THE 8th INTERNATIONAL SYMPOSIUM ON ADVANCED TOPICS IN ELECTRICAL ENGINEERING May 23-25, 2013

Bucharest, Romania

Analysis of the Activation of Spinal Nerves during Magnetic Stimulation of the Lumbar Area

Laura Darabant, Mihaela Cretu, Claudiu Aciu

Technical University of Cluj-Napoca Laura.Darabant@et.utcluj.ro, Mihaela.Cretu@et.utcluj.ro, Claudiu.Aciu@cct.utcluj.ro

Abstract- This paper aims in estimating which nerve fibers can be activated during magnetic stimulation of the lumbar area of the thorax. For this purpose, we compute the electrical field and the activation function induced in the tissue. We used a simplified model, where the spinal cord is modeled as a continuous cylinder, while the vertebral column is also represented by a concentric interrupted cylinder. The general thorax is also modeled as a parallelipipedic domain that includes all these structures. The coil used for magnetic stimulation is a figure of eight, whose center is placed above L1-L2 vertebras.

I. INTRODUCTION

Recent studies in the field of spinal cord injury are investigating new methods, aiming to restore functional motor activities, in paralyzed people. Magnetic stimulation is one non-invasive technique that could directly stimulate the spinal cord, because the electromagnetic field can pass layers of high resistivity, as the vertebral bone. This paper aims to establish if transcutaneous magnetic stimulation is able to directly stimulate the spinal cord by computing the electrical field and the activation function induced in the human body during magnetic stimulation.

II. THEORETICAL BACKGROUND

We adopted a simplified model of the spinal cord, where this one is modeled as a continuous cylinder. The vertebral column is represented by a concentric interrupted cylinder. We considered 6 vertebras of the lumbar area, from T11 to L4. The stimulation coil was placed with its center above L1-L2 vertebras. Beside these components, we also considered the general thorax [1]. Our simulations aimed in resuming real stimulation conditions, for a possible subject, a male of average height (175 cm).

The lumbar area of the thorax is modeled as a parallelepiped, with a considered height of h=220 mm, width L=360 mm, and depth l=240 mm. The model and its geometrical dimensions are given in [2].

According to the electromagnetic field theory, the electric field inside the tissue can be computed by means of the scalar electric potential and the vector magnetic potential [2],[3],[4]:

.43421

321V

AE

E

gradVtAE −∂∂

−= (1)

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 [3].

According to (1), the computation of the electric field due to electromagnetic induction is done by means of the magnetic vector potential [1],[2]:

.4

)(),( 0 ∫⋅⋅

=coil r

dltINtrAπ

μ (2)

where the vector dl represents the differential element of the coil, the vector 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 skin-air, skin-bone and bone-white matter interface. This term is computed knowing that on the surface, the boundary condition to be fulfilled is[1],[2]:

0)11( =⋅

++− niEi

iEi σσ (3) (Continuity of the normal component of the current density vector, valid considering the fact that the regime of the electromagnetic field is quasistatic (f < 1000 Hz) and therefore the time variation of the charge accumulated on the tissue-air boundary is zero). The electric potential inside this domain, V, is numerically evaluated by solving Laplace equation ( )0=ΔV with Neumann boundary conditions inside the tissue. In order to solve this problem we implemented a Matlab routine based on the Finite Difference Method. The system of equations created is solved using Gauss elimination algorithm. For computing the electric field induced in a cylindrical volume conductor, the computation domain is divided into a certain number of points [1],[2].

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 circuit works in an overdamped

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transient state and the condition in this case is: LCL

R 12

2

>⎟⎠⎞

⎜⎝⎛ .

The current waveform through the discharging of a capacitor, with an initial voltage U0, to the coil is [6]: ( ) ( ) .expsinh0 ttLUI αωω −⋅= (4)

where )2/( LR=α , LC/12 −= αω C is the capacitance, and R and L are the resistance and inductance of the coil, respectively.

III. RESULTS

The magnetic coil considered in our first simulation is a figure of eight, with 64 turns, radius 25 mm and the wire’s radius is 1mm. The computed inductance of this coil is L=31.85 μH. Fig. 1 shows the geometry of the problem.

Fig. 1:Geometry of the problem [1]

To compute the electric field induced in the tissue during magnetic stimulation, we developed a Matlab routine based on the theoretical background presented above. The coil is placed directly on the skin, 20 mm away from the vertebra. The electric field was then computed on a "control line" situated 40 mm inside the thorax, inside the spinal cord. The maximum value of the electric current's derivative through the coil was computed for an initial charge of the stimulator's capacitor of 50 V, sA.LUtI μ5710 ==∂∂ .

To validate our routine, we used a professional software, COMSOL, to compute the same quantity using the same geometrical model. To specify the current running through the coil, we used a sinusoidal waveform, with an amplitude Imax=3200 A and a frequency of f=2500 Hz. The maximum value of this current's derivative is the same as the one used by our routine.

Fig. 2 shows the geometry of the COMSOL model and the distribution of the electric field induced in the tissue.

We computed the total electric field induced on the „control line” in fig. 1 that represents a spinal cord nervous pathway using both software packaes. The results are presented and compared in fig. 3. One can see that the results are in good agreement, but the discontinuous electrical properties of the tissues involved in the simulation affect more the Matlab routine results than the ones obtained with COMSOL. This is probably due to the refined mesh created by the professional software, leading to more precise results.

a)

b) Fig.2 COMSOL geometry (a), Electric field induced in the lumbar area during magnetic stimulation with the coil in the position M90 (b)

0,000

0,500

1,000

1,500

0,100 0,150 0,200 0,250 0,300 0,350 0,400

z [m]

Etot

[mVm

m

Fig. 3 Electric field induced along the control line emphasized in fig. 1.

Values computed with two different software tools.

The maximum value of the electric field induced is below the center of the coil. According to Ruohonen [5], for single pulse stimulation, the value of the electric field induced in the tissue that can trigger the activation of a nerve fibber should be between 100 and 150 mV/mm. From fig. 3 one can see that in our case, this value is not large enough to lead to activation of the spinal cord.

Our analytical routine has the advantage of offering some reliable information about the magnitude order of the solution, and therefore validating, in a way, the numerical solution. Considering the good agreement between the results obtained by both methods, we further used the Matlab routine.

A. SIMULATION CASE 1

Since the first results of our simulations showed no succes in activating the spinal cord, we decided to change the input

Control line

20mm Figure of 8 Magnetic coil (64 turns, 25 mm radius)

Spine diameter: 15mm Vertebra diameter: 50mm

20mm

Spinal nerves

rezultate COMSOL

rezultate model propriu

parameters of the coil. We used the specifications of a figure of eight coil, designed by the magnetic stimulators manufacturer, Magstim: diameter 70 mm, 18 turns - 9 on each leaf, and inductivity L= 16.35 μH.

We kept the same value of the initial voltage on the capacitor, U0= 50 V, since it was proved that a nerve fibber located inside a cylindrical body can be activated by using an initial voltage on the capacitor of U0= 35 V [4]. With these parameters, the maximum value of the coil current's derivative is sA.tI μ053=∂∂ , at the beginning of the capacitor's discharge.

The maximum value of the electric field induced on the control line in this case is 1.655 mV/mm, still much too small to induce activation. Therefore, even by considering the errors due to our simplified model, it’s unlikely that we could activate the spinal cord with this technique, since the electric field should be about 100 times larger!

Assuming an other activation mechanism - the activation function zEz ∂∂ [2], we concluded that activation through this mechanism is also impossible since the value of this quantity is always 0 along the considered control line.

Never the less, considering the two possible trajectories of spinal nerves represented in fig. 1, we evaluated the activation function along these two structures and represented them in fig. 4. We concluded that the value of these functions is negative enough to induce activation of both nerves (-0.095 mV/mm2 for the first one and -0.048mV/mm2 for the second).

0 50 100 150 200 250-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

z [mm]

dEz/

dz [

mV

/mm

2 ]

0 50 100 150 200 250-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

z [mm]

dEz/

dz [

mV

/mm

2 ]

Fig. 4 Activation function along the two spinal nerves in fig. 1

By solving the cable equation that characterizes the nerve

behavior, we represented in fig. 5 the response of both spinal nerves. For the first one, with a smaller value of zEz ∂∂ , the latency period is about t=1 ms, while for the second one, the latency is slightly larger, t=2.25 ms.

0 0.5 1 1.5 2 2.5 3-100

-50

0

50

timp (ms)

Vm

(mV

)

1

2

Fig. 5 Variation in time of the action potential during magnetic stimulation of

the two spinal nerves, for the initial position of the coil

We now conclude that magnetic stimulation with the coil in this position is only able to activate spinal nerves, but not the spinal cord.

B. SIMULATION CASE 2

Next, we changed the coil position with respect to the spinal cord as seen in fig. 6.

Fig. 6 Simplified model of the vertebral column and coil position - case 2

We took the same steps as in the first case, and started by evaluating the induced electric field along the control line. The maximum value of this field is Etot =4.35 mV/mm, still not enough to produce activation of the spinal cord. Next, we evaluated the activation function along the same line - fig. 7.

0 50 100 150 200-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

z [mm]

dEz/

dz [m

V/m

m2 ]

Fig. 7 Distribution of zEz ∂∂ along the control line in fig. 1

The minimum value of zEz ∂∂ is -0.02 mV/mm2, not

negative enough to induce activation of the spine. In the next step, we evaluated the activation function

along the two spinal nerves depicted in fig. 6. The minimum negative value of zEz ∂∂ along these structures is -0.05 mV/mm2 – fig. 8, and therefore the bilateral nerves will have an active behavior.

0 50 100 150 200 250-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

z [mm]

dEz/

dz [m

V/m

m2 ]

Fig. 8 Distribution of the activation function along the two nerve fibers in fig.

6 (identical for both nervous pathways)

Fig. 9 emphasizes the transmembrane response for both spinal nerves. After a latency of about 2.2 ms, this potential rises suddenly to a positive value, and then decreases again to its resting value, of -65 mV.

Spinal nerves

0 0.5 1 1.5 2 2.5 3-100

-50

0

50

timp (ms)

Vm

(mV

)

Fig. 9 Variation in time of the action potential during magnetic stimulation

of the spinal nerves, for the second position of the coil C. SIMULATION CASE 3

The final simulation case considers the coil in a lateral position with respect to the spinal cord. The above mentioned steps were followed again, with the coil above the same vertebras, L1-L2.

Fig. 10 Simplified model of the vertebral column and coil position - case 3

We first evaluated the total electric field along the control

line in fig. 1, in order to establish if the spinal cord can be stimulated for this third position of the magnetic coil. The maximum computed value of this quantity is 1.38 mV/mm, still not large enough to cause the activation of the spine.

Next, we assessed the activation function along the same line, now situated below the edge of the coil. One can notice that the variation of the electrical properties of the target tissue (vertebral bone, inter-vertebral space, etc.) strongly influences the shape of the activation function - fig. 11.

0 50 100 150-0.06

-0.04

-0.02

0

0.02

0.04

0.06

z [mm]

dEz/

dz [m

V/m

m2 ]

Fig. 11 Distribution of the activation function along the control line

emphasized in fig. 1

For this case, again, the activation function is unable to trigger the activation of the spine (minimum value is -0.042 mV/mm2).

Never the less, the activation function computed for the spinal nerve in fig. 10 activates this structure (for the first time, the stimulation only activates one nerve fiber, the one corresponding to the lateral position of the coil) - fig. 12.

0 50 100 150 200-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

z [mm]

dEz/

dz [m

V/m

m2 ]

Fig. 12 Distribution of the activation function along the spinal nerve

emphasized in fig. 10

Fig. 13 shows the action potential for the spinal cord (1) – as we stated before, the stimulation applied is not enough to induce activation of this nerve, and for the spinal nerve (2) in fig. 11 - activated after a latency period of about 1.1 ms.

Fig. 13 Variation in time of the action potential during magnetic stimulation

of the spine (1) and of a spinal nerve (2), for the third position of the coil

IV. CONCLUSIONS

Even considering the limitations of our simplified model, the simulation results emphasized that the spinal cord could not have been stimulated with the given conditions, because the amplitude of the induced electric field in the spine is two magnitude orders too small to achieve activation of a nerve fiber. Furthermore, the second activation mechanism, the activation function, was also unable to stimulate the spinal cord. Therefore, we showed that only the spinal nerves, situated bilaterally in first and second simulation case, and only on the same side as the magnetic coil for the third case, were activated by the activation function.

REFERENCES [1] L. Darabant, M. Cretu, D.D. Micu, R. Ciupa, and D. Stet, “Assessment

of the electric field induced in the human tissue during magnetic stimulation of the spinal cord,” COMPEL, vol. 31, pp. 1164–1172, 2012.

[2] L. Darabant, M. Cretu, A. Darabant, “Magnetic Stimulation of the Spinal Cord: Experimental Results and Simulations,” IEEE Trans on Magnetics, vol. 49, 2013, in press.

[3] D.D. Micu, L. Czumbil, G. Christoforidis, A. Ceclan and D. Stet, “Evaluation of Induced AC Voltages in Underground Metallic Pipeline,” COMPEL, Vol. 31, pp.1133-1143, 2012.

[4] B.J. Roth, P.J. Basser, “A Model of the Stimulation of a Nerve Fiber by Electromagnetic Induction”, IEEE Transactions on Biomedical Engineering, vol. 37, nr. 6, June, 1990, pp. 588-597

[5] J. Ruohonen, J. Virtanen, “Coil optimization for magnetic brain stimulation,” Annals of Biomed. Eng., vol. 25, 1997.

[6] D.D Micu, L. Czumbil, A. Polycarpou, A. Ceclan and D. Şteţ, “Analysis of electromagnetic interference problems through an innovative Monte Carlo-neural network method,” Proceed. of MedPower, November 2010.

Spinal nerve