferred 
embodiment described herein, but is instead defined in the following 
claims.