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Description  |
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BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to apparatus and a method for imaging the internal
structure of objects which are paramagnetic or diamagnetic using a
sensitive magnetometer, and particularly a superconducting quantum
interference device (SQUID) magnetometer, to measure local magnetic
susceptibility through the object.
Background Information
There is a continued interest in expanding the techniques available for
imaging the internal structure of opaque objects for such diverse purposes
as medical diagnosis and non-destructive evaluation (NDE) of inanimate
objects. Use of x-rays to image the internal features of opaque objects is
well known, especially in the field of medical imaging. X-ray tomography
wherein images along selected planes through a subject are constructed
from data gathered from x-rays passed through the object in multiple
directions is also well established. However, the ionizing radiation used
in x-ray imaging has long been recognized as posing a health risk. Also,
x-rays are best suited to imaging features defined by variations in
density, and therefore, are not helpful in imaging soft tissue features.
Furthermore, it is difficult to image the internal structure of objects
that are strong absorbers of x-rays. Regions in an object that are strong
absorbers can produce noticeable image artifacts in adjacent, less
strongly absorbing regions.
Eddy current testing, wherein the effects of cracks and other defects in an
electrically conductive object on the impedance of a coil which induces
eddy currents in the conductive object is widely used in industry for
non-destructive testing. However, the information obtained is very limited
and highly skilled technicians are required to interpret the signals
generated. Furthermore, eddy current techniques are primarily used to
detect surface-breaking flaws since the signal strength falls off as the
inverse 6th power of the distance between the flaw and the detector, and
because the high-frequency electromagnetic fields used typically penetrate
no more than a fraction of a millimeter into the conducting object.
Another technique for imaging the internal structure of opaque objects is
ultrasound in which high frequency sound is injected into the object and
the echoes are used to map internal features creating the echoes.
Difficulties with this technique include distortion created by internal
reflections of the sound waves. Also, many objects have to be immersed in
a water bath to couple the ultrasound. It is also difficult to obtain
quantitative measures of flaw size, and because of strong reflections from
surfaces, it is difficult to image flaws just beneath a surface. It is
also impossible to "see through" air pockets in an object with ultrasound.
More recently, magnetic resonance imaging (MRI) has become widely used for
medical imaging. A very intense magnetic field is applied to the subject
to align the nuclei in the area of interest. Pulses of rf energy are then
applied to disturb this orientation. Energy released by relaxation of the
nuclei between pulses is detected and then processed to produce the image.
Through control of the frequency of the rf energy and the magnetic field,
resonance can be induced at a desired location within the subject, and
thus images along a selected plane through the subject can be generated.
While MRI produces sharp images, the cost is very high and the intense
magnetic fields required for the technique have raised concerns.
The use of magnetic fields for non-invasive examination of biological
organisms and for non-destructive testing has also been investigated.
Superconducting quantum interference devices (SQUIDS), which are highly
sensitive detectors of magnetic fields, have led to the development of
several such techniques. While there are other sensitive detectors of weak
magnetic fields, such as flux gates, SQUIDs offer the unique combination
of sensitivity, wide dynamic range, linearity and frequency response, and
are thus able to detect very weak magnetic fields in the presence of much
stronger magnetic fields. The most sensitive small-sample superconducting
susceptometer on the commercial market manufactured by Quantum Design of
San Diego, Calif., utilizes SQUID magnetometers to measure the
sample-induced perturbations in an intense magnetic field that is produced
by a superconducting magnet. Such susceptometers are used to study the
properties of high temperature superconductors, biological molecules, and
other materials.
One of the most widely publicized applications of SQUID magnetometers is in
the measurement of the magnetic fields produced by bioelectric activity in
the human heart and brain. In these biomagnetometers, a SQUID is connected
to a superconducting pick-up loop to form a differential magnetometer that
is relatively immune to uniform magnetic fields, but is highly sensitive
to the magnetic fields produced by the electrical activity in the brain
and other organs. A number of these SQUIDs are arranged in an array which
is placed adjacent to the head or chest of the subject being studied. The
SQUIDs in the array are electronically scanned and the pattern of magnetic
signals detected is inverted mathematically in an attempt to map the
electrical activity generating the field. As the mathematical inversion
has an indeterminate solution for a random distribution of
three-dimensional current sources, the technique has had some success in
localizing single dipole sources, but much less success in mapping
multiple bioelectrical dipole sources. This technique is only sensitive to
systems that exhibit internal electrical current sources.
A lesser amount of activity has been directed to the use of sensitive
magnetometers to make magnetic measurements for non-destructive testing or
evaluation. One technique has been to apply an electric current, directly
or through induction, to a substantially geometrically regular object and
to observe, in the magnetic field generated by the current, perturbations
caused by flaws, such as cracks or holes in the object. Typically, the
magnetometer is scanned over the object to locate the perturbations
resulting from the flaw. Obviously the object must be electrically
conductive. Only limited information about the object and the flaw can be
obtained from such a technique, since the current distribution, and
therefore, the magnetic field, is dependent upon the distribution of
conductivity through the object. One flaw could perturb the distribution
throughout the object, and hence render other flaws less easily
detectable.
Work has also been done in mapping perturbations in a magnetic field
applied to ferromagnetic materials to locate flaws such as cracks and
holes. Such a technique can provide a map of the resulting perturbations
in the magnetic field external to the object, but can only provide a
qualitative evaluation of structural details of the flaw. The difficulty
lies in the fact that the orientation of the induced or remnant magnetic
dipoles in the ferromagnetic materials, which distort the applied field,
is unknown. Sequential measurement of both the induced and remnant
magnetic fields in ferromagnetic materials has been used to determine the
state of heat treatment to which the material has been subjected.
Techniques have been developed for non-invasive analysis of human organs
based upon measurements of variations in magnetic susceptibility. U.S.
Pat. Nos. 3,980,076 and 4,079,730, which share a common inventor with the
subject invention, disclose a method and apparatus for making a
quantitative evaluation of cardiac volume. The technique is based on the
fact that the magnetic susceptibility of blood is substantially different
from that of the heart and surrounding tissue. The heart is modeled as a
sphere which changes in volume and moves within the chest cavity between
the systole and diastole. Measurements are made at several points of
perturbations in an applied magnetic field caused by the beating heart.
These measurements are used to adjust the model until calculations made by
the model match the empirical measurements. Patterns of normal and
abnormal cardiac activity must be developed to evaluate the results.
Similar techniques have been applied to determine the amount of iron stored
in a human liver which is useful in the diagnosis of certain medical
conditions. A SQUID measures the perturbations to an applied magnetic
field caused by a bag of water placed on the patient's abdomen which
provides an indication of the bulk susceptibility of the water. The SQUID
is then advanced toward the patient thereby flattening the water bag until
the sensitive volume of the SQUID intersects the liver and additional
measurements of the perturbations to the applied magnetic field are made.
As in the case of determining cardiac volume, a physiological model is
used to determine the concentration of iron in the liver. In this model,
the liver is considered to be a continuous homogeneous mass, and hence,
the bulk susceptibility of the liver is presumed in the model.
Another medical imaging technique which has received attention is
electrical impedance tomography. This technique relies upon the
differences in electrical resistivity in different tissues within the
subject. As applied to cardiac imaging, a number of oppositely positioned
electrode pairs are distributed around the torso and as electric currents
are successively passed through the subject between electrode pairs, the
electric voltage drop is measured across the electrode pair. As the
current through tissue of any particular resistivity is dependent upon the
resistivity of parallel current paths, imaging with this technique is
difficult.
U.S. Pat. No. 4,982,158 suggests a non-destructive testing technique in
which currents are injected into an electrically conductive or
semiconductive object from two or more directions and the resulting
magnetic field is mapped with a SQUID array to detect inhomogeneities in
the object.
There remains a need for a system for imaging the surface and internal
structure of opaque objects which provides good detail without the need
for ionizing radiation or high field intensities, and at a reasonable
cost.
SUMMARY OF THE INVENTION
This and other objects are realized by the invention which is directed to a
system and a method of mapping local magnetic susceptibility to generate
an image of the surface or internal structure of an object. The invention
is based upon the principle that all materials, even diamagnetic materials
generally considered to be non-magnetic, can in fact be at least weakly
magnetized. The invention is useful for the imaging of objects made of
materials and combinations of materials which are diamagnetic and
paramagnetic. Such materials may be characterized by the fact that while
they may be magnetized, the magnetic dipoles are so weak that they do not
have any appreciable distorting effect upon the magnetizing field applied
to adjacent molecules of the object. Thus, the invention can also be used
to image features of diamagnetic and paramagnetic materials with embedded
superparamagnetic and even ferromagnetic materials as long as the
concentration of the superparamagnetic or ferromagnetic material is not so
great as to appreciably distort the applied magnetic field permeating the
features of the object to be imaged.
More particularly, the invention is directed to a system and method for
imaging the internal or surface features of objects composed substantially
of diamagnetic and/or paramagnetic materials which include applying a
magnetic field to the object with a known strength, orientation, and time
dependence. This includes utilization of the earth's magnetic field as the
magnetizing field with the object positioned in a known orientation with
respect to the field. Preferably, she applied magnetic field is generated
by an electromagnet. A superconducting magnet, which can be housed within
the same cold volume as houses a SQUID used as the magnetometer to detect
perturbations in the magnetic field due to local susceptibility in the
object, could be used to generate the applied magnetic field, but is not
essential. In fact, a uniform field such as is produced by a Helmholtz
coil pair is preferred. Perturbations in the applied magnetic field at an
array of locations across the object are measured by magnetometer means to
generate an array of perturbation signals. This array of perturbation
signals is processed to generate an array of signals representing a map of
local susceptibility across the object.
A single magnetometer or a small array of magnetometers may be scanned over
the object to generate the array of perturbation signals. This scan can be
generated by movement of either the magnetometer or the object.
Alternatively, the object is scanned electronically by a large array of
magnetometers.
The invention includes susceptibility tomography in which the orientation
of at least one of the magnetic field, the position of the array of
magnetometer measurements, and the position of the object is varied to
generate a plurality of arrays of perturbation signals which are used to
determine the local susceptibility at selected sites in the object. Thus,
the susceptibility at all locations through the object, or at selected
sites such as for example along a given plane representing a section
through the object, can be generated.
The variations in orientation needed to generate the plurality of arrays of
perturbation signals can be produced by physical movement of the elements,
or for instance by providing a plurality of sets of coils and/or
magnetometer arrays and operating these multiple elements to generate the
variable orientations.
BRIEF DESCRIPTION OF THE DRAWINGS
A full understanding of the invention can be gained from the following
description of the preferred embodiment when read in conjunction with the
accompanying drawings in which:
FIG. 1 is a schematic diagram of a susceptibility imaging system in
accordance with the invention.
FIG. 2 schematically illustrates a SQUID magnetometer suitable for use in
the system of FIG. 1.
FIG. 3 schematically illustrates a large SQUID array which electronically
scans perturbations in an applied magnetic field across an object.
FIG. 4 schematically illustrates a superconducting magnet which can be used
an an alternative to the warm magnets used in the system of FIGS. 1 and 2.
FIG. 5 is a schematic diagram of a portion of the system of FIG. 1 modified
to operate with an ac magnetic field.
FIG. 6 illustrates an alternative embodiment of the system of FIG. 1
utilizing three orthogonal sets of electromagnets and six perpendicular
planar SQUID arrays.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Broadly, the invention is directed to imaging the structure of objects made
substantially of diamagnetic or paramagnetic materials by mapping local
magnetic susceptibility in the object.
The magnetic susceptibility of a material is defined as the ratio of the
magnetization density to the intensity of the magnetizing field and thus
may be expressed by the relation:
.chi.=H.sup.M 1
where .chi. is the magnetic susceptibility, M is the magnetization density
and H is the magnetic field intensity.
The magnetization of the material has an effect on the magnetic field which
may be written as:
B=.mu..sub.0 (H+M) (2)
where B is the field and .mu..sub.0 is the permeability of free space. As
can be seen, the field B is the sum of the applied field H plus the field
contributed by magnetization M of the material that is produced by the
applied field. Substituting Equation 1 into Equation 2 results in:
B=.mu..sub.0 (1+.times.)H=.mu..sub.0 .mu..sub.r H=.mu.H (3)
where .mu..sub.r is the relative permeability and .mu. is the absolute
permeability and:
.mu..sub.r =1+.times. (4)
In ferromagnetic materials, .times. dominates so that:
.mu..sub.r (iron).perspectiveto..times.=10.sup.2 to 10.sup.5 (5)
On the other hand, in diamagnetic, and to a lesser extent paramagnetic
materials, .times. is very much smaller than one, and hence, has only very
small effect on relative permeability. Thus, for diamagnetic materials for
which .times..perspectiveto.-10.sup.-5 :
.mu..sub.r (diagramatic)=1-10.sup.-5 .perspectiveto.1 (6)
and for paramagnetic materials for which .times..perspectiveto.10.sup.-2 ;
.mu..sub.r (paramagnetic)=1+10.sup.-2 .perspectiveto.1 (7)
What this means is that in ferromagnetic materials, the magnetization M has
a strong effect on the field intensity, H, within the material and the
magnetic dipoles are coupled to the surrounding magnetic dipoles and hence
it is not readily possible to determine the field, B, within the
ferromagnetic material.
However, in diamagnetic and paramagnetic materials the magnetization has
neglible effect on the applied field, about five to eleven orders of
magnitude less than in ferromagnetic materials. Thus, in accordance with
the Born approximation, the effects of magnetic dipoles within diamagnetic
and paramagnetic materials on the magnetization of adjacent regions of the
material can be ignored. Still, perturbations to the applied field as a
result of magnetization of the diamagnetic or paramagnetic material can be
measured by a sensitive magnetometer.
In accordance with the invention, measurements of the perturbations to an
applied magnetic field across an object, or a selected portion of the
object are used to calculate the local susceptibility in the object. A map
of the local susceptibility represents an image of the structure of the
object, and thus can be be used, for instance, to image different tissues
within a biological sample, such as the organs, diseased tissue and
non-ferromagnetic foreign objects, and to image defects such as cracks,
voids and inclusions as well as internal structures in inanimate samples
and tracer materials introduced into a sample. The latter can include, for
example, introducing a tracer into the blood stream of a living subject,
applying a tracer material over the surface of a sample to detect surface
defects, and mixing the tracer material with a bulk material. The tracer
material should have a susceptibility substantially different from the
surrounding material. This can include the use of superparamagnetic
materials and even ferromagnetic materials as tracers, so long as the
concentration is not so great as to mask the features sought to be imaged.
In imaging the local susceptibility across a two-dimensional object made
primarily of a paramagnetic or diamagnetic material, or imaging only the
surface of a three dimensional object of such a material, the local
perturbation measurements are converted to local susceptibility. This is
possible because paramagnetic and diamagnetic materials are, in general,
linear and non-hysteretic, and at the applied field strengths used, the
susceptibility is independent of the applied field. As a result of
Equations 6 and 7, we can usually work in the weak-field limit, which is
also known as the Born approximation: at any point in the material we can
ignore the contributions to the applied field from the magnetization
elsewhere in the object and we are left only with the initial fields
B.sub.0 and H.sub.0 and the local field B.sub.1 due to the local
magnetization M(r') at the source point r':
B.sub.o (r')+B.sub.1 (r')=.mu..sub.o H.sub.o (r')+.mu..sub.o M(r) (8)
This is truly a local equation that is independent of the magnetization
elsewhere in the sample. Because M is so weak, if we know H.sub.0
everywhere, we will then know B.sub.0 to at least one part in 10.sup.2 for
a paramagnetic material with .times.=10.sup.-2, and to 1 part in 10.sup.5
for a diamagnetic material with .times.=10.sup.-5. Thus, we have
eliminated a major problem in obtaining a self-consistent, macroscopic
solution that is based upon the microscopic constructive equation given by
Eq. 2. Since we know B.sub.0 (r') and H.sub.0 (r'), we can thereby
eliminate B.sub.0 and H.sub.0 from the problem and Eq. 8 becomes:
B.sub.1 (r')=.mu..sub.o M(r') (9)
If H.sub.0 is uniform, the spatial variation of M(r') is determined only by
.chi.(r'), so that:
B.sub.1 (r')=.mu..sub.o .chi.(r')H.sub.o (r') (10)
For isotropic materials, x is a scalar and the direction of M is the same
as that of H.sub.0 ; otherwise a tensor susceptibility is required.
In most cases, the magnetometer does not measure the magnetic field at the
source point r', but instead at a distant "field" point r. If the
magnetometer measures only the very small perturbations B.sub.1 that are
superimposed upon the stronger applied B.sub.0 the weak paramagnetism and
diamagnetism can be detected. Hereafter, we will assume that the
measurements are only of the perturbation field from either diamagnetic or
paramagnetic materials. Suppose that a small element of volume dv' within
the object has a magnetization M(r'), with the element being small enough
that the magnetization can be assumed to be uniform throughout the
element. The dipole moment of this elemental volume is simply:
dm=Mdv' (11)
The contribution of the magnetic field at the field point r from this
element at r' is given by the dipole field equation:
##EQU1##
The total magnetic field B(r') is obtained by integrating the field from
the magnetization associated with each elemental dipole:
##EQU2##
The general inverse problem involves solving Equation 13 for the vector
magnetization M(r') throughout the sample given a set of measurements of
B(r). In general, this equation has no unique solution. However, if we
know that M(r')=x(r')H.sub.0 (r') we know the orientation of M everywhere.
If we assume that M(r')=M.sub.z (r')z, then we need only to solve for the
scalar magnetization M.sub.z (r') in the slightly simpler equation:
##EQU3##
The reduction of the number of unknowns from the three components of M(r')
to the single component M.sub.z (r') represents a major simplification of
the problem. Within the constraints imposed by measurement noise and
computational accuracy, this allows us, in general, to obtain a solution
of Equation 14. Except is cases with special geometrical constraints, any
single component of M could be used equally effectively.
If the source is restricted to two-dimensions, such as a thin sheet of
diamagnetic or paramagnetic material, Equation 14 reduces to a
two-dimensional surface integral. We will for now assume that we are
applying only z-component field H.sub.0 z, and are measuring only the
z-component of the sample-induced magnetic field B at a height (z--z')
above the two-dimensional sample, so that we have:
##EQU4##
In practice, the integrals need not extend beyond the boundary of the
source object, outside of which M.tbd.0. In order to solve this equation
for M.sub.z (r'), we first compute the two-dimensional spatial Fourier
Transform (FT) of the magnetic field:
b.sub.z (k.sub.z,k.sub.y,z)=FT{B.sub.z (x,y,z)} (16)
so that we can use the convolution theorem to express Equation 15 in the
spatial frequency domain:
b.sub.z (k.sub.z, k.sub.y, z)=g.sub.z (k.sub.z,k.sub.y,z)m.sub.z
(k.sub.z,k.sub.y), (17)
where g.sub.z (k.sub.x,k.sub.y,z) is the spatial Fourier transform of the
Green's function:
##EQU5##
and m.sub.z (k.sub.x,k.sub.y) is the spatial Fourier transform of the
magnetization M.sub.z (x',y'). The inverse problem then reduces to a
division in the spatial frequency domain:
##EQU6##
It may be necessary to use windowing techniques to prevent this equation
from blowing up because of zeros in the Green's function occurring at
spatial frequencies for which there is a contribution to the magnetic
field from either the sample or from noise. Typically, the window
w(k.sub.x,k.sub.y) is a low-pass filter which attenuates high-frequency
noise in the vicinity of the zeros of g.sub.z, so that Equation 19
becomes:
##EQU7##
As the final step, we use the inverse Fourier Transform (FT.sup.-1) to
obtain an image of the magnetization distribution:
M.sub.z (x',y')=FT.sup.-1 {m.sub.z (k.sub.z,k.sub.y)}, (21)
which can then be used to obtain the desired susceptibility image:
##EQU8##
This provides one approach to two-dimensional magnetic susceptibility
imaging. It is important to note that once the Green's function and the
window have been specified, it is possible to proceed directly from
B.sub.z (x,y) to x(x',y') by evaluating the appropriate convolution
integral in xy-space.
This embodiment of the invention is useful for instance in non-destructive
evaluation of semi-conductor devices. The susceptibility of the various
metals and insulators used in fabrication of silicon-based integrated
circuits has a sufficiently high range of values to provide adequate
contrast for susceptibility imaging of such devices. The following
susceptibilities (10.sup.-6 SI) are relevant to the semiconductor
industry:
##EQU9##
For mapping the surface of a three dimensional object, such as for locating
and quantifying surface defects, the sensitive volume of the magnetometer
can be adjusted such as through selection of known designs for gradient
pick-up coil arrangements, and/or, through shaping of the applied field.
An important aspect of the invention is susceptibility tomography.
Where, as in susceptibility tomography, the source is three-dimensional, a
somewhat more general approach must be taken. We can start with Equation
12, the dipole field equation for the magnetic field dB(r) produced by a
single magnetic dipole dm(r'). If the dipole moment arises from the
magnetization of an incremental volume dv in an applied field H(r'), we
have that:
dm(r')=.chi.(r')H(r')d.upsilon.' (23)
The dipole field equation then becomes:
##EQU10##
This equation can be written as:
dB(r)=G(r,r,H).chi.(r')d.upsilon.' (25)
where we introduce a vector Green's function:
##EQU11##
The components of G are simply:
G=G.sub.z x+G.sub.y y+G.sub.z z (27)
where:
##EQU12##
The three components of the magnetic field in Equation 25 can now be
written as:
dB.sub.z (r)=.chi.(r')G.sub.z (r, r', H) (31)
dB.sub.y (r)=.chi.(r)G.sub.y (r, r', H) (32)
dB.sub.y (r)=.chi.(r')G.sub.y (r, r',H) (33)
Note that H may in turn be a function of x', y', and z'. In contrast, the
Green's function in Equation 18 did not contain H. However, H is assumed
to be known and only adds a geometrically-variable scale factor into the
Green's function. The increased complexity of Equations 28 through 30
arises from our desire to include H as a vector field with three
independently-specified components.
If we know both the location r' of a source that is only a single dipole,
and also the strength and direction of H at that point, we can make a
single measurement of the magnetic field at r to determine .chi.(r'). It
is adequate to measure only a single component of B(r') as long as that
component is non-zero. The problem becomes somewhat more complex when
there are either multiple dipoles or a continuous distribution of dipoles.
In that case, we need to summate or integrate Equation 25 over the entire
source object:
##EQU13##
To proceed numerically, we will assume that we can discretize the source
object into m elements of volume v.sub.j where 1.ltoreq.j.ltoreq.m. The
field from this object is then:
##EQU14##
A single measurement of B will be inadequate to determine the
susceptibility values for | | |
