Download Energy Efficient Coils for Magnetic Stimulation of Peripheral Nerves

1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
Energy Efficient Coils for Magnetic Stimulation of
Peripheral Nerves
Laura Cret, Mihaela Plesa, Radu V. Ciupa, Dan D. Micu and Lucian Man
Abstract — The preoccupation for improving the quality of life, for persons with different handicaps, led to extended research in the
area of functional stimulation. Due to its advantages compared to electrical stimulation, the technique of magnetic stimulation of
nerve fibers represents a new direction of research in modern medicine. This study starts from a major limitation of the coils used for
magnetic stimulation: their inability to specifically stimulate the target tissue, without activating the surrounding areas. The first
goal of this study was to determine the optimal configuration of coils for some specific applications and evaluate the distribution of
the electric field induced. Once the coil configuration established, we address other issues that need to be solved: achieving smaller
coils, reducing power consumption (the low efficiency of power transfer from the coil to the tissue is a major drawback) and
reducing coil heating.
Keywords: biomedical engineering, biotechnology.
1. INTRODUCTION
where V is the transmembrane voltage, Ex the axial
component of the induced electric field, λ the space
constant of the cable and τ the time constant.
The term on the right of the equation represents the
activation function, equal to the spatial derivative of the
electric field induced along the nerve fiber.
Never the less, for finite cables, two additional difference
equations are required to describe the transmembrane
potential at the boundaries of the cable. This is because
there is no current flows through the high impedance of
the membrane termination. Therefore, if L is the length of
the nerve fiber, d the fiber diameter, ρa the axoplasmatic
resistivity, cm the membrane capacitance per unit area and
Iionic the membrane current due to other (than capacitive)
channels, the following equation has to be considered, for
both x=0 and x=L boundary conditions (Δx → 0) [1]:
I
dV
d
dV
d
+ ionic +
=
E x (2)
dt c m πdΔx 4ρ a c m Δx dx 4ρ a c m Δx
Thus, for the ends of the nerve fiber, the activation
function is given by the magnitude of the axial
component of the electric field and not the magnitude of
the gradient of this field as in (1).
This electric field can be produced by electrical or
magnetic means. The second method is presented in this
paper. The principle of magnetic nerve stimulation is
emphasized in figure 1.
The preoccupation for improving the quality of life, for
persons with different handicaps, led to extended research
in the area of functional stimulation. Due to its
advantages compared to electrical stimulation, magnetic
stimulation of the human nervous system is now a
common technique in modern medicine.
A disadvantage consists, however, in the fact that the
need of focal stimulation can not always be fulfilled. This
is why the design of coils with special geometries can
help achieving this goal. Another drawback consists of
the low efficiency of power transfer from the coil to the
tissue.
The present paper starts by emphasizing the theoretical
background of magnetic stimulation. Then, a computer
model with all its characteristics is presented. Finally, the
total electric field and energy transfer parameters are
computed for coils with special design and important
conclusions are drawn.
2. THEORETICAL BACKGROUND
Neuronal structures can be modeled in the form of a cable
and the membrane response can be computed by solving
the equations describing the transmembrane potential
across the membrane of the cable in the presence of
induced electric fields. The relation between the
transmembrane potential along an infinitely long nerve
fiber in the presence of induced electric fields is given by
[1]:
∂E x
∂V
∂ 2V
+ V − λ2
= − λ2
τ
(1)
2
∂t
∂3
x
∂x
1424
M a g n e ti c f i e l d
M a g n e t ic
c o il
N erv e
fib re
Eddy
c u rre n ts
= f ( x)
Figure 1. An alternating current flowing through the
magnetic coil causes a time varying magnetic field, which
induces an electric field.
L. Cret, M. Plesa, R.V. Ciupa, D.D. Micu and L. Man are with the
Technical University of Cluj-Napoca, Romania, phone: +40-264-401462; fax: +40-264-592-903; e-mail: [email protected].
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1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
(3)) with dI / dt t =0 = U 0 / L . Assuming that the activation
of the nerve fiber occurs for a preset value of the electric
field E, we obtain U0, the necessary initial voltage on the
capacitor that would lead to activation.
The energy dissipated in the circuit during one pulse of
duration Δt is [4]:
According to the electromagnetic field theory, the electric
filed inside the tissue can be computed by means of the
scalar electric potential and the vector magnetic potential
[2]:
∂A
(3)
E=−
− gradV
424
3
∂3
t1
12
EV
Δt
∫
EA
WJ = R I 2 (t )dt
The first term of the electric field is called “primary
electric field”, and it is due directly to the
electromagnetic induction phenomenon, while the second
term represents the “secondary electric field”, due to
charge accumulation on the tissue-air boundary.
According to formula (3), the computation of the electric
filed due to electromagnetic induction is done by means
of the magnetic vector potential [2]:
μ ⋅ N ⋅ I( t )
dl
(4)
A( r , t ) = 0
∫ r
4π
(6)
0
The peak magnetic energy in the coil WB required to
induce a given electric field is [4]:
WB =
1 2
LI peak
2
(7)
The temperature rise in the coil after one pulse of
duration Δt is (assuming there is no cooling) [4]:
η Δt 2
(8)
ΔT =
I (t )dt
cσA2 ∫0
where η is the resistivity, σ the density, c the specific
heat and A the cross-sectional area of the copper wire of
the coil.
Those three quantities are evaluated to establish the
parameters of energy transfer from the coil to the target
tissue.
coil
where the vector dl represents the differential element of
the coil, r is the distance from the coil element to the
field point, and N is the number of turns of the coil.
For coils of different shapes, one can compute A using
the following technique: The contour of the coil is
divided into a certain number of equal segments, and the
magnetic vector potential in the calculus point is obtained
by adding the contribution of each segment to the final
value.
The secondary electric field depends on the geometry of
the tissue-air interface, considered a cylindrical surface.
This term is computed knowing that on the surface, the
3. RESULTS
Magnetic coil design is one of the most important aspect
of the technique of magnetically stimulating the nervous
system. Recent advances in this area have improved
significantly the focality. The planar “butterfly” coil was
the first major improvement over the conventional planar
circular coil. But more recently, 3-D design has shown
even greater promise. This paper presents the electric
field generated by a family of “slinky” coils, and some
studies regarding the electromagnetic field penetration in
the tissue for different parameters of these coils (radius,
configuration, number of turns per loop).
Considering a coil with N turns, the “slinky-k” coils are
generated by spatially locating these turns at successive
angles of i x 180 / (k − 1) degrees, were i = 0, 1, …, k-1
[5]. If the current passing through this coil is I, then the
central leg carries the total current N x I. With this
definition, the circular coil is considered a “slinky-1”
coil, and the figure of eight is a “slinky-2” coil.
The geometry of the problem is given in figure 2.
boundary condition to be fulfilled is: n ⋅ E A = −n ⋅ E V
(continuity of the normal component of the current
density vector). The electric potential inside this domain,
V, is numerically evaluated by solving Laplace equation
(ΔV = 0) with Neumann boundary conditions inside the
tissue.
The electric current required to induce the electric field
( A is proportional to I – see (4)) is delivered by a
magnetic stimulator (LRC circuit). The current waveform
through the discharging of a capacitor, with an initial
voltage U0, to the coil is [3,4]:
(5)
I = U 0 ωL ⋅ sin (ωt ) exp(− αt )
where α = R /( 2 L ) , ω = 1 / LC − α 2 , C is the
capacitance, and R and L are the resistance and
inductance of the coil, respectively.
We focus on stimulators with a fixed rise time of the
current I(t) from 0 to peak, which is sufficient for
comparing relative figures of merit of the stimulators.
The inductance is evaluated by taking the line integral of
the vector potential around the coil, for unit current [2]:
Slinky_5
coil
10 [mm]
Cylinder radius:
25 [mm]
6,25 [mm]
nerve
L = ∫ A ⋅ dl . This technique permits the computation of
tissue
inductances of the special coils we designed to improve
focality. Given the values of L and R, capacitance C is
obtained from requiring that the rise time of the current is
fixed.
Because of the same requirement, we may substitute dI/dt
(the derivative of the current, that comes up in equation
Figure 2. Slinky coil over a cylindrical surface modeling the
hand
First, we considered a set of 5 coils with the same number
of turns (8) and the same radius of the loop (25[mm]).
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1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
The number of turns per loop is 8, for “slinky-1”, 4-4 for
“slinky-2”, 2-4-2 for “slinky-3”; 2-2-2-2 for “slinky-4”
and 1-2-2-2-1 for “slinky-5”. For each one, we computed
the total electric field along the nerve fiber (figure 2). The
results obtained are given in figure 3.
Another issue investigated is the impact that the loops
radius may have on the electromagnetic field distribution
along the nerve fiber and its penetration inside the tissue.
For a point situated exactly under the symmetry center of
a Slinky_3 coil and different values of the radius raza, the
total electric field is plotted in figure 4.
90
Slinky1
Slinky2
Slinky3
Slinky4
Slinky5
80
70
70
50
60
40
30
20
50
40
30
10
0
raza=25mm
raza=15mm
raza=38.1mm
80
E-total [V/m]
E-total [V/m]
60
90
20
0
50
100
150
200
250
z [mm]
300
350
400
450
10
500
Figure 3. Electric field strength induced by a set of 8 turns
(25 mm radius) Slinky coils along the nerve fiber – the
underlined configurations in table 1
0
0
50
100
150
200
250
z [mm]
300
350
400
450
500
a)
140
The focality criterion is given by the ratio between the
primary and secondary peak of the total electric field.
Therefore, focalization requires the maximization of E in
the center of the coil and minimization of the same
quantity along the periphery. Table 1 gives the value of
this ratio for some coils of different configurations and
radii of the turns.
120
E-total [V/m]
100
Table1. Evaluation of focality ratio for a set of Slinky coils
over a cylindrical surface
Coil type
Number of
Turn Inductivity Focality
turns
Radius
[μH]
ratio
(Configuration) [mm]
Slinky_1
8
25
3,9
1
Slinky_2
8 (4-4)
15
25
38,1
25
25
15
25
38,1
25
25
15
25
38,1
25
15
25
38,1
25
25
1,4
2,8
4,9
0,87
5,4
1,2
2,5
4,4
2,4
2,8
1,2
2,2
3,8
2,3
1,1
2,2
3,7
7,4
3,8
2,512
2,19
1,99
2,19
2,19
3,21
2,83
3
2,47
3,57
3,28
2,85
2,76
3,55
4,04
3,5
3,71
3,5
3,71
Slinky_3
Slinky_4
4 (2-2)
12 (6-6)
8 (2-4-2)
8 (3-2-3)
8 (1-6-1)
8 (2-2-2-2)
8 (1-3-3-1)
Slinky_5 8 (1-2-2-2-1)
16 (2-4-4-4-2)
11 (1-3-3-3-1)
raza=25mm
raza=15mm
raza=38.1mm
80
60
40
20
0
0
5
10
15
Depth [mm]
20
25
b)
Figure 4. The electric field induced by a Slinky_3 coil, for
different values of the turn radius a) Along the nerve fiber
and b) For different depths inside the tissue
For the same depth, one can see that a lower value of the
loop radius would lead to a larger electric field induced in
the tissue. On the other hand, the decreasing slope of the
field is more significant for lower values of the radius.
Finally, we changed the configuration of the initial
Slinky_3 coil by modifying the number of turns per loop:
configuration 1 is the initial one, with 2-4-2 turns/loop,
configuration 2 has 3-2-3 turns/loop and configuration 3
has 1-6-1 turns/loop. Figure 5 shows the impact of these
new configurations on the field penetration inside the
tissue.
341
1st International Conference on Advancements of Medicine and Health Care through Technology, MediTech2007,
27-29th September, 2007, Cluj-Napoca, ROMANIA
90
increase for the case of the cylindrical surface (more
than 10 times for Slinky_3, 4 and 5 coils)!
2-4-2
3-2-3
1-6-1
80
70
Table 2. Energetic parameters of a set of Slinky coils
(turn radius 25 mm)
E-total [V/m]
60
Coil Slinky1 Slinky2
L(μH)
3.9
2.8
WJ (J)
181
162.55
WB (J) 105.04 70.51
3.07
ΔT (°C) 3.42
Ipeak (A) 7339.7 7096.9
C (mF)
0.51
0.71
50
40
30
20
10
0
0
50
100
150
200
250 300
z [mm]
350
400
450
500
a)
2-4-2
3-2-3
1-6-1
E-total [V/m]
80
60
40
20
0
0
5
10
15
Depth [mm]
20
4. CONCLUSIONS
The main conclusions arising from the work presented in
this paper are:
•
The focality of a slinky coil improves when the
turns are distributed over a larger number of
directions (the Slinky_5 coil generates the largest
ration between the primary and secondary peaks of
the induced electric field);
•
The total electric field decreases as it penetrates
inside the tissue. This phenomenon is affected by
the radius of the loops for a slinky coil, and there
would also be an optimum configuration of the coil
to minimize this effect;
•
From the point of view of energy transfer from the
coil to the target tissue, the figure of 8 coil is the
most efficient of the set.
One can conclude that the right choice of the turns
dimensions and geometrical disposition has a strong
influence on the magnetic stimulation efficiency, even if
it’s the application that finally establishes the relative
importance of the parameters.
120
100
Slinky3 Slinky4 Slinky5
2.5
2.2
2.2
284.47 261.03 331.4
111.99 92.27 117.14
5.38
4.93
6.27
9465.3 9158.9 10320
0.8
0.9
0.9
25
b)
Figure 5. Electric field induced by a Slinky_3 coil with
different configurations of the turns a) Along the nerve fiber
and b) For different depths inside the tissue
5. REFERENCES
The second configuration seems to show the most rapid
drop of the induced electric field along with the distance
– the worst case, while the initial configuration represents
the best case.
Finally, we compare the energetic efficiency of the set of
8 turns Slinky coils. The coil’s wire has a circular crosssectional area, with 1[mm] radius; the insulation gap
between turns is 0,2[mm] and the target point is situated
16,25[mm] below the center of the coil (there are 10[mm]
between the coil and the tissue, and the target point is
6,25[mm] inside the tissue – figure 2). We require that
the value of the induced electric field in the target point is
equal to 100[V/m].
Table 2 gives the energetic parameters for these coils of
25[mm] radius. One can observe that the Slinky_2 coil is
the most efficient one in the set! If one would compare
the necessary amount of energy required to produce the
same field value in the target point (100[V/m]) for a flat
tissue-air interface [3], one can see a very significant
[1] Nagarajan S., Durand D., Warman E., Effects of Induced
Electric Fields on Finite Neuronal Structures: A
stimulation Study, IEEE Transactions on Biomedical
Engineering, vol. 40, nr. 11, November, 1993;
[2] Roth B.J., Basser P.J., A Model of the Stimulation of a
Nerve Fiber by Electromagnetic Induction, IEEE
Transactions on Biomedical Engineering, vol. 37, nr. 6,
June, 1990;
[3] Cret L., Ciupa R., Remarks on the Optimal Design of Coils
for Magnetic Stimulation, ISEM, Bad Gastein, Austria,
2005.
[4] Ruohonen J., Virtanen J., Ilmoniemi R., Coil Optimisation
for Magnetic Brain Stimulation, Annals of Biomedical
Engineering, Vol 25, 1997;
[5] Lin V., Hsiao I., Dhaka V., Magnetic Coil Design
Considerations for Functional Magnetic Stimulation, IEEE
Transactions on Biomedical Engineering, vol. 47, nr. 5,
May, 2000;
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