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
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 cm being the dipole, quadrupole and octopole moments of the field expansion.
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 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
by
It follows
and
The tensors of 3
rd order
and of 4
th 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 2
nd, 3
rd and 4
th 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 (
Nsen/5) with the rows belonging to quadrupole moments with indices (1,1; 3,3; 1,2; 2,3; 3,1).
Matrix
and
are the forward matrices for inner and outer octopoles. The size of
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).
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 2
nd 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 2
nd order, which has no spatial derivations of 3
rd or higher order and whose field strength and spatial derivatives of 1
st 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 2
nd 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 [
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:
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. ). The localization step must be monitored, since it contains information about the marker speed and the noise dependent and speed dependent localization errors.
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. ).
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.
1 and Jens Haueisen1
