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Description  |
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BACKGROUND OF THE INVENTION
(1) Field of the Invention
This invention relates to a method and apparatus for deducing positions,
orientations and sizes of bioelectric current sources.
(2) Description of the Related Art
A stimulus given to a living body breaks polarization across cell membranes
and generates bioelectric currents. Such bioelectric currents take place
in the brain and the heart, and are recorded as an electro-oencephalogram
or an electrocardiogram. The magnetic fields formed by such bioelectric
currents are recorded as a magnetoencephalogram or a magnetocardiogram.
In recent years, a senor using a SQUID (Super-conducting Quantum Interface
Device) has been developed as a device for measuring minute magnetic
fields in the living body. This sensor may be placed outside the head to
measure, in a painless and harmless way, minute magnetic fields formed by
current dipoles (hereinafter simply called current sources also) which are
bioelectric current sources occurring in the brain. The positions,
orientations and sizes of the current sources relating to a lesion are
deduced from the magnetic field data thus gained. The current sources
deduced are superposed on sectional images obtained from a radiographic CT
apparatus or MRI apparatus, to determine a physical position and other
features of a disease or the like.
One example of conventional methods for deducing current sources uses a
least norm method (see, for example, W. H. Kullmann, K. D. Jandt, K. Rehm,
H. A. Schlitte, W. J. Dallas and W. E. Smith, Advances in Biomagnetism,
pp. 571-574, Plenum Press, New York, 1989).
The conventional method of deducing current sources using the least norm
method will be described hereinafter with reference to FIG. 1.
As shown in FIG. 1, a multichannel SQUID sensor 1 is disposed adjacent an
examinee M. The multichannel SQUID sensor 1 has a multiplicity of magnetic
sensors (pickup coils) S1 to Sm immersed in a coolant such as liquid
nitrogen within a vessel called a Dewar.
On the other hand, a multiplicity of lattice points "1" to "n" are set in a
region to be diagnosed, e.g. the brain, of the examinee M. Unknown current
sources (current dipoles) are assumed for the respective lattice points,
which are expressed by three-dimensional vectors VPj (j=1 to n). Then, the
respective magnetic sensors S1 to Sm of the SQUID sensor 1 detect magnetic
fields B1 to Bm which are expressed by the following equations (1):
##EQU1##
In the equations (1), VPj=(Pjx, Pjy, Pjz), and .alpha.ij=(.alpha.ijx,
.alpha.ijy, .alpha.ijz). .alpha.ij is a known coefficient representing
intensity of a magnetic field detected in the position of each magnetic
sensor S1 to Sm, where the current sources of unit sizes in X, Y and Z
directions are arranged on the lattice points.
If [B]=(B1, B2, . . . Bm), and [P]=(P1x, P1y, P1z, P2x, P2y, P2z, . . .
Pnx, Pny, Pnz), then the equations (1) are rewritten as the following
linear relationship (2):
[B]=A[P] (2)
In the equation (2), A is a matrix having 3n.times.m elements expressed by
the following equation (3):
##EQU2##
If the inverse matrix of A is expressed by A.sup.-, [P] is expressed by the
following equation (4):
[P]=A.sup.- [B] (4)
The least norm method is based on the premise that the number of unknowns
3n (where the sizes in X, Y and Z directions of the current sources
assumed for the respective lattice points are taken into account) is
greater than the number of equations m (the number of magnetic sensors S1
to Sm). This method finds solutions for current sources [P] by applying
the condition that norm .vertline.[P].vertline. of current sources [P] is
minimized. The solutions could be obtained uniformly by equalizing the
number of equations m and the number of unknowns 3n, but such solutions
would be very unstable. For this reason, the least norm method is
employed.
By applying the condition that norm .vertline.[P].vertline. of current
sources [P] is minimized, the above equation (4) is rewritten as the
following equation (5):
[P]=A.sup.+ [B] (5)
where A.sup.+ is a general inverse matrix expressed by the following
equation (6):
A.sup.+ =A.sup.t (AA.sup.t).sup.-1 ( 6)
where A.sup.t is a transposed matrix of A.
The orientatins and sizes of the current sources VPj on the respective
lattice points are deduced by solving the above equation (5). The current
source having the greatest value thereamong is regarded as the closest to
a true current source. This is the principle of the current source
deducing method based on the least norm method.
In order to improve the position resolving power of the least norm method,
proposals have been made to gain least norm solutions repeatedly while
subdividing the lattice points (see, for example, Y. Okada, J. Huang and
C. Xu, 8th International Conference on Biomagnetism, Munster, August
1991). This method will be described briefly with reference to FIG. 2.
FIG. 2 is an enlarged view of part of the lattice points N shown in FIG. 1.
Reference J in FIG. 2 denotes the lattice point having the current source
deduced by the above least norm method as being close to the true current
source. A group of subdivided lattice points M (shown in small black spots
in FIG. 2) is additionally established around this lattice point J. The
technique described above is applied to the newly established group of
lattice points M as included in the initially established group of lattice
points N, to deduce a current source still closer to the true current
source.
The prior art described above has the following disadvantage.
The conventional method illustrated in FIG. 2 involves an increased number
of lattice points since the subdivided lattice points M are newly
established in addition to the initially established lattice points N.
Consequently, vector [P] in equation (5) has a large number of elements
which lowers the precision in computing the least norm solutions.
SUMMARY OF THE INVENTION
This invention has been made having regard to the state of the art noted
above, and its primary object is to provide a method and apparatus for
deducing bioelectric current sources with high precision.
The above object is fulfilled, according to this invention, by a method of
deducing physical quantities such as positions, sizes and orientations of
bioelectric current sources, comprising:
a magnetic field measuring step for measuring minute magnetic fields formed
by the bioelectric current sources in a region under examination of an
examinee, with a plurality of magnetic sensors arranged adjacent the
region under examination;
a lattice point setting step for setting a plurality of lattice points in
the region under examination;
a current source computing step for deriving physical quantities of the
current sources by solving a relational expression of unknown current
sources at the lattice points and field data provided by the magnetic
sensors, with a condition added thereto to minimize a norm of a vector
having the current source at each of the lattice points;
a lattice point rearranging step for moving the lattice points toward a
lattice point having a large current value among the current sources
computed;
a checking step for checking whether a minimum distance among the lattice
points having been moved is below a predetermined value; and
a current source identifying step for repeating the current source
computing step to the checking step for the lattice points having been
removed, when the minimum distance exceeds the predetermined value, and
regarding as a true current source the current source corresponding to a
magnetic field occurring when the minimum distance is determined to be
below the predetermined value at the checking step.
This invention has the following functions.
The lattice point having a large current value among the current sources
computed at the current source computing step is not a true current source
but a current source close to the true current source. Thus, at the
lattice point rearranging step, the other lattice points set at the
lattice point setting step are moved toward the lattice point having a
large current value. Current sources are deduced similarly for the
rearranged lattice points. That is, according to this invention, a true
current source is deduced by moving the lattice points without varying the
number of lattice points. Consequently, a true current source is deduced
with precision while maintaining the computing precision of the least norm
method.
Where a plurality of true current sources (current sources having a large
value) exist, the above method poses a question which lattice point should
be selected as one toward which the other are to be moved. It is
preferred, in such a case, that likelihood of current sources being
present at the lattice points is derived from the physical quantities of
the current sources at the lattice points deduced, and the lattice points
are divided into a plurality of groups based on the likelihood derived.
Then, current sources may be deduced with precison even where a plurality
of true current sources are present.
In the above technique, the physical quantities of the current sources for
determining likelihood of current sources being present at the respective
lattice points are, for example, the size of the current source at each
lattice point and density of lattice points around that lattice point. It
is then necessary to determine empirically a parameter representing the
degree of influence of the lattice point density on the likelihood of
current sources. However, this parameter setting is not necessarily easy,
and an improper value selected will lower the current source deducing
precision.
To obviate such parameter setting, it is preferable to measure
simultaneously three orthogonal components (vector measurement) of the
minute magnetic fields formed by the bioelectric current sources in the
region under examination, and to deduce current sources based on measured
field data of the three orthogonal components. With such vector
measurement, the measured field data have a high degree of mutual
independence, resulting in improved spatial resolving power. Since this
eliminates the need to consider lattice point density around each lattice
point as a factor applied to the likelihood of a current source being
present at each lattice point, the above parameter setting is made
unnecessary.
For example, a group function showing the influence of a current source at
a certain lattice point on the other lattice points is used in dividing
the lattice points into a plurality of groups based on the likelihood of
current sources being present at the lattice points. It is then necessary
to determine empirically a parameter (moving parameter) determining a form
of the group function. However, this parameter setting is not necessarily
easy either, and an improper value selected will lower the current source
deducing precision. Preferably, this moving parameter is automatically
optimized with a condition to minimize a norm of a solution (a vector
having the current source at each lattice point as an element).
The deducing method using the least norm method described above is based on
the premise that the number of unknowns 3n (n being the number of lattice
points), where the sizes in X, Y and Z directions of the current sources
assumed for the respective lattice points are taken into account, is
greater than the number of magnetic sensors m (the number of equations),
i.e. 3n>m. Consequently, the coefficient matrix representing the
relationship between the unknown current sources at the lattice points and
measured magnetic fields could be lowered in level to render the solutions
unstable. Further, at the step of identifying an optimal current source,
whether a minimum distance between lattice points is below a predetermined
value (convergent criterion) is used as a determination condition. Thus,
deduction results could vary with the predetermined criterion. This
problem is solved by a method according to a further aspect of this
invention.
Thus, this invention provides a method of deducing physical quantities such
as positions, sizes and orientations of bioelectric current sources,
comprising:
a magnetic field measuring step for measuring minute magnetic fields formed
by the bioelectric current sources in a region under examination of an
examinee, with a plurality of magnetic sensors arranged adjacent the
region under examination;
a lattice point setting step for setting a plurality of lattice points in
the region under examination, the lattice points being smaller in number
than the magnetic sensors;
a first current source computing step for deriving unknown current sources
by adding a condition to minimize a square error of a magnetic field
formed by an unknown current source at each of the lattice points and a
magnetic field measured by each of the magnetic sensors;
a checking step for checking whether the square error of the magnetic field
computed from the current source derived and the magnetic field actual
measured by each of the magnetic sensors is a global minimum;
a lattice point rearranging step for moving the lattice points toward a
lattice point having a large current value among the current sources
computed at the first current source computing step, when the square error
is determined to differ from the global minimum;
a current source identifying step for repeating the first current source
computing step to the lattice point rearranging step, and regarding as a
true current source the current source corresponding to a magnetic field
occurring when the square error is determined to be a global minimum at
the checking step.
According to this method, the number of magnetic sensors is larger than the
number of unknowns for the lattice points set, to obtain stable solutions
(current sources). The current sources may be deduced with increased
precision by adopting the condition to minimize a square error of a
magnetic field formed by an unknown current source at each of the lattice
points and a magnetic field actually measured. Further, since the current
source occurring when the square error is determined to be a global
minimum is regarded as a true current source, the convergent determination
value need not be set at the step of deducing a final current source.
Thus, the final current source deduction may be effected uniformly.
When the above group function is used in rearranging the lattice points at
the above lattice point rearranging step, a troublesome operation is
involved such as for setting parameters. Further, since the lattice points
is moved little by little within each group, a considerable time is
consumed before results of the deduction are produced. To overcome such
disadvantages, it is preferred that current sources at the lattice points
are newly derived, when the square error is determined to differ from the
global minimum at the checking step, by adding a condition to minimize a
sum of the square error and a weighted sum of squares of the current
source, and the lattice points are moved toward a lattice point having a
large current value among the current sources. This technique employs the
square error combined with a penalty term which is a weighted sum of
squares of the current source as an evaluation function for moving the
lattice points. Consequently, stable solutions are obtained even where the
lattice points are not in the true current source. This allows the lattice
points to be moved at a time to the vicinity of the greatest current
source, thereby to shorten the time consumed in deducing the current
sources. The lattice points are rearranged without using a group function,
which dispenses with an operation to set parameters.
In the above technique of determining current sources at the respective
lattice points by the linear least squares method, if noise mixes into the
magnetic fields measured, noise components may also be calculated as
solutions (current sources). This results in the disadvantage that the
position of each current source deduced tends to vary. To overcome this
disadvantage, it is preferred that the first current source computing step
is executed to derive current sources at the lattice points by adding a
condition to minimize a sum of the square error of the magnetic field
formed by the unknown current source at each lattice point and the
magnetic field actually measured, and a weighted sum of squares of the
current source, the checking step is executed to check whether the sum of
the square error and the weighted sum of squares of the current source
computed is a global minimum, and when the sun is determined to differ
from the global minimum, the lattice points are moved toward a lattice
point having a large current value among the current sources computed. At
the checking step for deducing an optimal current source, this technique
evaluates the function having a weighted sum of squares of the current
source (penalty term) added to the square error. The penalty term has the
smaller value, the closer the current sources lie to one another.
Consequently, noise components occurring discretely have little chance of
being adopted as solutions.
The influence of noise components is avoided by evaluating, at the checking
step for deducing an optimal current source, the function having a
weighted sum of squares of the current source (penalty term) added to the
square error, as noted above. However, the current sources at the lattice
points deduced tend to consolidate. This results in the disadvantage that,
where current sources are distributed over a certain range, a true current
source could be difficult to deduce correctly. In such a case, for the
condition added at the first current source computing step, i.e. the
condition to minimize a sum of the square error and a weighted sum of
squares of the current source, a weight for the current source is set to
have the smaller value the smaller a distance is between the lattice
points. At the checking step, the penalty term is excluded to check
whether the square error between the magnetic fields formed by the current
sources obtained at the first current source computing step and the
magnetic fields actually measured is a global minimum or not. If the
square error is found not to be a global minimum, the lattice points are
moved toward a lattice point having a large current value among the
current sources computed. According to this technique, when the lattice
points concentrate locally, the influence of the penalty term diminishes.
Further, since the penalty term is excluded from the criterion for
identifying optimal current sources, the current sources deduced are not
unnecessarily concentrated. Thus, current sources distributed over a
certain range may be deduced correctly.
BRIEF DESCRIPTION OF THE DRAWINGS
For the purpose of illustrating the invention, there are shown in the
drawings several forms which are presently preferred, it being understood,
however, that the invention is not limited to the precise arrangements and
instrumentalities shown.
FIG. 1 is an explanatory view of a conventional method of deducing
bioelectric current sources, using the least norm method;
FIG. 2 is an explanatory view of another conventional method of deducing
current sources;
FIG. 3 is a block diagram showing an outline of an apparatus embodying the
present invention;
FIG. 4 is a flowchart of current source deduction processing in a first
embodiment;
FIG. 5 is an explanatory view of lattice point movement in the first
embodiment;
FIG. 6 is a flowchart of current source deduction processing in a second
embodiment;
FIG. 7 is an explanatory view of group functions;
FIG. 8 is an explanatory view of lattice point movement in divided groups
in the second embodiment;
FIG. 9 is an explanatory view of lattice point movement in further divided
groups in the second embodiment;
FIG. 10A is an explanatory view of a model used in a simulation of the
second embodiment;
FIG. 10B is a schematic view of a magnetic sensor used in the simulation of
the second embodiment;
FIG. 11A is a view showing a setting of current sources in the simulation
of the second embodiment;
FIG. 11B is a view showing a reconstruction of the current sources shown in
FIG. 11A;
FIG. 12A is a view showing a different setting of current sources in the
simulation of the second embodiment;
FIG. 12B is a view showing a reconstruction of the current sources shown in
FIG. 12A;
FIG. 13 is a flowchart of current source deduction processing in a third
embodiment;
FIG. 14 is a schematic view of a magnetic sensor used in the third
embodiment;
FIG. 15A is a view showing a reconstruction of current sources obtained in
a simulation of the third embodiment;
FIG. 15B is a view for comparison with FIG. 15A, and showing a
reconstruction of current sources based on measurement in radial
directions;
FIGS. 16A and 16B are views showing norm variations of solutions
corresponding to moving parameter values;
FIG. 17A is a view showing a reconstruction of current sources
corresponding to FIG. 16A;
FIG. 17B is a view showing a reconstruction of current sources
corresponding to FIG. 16B;
FIG. 18 is a flowchart of current source deduction processing in a fourth
embodiment;
FIG. 19 is a view showing a reconstruction of current sources obtained in a
simulation of the fourth embodiment;
FIG. 20 is a flowchart of current source deduction processing in a fifth
embodiment;
FIGS. 21, 22 and 23 are views showing reconstructions in different stages
of current sources obtained in a simulation of the fifth embodiment;
FIG. 24 is a flowchart of current source deduction processing in a sixth
embodiment;
FIG. 25 is a view showing a reconstruction of current sources obtained by
improper parameter setting, for comparison with the sixth embodiment;
FIG. 26 is a view showing a reconstruction of current sources obtained
without using a penalty term, for comparison with the sixth embodiment;
FIGS. 27, 28 and 29 are views showing reconstructions in different stages
of current sources obtained in a simulation of the sixth embodiment;
FIG. 30 is a flowchart of current source deduction processing in a seventh
embodiment;
FIG. 31 is a view showing a reconstruction of current sources obtained from
the sixth embodiment, for comparison purposes;
FIG. 32 is a view showing a reconstruction of current sources obtained in a
simulation of the seventh embodiment;
FIG. 33 is a flowchart of current source deduction processing in an eighth
embodiment;
FIG. 34 is a view showing a reconstruction of current sources obtained from
the seventh embodiment, for comparison purposes; and
FIG. 35 is a view showing a reconstruction of current sources obtained in a
simulation of the eighth embodiment.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Preferred embodiments of this invention will be described hereinafter with
reference to the drawings.
First Embodiment
An outline of an apparatus embodying this invention for deducing
bioelectric current sources will be described with reference to FIG. 3.
Numeral 2 in FIG. 3 denotes a magnetic shield room. The magnetic shield
room encloses a bed 3 for supporting an examinee M lying thereon, and a
multi-channel SQUID sensor 1 disposed adjacent the brain of the examinee
M, for example, for measuring, in a painless and harmless way, minute
magnetic fields formed by bioelectric current sources occurring in the
brain. As noted hereinbefore, the multichannel SQUID sensor 1 has a
multiplicity of magnetic sensors immersed in a coolant within a Dewar. In
this embodiment, each magnetic sensor consists of a pair of coils for
detecting a magnetic field component in a radial direction, with the brain
being regarded as a spherical body.
Field data detected by the multichannel SQUID sensor 1 are applied to a
data converting unit 4 for conversion to digital data to be stored in a
data collecting unit 5. A stimulator 6 applies electric (acoustic, optical
or other) stimulation to the examinee M. A positioning unit 7 determines a
positional relationship of the examinee to a three-dimensional coordinate
system based on the multichannel SQUID sensor 1. For example, small coils
are attached to a plurality of sites on the examinee M, and the
positioning unit 7 supplies power to these small coils. Then, the coils
generate magnetic fields to be detected by the multichannel SQUID sensor
1, thereby enabling determination of the position of the examinee M
relative to the multichannel SQUID sensor 1. Other methods may be used to
determine the position of the examinee M relative to the SQUID sensor 1.
For example, a projector may be attached to the Dewar to emit a light beam
to the examinee M to determine the positional relationship. Various other
methods are available as disclosed in Japanese Patent Publications
(Unexamined) No. 5-237065 and No. 6-788925.
A data analyzing unit 8 is used to deduce current sources in a region to be
diagnosed of the examinee M, from the field data stored in the data
collecting unit 5. A magneto-optical disk 9 associated with the data
analyzing unit 8 stores sectional images obtained from a radiographic CT
apparatus or MRI apparatus, for example. The current sources deduced by
the data analyzing unit 8 may be superposed on these sectional images for
display on a color monitor 10 or for printing by a color printer 11. The
sectional images obtained from the radiographic CT apparatus or MRI
apparatus may be transmitted directly to the data analyzing unit 8 through
a communication line 12 shown in FIG. 3.
A sequence of current source deduction executed by the data analyzing unit
8 will be described hereinafter with reference to the flowchart shown in
FIG. 4.
As noted above, a positional relationship of the examinee M to the
three-dimensional coordinate system based on the multichannel SQUID sensor
1 is measured and stored first. Then, as in the prior art illustrated in
FIG. 1, three-dimensional lattice points N are set evenly in a region to
be diagnosed, e.g. the brain, of the examinee M (step S1).
The respective coefficients in the matrix A expressed by equation (3) are
computed by Biot-Savart's law (the coefficients in matrix A being computed
each time the lattice points are moved as described later). Subsequently,
a current source (least norm solution) at each lattice point is determined
by the least norm method (step S2).
Next, the lattice points are moved toward a lattice point having a current
source of large value among the current sources determined at step S2
(step S3). FIG. 5 shows how this step is taken. Reference N in FIG. 5
denotes the group of lattice points initially set at step S1. The lattice
point marked "x" is the lattice point having a current source of large
value among the current sources determined at step S2. The other lattice
points are moved toward this lattice point, to form a group of lattice
points N1 corresponding in number to the group of lattice points N but
lying closer together.
Step S3 of moving the other lattice points toward the lattice point having
a current source of large value may be executed by any suitable method,
and the following is one example. Assume that, by regarding the size of
the current source at each lattice point determined at step S2 as a mass,
and attractive forces due to gravity act among the lattice points. Then,
each lattice point moves toward a lattice point of greater mass. The
lattice points are collected with the higher density, the closer they are
to a lattice point having a large mass. The moving distance of each
lattice point is set as appropriate.
Step S4 is executed to check whether a minimum distance between lattice
points in the group of lattice points N1 formed after the movement made at
step S4 is less than a predetermined distance. This distance is determined
as appropriate, based on the precision of deduced positions of the current
sources.
If the minimum distance between lattice points exceeds the predetermined
value, the operation returns to step S2 to determine, by the least norm
method, the current source of each lattice point in the group of the
lattice points N1 formed by moving the original lattice points. As noted
above, the number of lattice points N1 is the same as the number of
original lattice points N. In the linear equation (5) (set out hereunder
again) used in the least norm method,
[P]=A.sup.+' [B] . . . (5) (5)
the number of elements in vector [P'] is not increased but is fixed. This
means that the computing precision of the least norm solutions is
maintained. On the other hand, since the lattice points have been moved,
the least norm method executed a second time disregards presence of
current sources in hatched regions in FIG. 5. However, these regions are,
after all, separate from a position expected to include a true current
source. The lattice points in these regions have hardly any chance of
including the true current source. Thus, there is no fear of lowering the
precision of deduction by excluding these regions.
As before, a current source at each of the lattice points N1 is determined
by the least norm method (step S2). It is presumed that a current source
of large value is close to the true current source. Toward the lattice
point having this current source, the other lattice points are moved to
form a new group of lattice points N2 (step S3).
When, after repeating the above process, a minimum distance between lattice
points is found to be below the predetermined value at step S4, the
current sources of the group of lattice points determined at step S2
executed the last time are regarded as corresponding to the true current
source.
According to this embodiment, as understood from the foregoing description,
the other lattice points are moved toward the lattice point having a
current source of large value deduced by the least norm method executed
first. Current sources are deduced by the least norm method executed next,
with the number of lattice points remaining unchanged from the previous
time, and with only the distances between the lattice points diminished.
Thus, the current sources may be deduced with high precision while
maintaining the precision in computing the least norm solutions.
Second Embodiment
Where a plurality of true current sources are present, the first embodiment
poses a question which lattice point should be selected as one toward
which the other are to be moved. The second embodiment determines
likelihood of a current source being present at each lattice point from
deduced physical quantities of the current source. Based on the
likelihood, the lattice points are divided into a plurality of groups. For
each group the lattice points are moved toward the lattice point having
the greatest current source.
The outline of the apparatus and the multichannel SQUID sensor 1 in this
embodiment are the same as in the first embodiment, and will not be
described again. A sequence of current source deduction will be described
hereinafter with reference to the flowchart shown in FIG. 6.
As in the first embodiment, three-dimensional lattice points N are set
evenly in a region to be diagnosed, e.g. the brain, of the examinee M
(step S11).
Then, a current source (least norm solution) at each lattice point is
determined by the least norm method (step S12).
Next, where the position of the "j"th lattice point is regarded as vector
Vrj, the deduced current source thereof as vector VPj, the position of the
"k"th (k.noteq.j) lattice point as vector Vrk, and the deduced current
source thereof as vector VPk, likelihood Q of a current source being
present at the "j"th lattice point is expressed by the following equation
(7), for example. This equation is used to determine the likelihood of
presence at each lattice point of the current source obtained at step S12
(step S13).
##EQU3##
In equation (7), B is a parameter for adjusting the degree of likelihood
relative to distances between the lattice points, and .gamma. is a
parameter for determining a weight of the second term. These parameters
are selected empirically. Further, in the above equation, "e" is the base
of natural logarithm (e=2.71828 . . . ), and "n" is a total number of
lattice points.
The first term in equation (7) indicates that the greater the size of the
current source at the "j"th lattice point, the greater the likelihood of
the current source being present at this lattice point. The second term
indicates that the higher the density of lattice points around the "j"th
lattice point, the greater the likelihood of the current source being
present.
In the subsequent processing, lattice points having less likelihood are
moved toward lattice points of greater likelihood, to deduce current
sources from a more appropriate arrangement of lattice points. To effect
such movement of the lattice points, the lattice points N are divided into
groups by using group function .phi.j expressed by the following equation
(8), for example (step S14):
##EQU4##
Group function .phi.j indicates influences of the current source at the
"j"th lattice point on the other lattice points. Vr in equation (8) is a
position vector of a given point. .alpha. is an empirically selected
parameter for determining a form of the functions in equation (8).
A method of dividing the lattice points N into groups by using group
function .phi.j will be described with reference to FIG. 7. In the graph
shown in FIG. 7, the vertical axis represents group function .phi.j, and
the horizontal axis a given position vector Vr. References A, B and C on
the horizontal axis denote lattice points in the group of lattice points
N. References .phi.A, .phi.B and .phi.C are group functions .phi.j of
lattice points A, B and C, respectively. In the example shown in FIG. 7,
the lattice point that gives the greatest function value at lattice point
A is B. In this case, therefore, lattice point A belongs to the same group
as lattice point B. On the other hand, the greatest function value at
lattice point C is given by lattice point C itself. Thus, lattice point C
belongs to a different group to lattice points A and B. In this way, the
lattice points N are divided into a plurality of groups.
The lattice points in each group are moved toward the lattice point having
the greatest function value (current source size) (step S15). FIG. 8 shows
how this step is taken. Reference N in FIG. 8 denotes the group of lattice
points initially set at step S11. The lattice points marked "x" are the
lattice points having a current source of the greatest value in the
respective groups. The other lattice points in each group are moved toward
this lattice point, to form a group of lattice points N1 or N2 lying
closer together. The number of lattice points in the initial group N
equals the total number of lattice points in the groups of lattice points
N1 and N2. Step S15 of moving the other lattice points toward the lattice
point having the greatest current source in each group is executed in the
same way as in the first embodiment.
Step S16 is executed to check whether a minimum distance between lattice
points in each group of lattice points N1 or N2 formed after the movement
made at step S15 is less than a predetermined distance. This distance is
determined as appropriate, based on the precision of deduced positions of
the current sources.
In the first stage of group division, moving distances of the lattice
points are set so that the minimum distance between the lattice points
exceeds the predetermined value. Thus, the operation returns to step S12.
The current sources of the rearranged lattice points are determined by the
least norm method, regarding the groups of lattice points N1 and N2 as a
new group of lattice points.
After the current source of each lattice point in the respective groups N1
and N2 is determined, the likelihood of presence at each lattice point of
the current source is determined for the respective groups N1 and N2 as
before (step S13). Then, the lattice points in each group N1 or N2 are
further divided into groups (step S14). The lattice points in each group
are moved (step S15). FIG. 9 shows new groups of lattice points N3 to N7
resulting from the above steps.
When, after repeating the above process, a minimum distance between lattice
points is found to be below the predetermined value at step S16, the
current sources of the group of lattice points determined at step S12
executed the last time are regarded as corresponding to the true current
sources.
<simulation>
A simulation was carried out to ascertain validity of the above technique.
A ball having an 80 mm radius was conceived as the head acting as the
region for which current sources are deduced. As the magnetic sensors S,
axial type linear differential gradiometers (see FIG. 10B) having a 30 mm
base line were arranged in 37 channels over a spherical surface having a
117 mm radius. All the gradiometers had axes extending to the origin.
Current dipoles (current sources) were set in the ball of the head, and
the magnetic fields formed by the dipoles were calculated by means of
Sarvas's equation (J. Sarvas, Phys. Med. Biol., vol. 32, pp 11-22, 1987)
taking the effect of volume current into account. These magnetic fields
were regarded as measured magnetic fields. FIG. 10A shows this model.
Next, the space was divided into lattices of 20 cubic millimeters, and 257
lattices lying within the ball of the head were used as objects of
reconstruction. The sensors S used in this simulation extend in radial
directions "r" of the ball, and therefore cannot detect magnetic fields
formed by current components in the radial directions. Thus, deduction
parameters were current components in .theta. directions and .phi.
directions.
The magnetic fields were computed on the assumption that two current
dipoles acting as current sources had the same depth (20,0,50),
(-30,0,50). The moment of both current dipoles were set to (0,10,0). The
units are mm for the position, and nAm for the moment. The computed
magnetic fields are regarded as measured values, and the results of
deduction carried out in the above embodiment are shown in FIGS. 11A and
11B. FIG. 11A shows the setting, and FIG. 11B shows a reconstruction.
While there are isolated current dipoles, the lattices gather around the
true values, and the current dipoles are reconstructed in the right
direction.
Next, a case of setting the two current dipoles to different depths will be
described. The current dipoles were set to positions (20,0,50) and
(-20,0,30), and the moment was (0 | | |
