Simultaneous suppression of disturbing
fields and localization of magnetic markers by means of multipole expansion
Address: Biomagnetic Center, Department of Neurology,
Friedrich Schiller University Jena, Germany
Email: Bernd Hilgenfeld* - bhi@biomag.uni-jena.de;
Jens Haueisen - haueisen@biomag.uni-jena.de
* Corresponding author
Background: Magnetically marked capsules serve for the
analysis of peristalsis and throughput times within the intestinal tract.
Moreover, they can be used for the targeted disposal of drugs. The capsules get
localized in time by field measurements with a superconducting quantum
interference device (SQUID) magnetometer array. Here it is important to ensure
an online localization with high speed and high suppression of disturbing
fields. In this article we use multipole expansions for the simultaneous
localization and suppression of disturbing fields.
Methods: We expand the measurement data in terms of
inner and outer multipoles. Thereby we obtain directly a separation of marker
field and outer disturbing fields. From the inner dipoles and quadrupoles we
compute the magnetization and position of the capsule. The outer multipoles get
eliminated.
Results: The localization goodness has been analyzed
depending on the order of the multipoles used and depending on the systems
noise level. We found upper limits of the noise level for the usage of certain
multipole moments. Given a signal to noise ratio of 40 and utilizing inner
dipoles and quadrupoles and outer dipoles, the method enables an accuracy of 5
mm with a speed of 10 localizations per second.
Conclusion: The multipole localization is an effective
method and is capable of online-tracking magnetic markers.
The transport of capsules in the alimentary tract
underlies complex influencing factors like the patients peristalsis, the
hydration and the form and size of the capsules. A procedure which allows the
instantaneous localization of the capsules supports a number of patient examinations
as well as examinations of new drug forms [1-4]. Capsules can be marked
radioactively (scintigraphy) or magnetically. The scintigraphy [5] has a lower
time resolution compared to the magnetic localization, and due to radia-
tion it is not appropriate for examinations with
healthy probands.
The localization of magnetically marked capsules (magnetic
markers) must be spatially accurate and with high temporal resolution. For the
spatial localization the marker field must be separated from the external magnetic
disturbing fields. This separation can be achieved by splitting the magnetic
field in multipole moments [6]. The method proposed utilizes the multipole
moments directly
for the determination of the position and the magnetic
moment of the marker. Thus, the separation of disturbing fields and the
localization are integrated numerically effective into one procedure. This
allows a fast online-localization of the marker capsules.
Multipole expansions are used also to model spatially
distributed biological sources such as brain currents [7,8].
The application of multipoles for the localization of
magnetic dipoles is described in [9,10], and is used in other technical areas
without disturbing field suppression [11].
Marking of capsules and pills takes place by partially
filling them with black iron oxide (Fe3O4) which is
subsequently magnetized up to saturation. The magnetic field measurement is
performed within magnetically shielded rooms by the use of highly sensitive
SQUID arrays. For the investigation at hand we conduct simulation runs to
determine the performance of the multipole localization.
The field of a magnetic marker located adjacent to the
point of origin can be expressed by a multipole expansion in Cartesian
coordinates {xv x2, x3). If the distance r
between marker and origin is small compared to the distance r between a
magnetic sensor and the origin, the field of the marker at the sensor position
is given by the first elements of the multipole expansion. With the notation
follows
with c being the
dipole, quadrupole and
octopole moments of the field expansion.
The form functions
Fm(f) arise from a
Taylor series expansion in the
parameter
and
It holds
With the Kronecker delta follows
and
Conversely, a Taylor series expansion of - compared to
the sensor coordinates - far away located field sources in the parameter
yields a multipole expansion of external disturbing
fields:
We denote the multipole moments c of the expansion of
fields of external sources as "outer moments" to distinguish them
from the "inner moments" c.
To get the same normalization and symmetry properties
for the outer and inner form functions, we define the outer form functions
It follows
and
We combine the resulting 3, 5 and 7 linearly
independent components of the tensors of 2nd, 3rd and 4th
order to one vector for the marker field Bm and one vector for the
external disturbing field B
The summation of equation (1) and equation (6) yields
the field expansion for a magnetic marker with disturbing fields. We truncate
this expansion after the octopole terms, and transcript it into a linear
equation system for the determination of equivalent multipole moments c for a
measurement Bmeas:
The structure of the vectors Bmeas and c
and the matrix F is given below in the formulas (15...24). The residuals 0( •)
are sufficiently small, if the coordinates of the marker are small compared to
the coordinates of the field sensors x'im < <, and if the
coordinates of the field sensors are small compared to the coordinates of the
external disturbing field sources .xr i << e
Bmeas is a vector with the measurement
values of the magnetometer sensor field in the positions rj- with the directions
:d i
The matrix F is built from the linearly independent
form functions for inner and outer field sources given in equation (13). Their
scalar product with the sensor normal
directions d, yields one row for every sensor:
The number of columns of F is the sum of the numbers
of inner and outer field functions used. Each column describes the field of one
specific magnetic moment with unit strength measured by the sensor system. The
Matrix F is called the forward matrix of all moments considered. The Matrix F
is structured into submatrices for different moments:
Matrix Fm and Fex are the
forward matrices for inner and outer dipoles:
Matrix F and F are the forward matrices for inner and
outer quadrupoles. The size of F, is with the rows belonging to quadrupole
moments with indices (1,1; 3,3; 1,2; 2,3; 3,1).
Matrix F and F are the forward matrices for inner and
outer octopoles. The size of is is with the
rows belonging to octopole moments with indices
(1,2,2; 2,3,3; 3,1,1; 1,3,3; 2,1,1; 3,2,2; 1,2,3).
The vector of multipole moments c is composed of inner
and outer dipole moments c, quadrupole moments c and octopole
moments c:
The inner dipole moments cjf describe a dipole at the
point of origin, the outer dipole moments c describe a homogeneous disturbing
field:
The c™ represent a quadrupole at the point of origin.
The c describe an external gradient
field, whose field strength vanishes at the origin and which has no spatial
derivations of 2nd or higher order. This field can be measured by
five ideal gradiometers at the origin and can be compared with the creation of
software gradiometers.
The c represent an octopole at the origin. The cm
describe an external gradient field of 2nd order, which has no
spatial derivations of 3rd or higher order and whose field strength
and spatial derivatives of 1st order vanish at the origin. This
field could be measured by 7 ideal second order gradiometers at the origin, it
can be compared with the creation of software gradiometers of 2nd
order:
Due to its small spatial extension, the magnetic
marker can be described as a dipole of strength m at position r' as a good
approximation. The field of this dipole is
With the Taylor series expansion
follows in analogy to equation (1)
A comparison of coefficients of (1) and (27) yields
and
The dipole strength m can be determined by the dipole
moment c. An equation system for the adjacent calculation of the dipole
position f from the dipole strength m and the quadrupole moment c™ follows from
(29) and (23):
with
This equation system is named shift equation in
analogy to [10]. It is overdetermined, and can be solved by means of the pseudo
inverse of m.
We get the multipole moments c, which are required for
the localization of the marker dipole, from solving the overdetermined equation
system (14) by means of the pseudo inverse of F:
Here, the matrix of form functions F must contain columns
at least for the moments c and c
Iterative dipole localization for a fixed dipole (e.g.
one time point) is achieved by using the localization position as a new point
of origin. The step length of the last localization step serves as a stop
criterion for the iterative localization procedure. This is justified by
considering the residuals of equation (14) within the convergence range of the
procedure, and will practically be shown by the results of the following
simulations.
The tracking of a moved dipole based on measurements
at consecutive time steps works by updating both the point of origin and the
measurement data set after each localization step (Fig. 1). The localization
step must be monitored, since it contains information about the marker speed
and the noise dependent and speed dependent localization errors.
The simulations to determine the performance of the
algorithm use the sensor geometry of the multi channel SQUID system Argos 200
from AtB (Advanced Technologies Biomagnetics, Pescara, Italy). The ARGOS 200
system contains fully integrated planar SQUID magnetometers produced using Nb
technology with integrated pick-up loops. The sensing area is a square of 8 mm
side length. The intrinsic noise level of the built in 195 SQUID sensors is
below 5 fT Hz-1/2 at 10 Hz. Three sensors form one orthogonal
triplet in each case. The measurement plane with a diameter of 23 cm consists
of 56 of those triplets. The reference array consists of seven SQUID sensor triplets
located in the second level in a plane which is positioned parallel to the measurement
plane at a distance of 98 mm. The third (196 mm above the first plane) and
fourth (254 mm above the first plane) levels contain one triplet each (Fig. 2).
Figure 1
Flow chart of the algorithm for online
localization.
This algorithm is meant for online localization, and
therefore comprises only one iteration. A high signal to noise ratio and a high
computing speed render 2-3 iterations per measurement cycle possible.
Figure 2
SQUID Array Argos 200. The ATB SQUID Array
Argos 200 consists of 195 magnetometers which are arranged in orthogonal sensor
triplets in four levels. The measurement area of each sensor is a square of 8
mm edge length.
The measurement system is positioned within a magnetically
shielded room, consisting of 3 highly permeable shieldings and one eddy current
shielding. The shielding performance is 38 db at 1 Hz, 55 db at Hz and 80 db at
20 Hz.
The sensor arrangement in orthogonal triplets
facilitates the measurement of all 3 spatial components of the magnetic field.
Thus, the required field coverage for the localization of a magnetic marker
with unknown dipole strength is achieved.
The subdivision into 168 measurement and 27 reference
sensors is meant for the creation of software gradiometers. We can use all
sensors simultaneously for the multipole method which integrates the
suppression of disturbing fields.
With the above described measurement system we performed
simulations with different signal to noise ratios.
We examined the localization characteristics of the
multipole method by means of simulation runs at the sensor geometry of the
measurement system Argos 200 (Fig. 2). All simulations performed are based on a
dipole at position (x, y, z) = (0, 0, -300 mm), i.e. 30 cm below the
measurement plane, with a dipole strength of 20 Amm2. This is a
realistic dipole position for an examination within the digestive tract. The
dipole field was superimposed by uncorrelated, Gauss distributed noise. The
noise level in fT is also given as signal-to-noise-ratio (SNR), based on the
channel with the strongest amplitude of the dipole field.
The average localization accuracy over 100 simulations
has been determined depending on the noise level and on the number of the
multipole moments used in the vector c (21) (Fig. 3).
The localization was run up to a stable point. We
define the localization error as the mean quadratic error of the 100 stable
points based on the true dipole position. The localization error increases if
we use higher order multipole moments. This holds true for the inner moments cm
and for the outer moments cex as well. As a good approximation the
interrelationship between noise level and localization error is linear, with
raising proportionality factor for higher mode numbers. This corresponds to
parallel translation of the curves in double logarithmic plotting.
We examined the localization speed depending on the
distance of the starting point to the dipole position. For any tested distance
the starting point has been moved from the dipole position into 100 random
directions. The remaining mean distance to the dipole position after one
localization step is depicted in Fig. 4. The localization speed turns to be
significantly higher when using inner octopoles. It gets higher with a shorter
starting distance in both an absolute and a relative manner based on the starting
distance. Both effects are to be expected directly from the residuals of
equation (14). The influence of the outer multipoles on the localization speed
is low. The convergence radius at which the dipole was found from all 100
directions decreases slightly with the raising number of outer multipoles used,
and increases slightly if inner octopoles are used (unequal right ends of the
respective curves in Fig. 4). The convergence radius ranges between 6 and 10
cm. The maximum number of iterations for a target accuracy of 1 mm can be
estimated from Fig. 4 as 3.
In the following we examined the interrelationship
between the convergence distance at y-direction and the noise level. The
maximum y-distance of the starting position to the dipole, at which the dipole
could be found with 100 random noise distributions, is depicted in Fig. 5. It
shows that the convergence distance remains unchanged almost up to the point of
critical noise level (see Fig. 3) at which localization becomes impossible. The
convergence distance, also compare the maximum convergence radius from the
curve ends of plot (Fig. 4), depends only marginally on the choice of the inner
ansatz functions. It decreases slightly when using outer multipoles. Having a
convergence radius of at least 6 cm for the dipole position tested, the choice
of the starting position can be regarded as noncritical.
The computing time used for one localization step is 5
ms with an implementation in Matlab at a standard Windows PC with a 2 GHz clock
frequency. With maximum 3 iterations per localization step and additional
computing time needed for data transfer and a basic visualization, 10
localizations per second are possible. This rate is normally sufficient for marker
localizations.
Figure 3
Noise-dependent localization error. The mean
squared localization
error err( f') over 100 simulations has been determined depending on the noise
level and on the different number of multipole moments used. The curves are plotted
up to the noise level, where all simulations still produced a stable
localization result. We used the inner moments up to the 3rd order
cjf, c™, plotted in curves 3x and the inner moments up to the 4th
order cjf . cj? > c™ > plotted in curves 4χ. The outer moments which
were used to model the disturbing fields were none (curves χ0), 2nd
order moments (curves χ2: homogeneous fields), 2nd and 3rd
order moments (curves χ3: homogeneous and gradient fields), and 2nd
to 4th order moments (curves χ4: external fields up to 2nd
order). The simplest disturbing field to model is a homogeneous field
having index χ2. The dipole field of a dipole with a strength of 20 Amm2
at position (x, y, z) = (0, 0, -300 mm) is superimposed by white, Gaussian
distributed noise, which is given in fT and as the signal to noise ratio (SNR).
Figure 4
Localization error depending on the starting
point distance for
one iteration. For each distance dsthe start position has been moved
from the dipole into 100 random directions. The mean remaining distance dr
after one localization step is shown. The curves are plotted until the
starting distance dr where the localization was still stable from
all 100 directions. To get the result after multiple iterative localization
steps, the dr-value has to be taken as the starting distance ds of
the following step. We used the inner moments up to the 3rd order
cjf , cq , plotted in curves 3x and the inner moments up to the 4tn
order cjf , c™ , c™ , plotted in curves 4χ. The outer moments which were
used to model the disturbing fields were none (curves χ0), 2nd
order moments (curves χ2: homogeneous fields), 2nd and 3rd
order moments (curves χ3: homogeneous and gradient fields), and 2nd
to 4th order moments (curves χ4: external fields up to 2nd
order).
The localization speed rises when using inner
octopoles cm2, but this is associated with a higher localization
error. At a signal to noise ratio lower than 103 inner octopoles
cannot be used. An SNR of at least 102 is required for a source positioned
30 cm below the measurement plane.
The outer moments cexused enlarge the localization
error depending on the uncorrelated sensor noise, as shown in Fig. 3. Contrary,
the localization error depending on the spatially correlated residual field within
the measurement room lowers when using outer moments. Depending on the ratio
between correlated and uncorrelated noise which has to be found with practical
test series.
Figure 5
Convergence distance depending on noise
level. The maximum y-distance dy between starting position and dipole position,
at which for 100 random noise distributions the dipole could still get
localized, is plotted. We used the inner moments up to the 3rd order
cjf, c™, plotted in curves 3c and the inner moments up to the 4tn
order ,, , cjj1, cm o plotted in curves
4χ. The outer moments which were used to model the disturbing fields were
none (curves χ0), 2nd order moments (curves χ2:
homogeneous fields), 2nd and 3rd order moments (curves
χ3: homogeneous and gradient fields), and 2nd to 4th
order moments (curves χ4: external fields up to 2nd order). The
dipole field of a dipole with a strength of 20 Amm2 at position (x,
y, z) = (0, 0, -300 mm) is superimposed by white, Gaussian distributed noise
given in fT and as the signal to noise ratio (SNR).
This work has been partially supported by the joint
research project "Mag-Mon/NET0158" within the "InnoNet"
program of the German Federal Government Department of Research (BMWA).