Induced current distributions in inductively stimulated bone growth
by Robert Royal McLeod
A thesis submitted in partial fulfillment of the requirements fo the degree of Master of Science in
Electrical Engineering
Montana State University
© Copyright by Robert Royal McLeod (1985)
Abstract:
Recently, use of EM-stimulated bone healing has gained wide clinical use. As yet, however, no
experimental data for the induced field distributions in an injured limb have been published. This
research presents a method for finding such distributions, consisting of a large-scale, PVC model filled
with saline "flesh". A dipole E-field probe is used in conjunction with a high-gain amplification and
filtering circuit to measure the horizontal, induced E-field at any point in the model.
Data is presented for several simple bone models. The results show that traditional Helmholtz coil
configurations do not give 2-D horizontal currents, as has been assumed. Currents are found to be large
near smooth bone faces and small near corners and edges.
An examination of current theoretical methods, based on the experimental information, is performed.
Prior work based on relaxation methods is shown to be incomplete and a complete formulation is
derived. This formula is shown to contain an extraneous solution which cannot be avoided,.
Multi-variable, first-order methods are examined, and it is demonstrated that the full power of
finite-element or boundary-integral methods is necessary to solve the stimulated bone problem. INDUCED CURRENT DISTRIBUTIONS
IN INDUCTIVELY STIMULATED
BONE GROWTH
by
Robert Royal McLeod
A thesis submitted in partial fulfillment
of the requirements fo the degree
of
Master of Science
in
Electrical Engineering
MONTANA STATE UNIVERSITY
Bozeman, Montana
November 1985
MMN UB.
A/373
H Tlsij&op. SL
ii
APPROVAL
of a thesis submitted by
Robert Royal McLeod
This thesis has been read by each member of the thesis committee
and has been found to be satisfactory regarding content, English usage,
format, citations, bibliographic style, and consistency, and is ready
for submission to the College of Graduate Studies.
X c y p - 6? —
' CXy.*?- ■
Date
Chairperson, Graduate Committee
Approved for the Major Department
/ 2 - J ' F
Date
j
Head, Major/De^Aptment
Approved for the College of Graduate Studies
Date
Graduate Dean
iii
STATEMENT OF PERMISSION TO USE
In
presenting
requirements
that
the
the
thesis
in
partial
fulfillment
of
the
for a master’s degree at Montana State University, I agree
Library
Library.
special
this
shall
Brief
permission,
make it available to borrowers under rules of
quotations from this thesis are allowable without
provided that accurate acknowledgment of source is
made.
Permission
thesis
may
Director
of
the
material
of
for
quotation
from or reproduction of this
be granted by my major professor, or in his absence, by the
Libraries
material
when, in the opinion of either, the proposed us
is for scholarly purposes.
Any copying or use of the
in this thesis for financial gain shall not be allowed without
my written permission.
Date
extensive
IAllhfG
iv
TABLE OF CONTENTS
1. INTRODUCTION................
EXPERIMENT ................
lO CTi
Model Structure ..........
Coils.......................
Probe .....................
Filtering . . . . ........
System Verification . . . .
IO
2.
CM tO
The Device ................
Method . ..................
Page
I
10
11
14
....
18
Complete Tartk. . . . . . . .
Short T a n k .......... ..
Two-D Geometry ............
22
4. NUMERICAL METHODS ..........
30
3. EXPERIMENTAL RESULTS
One Variable Solutions . . .
First-Order Solutions. . . .
More Sophisticated Solutions
19
26
30
34
36
5. CONCLUSIONS ................
37
6. REFERENCES CITED
40
V
LIST OF FIGURES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Sketch of coils and leg......................
Sketch of model with b a f f l e s .................................
Two views of experiment.................. ....................
View of Helmholtz coils. .................
Dipole p r o b e .................................................
Filter circuit block diagram .................................
Filter circuit d i a g r a m .................. v ..................
System block d i a g r a m .........................................
Probe calibration.............................................
Standard deviation of three s e t s .............................
Sketch of model with horizontal baffles....................
Current density, complete tank .............. . . . . . . . .
Second view of current density,complete tank. . .............
Current flow, complete tank...................................
Global current flow...........................................
Tank with baffles in p l a c e ...................................
Current density, short tank............................... , - .
Second view of current density,short tank . . ................
Current f l o w ..........
Sketch of tank with horizontal b a f f l e s .......................
Current density, 2-D case.....................................
Second view of current density, 2-D c a s e .....................
Current flow, 2-D case ........................................
Solution of equation 8 for complete t a n k .....................
Page
3
6
8
9
11
12
13
14
16
17
18
20
20
21
22
23
24
24
25
27
28
28
29
33
vi
ABSTRACT
Recently, use of EM-stimulated bone healing has gained wide
clinical use.
As yet, however, no experimental data for the induced
field distributions in an injured limb have been published. This
research presents a method for finding such distributions, consisting
of a large-scale, PVC model filled with saline "flesh". A dipole
E-field probe is used in conjunction with a high-gain amplification and
filtering circuit to measure the horizontal, induced E-field at any
point in the model.
Data is presented for several simple bone models. The results
show that traditional Helmholtz coil configurations do not give 2-D
horizontal currents, as has been assumed. Currents are found to be
large near smooth bone faces and small near corners and edges.
An examination of current theoretical methods, based on the
experimental information, is performed. Prior work based on relaxation
methods is shown to be incomplete and a complete formulation is
derived.
This formula is shown to contain an extraneous solution which
cannot be avoided,. Multi-variable, first-order methods are examined,
and it is demonstrated that the full power of finite-element or
boundary-integral methods is necessary to solve the stimulated bone
problem.
I
INTRODUCTION
In
hospitals
fields
are
being
of
non-union,
EM
fields
clinics
throughout
used to heal broken bones.
where
can
amputation.
and
the world, electromagnetic
Employed mainly in cases
the fracture shows virtually no sign of regrowth,
stimulate healing and are often the only alternative to
The
technique
has
a
reported success rate of about 10%
III, is non-invasive, and has no apparent side effects..
That
there
possibility
that
may
be
concerns
unapparent
many
side
doctors.
effects,
however,
is
a
Using similar or identical
techniques
on various animals, researchers have shown changes in mental
functions
121, cancer stimulation 133, and increased birth defects 141.
As
yet,
there is no complete explanation of either these effects or of
the stimulated bone healing process.
It
is
the
understanding
scientific
system
these
be
EM
purpose of this thesis to make a first step towards an
of EM-stimulated bone growth.
investigation,
is
to
completely
This first step, as in any
define
the input to the
the actual field distributions inside the injured limb.
fields
possible
Once
in and around the broken bone are known, it will then
to proceed to propose a model that will explain the direct
effect of these fields on bone cell growth.
2
The Device
The
use
most
common, commercially available bone growth stimulator in
is
the
Bio-ostogen [TM] unit, sold by Electro-biology Inc.
devices
are
so prevalent in hospitals and research labs that they will
be
taken
since
as the default for the study.
many
These
This is not overly restrictive
other devices on the market are similar or identical to the
Bio-ostogen [TM] device.
The
shown
configuration
in
figure
configuration,
in
the
Gauss
area
one.
for
The
an
actual broken leg under stimulation is
coils,
placed
in
a
nearly
Helmholtz
produce a time-varying, spatially uniform magnetic field
of
the
break.
This magnetic field, on the order of 20
in peak strength, induces currents in the leg - resulting in both
electric
variation
and
of
magnetic
the
field
distributions
that
depend
on the time
driving current plus the boundary conditions imposed
by the bone, flesh, and air interfaces.
3
Figure I.
The
exact
form
considerable
debate.
flesh
is
linear
input
waveform
sinusoidal
function
a
be
the
best
However,
material,
into
current
can
of
Sketch of coils and leg.
its
is
time
given
variation
the
is
a
subject of
reasonable assumption that
it is possible to Fourier transform any
sinusoidal
spectrum.
Once the response to
known, the response to the original complicated
determined.
Therefore,
this
research
will
use a
single-frequency, sinusoidal driving current.
Method
Electromagnetic
different
ways;
distributions in the flesh may be found in several
exact
solutions
of
Maxwell’s equations, approximate
4
numerical
descriptions,
solutions
of
the
or experimental measurements of models.
field
equations
are impossible for even moderately
realistic
biological
boundaries
involved.
Numerical computer solutions can be flexible and
convenient
but
the
have
systems
Exact
due
to
disadvantage
the
that
difficult
to
interpret and may not be unique.
difficult
to
construct
complicated
these
geometrical
results
are more
Experimental models are
and alter but, being actual physical analogues
of the problem, offer conclusive results.
As
explanations
features
of
change.
should
by
a
such
To
most
that the healing device must address will
useful,
then,
be completely flexible.
computerized
and
numerical
any practical method of solution
This requirement can really only be met
routine.
the fields measured.
compared.
results,
Thus
be
bone
However, since the results of
as a check, a physical model of a simple broken bone should also be
built
the
broken
the bone healing process evolve, the critical
a routine can not be entirely trusted without experimental data to
use
be
a
of
the
computer
the
If
the
computer
The results from both methods can then
solution agrees with the experimental
"correctness" of the computer results is assured and then
solution
could be generalized to more complex problems.
first step, and the main thrust of this thesis, is to produce
an accurate set of experimental data.
5
EXPERIMENT
The
model
perhaps
the
perfect,
ends
of
a broken bone structure used for this experiment is
simplest possible view of a broken bone, consisting of two
non-conducting
are
cylinders
perfectly
flat,
separated by a small gap.
and
all
other
tissue
is
The bone
considered
homogeneous.
Although
compared
the geometry of this structure is obviously simplified as
to
a
approximation
clinical
however
will
can
human
to
large
meter)
and
induced
data
will
currents
conductivity
the
taken
electrically
the
give
near
a
the
reasonable first
break in an actual
In reality, neither the bone nor the flesh is uniform,
dominate
be
the
case.
the
leg,
as
entire
the
modeling
the
system and, as a first-order approximation,
only
the
bone
change across the bone/flesh boundary
feature
flesh
present.
These assumptions for
(conductivity equal to one Semen per
(conductivity equal to zero) are consistent with
the latest physiological data available (SI.
A
"place
healing
the
problem
the
simplified
is
an
change.
ends".
coils
other.
with
and
These
model
artificial
this simplified model of a broken leg is where to
A
ends
human
is
quite long in comparison to the
in torso on one side and knee or lower leg on
features
must
leg
are
not
practical to build and thus the
stop at some point.
boundary
that
might
This unnatural termination
cause
the
field solution to
For example, if the induced currents normally flow up into the
6
lower
torso,
change,
"cutting"
possibly
problem,
model
that
two).
leg at the hip will force the currents to
throughout
potential
so
the
a
the
set
the system.
of
position
In order to investigate this
movable baffles were built into the lab
of
the ends could be varied (see figure
Data was taken with the "end boundaries" in several positions to
determine the effect of the artificial boundary location.
Figure 2.
Sketch of model with baffles.
Model Structure
A
is
major experimental design consideration in this and similar work
that the field probe is usually too large to give data that is truly
"data
at
constructed
actual
probe
a
point".
is
on
The
the
smallest
order
of
one
reasonable
probe
that
centimeter in size.
can
be
Since the
bone may be about five centimeters in diameter, a one centimeter
can
not
take
"point"
measurements
or
enough
data points to
7
adequately
describe
the
cannot
probe
where
a
scaled
of
one
up
the
be
fields.
reduced,
The solution to this problem, since
is to enlarge the bone model to a size
centimeter probe can be considered a point.
by
Our model is
a factor of approximately 12. This allows a 45 x 25 grid
points across the model at a one inch spacing, and the maximum probe
dimension
is
at
least 30 times smaller than the smallest dimension of
the model.
The
PVC
model,
"bones".
conductivity
are
then, is a PVC tank 4 ’ x 4 ’ x 2 ’ with 15 inch diameter
The
that
constructed
closely
seal
to
"flesh"
fills
the
is
saline solution of one Semen per meter
rest of the model.
The moveable baffles
of one inch styrofoam that can deform slightly to more
the
tank
walls.
Around this tank structure are the
driving Helmholtz coils (see figure three).
O
Figure 3.
Two views of experiment.
9
Coils
Increasing
Helmholtz
the size of the model in this fashion requires a set of
coils five feet in diameter
set
(see figure four).
Each coil in
the
particular
used in this experiment is made using 288 turns of
#16
magnet wire with the set having an inductance of approximately .350
Henries.
Figure 4.
The
use
View of Helmholtz coils.
Bio-ostogen (TM! clinical device and many other clinical units
complicated,
asymmetrical
or
square
wave
shapes to drive their
10
coils.
Although this apparently works well for healing, it causes many
difficulties
in
generating
and
shape,
is
this
it
model
reasonable
be
done
experimental situation.
delivering
difficult
to
to
Besides the difficulty of
the coils a high voltage wave of such a
filter or measure the output wave.
Since
is made up entirely of electrically linear materials, it is
to
assume a Fourier decomposition of the input waveform can
and then simply use a single frequency sine wave to obtain the
response
in
an
of the system.
the
frequency
Sixty cycles, being both readily available and
range
of
interest,
field
generated
was
chosen
to
be the signal
frequency.
Probe
The
flow
magnetic
in
the
saline
picked
up
by
figure
five).
a
and
by these coils induces currents to
creates a measurable E-field.
This field is
standard dipole antenna made of braided aluminum (see
The
use
and slows the electric polarization of the probe.
input
impedance
transmitted
to
also
(approximately
the
gives the probe a much
surface
is
lowers
aluminum
surface
polarization,
which
braided
greater
therefore
area
of
current density at the probe
inhibited
one
Meg
by
ohm).
Current, and
the probe's very high
The
probe signal is
the measurement circuitry by a very long coaxial cable,
running vertically out of the tank area to reduce cable pick-up.
11
Figure 5.
Dipole probe.
Filtering
This
(see
to
probe
figures
raise
the
60
six
the
high-gain
signal goes into a filtering and amplification circuit
incoming
Tow-Thomas
hertz
because
which
much,
where
is
to
it
it
much
detectable
would
The first circuit block is an amplifier
signal
level above the power-supply noise.
A
pre-filter then removes most of the noise outside
bandwidth
amplified
signal
and seven).
of
interest.
contains
a
This
constant
signal
can
not yet be
60 hertz background level
greater than the desired signal.
To amplify the
levels would amplify the background to the point
saturate
the
amplifier.
This then requires that the
12
constant
added)
the
portion
by
means
signal
induced
reference
to
the
signal be subtracted out (phase inverted and
of a 60 hertz reference.
The phase shifter (to align
signals in time) and the reference level adjust are used to set the
reference
the
of
zero
signal
at
any desired level and phase.
current
must
be
zero
at
the
Since, by symmetry,
center
of the tank, the
adjustments may be used to set the level of the output signal
at
that
remaining
this
point, calibrating the system.
remains
spectral
is
noise.
amplified
and
a
Finally, the induced
post-filter
removes
any
The output signal is then fed to a digital
volt-meter,
allowing direct measurement of the induced signal strength,
and
oscilloscope
to
an
which
is used to obtain phase data.
diagram of the entire system is shown in figure eight.
Post -
Figure 6.
Filter circuit block diagram.
A block
13
Rer
Filter circuit diagram.
6
Figure 7.
14
Dipole Signal
in
--
Dipole Probe Signal
Out ^
Arrme ter
60 Hertz Line
Figure 8.
System block diagram.
System Verification
Several
that
system
it
was
tests were performed on the experimental set-up to confirm
operating properly.
Specifically, in one experiment, the
was tested for linearity and the calibration constant was found.
In a second test, the repeatability of the measurements was verified.
15
The
should
of
probe circuit gives readings in volts which, theory indicates,
be
the
line
proportional to the E-field at the test point.
input
E-field
through
constant,
the
the
versus
origin.
Thus a graph
the output voltage should be a straight
The
slope
of the line is the calibration
multiplier necessary to translate output voltage back to
the actual measured E-field value.
To
A
beaker
coils.
test
this assumption, the following experiment was performed.
of
saline
solution
This
system
closely approximates a two-dimensional, circular,
saline-filled
boundary
in
a
was
placed in a small set of Helmholtz
uniform magnetic field.
The E-field for
this problem may easily be calculated and compared to the probe output.
The
in
The
the
probe was placed to measure the induced, circular current flow
saline
results
function
is
are
and a number of data points were taken along a radius.
shown in figure nine.
linear
and
As was predicted, the transfer
intercepts the origin.
gives the calibration constant as .00836.
The slope of the line
16
O
•.SO
0.00
% SO
I. so
RADIUS (CM)
Figure 9.
The
linear,
data
must
accurate,
feature
from
also
the
be
repeatable
of
Probe calibration.
experimental bone model system, besides being
repeatable.
results,
the
Without
data
the
would
ability
to produce
be meaningless.
This
the test system was checked by simply taking three complete
sets
of
data and comparing them.
The curves were nearly identical - a
plot
of
the
is shown in figure ten.
standard
deviation
The average
17
deviation
the
in the data over the tank for the three data sets is .007 and
maximum
deviation
these
correspond
noise
level
system
is
to
causes
linear,
is
.07.
zeros
the
in
Note
the E-field where the minimum constant
deviation
calibrated,
the spikes in the plot below
to
become
and repeatable.
very large.
Confident results may
now be presented.
"gy- «r
Figure 10.
Thus, the
Standard deviation of three sets.
18
EXPERIMENTAL RESULTS
Results
tank,
presented
for three configurations - I) the complete
2) the "short" tank with the end baffles in place, and 3) a "2-D"
set-up
a
are
thin
strictly
where lower water and horizontal baffles limited current flow to
plane
(see
figure
two-dimensional
11).
currents.
Case
three
was performed to force
As will be shown, it is necessary
to force this condition - it is not the normal mode of operation.
Figure 11.
Sketch of model with horizontal baffles.
19
Complete Tank.
Figures
throughout
12-14
present
the
magnitude
full
tank.
First,
current
does
not
fact),
resulting in zeros at all corners including the edge of the bone
face.
Also,
expected.
have
the
and
disappear
expected.
flow into the insulating walls (a comforting
current
would
flow
other.
behavior
many features are as expected.
flow
between
the
bone faces is nearly as
It flows along the faces, not across as some clinical people
assumed
center
that
the horizontal current
the
the
Note
of
A
of
the
here
This
be
the
case.
The expected zero was found in the
was one way along one face and the opposite way along
feature
which
does
not seem correct, however, is the
currents at the end walls.
instead
of
reversing
The current seems to just
and flowing back up the bone as
is a major departure from the expected two-dimensional
flow.
I
20
Figure 12.
Figure 13.
Current density, complete tank.
Second view of current density, complete tank.
?
:
5
5
5
'S
2
21
Figure 14.
Due
to
the
nature
horizontal
currents
that
would
this
two-dimensional
vertical
horizontal
between
flow
magnetic
the
of the experiment (primarily the probe), only
could
be
be
measured,
sufficient
(i.e.
pattern).
It
field
currents
Current flow, complete tank.
-
from
but it was initially assumed
that
was
there
reasoned
would
that
be
the
only a
uniform
the Helmholtz coils should produce only
especially
in
the
center
plane of the bones
end faces of the break where the geometry is constant in z
to first-order.
As
wrong.
can
be
seen
from the flow diagram, these assumptions must be
The current flowing down beside the bone does not flow back out
22
and
is
must,
that
and
indicate
along
that
the
must
flow up or down.
the
bottom
somewhere .
current
flows
The only conclusion
Experiments with various
in
a circuit down one end
parallel to the bone, up the other end plate
the surface of the saline (see figure 15).
by
the
vertical,
region
near
non-uniform
have
current
along
forced
and
current continuity, go
the
baffles
plate,
by
This behavior is
fact that the Helmholtz field is not everywhere uniform
but
is
instead quite non-uniform outside the Helmholtz
the center of the coils.
Outside of the Helmholtz region,
magnetic fields are set up; hence, the induced current must
vertical
components which will effect the current flow throughout
the entire leg.
Figure 15.
Global current flow.
Short Tank.
If
magnetic
the
vertical currents are due to the non-uniform nature of the
field,
a
reasonable
move
would
be to restrict the problem
23
entirely
coils.
in
tank
to
the
Using
figure
two
16),
feet
uniform
the
magnetic field region near the center of the
styrofoam end-baffles discussed before (shown again
this was done.
on
a
side
The problem then consisted of a square
centered
in
the
coils.
presented below.
Figure 16.
Tank with baffles in place.
The results are
24
25
Figure 19.
The
major
complete
tank
solution,
though
the
magnetic
uniformity
The
did
not
vertical
Helmholtz
in
the
change
-
currents.
coils
data
as
discussed
above
for the
no currents into walls, symmetry of
It
is
thus
concluded that, even
produce a nearly-uniform, nearly-vertical
field
in the Helmholtz region, slight
deviationsfrom perfect
are
important and in fact appear to
dominate the solution.
fact that the magnetic field
magnitude
is
and
features
Current flow, short tank.
has
significant
distributions.
is not quite all vertical and that the
decreased by only fractions of a
percent atthe box edge
and does strongly effect the induced currents and their
26
A
the
first
conclusion that can be drawn from this is that the use of
Helmholtz
coil
configuration
special
significance.
special
or
geometry.
produce
does
qualitatively
The
an arbitrary choice that has no
general,
this
type
different
from
any
of
coil is in no way
other
particular coil
Helmholtz configuration is simple for analysis and does
two-D
not
In
is
currents
yield
between
the bone faces.
two-dimensionality
Overall, however, it
or any other special properties in
the induced current.
Secondly,
boundary
still
the
it
does
have
be seen that the position of the artificial end
not significantly change the current mode.
the
magnitudes
boundary
can
same
are
The currents
overall shape and, especially in the break area,
nearly
the
same.
Thus the introduction of this
into the problem has probably not strongly changed the induced
fields, particularly in the region of the bone break.
Two-D Geometry.
In
order to confirm the fact that the currents were indeed flowing
vertically
only
in
the
horizontal
between
this
preceding
cases, the geometry was changed to force
(two-dimensional)
case
and
current
flow.
A large difference
the others would confirm the three-dimensional
nature of the overall problem.
A
level
from
two-dimensional geometry was created by first lowering the water
until it came below the top of the bone - this prevented currents
flowing
"over
the
top"
of
the bone.
Next, a set of styrofoam
27
baffles
the
was
placed horizontally about four inches below the surface of
water - restricting currents to a thin flat volume (see figure 20).
Data was then taken in this restricted area.
Figure 20.
Sketch of tank with horizontal baffles.
28
Current density, 2-D case.
J(A'
Figure 21.
*-S)
Figure 22.
Second view of current density, 2-D case.
29
Figure 23.
As
can
current
cases
seen
continuity
currents
they
be
flow
along
Current flow, 2-D case.
by the flow diagram (figure 23), two-dimensional
now
the
appears
side
to hold where it did not before.
walls and then back along the bones as
must in a two-dimensional system.
above
demonstrates
The
This strong change from the two
that the current normally flows in all three
dimensions and cannot be considered two-dimensional.
NUMERICAL METHODS
As
were
in
described in the introduction, a number of numerical algorithms
formulated
Fortran
77
and tested.
on
The routines described below were written
a VAX 11/750.
The IMSL scientific library was used
for advanced mathematics and matrix manipulation.
One Variable Solutions
The
problem
experimental
is
not
results
above
two-dimensional,
show
that
the
simplified bone
but must really be analyzed in three
dimensions.
Prior
work 161 has used a two-D Helmholtz equation which,
as
shown
below,
will
limited
to
however,
will
be
be
two
can
dimensions.
incorrectly
The
simpler
formulated,
besides being
two-dimensional
approach,
yield some insights into the mathematics of the model and
used
two-dimensional
is
for
that
problem,
the
purpose
following
solved over the two-dimensional region:
here.
In
general,
for
the
two Maxwell equations must be
,
31
V x H =
(o + Jcue ) E
(1)
V x E =
-juvH
(2 )
with the following two-dimensional restrictions:
H = Hz only,
E = curl H = Ex and Ey only,
all derivatives with respect to z = 0.
Although
frequency
regions
Starting
similar
(E
or
the
in
of
conductors,
zero
from
to
H).
displacement-current
is
quite
small
at low
it will be included since the bone model has
conductivity
the
term
where
this
term
may
be
important.
equations above, one may proceed with a development
the Helmholtz wave equation in order to remove one variable
We
will
choose to work with H in this case since it will
have only one component (vertical).
V x V x H
= Vx
-v H + v (v
(o + Joie) E
H) = V (a + jme) X E
+ (a + jwc) V x E
(4)
32
Note
that
the
conductivity
pulled
outside
the
curl operation since in this case the conductivity
varies
in
space
(i.e.
across
term
in
equation
three may not be
the flesh, bone boundary).
As will be
seen, this term causes some severe problems.
Continuing the development, then, we have:
v (v • H) = ( Vu) v (v • B) = 0
(5)
(iff mu = constant)
(6 )
(a + jcuc) V x E = (a + Ju e ) (-juy) H
Simplifying
(equation
material
five).
is
terms, the div of H may be removed if mu is a constant
This is a reasonable assumption here since biological
non-magnetic.
substituting
for
restrictions
stated
zero
everywhere
epsilon
change.
E
from
Ampere’s
above.
except
Equation
at
Combining
seven may be simplified by first
law
and using the two-dimensional
This results in a loss-1 ike term which is
interfaces
these
terms
between regions where sigma or
with
the
original equation
yields:
I
(a+ Jm e )
[v(a+
Jcue)-V]
Hz
(8 )
33
This
any
equation
problem
described
currents
may
be
geometry.
solved by finite difference techniques for
Sample
solutions
above
are
shown in figure 24.
inside
the
bone
-
an
of
the
bone model problem
Note that there appear to be
insulator!
The
reason for this is
discussed in the next paragraphs.
Figure 24.
Imagine
possible
terms)
letting
find
which
equation
conditions
\
\
\
\
\
Solution of equation 8 for complete tank.
the closed form solution of the above equation (if it were
to
the
'
one)
vary
in
frequency
eight,
are
one
-
it Hsol.
Hsol may contain a term (or
space but are not induced.
This can be seen by
approach
time
is
applied
call
zero.
With
no
equation.
variation
in
left
with
Laplace’s
If boundary
which
vary
in space, so will Hsol; a classic
34
example
sides
of
and
vary
in
over
this
is
a
H non-zero on the fourth.
entire
problem.
frequency-independent,
boundary
extraneous
equation
for
term
Laplace’s
must
equation,
this
go to zero when constant
This same result is also true for the
- the left-hand side of equation eight.
However,
the inhomogeneous term is present (the right-hand side of equation
eight),
the
boundary
sigma
extraneous
conditions.
changes,
Hsol
of
So,
conditions are applied,.
wave
when
If the boundary conditions do not
space and all boundaries are constant, Hsol will be a constant
the
lossy
square area with H equal zero applied on three
part
This
alters
of Hsol does not go to zero for constant
extra
the
term,
which is non-zero only where
solution and causes the extraneous part of
to remain even for constant boundary conditions.
the
equation by means of a computer implementation reveals that the
"improper"
part
An investigation
part
and,
solution.
single
the
therefore,
It
"currents"
of
in
is
the
solution
cannot
this
spurious
insulating
second-order
be
equation
bone
is not simply added onto the proper
separated
from
information
above.
the
that
real
results
induced
in
the
This demonstrates that a
in E or H applied to the entire geometry
is insufficient to solve the problem.
First-Order Solutions
the
The
error
two
first-order
second-order
above
was
introduced by eliminating one variable from
differential
formulation.
It
can
equations
to
arrive
at a single,
be seen that in taking the curl of
35
Ampere’s
law,
alternative,
the
then,
extraneous
is
to
solution
consider
was
allowed.
A
logical
solving the first-order equations
directly.
The set of first-order equations to be solved then, is
V x E =
-JwuH
(9)
( 10)
V x H =
(a + J w
e
) E
with the boundary conditions of
It
a
Htl = Ht2
(11)
Etl = Et2.
(12)
has been proven [71 that this formulation has a unique solution
reassuring
other
two
both
beginning.
Maxwell
An
indication of this result is that the
equations can be determined from those above.
With
the curl and the divergence of the fields specified, the solutions
are unique.
A
problem
displacement
flow
and
impossible
currents
the
insulators.
arises,
curl
however,
are
of
derive
the
practically
H
This formulation
to
when
the
frequency
zero.
(equation
ten)
is so low that
Now only Ohmic currents
is
essentially
zero
does not have a unique solution .
divergence
of
E
from
these
in
It is
equations
and,therefore, E is incompletely specified.
An
optimist
at
this
point
will
proceed
and
include
displacement current terms in the equations even though they are very
the
36
small,
and
and
it
will
attempt to solve the problem anyway.
doesn’t work.
strongly
near zero.
It
considered.
an
then,
that
these simpler methods must discarded and a
approach
overwhelming
importance
have
matrix solutions have determinants very
with
guaranteed
uniqueness
must
be
A survey of current literature on this more general problem
great
and
and
Obviously, yet another method must be tried.
three-dimensional
of
An iterative solution does not converge (indeed,
diverge)
seems,
yields
This attempt was made
been
amount of material.
Eddy current problems are
in electric machine design (and many other areas)
extensively researched.
literature
is
paragraphs
sketch the general line of attack of the most common methods
used
solve
to
beyond
the
scope
of
A comprehensive survey of this
three-dimensional
this
thesis,
induced
current
but the following
(eddy
current)
problems.
More Sophisticated Solutions
In
lost
general,
additional
uniqueness.
Solving
inefficient
real
to
(and
since
values.
store
the
potential
these
are placed on E to regain the
equations
directly,
however,
is
and H consist of six complex components or twelve
On a four-byte real system this requires nearly 50 bytes
fields
associated
problems.
E
restraints
These
at
a single node.
computing
memory
formulations.
time)
and
Thus a huge amount of memory
is required for even the simplest of
efficiency problems are avoided by using
Mayergoyz [7] has shown that, with appropriate
37
boundary
single
conditions, a scalar problem may be solved in insulators and a
vector
potential CA) used in conductors.
This reduces both the
storage and computation to a workable level.
The
solution
formulated,
of
these
can
are
difference
be
these
elements
formulations
generally
No
field
equations,
however
accomplished in a number of ways.
finite
inflexibility.
are
direct
they
The most common
and boundary integral methods.
not
generally
comparison
of
the
used
other
are
due
two
to
Finite
their
methods is
possible - each has advantages and disadvantages for specific
problems.
Of the two, finite elements is generally more popular.
these
formulations
shown
to
from
of
have
been implemented for many cases and have been
give accurate, reliable results.
being
Both
The broken bone system, far
the simple question assumed before, is a complex, intricate
problem requiring the full power of such methods.
CONCLUSIONS
The
overall
These
experimental
presented
in
this
paper reveal both the
mode and magnitude of the currents in a first-order bone model.
results,
none-the-less
complicated
currents
found
scale
for
generally
models.
inside
approximate
large
data
the
The
the
simplest
applicable
as_ a
currents
possible bone structure, are
first approximation to more
measured
are
the
actual induced
the boundaries of the model and they can be assumed to
currents
modes,
both
inside
a
the
and
E
human leg.
H
fields
In the broad sense of
are
now
known in a
38
stimulated
broken limb.
This information will be essential in building
a theoretical model of the healing process.
Besides
been
presented
will
again
currents
and
small
modes, specific plots of the current density have
the simplified bone model.
invaluable
to
future
Features of these data
investigators.
For
example,
shown to be large near flat, cdntinuous parts of the bone
at
corners
or
discontinuities.
This
fact,
perhaps not
could strongly influence the design of a healing device based
current
extended
for
be
are
expected,
on
general
stimulation
to
more
near
realistic
bone
bone
shards.
models
These results can also be
by
employing
the verified
experimental set-up created for this work.
The
theoretical
solution
and
methods
cannot
stimulated
complete
these
to
broken
produce
have
and
literature.
A
general
this
thesis
has
shown
that previous
accurate
results.
The
magnetically
bone problem is fully three-dimensional and requires
three-dimensional
algorithms
of
were both oversimplified and incorrect mathematically
hope
methods
solve
part
eddy
current
formulations.
Fortunately,
been investigated for many years and many complete
mathematical
number
problems
of
developments
commercial
of this nature.
packages
are
available
in
the
are also available to
It should be possible to find
such a package to apply to this problem.
In
inside
never
the
It
summary,
the goals of this research have been met.
The fields
an inductively stimulated limb have been measured, an experiment
before
successfully
completed.
The theoretical foundations of
problem have been investigated and prior solution methods rejected.
has
been shown that a fully three-dimensional eddy current solution
39
is
necessary
such
problems.
tool
in
healing.
the
and
that
With
methods are perhaps already available to solve
these results, researchers now have an essential
quest for the basic mechanisms of electrically stimulated
I
40
REFERENCES CITED
1.
Downes, E . M., and Watson, J., “A Clinical Study of the Efficacy of
~\
a Battery-Operated Orthopaedic Stimulator," presented at the
1985 Bioelectric Repair And Growth Society meeting.
2.
Liboff, A. R., Thomas, J. R-, and Schrot J., "Magnetically Induced
Behavior Modification in Rats," presented at 7th Annual
Bioelectromagnetics Society Meeting, 1985.
3.
Winters, W. D., and Phillips, J. L., "Monoclonal Antibody Detection
of Tumor Antigens in Human Colon Cancer Cells Following EM
Field Exposure,"
Proceedings of the Bioelectromagnetics
Society , 6, 1984, p. 46.
4.
Sisken, Betty F., Fowler, Ira, and Kryscio, Richard, "The Effects
of Pulsed EM Fields (PEMF) on Chick Embryos After Limb
Amputation,"
presented at 5th Annual Bioelectromagnetics
Society Meeting, 1983.
5.
Kosterich, Jeffrey D., Foster, Kenneth R., and Pollack, Solomon R.,
"Dielectric Properties of Fluid-Saturated Bone - The Effect
of Variation in Conductivity of Immersion Fluid,"
IEEE
Transactions on Biomedical Engineering , 31, April 1984, pp.
369-373.
41
6.
Parker, Reed A . , “Computer Solutions of Complex Biological
Boundary-Value Problems,"
Master’s Thesis - Montanta State
University, March 1984.
7.
Mayergoyz, I. D., "3-D Eddy. Current Problems and the Boundary
Integral Equation Method.", available from the author at EE
Department, University of Maryland, College Park, MD
20724.
___.,tv/ ITBBiRIES
3 1762 10014836 8
!'a.i n
N378
M22U
cop. 2
McLeod, Robert Royal
Induced current dis­
tributions in inductive­
ly. . .
Vain
N378
M224
cop. 2