Method for determining a position of a magnetic source
A method for determining a position of a magnetic source includes measuring a magnetic field generated by the magnetic source and determining components of the three-dimensional magnetic flux density of the magnetic field at a plurality of points in space based on such measurement. The method also includes estimating a position of the magnetic source and determining components of a theoretical three-dimensional magnetic flux density at the plurality of points in space based on the estimated position. The position of the magnetic source may then be determined by minimizing the difference between the components of the measured and corresponding theoretical three-dimensional magnetic flux density components.
1. A method for determining a position of a magnetic source coupled to a bone of a patient, the method comprising:
(i) measuring a magnetic field generated by the magnetic source with a magnetic sensor array, the magnetic field being defined by a plurality of three-dimensional magnetic flux densities;
(ii) generating first data indicative of components of the three-dimensional magnetic flux density of the magnetic source at a plurality of points in space based on the measuring step;
(iii) determining an estimated position of the magnetic source;
(iv) calculating second data indicative of a theoretical value of the components of the three-dimensional magnetic flux density of the magnetic source based on the estimated position;
(v) calculating a difference between the first data and the second data; and
(vi) repeating steps (iii)-(v) until the difference between the first data and the second data is less than a predetermined minimum threshold value,
wherein calculating second data comprises calculating a theoretical value of an X-component of the three-dimensional magnetic flux density of the magnetic field using the following equation:
B
x
=
μ
m
[
3
x
(
x
sin
(
Θ
)
cos
(
Φ
)
+
y
sin
(
Θ
)
sin
(
Φ
)
+
z
cos
(
Θ
)
)
r
2
-
sin
(
Θ
)
cos
(
Φ
)
]
4
π
r
3
wherein B x is the theoretical value of the X-component, x is an X-coordinate value of the estimated position of a magnetic sensor of the magnetic sensor array with respect to the magnetic source, y is a Y-coordinate value of a magnetic sensor with respect to the magnetic source, z is a Z-coordinate value of the estimated position of the magnetic sensor with respect to the magnetic source, Θ is a rotational value about the X-axis of the estimated position of the magnetic source, Φ is a rotational value about the Y-axis of the estimated position of the magnetic source, μ is the permeability of free space, m is the magnetic strength of the magnetic source, and r=√{square root over (x 2 +y 2 +z 2 )}.
2. A method for determining a position of a magnetic source coupled to a bone of a patient, the method comprising:
(i) measuring a magnetic field generated by the magnetic source with a magnetic sensor array, the magnetic field being defined by a plurality of three-dimensional magnetic flux densities;
(ii) generating first data indicative of components of the three-dimensional magnetic flux density of the magnetic source at a plurality of points in space based on the measuring step;
(iii) determining an estimated position of the magnetic source;
(iv) calculating second data indicative of a theoretical value of the components of the three-dimensional magnetic flux density of the magnetic source based on the estimated position;
(v) calculating a difference between the first data and the second data; and
(vi) repeating steps (iii)-(v) until the difference between the first data and the second data is less than a predetermined minimum threshold value,
wherein calculating second data comprises calculating a theoretical value of a Y-component of the three-dimensional magnetic flux density of the magnetic field using the following equation:
B
y
=
μ
m
[
3
y
(
x
sin
(
Θ
)
cos
(
Φ
)
+
y
sin
(
Θ
)
sin
(
Φ
)
+
z
cos
(
Θ
)
)
r
2
-
sin
(
Θ
)
sin
(
Φ
)
]
4
π
r
3
wherein B y is the theoretical value of the Y-component, x is an X-coordinate value of the estimated position of a magnetic sensor of the magnetic sensor array with respect to the magnetic source, y is a Y-coordinate value of the estimated position of the magnetic sensor with respect to the magnetic source, z is a Z-coordinate value of the estimated position of the magnetic sensor with respect to the magnetic source, Θ is a rotational value about the X-axis of the estimated position of the magnetic source, Φ is a rotational value about the Y-axis of the estimated position of the magnetic source, μ is the permeability of free space, m is the magnetic strength of the magnetic source, and r=√{square root over (x 2 +y 2 +z 2 )}.
3. A method for determining a position of a magnetic source coupled to a bone of a patient, the method comprising:
(i) measuring a magnetic field generated by the magnetic source with a magnetic sensor array, the magnetic field being defined by a plurality of three-dimensional magnetic flux densities;
(ii) generating first data indicative of components of the three-dimensional magnetic flux density of the magnetic source at a plurality of points in space based on the measuring step;
(iii) determining an estimated position of the magnetic source;
(iv) calculating second data indicative of a theoretical value of the components of the three-dimensional magnetic flux density of the magnetic source based on the estimated position;
(v) calculating a difference between the first data and the second data; and
(vi) repeating steps (iii)-(v) until the difference between the first data and the second data is less than a predetermined minimum threshold value,
wherein calculating second data comprises calculating a theoretical value of an Z-component of the three-dimensional magnetic flux density of the magnetic field using the following equation:
B
z
=
μ
m
[
3
z
(
x
sin
(
Θ
)
cos
(
Φ
)
+
y
sin
(
Θ
)
sin
(
Φ
)
+
z
cos
(
Θ
)
)
r
2
-
cos
(
Φ
)
]
4
π
r
3
wherein B z is the theoretical value of the Z-component, x is an X-coordinate value of a magnetic sensor of the magnetic sensor array with respect to the magnetic source, y is a Y-coordinate value of the estimated position of the magnetic sensor with respect to the magnetic source, z is a Z-coordinate value of the estimated position the magnetic sensor with respect to the magnetic source, Θ is a rotational value about the X-axis of the estimated position of the magnetic source, Φ is a rotational value about the Y-axis of the estimated position of the magnetic source, μ is the permeability of free space, m is the magnetic strength of the magnetic source, and r=√{square root over (x 2 +y 2 +z 2 )}.
4. A method for determining a position of a magnetic source coupled to a bone of a patient, the method comprising:
(i) measuring a magnetic field generated by the magnetic source with a magnetic sensor array, the magnetic field being defined by a plurality of three-dimensional magnetic flux densities;
(ii) generating first data indicative of components of the three-dimensional magnetic flux density of the magnetic source at a plurality of points in space based on the measuring step;
(iii) determining an estimated position of the magnetic source;
(iv) calculating second data indicative of a theoretical value of the components of the three-dimensional magnetic flux density of the magnetic source based on the estimated position;
(v) calculating a difference between the first data and the second data; and
(vi) repeating steps (iii)-(v) until the difference between the first data and the second data is less than a predetermined minimum threshold value,
wherein calculating a difference between the first data and the second data comprises calculating a sum of the squared difference between the first data and the second data,
wherein calculating the sum of the squared difference between the first data and the second data comprises calculating the sum of the squared difference between the first data and the second data using the following equation:
F
=
∑
i
=
0
n
w
i
(
B
th
i
-
B
me
i
)
2
wherein F is a weighted sum, n is the number of magnetic field strength components measured, w i is a weighting factor for the ith measured component, B th is a theoretical value of the ith measured component, B me is a measured value of the ith measured component.
5. The method of claim 4 , wherein determining the position of the magnetic source comprises using a local optimization algorithm to determine a minimum value of the sum of the squared difference between the first data and the second data by adjusting the estimated position.
6. The method of claim 4 , wherein determining the position of the magnetic source comprises using a global optimization algorithm to determine a minimum value of the sum of the squared difference between the first data and the second data by adjusting the estimated position.