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Biomagn Res Technol. 2004; 2: 6.
Published online 2004 September 1. doi:  10.1186/1477-044X-2-6
PMCID: PMC519033

Simultaneous suppression of disturbing fields and localization of magnetic markers by means of multipole expansion

Abstract

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.

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 [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 radiation 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.

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 (x1, x2, x3). If the distance An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i1.gif between marker and origin is small compared to the distance An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i2.gif 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

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i3.gif

follows

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i4.gif

with cm being the dipole, quadrupole and octopole moments of the field expansion.

The form functions An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i5.gif arise from a Taylor series expansion in the parameter An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i6.gif with An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i7.gif and An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i8.gif. It holds

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i9.gif

With the Kronecker delta An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i10.gif follows

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Object name is 1477-044X-2-6-i11.gif

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i12.gif

and

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i13.gif

Conversely, a Taylor series expansion of – compared to the sensor coordinates – far away located field sources in the parameter An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i14.gif with An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i15.gif yields a multipole expansion of external disturbing fields:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i16.gif

We denote the multipole moments cex of the expansion of fields of external sources as "outer moments" to distinguish them from the "inner moments" cm.

To get the same normalization and symmetry properties for the outer and inner form functions, we define the outer form functions An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i17.gif by

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Object name is 1477-044X-2-6-i18.gif

It follows

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Object name is 1477-044X-2-6-i19.gif

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i20.gif

and

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i21.gif

The tensors of 3rd order An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i22.gif and of 4th order An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i23.gif own the following symmetry features which are identical for the inner and outer multipole expansion:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i24.gif

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i25.gif

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 An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i26.gif and one vector for the external disturbing field An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i27.gif:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i28.gif

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:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i29.gif

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 An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i30.gif, and if the coordinates of the field sensors are small compared to the coordinates of the external disturbing field sources An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i31.gif.

Bmeas is a vector with the measurement values of the magnetometer sensor field in the positions An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i32.gif with the directions An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i33.gif:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i34.gif

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 An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i33.gif yields one row for every sensor:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i35.gif

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:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i36.gif

Matrix An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i37.gif and An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i38.gif are the forward matrices for inner and outer dipoles:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i39.gif

Matrix An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i40.gif and An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i41.gif are the forward matrices for inner and outer quadrupoles. The size of An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i42.gif is (Nsen/5) with the rows belonging to quadrupole moments with indices (1,1; 3,3; 1,2; 2,3; 3,1).

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i43.gif

Matrix An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i44.gif and An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i45.gif are the forward matrices for inner and outer octopoles. The size of An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i46.gif is (Nsen/7) 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).

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i47.gif

The vector of multipole moments c is composed of inner and outer dipole moments cd, quadrupole moments cq and octopole moments co:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i48.gif

The inner dipole moments An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i49.gif describe a dipole at the point of origin, the outer dipole moments An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i50.gif describe a homogeneous disturbing field:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i51.gif

The An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i52.gif represent a quadrupole at the point of origin. The An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i53.gif 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.

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i54.gif

The An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i55.gif represent an octopole at the origin. The An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i56.gif 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:

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i57.gif

Due to its small spatial extension, the magnetic marker can be described as a dipole of strength An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i58.gif at position An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i1.gif as a good approximation. The field of this dipole is

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i59.gif

With the Taylor series expansion

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i60.gif

follows in analogy to equation (1)

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i61.gif

A comparison of coefficients of (1) and (27) yields

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i62.gif

and

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i63.gif

The dipole strength An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i58.gif can be determined by the dipole moment An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i49.gif. An equation system for the adjacent calculation of the dipole position An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i1.gif from the dipole strength An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i58.gif and the quadrupole moment An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i52.gif follows from (29) and (23):

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i64.gif

with

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i65.gif

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.

An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i66.gif

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 An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i49.gif and An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i52.gif.

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. (Fig.1).1). The localization step must be monitored, since it contains information about the marker speed and the noise dependent and speed dependent localization errors.

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 ...

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. (Fig.22).

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.

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. (Fig.2).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. (Fig.33).

Figure 3
Noise-dependent localization error. The mean squared localization error err(An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i1.gif) 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 ...

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. Fig.4.4. 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. Fig.4).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. Fig.44 as 3.

Figure 4
Localization error depending on the starting point distance for one iteration. For each distance ds the start position has been moved from the dipole into 100 random directions. The mean remaining distance dr after one localization step is shown. The ...

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. Fig.5.5. It shows that the convergence distance remains unchanged almost up to the point of critical noise level (see Fig. Fig.3)3) at which localization becomes impossible. The convergence distance, also compare the maximum convergence radius from the curve ends of plot (Fig. (Fig.4),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.

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 3 ...

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 An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i55.gif, 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. Fig.3.3. 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, noise suppression of homogeneous disturbing fields using An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i50.gif and possibly noise suppression of gradient fields using An external file that holds a picture, illustration, etc.
Object name is 1477-044X-2-6-i53.gif are applicable. To ensure convergence, the starting point for the algorithm has to be within the convergence radius given in Fig. Fig.4.4. With typical measurement conditions, thus, a starting point 10 cm below the center of the measurement system will suffice.

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.

Acknowledgements

This work has been partially supported by the joint research project "MagMon/NET0158" within the "InnoNet" program of the German Federal Government Department of Research (BMWA).

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