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Claims  |
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What is claimed is:
1. A system for providing a multi-dimensional magnetic susceptibility image
of a body, comprising:
a first radiation source for applying a first radiation field having a
magnetic field component to magnetize the body,
a first detector for measuring said magnetic field in a volume to be
occupied by the body,
means for applying an electromagnetic second radiation field to the body
when disposed in the magnetic field to produce a third radiation field
emanating from the body,
a second detector producing output signals in response to said third
radiation field, and
a processor coupled to said second detector for receiving said output
signals to produce a multi-dimensional magnetic susceptibility image of
the body.
2. The system of claim 1, wherein said first radiation source comprises a
magnet for producing a DC magnetic field.
3. The system of claim 2, wherein said magnet comprises a superconducting
magnet.
4. The system of claim 1, wherein said first detector comprises a
magnetometer.
5. The system of claim 1, wherein said means for applying a second
radiation field comprises an RF generator.
6. The system of claim 1, further comprising an analog to digital converter
coupled to said second detector for digitizing said output signals.
7. The system of claim 1, wherein said electromagnetic second radiation
field polarizes nuclei having non-zero spin in the body, said polarized
nuclei precessing to produce said third radiation field.
8. The system of claim 1, wherein said processor includes a Fourier
transform stage for obtaining Fourier spectra of said output signals of
said second detector.
9. The system of claim 1, further comprising a plurality of detectors for
detecting said third radiation field, said plurality of detectors being
disposed in a plane with respect to each other and with respect to said
second detector, said plurality of detectors being configured to move in a
direction perpendicular to the plane to sample said third radiation field
over a three-dimensional volume external to the body.
10. The system of claim 1, further comprising a plurality of detectors
forming a three-dimensional array to sample said third radiation field
over a three-dimensional volume external to the body.
11. The system of claim 9, wherein said processor comprises a Fourier
transform stage for obtaining frequencies and intensities of Fourier
components of said sampled third radiation, said frequency components
indicating the magnetic susceptibility corresponding to voxels of the body
and said intensity components indicating positions of said voxels.
12. The system of claim 1, wherein the means for applying includes means
for applying an electromagnetic second radiation field to the body when
disposed in the magnetic field to produce a third radiation field
emanating from a voxel in the body that approximates a dipole near field
having a maximum amplitude given by the following formula:
B.sup.1 =m.sub.z 2z.sup.2 -x.sup.2 -y.sup.2 /(x.sup.2 +y.sup.2
+z.sup.2).sup.5/2
wherein x,y, and z are Cartesian coordinates, and m.sub.Z is a bulk
magnetization of the voxel along the z-coordinate, and B.sup.1 is the
maximum amplitude of the near field.
13. The system of claim 1, wherein said processor includes a Fourier
transform processor for obtaining Larmor frequencies corresponding to
voxels of the body from which said third radiation field emanates.
14. The system of claim 13, wherein said display element generates said
anatomical images based on selected physiological parameters, said
anatomical images being substantially free of motion artifacts.
15. The system of claim 1, further comprising a display element for
generating an anatomical image of the body from the multi-dimensional
susceptibility image.
16. System for providing a multi-dimensional image representation of
spatial variations of magnetic susceptibility of a body including a
paramagnetic or a diamagnetic substance, comprising:
a first generator for generating an excitation field in the body to
polarize paramagnetic or diamagnetic substances within the body such that
said polarization creates a local magnetic field in voxels of the body
according to the magnetic susceptibility of each voxel,
a second generator for generating a primary electromagnetic radiation field
in the body to induce a secondary radiation field to emanate from the
body, each voxel of the body contributing to said secondary radiation
field at a frequency determined by the magnetic susceptibility of said
voxel,
a detector positioned to receive one of said fields for producing an output
signal in response to said secondary radiation field, and
a processor for receiving and processing the output signal of said detector
to create the multi-dimensional image.
17. System for providing a multi-dimensional image representation of
spatial variations of magnetic susceptibility of a body having at least
one type of nuclei with non-zero spin, and a paramagnetic or a diamagnetic
substance, comprising:
a first generator for generating an excitation field in the body to
polarize the paramagnetic or the diamagnetic substances such that said
polarization creates a local excitation field in the body, said local
excitation field generating a shift in the Larmor frequency of nuclei in
the body,
a second generator for generating a primary radio frequency field in the
body to create a transverse magnetization of the nuclei,
a detector for producing output signals in response to detecting a
component of a secondary radio frequency field generated by the polarized
nuclei, and
a reconstruction processor for receiving the output signals of said
detector to create the multi-dimensional image.
18. System for generating a three-dimensional magnetic susceptibility map
of a body, comprising:
means for producing a magnetic field for magnetizing the body;
means for measuring the magnetic field in a volume to be occupied by the
body;
a radiation generator and a transmitter for producing a primary
electromagnetic radiation field for causing the body to produce a
secondary radiation field by excitation;
a three-dimensional or planar array of detectors movable in a direction
perpendicular to the plane of the detectors for producing output signals
by spatially sampling a selected component of said secondary radiation
field over a three-dimensional volume at least at the Nyquist rate;
an analog to digital converter for digitizing said output signals;
a Fourier transform stage for converting the output signals of the
detectors acquired over a period of time into frequency spectra; and
a processor for converting the frequency spectra into a map of magnetic
moments in the volume and for converting the magnetic moment map into a
three-dimensional magnetic susceptibility map.
19. The system of claim 18, further comprising an image processor for
displaying said magnetic susceptibility map, said processor being
configured to display said map from any perspective as a two-dimensional
or a three-dimensional image.
20. The system of claim 18, further comprising coils for providing a
magnetic field gradient to alter intensity of the said secondary radiation
field when said radiation generator provides T.sub.1 and T.sub.2 radio
frequency pulse sequences.
21. The system of claim 18, wherein said radiation generator comprises a
radio frequency generator providing a radio frequency field for causing
precession of protons in said body for producing said secondary radiation
field.
22. The system of claim 18, wherein said means for measuring the magnetic
field comprises a magnetometer that employs NMR frequency of protons in
water to detect magnitude of said magnetic field in the volume to be
occupied by the body.
23. The system of claim 18, wherein the protons are substantially free of
ferromagnetic influences.
24. The system of claim 18, wherein the radiation generator includes an
amplifier for increasing magnitude of said primary field in portions of
the body distal to said array of detectors.
25. The system of claim 18, wherein said means for producing a magnetic
field produces a magnetic field varying as a quadratic function of
distance from said detectors in a direction perpendicular to the plane of
the detectors.
26. The system of claim 18, wherein said magnetic field has a component
B.sub.z in a voxel along a Cartesian coordinate perpendicular to said
array of detectors, said component B.sub.z being given by:
B.sub.z =B.sub.o [a.sup.2 +y.sup.2.sub.m ].sup.3/2
where a.sub.0 and B.sub.o are constants and Y.sub.n is the distance from
the plane of the detectors to the voxel.
27. The system of claim 18, wherein said magnetic field is substantially
constant.
28. The system of claim 18, wherein said magnetic field is substantially
confined to a volume of the body to be imaged.
29. The system of claim 18, wherein said means for producing a magnetic
field produces a magnetic field having a magnitude that varies within the
body, said varying field providing a range of Larmor frequencies.
30. The system of claim 18, wherein said array of detectors comprises an
array of antennae responsive to a selected component of said secondary
radiation field.
31. A method for providing a multi-dimensional magnetic susceptibility
image of a body, comprising the steps of
measuring a magnetic field component in a volume to be occupied by the
body,
applying a first radiation field having the magnetic field component to the
body to magnetize the body,
applying an electromagnetic second radiation field to the body when
disposed in the magnetic field to produce a third radiation field
emanating from the body,
detecting the third radiation field emanating from the body, and,
producing the magnetic susceptibility image of the body from said detected
third radiation field.
32. The method of claim 31, wherein the step of detecting comprises the
step of positioning an array of detectors movable in a direction
perpendicular to the body to sample said third radiation field over a
three-dimensional volume.
33. The method of claim 32, wherein said step of producing the magnetic
susceptibility image further comprises the steps of
determining the frequency components of voxels of said third radiation
field,
determining the magnetic susceptibility of said voxels at each said
frequency component, and
obtaining the positions of said voxels by utilizing intensity variation of
said frequency components over the sampled three-dimensional volume.
34. The method of claim 31, wherein said step of applying a first radiation
field comprises the step of applying a DC magnetic field to the body.
35. The method of claim 31, wherein the step of applying an electromagnetic
second radiation field comprises the step of applying an RF pulse to the
body.
36. A method for providing a multi-dimensional image representation of the
spatial variations of magnetic susceptibility of a body having a
paramagnetic or a diamagnetic substance, comprising the steps of
generating an excitation field in the body to polarize the paramagnetic or
the diamagnetic substance to create local magnetic fields in voxels in the
body according to the magnetic susceptibility of each voxel,
generating an electromagnetic primary radiation field in the body to induce
a secondary radiation field from the body, each voxel of the body
contributing to said secondary radiation field at a frequency determined
by the magnetic susceptibility of said voxel,
generating an output signal in response to said secondary radiation field
and indicative of said radiation field, and
creating the multi-dimensional susceptibility image from said output
signal.
37. A method for generating a three-dimensional magnetic susceptibility map
of a body, comprising the steps of
producing a magnetic field for magnetizing the body,
measuring the magnetic field in a volume to be occupied by the body,
producing an electromagnetic primary radiation field for causing the body
to produce a secondary radiation field by excitation,
disposing a three-dimensional or planar array of detectors external to said
body, said three-dimensional or planar array of detectors being movable in
a direction perpendicular to the plane of the detectors for producing
output signals by spatially sampling a selected component of said
secondary radiation field over a three-dimensional volume at least at a
Nyquist frequency, and
producing the magnetic susceptibility image of the body from said sampled
selected component of said secondary radiation field.
38. A method for generating a three-dimensional magnetic susceptibility map
of a body, comprising the steps of
producing a magnetic field for magnetizing the body,
measuring the magnetic field in a volume to be occupied by the body,
producing an electromagnetic primary radiation field for causing the body
to produce a secondary radiation field by excitation,
disposing an array of detectors external to said body, said array of
detectors being three-dimensional or movable in a direction perpendicular
to the detectors for producing output signals by spatially sampling a
selected component of said secondary radiation field over a
three-dimensional volume at least at a Nyquist rate,
digitizing said output signals,
converting said output signals acquired over a period of time into
frequency spectra,
converting said frequency spectra into a map of magnetic moments in the
volume and converting the magnetic moment map into a three-dimensional
magnetic susceptibility map of the body.
39. A method for producing a multi-dimensional image representation of the
spatial variations of the magnetic susceptibility of a body having a
paramagnetic or a diamagnetic substance, comprising the steps of
exciting said paramagnetic or diamagnetic substance by applying an
electromagnetic excitation field to said body,
detecting a radiation field originating in said body from said excited
paramagnetic or diamagnetic substances, wherein frequency components of
said radiation field provide information regarding the variations of the
magnetic susceptibility within said body,
constructing the multi-dimensional susceptibility image according to the
information provided by the frequency components of said radiation field.
40. A method for providing a multi-dimensional image representation of
spatial variations of the magnetic susceptibility of a body, comprising
the steps of:
providing a primary magnetic field to magnetize said body,
measuring said primary magnetic field in a volume to be occupied by said
body,
applying a primary radio frequency field to said body to rotate at least
one type of magnetic moments in said body by 90 degrees such that said
rotated moments point in a direction transverse to said primary magnetic
field,
detecting a secondary radio frequency field produced by a free induction
decay of said rotated magnetic moments by a plurality of detectors
arranged in a plane and movable in a direction perpendicular to said
plane,
obtaining Fourier transform of said secondary radio frequency to derive
Larmor frequencies of said rotated magnetic moments,
determining the magnetic moment corresponding to each Larmor frequency,
determining maximum intensity of each Larmor frequency at positions of said
detectors, and
constructing the multi-dimensional susceptibility image according to said
derived Larmor frequencies and according to spatial variations of said
secondary radio frequency field.
41. The method of claim 40, wherein said constructing step comprises the
step of employing a reconstruction algorithm.
42. A method for determining a correspondence between detected voltages in
an array of detectors to a radiation distributed within a body, comprising
the steps of:
applying a magnetic field to the body,
applying an electromagnetic first radiation field to the body to cause the
body to cause a radiation distribution within the body,
detecting the radiation distribution within the body with an array of
detectors, and
determining the correspondence between voltages of the array and the
radiation distribution such a set of detected voltages corresponds to a
unique radiation distribution within the body.
43. The method of claim 42, further comprising the step of constructing a
magnetic susceptibility map of the body in response to one or more
voltages detected by said array of detectors.
44. The method of claim 42, further comprising the step of correlating
detected voltages with a particular frequency of a magnetic field
emanating from the body and induced by said applied field, and determining
the spatial locations within the body contributing to each detected
frequency.
45. The method of claim 44, further comprising the step of constructing a
magnetic susceptibility map of the body as a function of said detected
voltages. |
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Claims |
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Description |
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BACKGROUND
This invention relates to method and apparatus for imaging a body. In
particular, the invention provides a magnetic susceptibility image of an
animate or inanimate body. Many imaging techniques exploit some natural
phenomenon which varies from tissue to tissue, such as acoustic impedance,
nuclear magnetic relaxation, or x-ray attenuation to provide a contrast
image of the tissue. Alternatively, some imaging techniques add a
substance such as a positron or gamma ray emitter to the body to construct
an image of the body by determining the distribution of the substance.
Each imaging technique possesses characteristics which result in certain
advantages relative to other imaging techniques. For example, the short
imaging time of x-ray contrast angiography reduces motion artifacts. In
addition, the high resolution of x-ray contrast angiography renders this
technique superior to many prior known imaging techniques for high
resolution imaging of veins and arteries. However, x-ray contrast
angiography is invasive, requires injection of a noxious contrast agent,
and results in exposure to ionizing radiation. Thus, it is not typically
employed except for patients with severe arterial or venous pathology.
Nuclear Magnetic Imaging (NMR) which is commonly called magnetic resonance
imaging (MRI) entails magnetizing a transverse tissue slice with a
constant primary magnetic field in a direction perpendicular to the slice,
and further magnetizing the slice by applying a gradient in the plane of
the slice. A radiofrequency pulse excites selected nuclei of the slice.
The excited nuclei relax and emit energy, i.e., radio signals, at
frequencies corresponding to local magnetic fields determined by the
gradient. A Fourier analysis of the emitted signals provides the signal
intensity at each frequency, thereby providing spatial information in one
dimension. Repeating the excitation of the nuclei and obtaining the
Fourier spectrum of the emitted signals, as the gradient rotates in the
plane of the slice, provides a two-dimensional image.
MRI is of primary utility in assessing brain anatomy and pathology. But
long NMR relaxation times, a parameter based on how rapidly excited nuclei
relax, have prevented NMR from being of utility as a high resolution body
imager. The most severe limitation of NMR technology is that for spin echo
imaging n, the number of free induction decays ("FIDs"), a nuclear radio
frequency energy emitting process, must equal the number of lines in the
image. A single FID occurs over approximately 0.1 seconds. Not considering
the spin/lattice relaxation time, the time for the nuclei to reestablish
equilibrium following an RF pulse, which may be seconds, requires an
irreducible imaging time of n times 0.1 seconds, which for 512.times.512
resolution requires approximately one minute per each two dimensional
slice. This represents a multiple of 1500 times longer that the time that
would freeze organ movements and avoid image deterioration by motion
artifact. For example, to avoid deterioration of cardiac images, the
imaging time must not exceed 30 msec. A method for speeding NMR imaging
flips the magnetization vector of the nuclei by less than 90 degrees onto
the x-y plane, and records less FIDs. Such a method, known as the flash
method, can obtain a 128.times.128 resolution in approximately 40 seconds.
Another technique used to decrease imaging time is to use a field gradient
and dynamic phase dispersion, corresponding to rotation of the field
gradient, during a single FID to produce imaging times typically of 50
msec. Both methods produce a decreased signal-to-noise ratio ("SNR")
relative to spin echo methods. The magnitude of the magnetization vector
which links the coil is less for the flash case because the vector is
flipped only a few degrees into the xy-plane. The echo-planar technique
requires shorter recording times with a concomitant increase in bandwidth
and noise. Both methods compensate for decreased SNR by increasing the
voxel size with a concomitant decrease in image quality. Physical
limitations of these techniques render obtaining high resolution, high
contrast vascular images impractical.
Thus, it is an object of the invention to provide high resolution
multi-dimensional images of tissue.
It is another object of the invention to provide multi-dimensional magnetic
susceptibility images of an object.
It is yet another object of the invention to provide high resolution
multi-dimensional images of the cardiopulmonary system.
It is yet another object of the invention to provide a magnetic
susceptibility image of a body.
SUMMARY OF THE INVENTION
These and other objects of the invention are attained by providing an
apparatus for obtaining a multi-dimensional susceptibility image of a
body. The apparatus includes a radiation source for magnetizing the body
with a magnetic component of a first radiation field. The apparatus also
includes a first detector for measuring the magnetic component of the
first radiation field in the absence of the body in a volume to be
occupied by the body. The apparatus further includes a source for applying
a second radiation field to the body, to elicit a third radiation field
from the body. A second detector senses this third radiation field, and
produces a signal that a reconstruction processor employs to create the
magnetic susceptibility image of the body.
One practice of the invention provides a method for determining the
distribution of radiation within a magnetized body, emanating from the
body in response to an excitation radiation. The method includes the steps
of measuring the emanated radiation over a three-dimensional volume by an
array of detectors, and uniquely correlating each frequency component of
the detected radiation with locations within the body producing that
frequency component.
The invention is in part based on the realization that matter having a
permeability different from that of free space distorts a magnetic flux
applied thereto. This property is called magnetic susceptibility. An
object, herein called a phantom, can be considered as a collection of
small volume elements, herein referred to as voxels. When a magnetic field
is applied to the phantom, each voxel generates a secondary magnetic field
at the position of the voxel as well as external to the phantom. The
strength of the secondary magnetic field varies according to the strength
of the applied field, the magnetic susceptibility of the material within
the voxel, and the distance of the external location relative to the
voxel. For example, U.S. Pat. No. 5,073,858 of Mills, herein incorporated
by reference including the references therein, teaches that the net
magnetic flux at a point extrinsic to a phantom to which a magnetic field
is applied, is a sum of the applied field and the external contributions
from each of the voxels. The '858 patent further teaches sampling the
external flux point by point and employing a reconstruction algorithm, to
obtain the magnetic susceptibility of each voxel from the sampled external
flux.
Unlike the '858 patent that relies on a static response from a magnetized
body to determine the magnetic susceptibility of the body, a preferred
practice of the invention elicits a radiative response from a magnetized
body by subjecting the body to a resonant radiation field. One embodiment
of the present invention generates a three-dimensional magnetic
susceptibility image of an object including a patient placed in a magnetic
field from a three-dimensional map of a radio frequency (RF) magnetic
field external to the patient, induced by subjecting selected nuclei of
the body to a resonant RF field. Application of an RF pulse to the body
causes the body to emit the RF magnetic flux external to the body. A
Fourier transform of this external flux produces its frequency components
("Larmor frequencies"). Each Larmor frequency determines the magnetic
susceptibility of the voxels of the body producing that Larmor frequency.
Further, the intensity variation of the external RF field over a
three-dimensional volume of space is used to determine the coordinate
location of each voxel.
The radiation source for magnetizing a body to be imaged can be a direct
current ("DC") magnet, including a superconducting magnet. The radiation
sources and amplifiers for applying an RF pulse to the magnetized body are
well known in the art, and include, but are not limited to, klystrons,
backward wave oscillators, Gunn diodes, and Traveling Wave Tube
amplifiers. A preferred embodiment of the invention employs
superconducting quantum interference devices (SQUID) as detectors for
sensing the external RF field. SQUIDs advantageously allow nulling the
applied magnetic flux in order to measure small external fields produced
by precessing nuclei. Because the contributions of voxels to an external
field at a detector typically drops off as the cube of the distance
between the voxel and the detector, some embodiments of the invention
employ a magnetizing field whose amplitude increases as the cube of the
distance from the detectors. Such a spatial variation of the magnetizing
field "levels" the magnitude of the external RF radiation from voxels at
different locations relative to the detectors.
One practice of the invention obtains a three-dimensional magnetic
susceptibility map of a magnetized body from a three-dimensional map of a
secondary magnetic flux produced by the magnetized body, and detected over
a three-dimensional volume of space external to the body, herein referred
to as the "sample space." The extrinsic magnetic flux is sampled at least
at the Nyquist rate, i.e., at twice the spatial frequency of the highest
frequency of the Fourier transform of the magnetic susceptibility map of
the phantom, to allow adequate sampling of spatial variations of the
external magnetic flux. This practice of the invention preferably employs
a Fourier transform algorithm, described in Fourier Transform
Reconstruction Algorithm Section, to form the magnetic susceptibility map
of the object.
One embodiment of the present invention employs nuclear magnetic resonance
(NMR) to induce a magnetized body to emit an external radiation having a
magnetic field component. In particular, application of an RF pulse,
resonant with selected nuclei of a magnetized body, can polarize the
nuclei through rotation of their magnetic moments. The polarized nuclei
within a voxel precess about the local magnetic field in the voxel at a
Larmor frequency determined in part by the applied magnetic field at
position of the voxel and the susceptibility of the voxel. The
superposition of external RF fields produced by all the voxels of the body
creates the total external RF field. Thus, the external RF field contains
frequency components corresponding to precession frequencies of the nuclei
in different voxels of the body.
One practice of the invention detects the external RF field in the near
field region where the distance of a detector sensing radiation from a
voxel at a distance.sup.r from the detector is much smaller than the
wavelength .lambda. of the radiation emitted by the voxel, i.e.,
r<<.lambda. (or kr<<1). The near fields are quasi-stationary,
that is they oscillate harmonically as e.sup.-i.omega.t, but are otherwise
static in character. Thus, the transverse RF magnetic field of each voxel
is that of a dipole. In one embodiment, an array of miniature RF antennas
sample the external RF field over a three-dimensional volume of space that
can be either above or below the object to be imaged. The frequency
components of the detected RF signals determine the magnetic
susceptibility of voxels that give rise to the RF signals, and the
location of each voxel is determined through measurements of the spatial
variations of the intensity of the external RF field at a given frequency.
Thus, the frequency components of the external RF radiation, and the
intensity variations of the external RF radiation provide the necessary
information for providing a magnetic susceptibility map of the magnetized
phantom, such as a human body. Such a susceptibility map can be employed
to obtain anatomical images of a human body based on selected
physiological parameters.
Another practice of the invention employs paramagnetic and/or diamagnetic
substances present in a body to be imaged to cause variations of local
magnetic fields in different parts of the body, thereby providing a
susceptibility image of the body. In particular, an excitation field, such
a magnetic field, polarizes the paramagnetic and/or diamagnetic substances
of the body. The polarized substances contribute to the local magnetic
field at the positions of the voxels comprising the body. The amount of
this contribution varies from one voxel to another depending on the
variation of the concentration of the substances throughout the body. An
excitation of selected nuclei of the voxels cause the nuclei to provide an
external RF field through precession about the local magnetic fields. The
external RF field includes a range of frequencies relating to the local
magnetic field at position of each voxel. Thus, an analysis of the
frequencies lead to information regarding the distribution of the
paramagnetic or diamagnetic substances throughout the body.
Magnetic Susceptibility
All matter has a permeability different than that of free space that
distorts an applied magnetic flux. This property is called magnetic
susceptibility. An object, herein called a phantom, can be considered as a
collection of small volume elements (hereafter called "voxels"). When a
magnetic flux is applied to the phantom, each voxel generates a secondary
magnetic flux at the position of the voxel as well as external to the
phantom. The strength of the secondary magnetic flux varies according to
the strength of the applied flux, the magnetic susceptibility of the
material within the voxel, and the distance of the external location
relative to the voxel. In one embodiment described in Mills [1] which is
herein incorporated by reference including the references given therein in
the Reference Section, the net magnetic flux at a point extrinsic to the
phantom is a sum of the applied flux and the contributions of each of the
voxels within the object. This flux is point sampled over a three
dimensional space, and the magnetic susceptibility of each voxel is solved
by a reconstruction algorithm.
In an embodiment of the present invention, the three-dimensional magnetic
susceptibility map of an object including a patient placed in a magnetic
field is generated from a three-dimensional map of the transverse resonant
radio frequency (RF) magnetic flux external to the patient. The magnetic
susceptibility of each voxel is determined from the shift of the Larmor
frequency due to the presence of the voxel in the magnetizing field. The
intensity variation of the transverse RF field over space is used to
determine the coordinate location of each voxel. The RF field is the near
field which is a dipole that serves as a basis element to form a unique
reconstruction. The geometric system function corresponding to a dipole
which determines the spatial intensity variations of the RF field is a
band-pass for k.sub..rho. =k.sub.z. In the limit, each volume element is
reconstructed independently in parallel with all other volume elements
such that the scan time is no greater than the nuclear free induction
decay (FID) time.
Secondary Magnetic Field
The magnetic moment, m.sub.Z, of each voxel within the phantom is given by
the product of its volume magnetic susceptibility, .chi., the magnetizing
flux oriented along the z-axis, B.sub.Z, and the volume of the voxel,
.DELTA.V.
m.sub.Z =.chi.B.DELTA.V, (1)
The magnetic moment of each voxel is a magnetic dipole. And the phantom can
be considered to be a three-dimensional array of magnetic dipoles. At any
point extrinsic to the phantom, the z-component of the secondary flux, B',
from any single voxel is
##EQU1##
where x, y, and z are the distances from the center of the voxel to the
sampling point. It is shown below that no geometric distribution of
magnetic dipoles can give rise to Eq. (2). Therefore, the flux of each
magnetic dipole (voxel contribution) forms a basis set for the flux of the
array of dipoles which comprise the magnetic susceptibility map of the
phantom.
Eq. (2) is a system function which gives the magnetic flux output in
response to a magnetic dipole input at the origin. The phantom is an array
of spatially advanced and delayed dipoles weighted according to the
magnetic susceptibility of each voxel; this is the input function. The
secondary flux is the superposition of spatially advanced and delayed
flux, according to Eq. (2); this is the output function. Thus, the
response of space to a magnetized phantom is given by the convolution of
Eq. (2) with the series of weighted, spatially advanced and delayed
dipoles representing the susceptibility map of the phantom.
In Fourier space, the output function is the product of the Fourier
transform (FT) of the system function and the FT of the input function.
Thus, the system function filters the input function. The output function
is the flux over all space. However, virtually all of the spectrum
(information needed to reconstruct the magnetic susceptibility map) of the
phantom exists in the space outside of the phantom because the system
function is essentially a band-pass filter. This can be appreciated by
considering the FT, H[k.sub..rho., k.sub.Z ], of Eq. (2):
##EQU2##
where k.sub..rho. is the spatial frequency in the xy-plane or k.sub..rho.
-plane and k.sub.Z is the spatial frequency along the z-axis.
H[k.sub..rho., k.sub.Z ] is a constant for k.sub..rho. and k.sub.Z
essentially equal as demonstrated graphically in FIG. 1c.
Band-Pass Filter
A magnetic field with lines in the direction of the z-axis applied to an
object comprised of magnetically susceptible material magnetizes the
material so that a secondary field superposes the applied field as shown
in FIG. 9. The secondary field outside of the object (phantom) and
detected at a detector 301 is that of a series of magnetic dipoles
centered on volume elements 302 of the magnetized material. In Cartesian
coordinates, the secondary magnetic flux, B', at the point (x,y,z) due to
a magnetic dipole having a magnetic dipole moment m.sub.z at the position
(x.sub.0,y.sub.0,z.sub.0) is
##EQU3##
where i.sub.Z is the unit vector along the z-axis. The field is the
convolution of the system function h(x,y,z) or h(.rho.,.PHI.,z) (the
left-handed part of Eq. (5)), with the delta function (the right-hand part
of Eq. (5)), at the position (x.sub.0,y.sub.0,z.sub.0). A very important
theorem of Fourier analysis states that the Fourier transform of a
convolution is the product of the individual Fourier transforms [2]. The
Fourier transform of the system function, h(x,y,z) or h(.rho.,.PHI.,z) is
given in APPENDIX V.
The z-component of a magnetic dipole oriented in the z-direction has the
system function, h(x,y,z), which has the Fourier transform,
H[k.sub.x,k.sub.y,k.sub.z ], which is shown in FIG. 1c.
##EQU4##
The output function, the secondary magnetic field, is the convolution of
the system function, h(x,y,z)--the geometric transfer function for the
z-component of a z-oriented magnetic dipole with the input function--a
periodic array of delta functions each at the position of a magnetic
dipole corresponding to a magnetized volume element.
##EQU5##
The Fourier transform of a periodic array of delta functions (the
right-hand side of Eq. (8)) is also a periodic array of delta functions in
k-space:
##EQU6##
By the Fourier Theorem, the Fourier transform of the spatial output
function, Eq. (8), is the product of the Fourier transform of the system
function given by Eq. (7), and the Fourier transform of the input function
given by Eq. (9).
##EQU7##
In the special case that
k.sub..rho. =k.sub.z (11)
the Fourier transform of the system function (the left-hand side of Eq.
(10)) is given by
H=4.pi. (12)
Thus, the Fourier transform of the system function band-passes the Fourier
transform of the input function. Both the input function (the right-hand
part of Eq. (8)) and its Fourier transform (the right-hand part of Eq.
(10)) are a periodic array of delta functions. No frequencies of the
Fourier transform of the input function are attenuated; thus, no
information is lost in the case where Eq. (11) holds. Thus, the resolution
of the reconstructed magnetic susceptibility map is limited by the spatial
sampling rate of the secondary magnetic field according to the Nyquist
Sampling Theorem.
Reconstruction of the Magnetic Susceptibility Map
Superconducting Quantum Interference Devices (SQUIDs) allow the user to
null the ambient or applied magnetic flux and to measure very small
secondary magnetic fluxes (10.sup.-15 Tesla). The flux contribution from a
voxel of human tissue of millimeter dimensions is typically 10.sup.-9
Tesla. For example, representative parameters of Eq. (1) are
.chi.=10.sup.-4, B.sub.Z =10.sup.5 G, and .DELTA.V=10.sup.-3 cm.sup.3
which results in m.sub.z =10.sup.-2 Gcm.sup.3. Substitution of this voxel
magnetic moment into Eq. (2) with x=0; y=0; z=10 cm results in B=10.sup.-5
G=1 nT. Eq. (2) shows that the field at the detector drops off as the cube
of the distance from the voxel of tissue. By applying a voxel magnetizing
field that increases as the cube of the distance from the detector to the
voxel, the field at the detector becomes independent of the distance from
the voxel. In this case, the field is given by
B.sub.z =B.sub.0 [a.sup.2 +z.sub.n.sup.2 ].sup.3/2 (13)
where a.sub.0 and B.sub.0 are constants and z.sub.n is the distance from
the detector to the voxel. Thus, a magnetizing field that increases as the
distance cubed from the detector "levels" the magnitude of signals from
voxels at different locations relative to the detector. A
three-dimensional magnetic susceptibility map can be generated from a
three-dimensional map of the secondary magnetic flux of a magnetized
phantom--such as a human (the corresponding space of detection is
hereafter called the "sample space"). The magnetic susceptibility map is
reconstituted using a Fourier transform algorithm. The algorithm is based
on a closed-form solution of the inverse problem--solving the spatial
distribution of magnetic dipoles from the extrinsic secondary flux from an
array of magnetized voxels. The extrinsic magnetic flux is sampled at the
Nyquist rate (twice the spatial frequency of the highest frequency of the
Fourier transform of the magnetic susceptibility map of the phantom).
In an embodiment of the present invention, nuclear magnetic resonance (NMR)
is the means to measure the secondary magnetic field to provide the input
to the magnetic susceptibility reconstruction algorithm. In this case, the
measured secondary (RF) field is transverse to the magnetic flux including
the local contribution due to the magnetic susceptibility of the voxel.
The RF field is detected in the near zone where r<<.lambda. (or
kr<<1), and the near fields are quasi-stationary, oscillating
harmonically as e.sup.-i.omega.t, but otherwise static in character. Thus,
the transverse RF magnetic field of each voxel is that of a dipole, the
maximum amplitude of which is given by Eq. (2) wherein the Larmor
frequency of each voxel is shifted due to its magnetic susceptibility, and
m.sub.z, the magnetic moment along the z-axis, of Eq. (2) corresponds to
the bulk magnetization M of each voxel. In terms of the coordinates of Eq.
(2), an array of miniature RF antennas point samples the maximum dipole
component of the RF signal over the sample space such as the half space
above (below) the object to be imaged wherein each RF signal is
frequency-shifted by the perpendicularly oriented magnetic susceptibility
moment of each voxel. A three-dimensional magnetic susceptibility map is
generated from a three-dimensional map of the secondary (RF) magnetic flux
of a magnetized phantom--such as a human. The magnetic susceptibility of
each voxel(s) is determined from the shift of the Larmor frequency due to
the presence of the voxel(s) in the magnetizing field. The measurements of
the spatial variations of the intensity of the transverse RF field of a
given frequency is used to determine the coordinate location of each
voxel(s). In one embodiment, the magnetic susceptibility map is
reconstructed using a Fourier transform algorithm. The algorithm is based
on a closed-form solution of the inverse problem--solving the spatial
location(s) of a magnetic dipole(s) of known magnetic moment (via the
Larmor frequency) from the spatial variations of the extrinsic transverse
secondary (RF) flux from the magnetized v | | |
