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Description  |
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FIELD OF THE INVENTION
The present invention provides an image of the magnetic susceptibility function of inanimate and animate objects, including the human body.
BACKGROUND
The basis of all imaging modalities exploits some natural phenomenon which varies from tissue to tissue, such as acoustic impedance, nuclear magnetic relaxation, or x-ray attenuation; or, a substance such as a positron or gamma ray emitter is
added to the body and its distribution is reconstructed; or, a substance is added to the body which enhances one or more of acoustic impedance, nuclear magnetic relaxation, or x-ray attenuation. Each imaging modality possesses certain characteristics
which provide superior performance relative to other modalities of imaging one tissue or another. For example, x-ray contrast angiography has an imaging time less than that which would lead to motion artifact and it possesses high resolution which makes
it far superior to any prior known imaging modality for the task of 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; therefore, it is not indicated except for patients with severe arterial or venous pathology.
SUMMARY OF THE INVENTION
All tissues of the body possess a magnetic susceptibility which is diamagnetic or paramagnetic; therefore, magnetized tissue produces a secondary magnetic field. This field is that of a series of negative and positive dipoles spatially
distributed at frequencies representative of the magnetic susceptibility function of the tissue at a given level of resolution, where each dipole is representative of a volume element or voxel of dimensions equalling the limiting resolution, and where
the magnitude of any dipole is given by the product of the volume of the voxel, the magnetic flux strength at the voxel, and the magnetic susceptibility of the voxel.
The present invention herein disclosed as MSI (Magnetic Susceptibility Imaging) is directed to the reconstruction of the magnetic susceptibility function of the tissue at the said limiting resolution from measurements of the said secondary field,
where the signal-to-noise ratio of the said measurement determines the said resolution. MSI comprises 1) a generator to magnetize the tissue to be imaged, such as a first Helmholtz coil, 2) detector to record the secondary magnetic field produced by the
tissue, such as an array of Hall magnetic field detectors, 3) a nullifier to null the external magnetic field produced by the magnetizing means such that the secondary field can be recorded independently of the external magnetizing field, such as a
second Helmholtz coil which confines the magnetic flux of the said first and second Helmholtz coils to the plane of the said detector array, so that the nonzero perpendicular component of the secondary field may be recorded, and 4) a reconstruction
processor to reconstruct the magnetic susceptibility function of the tissue from recordings of the secondary magnetic field made over a sample space, such as a reconstruction algorithm which Fast Fourier Transforms the signals, divides the said transform
by the Fourier Transform of the system function which is the impulse response of the said detector array, Fast Fourier Inverse Transforms the said product, and evaluates the dipole values by applying a correction factor to each element of the resultant
matrix, where the formula for the correction factors is determined by the dimensions of the said sample space over which the signals of the said secondary magnetic field were recorded.
The resultant image is displayed three-dimensionally and can be further processed to provide enhancement or to be displayed from any three-dimensional perspective or as two-dimensional slices.
DETAILED DESCRIPTION OF THE DRAWINGS
The present invention is further described with respect to the drawings having the following, solely exemplary figures, wherein:
FIG. 1A shows the electron population diagram of the e.sub.g and t.sub.2g orbitals of a high spin d.sup.6 complex;
FIG. 1B shows the electron population diagram of the e.sub.g and t.sub.2g orbitals of a low spin d.sup.6 complex;
FIG. 2 shows the general process of reconstruction by reiteration;
FIG. 3 shows a coordinate system and distances from a voxel to a point detector;
FIG. 4 shows a coordinate system of a two-dimensional detector array where the detectors generate a voltage along the length l in response to a magnetic field perpendicular to the plane;
FIGS. 5a and 5b are the coordinate system of the prototype; and
FIG. 6 is a block diagram of one embodiment of the system according to the present invention.
FIG. 7 shows a three dimensional detector array.
Further details regarding specific derivations, calculations and experimental implementation are provided in the attached appendices, wherein:
Appendix I is the derivation of the field produced by a ring of dipoles;
Appendix II is the derivation of the field produced by a shell of dipoles;
Appendix III is the derivation of the field produced by a sphere of dipoles;
Appendix IV is the derivation of the Fourier Transform of the System Function used in a reconstruction process according to the present invention;
Appendix V is the derivation of S=HF*U[K.sub.z ] convolution used in a reconstruction process according to the present invention;
Appendix VI is the derivation of the solution of Inverse Transform 1 used in a reconstruction process according to the present invention;
Appendix VII is the listing for the PSI Prototype LIS Program used to calculate experimental MSI results; and
Appendix VII is the listing for the PSI Prototype I LIS Program used to calculate experimental MSI results.
DETAILED DESCRIPTION OF THE INVENTION
Uniqueness
Linus Pauling demonstrated in 1936 that blood is a mixture of components of different magnetic susceptibilities. The predominant components are water and iron containing hemoglobin of red blood cells having magnetic susceptibilities of
-7.times.10.sup.-6 and 1.2.times.10.sup.-2, respectively, where blood corpuscles constitute about one-half of the volume of blood. Due to the presence of an iron atom, each hemoglobin molecule has a paramagnetic moment of 5.46 Bohr magnetons resulting
from four unpaired electrons. Hemoglobin in blood contributes a significant paramagnetic contribution to the net magnetic susceptibility of blood. The net susceptibility arises from the sum of noninteracting spin wave-functions and a state of uniform
magnetization is not achieved by magnetizing blood. In fact, there is no interaction between spin wave-functions or orbital wave-functions of any pure paramagnetic or diamagnetic material, respectively, or any paramagnetic or diamagnetic mixture,
respectively, including the constituents of human tissue. The divergence of the magnetization in magnetized blood or tissue is not zero, and the secondary magnetic field due to magnetized tissue has to be modeled as noninteracting dipoles aligned with
the imposed field It is demonstrated below that the field of any geometric distribution of dipoles is unique, and the superposition principle holds for magnetic fields; therefore, a unique spatial distribution of dipoles gives rise to a unique secondary
magnetic field, and it is further demonstrated below that this secondary field can be used to solve for the magnetic susceptibility map exactly. It follows that this map is a unique solution.
To prove that any geometric distribution of dipoles has a unique field, it must be demonstrated that the field produced by a dipole can serve as a mathematical basis for any distribution of dipoles. This is equivalent to proving that no
geometric distribution of dipoles can produce a field which is identical to the field of a dipole. By symmetry considerations, only three distributions of uniform dipoles need to be considered a ring of dipoles, a shell of dipoles, and a sphere of
dipoles The fields produced by these distributions are given as follows, and their derivations appear in Appendices I, II and III, respectively. ##EQU1## for R=0. ##EQU2## which is the field due to a single dipole.
Thus, a ring of dipoles gives rise to a field different from that given by a dipole, and the former field approaches that of a single dipole only as the radius of the ring goes to zero. ##EQU3## For R=0 ##EQU4## which is the field due to a
single dipole at the origin. Thus, a shell of dipoles gives rise to a field which is different from that of a single central dipole. The field in the former case is that of a dipole only when the radius of the shell is zero as would be expected.
##EQU5## For R=0, ##EQU6## which is the field due to a single dipole at the origin. Thus, a sphere of dipoles gives rise to a field which is different from that of a single central dipole. The field in the former case is that of a dipole only when the
radius of the sphere is zero.
These cases demonstrate that the field produced by a magnetic dipole is unique. Furthermore, the image produced in MSI is that of dipoles. Since each dipole to be mapped gives rise to a unique field and since the total field at a detector is
the superposition of the individual unique dipole fields, linear independence is assured; therefore, the MSI map or image is unique. That is, there is only one solution of the MSI image for a given set of detector values which spatially measure the
superposition of the unique fields of the dipoles. This map can be reconstructed using the algorithms described in the Reconstruction Algorithm Section.
The resulting magnetic susceptibility map is a display of the anatomy and the physiology of systems such as the cardiopulmonary system as a result of the large difference in the magnetic susceptibility of this system relative to the background
susceptibility.
Magnetic Susceptibility of Oxygen and Deoxyhemoglobin
The molecular orbital electronic configuration of O.sub.2 is
and by Hund's rule,
that is, unpaired electrons of degenerate orbitals have the same spin quantum number and O.sub.2 is therefore paramagnetic.
The magnetic susceptibility of O.sub.2 at STP is 1.8.times.10.sup.-6. Also, ferrohemoglobin contains Fe.sup.2+ which is high spin d.sup.6 complex, as shown in FIG. 1A, and contains four unpaired electrons. However, experimentally oxyhemoglobin
is diamagnetic. Binding of O.sub.2 to hemoglobin causes a profound change in the electronic structure of hemoglobin such that the unpaired electrons of the free state pair upon binding. This phenomenon is not seen in all compounds which bind
hemoglobin. Nitrous oxide is paramagnetic in both the bound and free state and NO--Hb has a magnetic moment of 1.7 Bohr magnetons.
Furthermore, oxyhemoglobin is in a low spin state, containing no unpaired electrons, as shown in FIG. 1B, and is therefore diamagnetic. However, the magnetic susceptibility of hemoglobin itself (ferrohemoglobin) corresponds to an effective
magnetic moment of 5.46 Bohr magnetons per heme, calculated for independent hemes. The theoretical relationship between .mu..sub.eff, the magnetic moment, and S, the sum of the spin quantum numbers of the electrons, is given by ##EQU7##
The magnetic moment follows from the experimental paramagnetic susceptibility X according to ##EQU8## where T is the absolute temperature and .theta. is the Curie-Weiss constant (assumed to be zero in this case) The experimental paramagnetic
susceptibility of hemoglobin/heme is 1.2403.times.10.sup.-2 (molar paramagnetic susceptibility calculated per gram atom of heme iron), and the concentration of Hb in blood is 150 g/l=2.2.times.10.sup.-3 M; 8.82.times.10.sup.-3 M Fe.
Magnetic Susceptibility of Paramagnetic Species Other Than Oxygen and Hemoglobin
The specific susceptibility of H.sub.2 O saturated with air and deoxyhemoglobin is -0.719.times.10.sup.-6 and 1.2403.times.10.sup.-2, respectively.
By Curie's law, the paramagnetic susceptibility is represented by
where N is the number of magnetic ions in the quantity of the sample for which .chi. is defined, .mu. is the magnetic moment of the ion, and k is the Boltzmann constant. .mu. can be expressed in Bohr magnetons:
which is the natural unit of magnetic moment due to electrons in atomic systems Thus, the Bohr magneton number n is given by n=n.mu..sub.B.
Assuming the magnetic moments come solely from spins of electrons, and that spins of f electrons are aligned parallel in each magnetic ion, then,
and substituting the resultant spin quantum number,
The free radical concentrations in human liver tissues measured with the surviving tissue technique by Ternberg and Commoner is 3.times.10.sup.15 /g wet weight. Furthermore, human liver would contain the greatest concentration of radicals, since
the liver is the most metabolically active organ.
The molal paramagnetic susceptibility for liver is calculated from Curie's law: ##EQU9## For any material in which the magnetization M is proportional to the applied field, H, the relationship for the flux B is B=.mu..sub.0 (1+4.pi..chi..sub.m)H,
where .chi..sub.m is the molal magnetic susceptibility, .mu..sub.0 is the permitivity, and H is the applied field strength The susceptibility of muscle, bone, and tissue is approximately that of water, -7.times.10.sup.-7, which is very small; therefore,
the attenuation effects of the body on the applied magnetic field are negligible. Similarly, since the same relationship applies to the secondary field from deoxyhemoglobin, the attenuation effects on this field are negligible. Furthermore, for liver
Therefore, the effect of the background radicals and cytochromes on the applied field are negligible, and the signal-to-noise ratio is not diminished by these effects. Also, any field arising from the background unpaired electrons aligning with
the applied field would be negligible compared to that arising from blood, because the magnetic susceptibility is seven orders of magnitude greater for blood.
Furthermore, the vascular system can be imaged despite the presence of background blood in tissue because the concentration of hemoglobin in tissue is 5% of that in the vessels.
The magnetic susceptibility values from deoxyhemoglobin and oxygen determines that MSI imaging is specific for the cardiopulmonary and vascular systems where these species are present in much greater concentrations than in background tissue and
serve as intrinsic contrast agents. The ability to construct an image of these systems using MSI depends not only on the magnitude of the differences of the magnetic susceptibility between tissue components, but also on the magnitude of the signal as a
function of the magnetic susceptibility which can be obtained using a physical instrument in the presence of parameters which cause unpredictable random fluctuations in the signal which is noise. The design of the physical instrument is described next
and the analysis of the contrast and resolution of the image are described in the Contrast and Limiting Resolution Section.
System Construction
The MSI scanner entails an apparatus to magnetize a volume of tissue to be imaged, an apparatus to sample the secondary magnetic field at the Nyquist rate over the dimensions which uniquely determine the susceptibility map which is reconstructed
from the measurements of the magnetic field, and a mathematical apparatus to calculate this magnetic susceptibility map, which is the reconstruction algorithm.
Magnetizing Field
The applied magnetizing field (provided by coils 57, FIG. 6) which permeates tissue is confined only to that region which is to be imaged. The confined field limits the source of signal only to the volume of interest; thus, the volume to be
reconstructed is limited to the magnetized volume which sets a limit to the computation required, and eliminates end effects of signal originating outside of the edges of the detector array.
A magnetic field gradient can be applied in the direction perpendicular to the plane of the detector array described below to alter the dynamic range of the detected signal, as described in the Altering the Dynamic Range Provided by the System
Function Section.
The magnetizing field generating elements can also include an apparatus including coils 93 disposed on the side of the patient 90 having detector array 61, and energized by amplifier 91 energized by a selected d.c. signal, to generate a magnetic
field which exactly cancels the z directed flux permeating the interrogated tissue at the x,y plane of the detector array, as shown in FIG. 6, such that the applied flux is totally radially directed. This permits the use of detectors which produce a
voltage in response to the z component of the magnetic field produced by the magnetized tissue of interest as shown in FIG. 4, where the voltages are relative to the voltage at infinity, or any other convenient reference as described below in the
Detector Array Section.
The magnetizing means can also possess a means to add a component of modulation to the magnetizing field at frequencies below those which would induce eddy currents in the tissue which would contribute significant noise to the secondary magnetic
field signal. Such modulation would cause an in-phase modulation of the secondary magnetic field signal which would displace the signal from zero hertz, where white noise has the highest power density.
Detector Array
The MSI imager possesses a detector array of multiple detector elements which are arranged in a plane. This two dimensional array is translated in the direction perpendicular to this plane during a scan where readings of the secondary magnetic
field are obtained as a function of the translation. In another embodiment, the array is three dimensional comprising multiple parallel two dimensional arrays. See FIG. 7. The individual detectors of the array respond to a single component of the
magnetic field which is produced by the magnetized tissue where the component of the field to which the detector is responsive determines the geometric system function which is used in the reconstruction algorithm discussed in the Reconstruction
Algorithm Section. In one embodiment, the detectors provide moving charged particles which experience a Lorentz force in the presence of a magnetic field component perpendicular to the direction of charge motion. This Lorentz force produces a Hall
voltage in the mutually perpendicular third dimension, where, ideally, the detector is responsive to only the said component of the magnetic field.
Many micromagnetic field sensors have been developed that are based on galvanometric effects due to the Lorentz force on charge carriers. In specific device configurations and operating conditions, the various galvanomagnetic effects (Hall
voltage, Lorentz deflection, magnetoresistive, and magnetoconcentration) emerge. Semiconductor magnetic field sensors include those that follow.
1. The MAGFET is a magnetic-field-sensitive MOSFET (metal-oxide-silicon field-effect transitor). It can be realized in NMOS or CMOS technology and uses galvanomagnetic effects in the inversion layer in some way or another. Hall-type and
split-drain MAGFET have been realized. In the latter type, the magnetic field (perpendicular to the device surface) produces a current imbalance between the two drains. A CMOS IC incorporating a matched pair of n- and p-channel split drain MAGFET
achieves 10.sup.4 V/A.sup.. T sensitivity. Still higher sensitivity may be achieved by source efficiency modulation.
2. Integrated bulk Hall devices usually have the form of a plate merged parallel to the chip surface and are sensitive to the field perpendicular to the chip surface. Examples are the saturation velocity MFS, the DAMS (differential magnetic
field sensor) and the D.sup.2 DAMS (double diffused differential magnetic sensor) which can achieve 10 V/T sensitivity. Integrated vertical Hall-type devices (VHD) sensitive to a magnetic field parallel to the chip surface have been realized in standard
bulk CMOS technology.
3. The term mg magnetotransitor (MT) is usually applied to magnetic-field-sensitive bipolar transistors. The vertical MT (VMT) uses a two-or four-collector geometry fabricated in bipolar technology, and the lateral MT (LMT) is compatible with
CMOS technology and uses a single or double collector geometry Depending on the specific design and operating conditions, Lorentz deflection (causing imbalance in the two collector currents) or emitter efficiency modulation (due to a Hall-type voltage in
the base region) are the prevailing operating principles.
4. Further integrated MFS are silicon on saphire (SOS) and CMOS magnetodiodes, the magnetounijunction transistor (MUJT), and the carrier domain magnetometer (CDM). The CDM generates an electrical output having a frequency proportional to the
applied field.
5. (AlGa) As/GaAs heterostructure Van der Pauw elements exploit the very thin high electron mobility layer realized in a two-dimensional electron gas as a highly sensitive practical magnetic sensor with excellent linearity on the magnetic field.
6. The Magnetic Avalanche Transistor (MAT) is basically a dual collector open-base lateral dipolar transitor operating in the avalanche region and achieves 30 V/T sensitivity.
A broad variety of other physical effects, materials and technologies are currently used for the realization of MFS. Optoelectronic MFS, such as magnetooptic MFS based on the Faraday effect of optical-fiber MFS with a magnetostrictive jacket,
use light as an intermediary signal carrier. SQUIDS (Superconducting Quantum Interference Devices) use the effects of a magnetic field on the quantum tunneling at Josephson Junctions.
Metal thin film magnetic field sensors (magnetoresistive sensors) use the switching of anisotropic permalloy (81:19 NiFe) which produces a change in electrical resistance that is detected by an imbalance in a Wheatstone Bridge.
Integrated magnetic field sensors are produced with nonlinearity, temperature drift, and offset correction and other signal conditioning circuitry which may be integrated on the same chip. Hall type microdevices possess a nonlinearity and
irreproducibility error of less than 10.sup.-4 over a 2 Tesla range, less than 10 ppm/0.degree..sub.C temperature drift, and a high sensitivity to detect a magnetic flux of nanotesla range. And, micro Van der Pauw and magnetoresistive devices with high
linearity over 100 guass range have sufficient signal to noise ratio to measure a magnetic flux of lnT. Magnetooptic MFS and SQUIDS can detect a magnetic flux of 10.sup.-15 T.
The signal arising from the external magnetizing field can be eliminated by taking the voltage between two identical detectors which experience the same external magnetizing field, for example, where the magnetizing field is uniform Or, the
external magnetic field can be nulled (by coil 93, FIG. 6) to confine the flux to a single plane at the plane of the detector array where the detectors of the array are responsive only to the component of the magnetic field perpendicular to this plane.
The voltage of the detector can then be taken relative to the voltage at ground or any convenient reference voltage.
Exemplary circuitry according to the present invention to measure the secondary magnetic field using an array 61 of 10.sup.4 Hall effect sensors is shown in FIG. 6. For each magnetic field sensor, an a.c. constant-current source (oscillator 63)
is used to provide a sine-wave current drive to the Hall sensors in array 61. Alternately, the exciting and/or nulling coils 57 and 93 can receive an amplified oscillator 63 signal for the synchronous rectification (detection). Amplifier 59 energizes
the exciting coils 57 by selected signals including a d.c. signal on the a.c. oscillator signal, or a selected combination of both. A counter produces a sawtooth digital ramp which addresses a sine-wave encoded EPROM. The digital sine-wave ramp is
used to drive a digital to analogue converter (DAC) producing a pure voltage reference signal which is used as an input to a current-drive circuit 59 and the synchronous detector 56. After band-pass/amplification, the received detector signals are
demodulated to d.c. with a phase sensitive (synchronous) rectifier followed by a low pass filter in 56. In 112, a voltage-to-frequency converter (VFC) is used to convert the voltage into a train of constant width and constant amplitude pulses; the
pulse rate of which is proportional to the amplitude of the analogue signal. The pulses are counted and stored in a register which is read by the control logic. The use of a VFC and counter to provide analog to digital conversion 112 enables the
integration period to be set to an integral number of signal excitation cycles; thus, rejecting the noise at the signal frequency, and enabling indefinitely long integration times to be obtained. The element 112 also includes input multiplex circuits
which allow a simple A/D converter to selectively receive input signals from a large number (e.g., 100) of detectors. The voltages from 56 corresponding to the 10.sup.4 detectors are received by 100 multiplexer-VFC-12 bit analogue-to-digital converter
circuits in 112, where each multiplex-VFC-A/D converter circuit services 100 detectors. The conversion of a set of array voltages typically requires 10.sup.-4 seconds, and 100 sets of array voltages are recorded in the processor 114 and converted
requiring a total time of 10.sup.-2 seconds, where the detector array 61 is translated to each of 100 unique positions from which a data set is recorded during a scan. If the dynamic range of the data exceeds that of the 12-bit A/D converter, then a
magnetic field gradient of the magnetizing field is implemented as described in the Dynamic Range Section.
The processed voltages are used to mathematically reconstruct the magnetic susceptibility function by using a reconstruction algorithm discussed in the Reconstruction Algorithm Section to provide a corresponding image on display 116.
Dynamic Range
As described in the Nyquist Theorem with the Determination of the Spatial Resolution Section, the system function which effects a dependence of the signal on the inverse of the separation distance between a detector and a dipole cubed band-passes
the magnetic susceptibility function; however, the resulting signals are of large dynamic range requiring at least a 12-bit A/D converter.
A 12 bit A/D converter is sufficient with a magnetizing design which exploits the dependence of the signal on the strength of the local magnetizing field. A quadratic magnetizing field is applied in the z direction such that the fall-off of the
signal of a dipole with distance is compensated by an increase in the signal strength due to a quadratically increasing local magnetizing field. The reconstruction algorithm is as discussed in the Reconstruction Algorithm Section, however, each element
of the matrix containing the magnetic susceptibility values is divided by the corresponding element of the matrix containing the values of the flux magnetizing the said magnetic susceptibility. These operations are described in detail in the Altering
the Dynamic Range Provided by the System Function Section.
An alternate approach is to convert the analogue voltage signal into a signal of a different energy form such as electromagnetic radiation for which a large dynamic range in the A/D processing is more feasible.
Alternatively, the original signal or any analogue signal transduced from this signal may be further processed as an analogue signal before digitizing.
Reconstruction Algorithm
The reconstruction algorithm can be a reiterative, matrix inversion, or Fourier Transform algorithm. For all reconstruction algorithms, the volume to be imaged is divided into volume elements called voxels and the magnetized voxel with magnetic
moment .chi.B is modeled as a magnetic dipole where .chi. is the magnetic susceptibility of the voxel and B is the magnetizing flux at the position of the dipole.
A matrix inversion reconstruction algorithm is to determine a coefficient for each voxel mathematically or by calibration which, when multiplied by the magnetic susceptibility of the voxel, is that voxel's contribution to the signal at a given
detector. This is repeated for every detector and those coefficients are used to determine a matrix which, when multiplied by a column vector of the magnetic susceptibility values of the voxels, gives the voltage signals at the detectors. This matrix
is inverted and stored in memory. Voltages are recorded at the detectors and multiplied by the inverse matrix, to generate the magnetic susceptibility map which is plotted and displayed.
A reiterative algorithm is to determine the system of linear equations relating the voltage at each detector to the magnetic susceptibility of each voxel which contributes signal to the detector. Each of these equations is the sum over all
voxels of the magnetic susceptibility value of each voxel times its weighting coefficient for a given detector which is set equal to the voltage at the given detector. The coefficients are determined mathematically, or they are determined by
calibration. The magnetic susceptibility, .chi., for each voxel, is estimated and the signals at each detector are calculated using these values in the linear equations. The calculated values are compared to the scanned values and a correction is made
to .chi. of each voxel. This gives rise to a second, or recomputed, estimate for .chi. of each voxel. The signal value from this second estimate is computed and corrections are made as previously described. This is repeated until the correction for
each reiteration approaches a predefined limit which serves to indicate that the reconstruction is within reasonable limits of error. This result is then plotted.
The general process of reconstruction by reiteration is shown according to the steps of FIG. 2 (and is implemented in processor 114 in FIG. 6). The image displayed according to the process 200 is directly related to the magnetic susceptibility
of .chi. of voxel sections of the object examined, the image is merely a mapping of the magnetic susceptibility in three dimensions. Accordingly, signals produced by the magnetic sensors 110, in terms of volts, are a direct result of the magnetic
susceptibility .chi. of the voxel elements. Therefore, a reference voltage is generated at 210 from which the actual or measured sensor voltages is subtracted at 220. The reference voltages are modelled by assuming a signal contribution from each
voxel element to each sensor. Therefore, the signals are summed separately for each sensor according to a weighted contribution of the voxel elements over the x, y, and z axes according to a model of the tissue to be examined. The resulting modelled or
calculated voltage signals are compared at step 220, providing a difference or .DELTA. signal, weighted at step 230 to produce a weighted difference signal, which is then added to the previously estimated susceptibility value for each voxel element at
step 240. The resulting level, available in three dimensions corresponding to the axes x, y, and z, is selectively displayed on the display at step 250. Having adjusted the estimated susceptibility for each voxel, the calculated magnetic susceptibility
is recalculated at step 260, the resulting estimated sensor voltage is then compared to the actual sensor voltage at step 220, the process 200 being repeated until the difference is reduced to a tolerable value.
The reconstruction algorithm using Fourier Transforms involves exploiting the FFT to solve equation 4 given below. The FFT algorithm is given below and is followed by a discussion of the resolution of the magnetic susceptibility map in the
Nyquist Theorem with the Determination of the Spatial Resolution Section.
For the case that follows, data is acquired in the x, y, and z directions, but in general data is acquired over the dimensions which uniquely determine the magnetic susceptibility map. Also, the present analysis is for measuring the z magnetic
field component of a dipole oriented in the z direction; however, the analysis applies to the other two orthogonal components where the geometric system function for the z component of the z-oriented dipole is replaced by the geometric system function
for the x or y components of the magnetic field produced by the dipole where the geometric system function is defined below as the impulse response of the detector to the given component of the field of a dipole of given orientation.
The sample space, or space over which the secondary field is measured, is defined in the present example as the three-dimensional space comprising the entire x,y plane and the positive z axis, as shown in FIG. 4. Other sample spaces are valid
and each requires special consideration during the reconstruction as described below.
The discrete voltages recorded from an infinite detector array in the x,y plane which is translated to infinity along the z axis starting from the origin where the detector array is responsive to the z component of the secondary magnetic field is
given by Equation 2, where the voltage at any point in space produced by dipoles advanced in the z direction and advanced or delayed in the x and y directions relative to the origin is given by the following Equation 1, where the voltage response is
C.sub.o times the secondary magnetic flux strength. The flux magnetizing each voxel is given as unity. ##EQU10##
where the variables for Equations 1 and 2 are defined as follows:
______________________________________ x.sub.n.sbsb.1.sub.n.sbsb.2.sub.n.sbsb.3 = the magnetic susceptibility of the dipole located at .delta.(x - n.sub.1 k, y - n.sub.2 k, z + n.sub.3 k). k.sub.1, k.sub.2, k.sub.3 = dipole spacing in the x,
y, and z directions, respectively. l.sub.1, l.sub.2, l.sub.3 = the dimensions in x, y, and z, respectively, for which the magnetic susceptibility of the dipoles is nonzero. s.sub.1, s.sub.2 = the detector spacing in the x and y directions,
respectively. s.sub.3 = the distance the array is translated in the z direction between readings or the z interval between arrays of a multi-plane detector array (i.e., 3D detector array). ______________________________________
The voltage signal recorded at the detector array over the sample space is given by Equation 2 as follows: ##EQU11## Equation 2 can be represented symbolically as follows:
where C.sub.o is the constant which relates voltage to flux stength; S is the discrete function of the voltage signals recorded of the secondary magnetic flux over the sample space; where f is the secondary magnetic flux sampling function given
as follows: ##EQU12## where h is the system function which is also defined as the geometric system function given as follows ##EQU13## and it represents the impulse response of the detector array; where the external magnetizing field is set equal to one
(if the magnetizing field is nonuniform, then the magnetic susceptibility values determined by solution of Equation 4 must be divided by the strength of the magnetic flux magnetizing the corresponding susceptibility on a value-by-value basis as described
in detail in the Altering the Dynamic Range Provided by the System Function Section); where f is the magnetic susceptibility function given as follows: ##EQU14## and where u(z) is the unitary z function which is one for positive z and zero otherwise.
The function g discretizes the continuous voltage function, V, given by Equation 1, which is h convolved with f and multiplied by u(z).
The discrete voltages are used in a computer algorithm to reconstruct the magnetic susceptibility map. The algorithm follows from the following derivation which demonstrates that the magnetic susceptibility values of the dipoles can be recovered
from the voltage function defined over the sample space, which, in the present case, is defined as the x,y plane and the positive z axis. The voltage function of Equation 1 is defined over all space, but it can be defined to be zero outside of this
exemplary sample space via the operation given below of a multiplication by u(z). Other sample spaces are valid. For each case, the continuous voltage function defined over all space is multiplied by the function which results in the voltage function
being nonzero in the sample space and zero outside the sample space. In each case, the appropriate function which defines the sample space is substituted for u(z) in the analysis which follows. Furthermore, as described previously, the system function
of the present example is the geometric system function for the z component of a z-oriented dipole, which is given as follows: ##EQU15## A different geometric system function applies if a different component of the dipole field is recorded. In each
case, the appropriate system function is substituted for h in the analysis which follows.
Consider the function s of Equation 2, which is given as follows:
which is h convolved with f and multiplied by u(z). S, the Fourier Transform of s, is given as follows:
which is the resultant function of H multiplied by F convolved with U(k.sub.z), where H, F, and U(k.sub.z) are the Fourier Transformed functions of h, f, and u(z), respectively. The Fourier Transform of ##EQU16## where x.sub.n =n.sub.1 k;
y.sub.n =n.sub.2 k and z.sub.n =-n.sub.3 k.
The Fourier Transform of u(z)=1 for z>0 and zero for z<0 is
The Fourier Transform of the system function ##EQU17## is given as follows. The derivation appears in Appendix IV. ##EQU18##
The product of H and F is convolved with U(k.sub.z) as follows: ##EQU19## The result is given as follows and the derivation appears in Appendix V. ##EQU20## The function of S divided by H is given as follows: ##EQU21## The Inverse Fourier
Transform of S divided by H is given as follows, where the symbol .sup.-1 (Q) is defined as the Inverse Fourier Transform of the function Q. ##EQU22##
The solution of Inverse Transform 1 appears as follows and the derivation appears in Appendix VI. ##EQU23##
Combine Inverse Transforms and use the rule that a product in the frequency domain Inverse Transforms to a convolution integral in the Spatial Domain ##EQU24##
Evaluate at x.sub.n, y.sub.n, z.sub.n ##EQU25##
The solution of the magnetic susceptibility of each dipole of the magnetic susceptibility function follows from Equation 4. Discrete values of the voltages produced at the detector array due to the secondary magnetic field are recorded during a
scan which represent discrete values of function s (Equation 2); thus, in the reconstruction algorithm that follows, Discrete Fourier and Inverse Fourier Transforms replace the corresponding continuous functions of Equation 4 of the previous analysis.
Discrete values of H of Equation 3, the Fourier Transform of the system function replace the values of the continuous function. Furthermore, each sample voltage of an actual scan is not truly a point sample, but is equivalent to a sample and
hold which is obtained by inverting the grid matrices or which is read directly from a microdevice as described in the Finite Detector Length Section. The spectrum of a function discretely recorded as values, each of which is equivalent to a sample and
hold, can be converted to the spectrum of the function discretely recorded as point samples by dividing the former spectrum by an appropriate sinC function. This operation is performed and is described in detail in the Finite Detector Length Section.
From these calculated point samples, the magnetic susceptibility function is obtained following the operations of Equation 4 as given below.
Reconstruction Algorithm
1) Record the voltages over the sample space.
2) Invert the grid matrices defined by the orthogonal detector arrays described in the Finite Detector Length Section to obtain the sample and hold voltages which form Matrix A (if microdevices are used, form Matrix A of the recorded voltages
directly).
3) Three-dimensionally Fourier Transform Matrix A, using a Discrete Fourier Transform formula such as that which appears in W. McC. Siebert, Circuits, Signals, and Systems, MIT Press, Cambridge, Mass., 1968, p. 574, incorporated by reference, to
form Matrix A* of elements at frequencies corresponding to the spatial recording interval in each dimension.
4) Multiply each element of Matrix A* by a value which is the inverse of the Fourier Transform of a square wave evaluated at the same frequency as the element where the square wave corresponds to a sample and hold operation performed on the
continuous voltage function produced at the detector array by the secondary field. This multiplication forms matrix A** which is the discrete spectrum of the continuous voltage function discretely point sampled (see the Finite Detector Length Section
for details of this operation).
5. Multiply each element of Matrix A** by the value which is the inverse (reciprocal) of the Fourier Transform of the system function evaluated at the same frequency as the element to form Matrix B.
6. Inverse three-dimensionally Fourier Transform Matrix B using a Discrete Inverse Fourier Transform Formula such as that which appears in W. McC. Siebert, Circuits, Signals, and Systems, MIT Press, Cambridge, Mass., (1986), p. 574,
incorporated by reference, to form Matrix C whose elements correspond to the magnetic susceptibility of the dipoles at the points of the image space of spatial interval appropriate for the frequency spacing of points of Matrix B.
7) Divide each element of Matrix C by ##EQU26## to correct for the restriction that the sample space is defined as z greater than zero. This operation creates Matrix D which is the magnetic susceptibility map. ##EQU27## (If the magnetizing
field is not unity, then further divide each element by the value of the magnetizing field at the position of the corresponding magnetic susceptibility element.)
8) Plot the magnetic susceptibility function with the same spatial interval in each direction as the sampling interval in each direction.
(The above steps relate generally to the program implementation shown in the listings of Appendices VII and VIII as follows. The above steps 1) and 2) relate to the Data Statements beginning at lines 50; and step 3) relates to the X Z and Y FFT
operations of lines 254, 455 and 972, respectively Steps 4) and 5) are implemented by the processes of lines 2050, 2155 and and step 6) relates to the X, Y and Z inverse t | | |
