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Optimal realization of arbitrary forces in a magnetic stereotaxis system.
Optimal realization of arbitrary forces in a magnetic
stereotaxis system.
�
David C. Meeker
Eric H. Maslen
Rogers C. Ritter
Francis M. Creighton
Abstract
Very small implanted permanent magnets guided by large electromagnetic coils have been proposed
previously as a method for delivering hyperthermia to or guiding catheters through brain tissue. This
procedure is termed magnetic stereotaxis. Early efforts employed a single coil on a movable boom, a
design that proved logistically difcult to use on human patients. The present work deals instead with
a design where several stationary coils are employed to develop a force on the implanted magnet. The
coil current-to-force relationship is developed for this type of machine, and several optimal solutions for
realizing an arbitrary static force are presented for various constraints on the orientation of the implanted
permanent magnet. Costs of the different solutions are compared in several examples using a mathematical
model based on the Magnetic Stereotaxis System (MSS) developed by Stereotaxis, Inc., the University of
Virginia, and Wang NMR.
1
Introduction
Magnetic stereotaxis is a novel therapeutic methodology for the treatment of brain tumors and other neurolog-
ical problems. The fundamental idea of magnetic stereotaxis is that large electromagnetic coils can be used
to guide a small piece of implanted permanent magnetic material (a magnetic seed) along some arbitrary
trajectory through brain tissue. Incidental damage is reduced by selecting a path that avoids important brain
structures. Once the seed has been maneuvered into a tumor, the seed is heated inductively by high-frequency
magnetic elds. This heating results in highly localized cell death. By successive movements and heating, a
tumor could be destroyed with little damage to the surrounding tissue [1]. Alternatively, the magnetic seed
could be used to guide the tip of a catheter. This catheter would then be used to deliver drug treatments
directly to sites inside the brain [2][3].
Previous efforts in magnetic stereotaxis used a single movable coil to act upon the implanted magnet. This
arrangement proved successful in experiments using live dogs [4]. Scaling the device to work on humans,
however, has been difcult. The size and weight of the required coil interfere with the necessary bi-planar
uoroscopes and make precision manipulation impractical in an operating room environment [2] [5].
Subsequent efforts have focused, instead, upon an arrangement of stationary coils, as shown in Figure 1.
Since the coils are operating in a region of uniform, linear magnetic permeability, the elds of the coils are
additively superimposed upon one another. With a proper selection of currents, a wide range of forces can
be applied to the magnetic seed. One such device, the Magnetic Stereotaxis System (MSS) developed by the
University of Virginia, Stereotaxis, Inc., and Wang NMR, Inc., has been built and is currently in the process
of testing [6] [7]. This device uses a roughly cubic array (or helmet) of six superconducting coils.
Several problems preclude the closed-loop control of machines such as the MSS. First, the superconduct-
ing coils of such a machine have a large inductance. Second, the currents inside the coils cannot be changed
faster than a certain rate (making changes in current difcult and costly), or the coils will become heated and
�
Copyright 1996 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for ad-
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copyrighted component of this work in other works must be obtained from the IEEE. Published IEEE Transactions on Magnetics, V
OL
.
32, N
O
. 2, March 1996, pp. 320328.
1 Nomenclature Denotes transpose.
a
Magnetic vector potential.
b
Vector ux density.
b
Column matrix of ux density components.
B
Current-to-ux density matrix.
c
Transformed coil currents.
C
Cost function.
d
Column matrix of desired force and eld.
D
Matrix of ux density derivatives.
e
Unit vector dening frame aligned with desired force.
f
Vector force on dipole.
f
Column matrix of force components.
G
Dipole constraints matrix.
i
Coil currents.
j
Vector current density.
K
Current-to-force matrix.
m
Vector dipole moment.
M
Spherical seed current-to-force matrix.
m
Column matrix of dipole moment components.
n
Unit vector dening helmet-xed reference
frame.
p
Position vector.
P
Identity matrix lacking last column.
r
Vector from differential current volume to eld
point.
t
Torque on dipole.
T
Transformation matrix.
U
�
V
�
W
Sing. val. decomposition. of G.
v
Unrealizable vector of singular G.
x
�
y
�
z Components of p. Flux density to force ratio. force ux misalignment angle. Magnitude of m (const. for a given seed). Lagrange multiplier.
�
Magnetic permeability.
lose their superconductivity a catastrophic failure known as quench. Lastly, the tissue through which the
seed moves is moderately heterogeneous, leading to signicant model uncertainty. Due to these difculties,
an open-loop control scheme has been chosen.
In the open-loop control scheme, movement of the seed proceeds in small steps. The steps are short
enough that variations in the current-to-force relationships due to seed movement can be ignored and the
effects of modeling errors remain small. The static current solutions addressed in this work then can be
assumed to apply over the entire length of the seeds movement. The scheme consists of rst determining
a set of currents which applies a force in the desired direction of motion. The magnitude of this set of coil
currents is then ramped up and down once, resulting in a small movement of the seed. Magnitude and duration
of the current ramping are chosen so that the seed will typically move a centimeter or less during each pulse.
After each pulse, the seeds position is determined by uoroscopes and is assessed by the operator. A new
force direction is then chosen to continue progress toward the ultimate target. A similar open-loop scheme
consisting of successive small seed motions has been used successfully in surgical trials of the single movable
coil MSS [4] and in on-axis movements of the six-coil MSS [6].
Previously, the control scheme lacked predetermined constraints on the selection of currents leading to a
force in the desired direction of movement. Selection therefore proceeded somewhat heuristically, compli-
cating the machines operation. The present work addresses the problem of the selection of currents needed
to realize an arbitrary force in the general class of stationary-coil magnetic stereotaxis systems. The current-
to-force relationship for such devices is rst characterized. Three classes of solutions for the currents needed
to produce an arbitrary force are presented. For convenience, a six-coil MMS is specically addressed; how-
ever, these solutions can be easily extended to a machine with more than six coils. These solutions assume
different constraints on the orientation of the implanted magnet relative to the direction of the desired force.
Seed orientation is especially important when a catheter is being pulled; misalignment can cause unacceptable
damage to brain tissue during seed motion. It will be shown that the problem is usually underconstrained,
implying the existence of many solutions for any given force orientation. An optimal solution is chosen in
each case on the basis of a quadratic measure of the coil currents. Solutions under each set of alignment
2 coil
patients
head
magnetic
seed
superconducting
Figure 1: Schematic of multi-coil MSS.
constraints are then demonstrated through several illustrative examples using the specic conguration of the
Magnetic Stereotaxis System.
2
Governing Dipole Equations
Since the dimensions of the permanent magnet seed are very small compared to the size of the superconduct-
ing coils, the seed can be idealized as a point dipole. The magnetic properties of the seed are summarized by
the seeds dipole moment, m. The direction of this vector is the same as the direction of magnetization in the
seed. The magnitude of m is the product of the magnetization and volume of the seed.
Derivation of the forces and torques on a dipole due to an applied magnetic eld can be found in the
literature [8]. Dening p as a position vector locating the seed relative to the center of the helmet, the force
on the seed is
f
�
p
��� �
m
�
b
�
p
���
(1)
and the torque is
t
�
p
���
m

b
�
p
�
(2)
3
Formulation of Current-to-Force Relationships
Equations (1) and (2) specify the force and torque respectively on the dipole seed, but they do not imply
any particular basis in terms of which these vectors are represented. A computationally useful form of these
equations can be obtained