Background
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
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
Multipole expansions are used also to model spatially distributed biological sources such as brain currents
The application of multipoles for the localization of magnetic dipoles is described in
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.
Methods
Algorithm
The field of a magnetic marker located adjacent to the point of origin can be expressed by a multipole expansion in Cartesian coordinates ( between marker and origin is small compared to the distance
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
The form functions arise from a Taylor series expansion in the parameter
with
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 with
yields a multipole expansion of external disturbing fields:
We denote the multipole moments
To get the same normalization and symmetry properties for the outer and inner form functions, we define the outer form functions by
It follows
and
The tensors of 3rd order and of 4th order
own the following symmetry features which are identical for the inner and outer multipole expansion:
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 and one vector for the external disturbing field
:
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 , and if the coordinates of the field sensors are small compared to the coordinates of the external disturbing field sources
.
Bmeas is a vector with the measurement values of the magnetometer sensor field in the positions with the directions
:
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 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 and
are the forward matrices for inner and outer dipoles:
Matrix and
are the forward matrices for inner and outer quadrupoles. The size of
is (
Matrix and
are the forward matrices for inner and outer octopoles. The size of
is (
The vector of multipole moments c is composed of inner and outer dipole moments cd, quadrupole moments cq and octopole moments co:
The inner dipole moments describe a dipole at the point of origin, the outer dipole moments
describe a homogeneous disturbing field:
The represent a quadrupole at the point of origin. The
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 represent an octopole at the origin. The
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 at position
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 can be determined by the dipole moment
. An equation system for the adjacent calculation of the dipole position
from the dipole strength
and the quadrupole moment
follows from (29) and (23):
with
This equation system is named shift equation in analogy to
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:
c = (FT·F)-1·FT·Bmeas. (33)
Here, the matrix of form functions F must contain columns at least for the moments and
.
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.
Flow chart of the algorithm for online localization
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.
Measurement system
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.
SQUID Array Argos 200
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.
Results
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.
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.
Noise-dependent localization error
Noise-dependent localization error. The mean squared localization error e) 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
,
, plotted in curves 3χ and the inner moments up to the 4th order
,
,
, 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 (
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.
Localization error depending on the starting point distance for one iteration
Localization error depending on the starting point distance for one iteration. For each distance ,
, plotted in curves 3χ and the inner moments up to the 4th order
,
,
, 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).
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.
Convergence distance depending on noise level
Convergence distance depending on noise level. The maximum y-distance ,
, plotted in curves 3χ and the inner moments up to the 4th order
,
,
, 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 (
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.
Discussion
The localization speed rises when using inner octopoles , 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 cex used enlarge the localization error depending on the uncorrelated sensor noise, as shown in Fig. and possibly noise suppression of gradient fields using
are applicable. To ensure convergence, the starting point for the algorithm has to be within the convergence radius given in Fig.
Conclusions
The multipole localization is an effective algorithm because it unites a method for the suppression of disturbing fields with a localization method. It can be used iteratively and online for the tracking of magnetic marker timelines within the intestinal tract.