DESIGN OF COILS FOR MAGNETIC NEURAL STIMULATION.

EFFICIENCY CRITERIA AND TECHNICAL SOLUTIONS

 

 

Mihaela Morega

POLITEHNICA University of Bucharest

e-mail:morega@amotion.pub.ro

 

 

OBJECTIVE

 

MAGNETIC STIMULATION - medical technique applied for neuro-muscular rehabilitation, therapy and stimulation tests.

 

The efficiency of the medical procedure can be enhanced through an adequate design of the electrical equipment.

 

 

METHOD           

 
The MAGNETIC STIMULATION procedure

 

Ø     non-invasive, non-contact and painless alternative to the ELECTRIC STIMULATION (through implanted electrodes)

 

Ø     applied on the TARGET CELLULAR TISSUE (in this model: the peripheral nervous system and spinal cord)

 

Ø     consists of inducing electric field in relatively good conducting tissues

 

 

It is considered the equivalence between:

 

MAGNETIC STIMULATION

Û

ELECTRIC STIMULATION

the ACTIVATING FUNCTION (AF) created through electromagnetic induction

the ELECTRIC CURRENT injected through electrodes in a SPECIFIC ELECTRIC RESISTANCE (of the tissue)

 

                        ß

ACTIVATING FUNCTION

(AF)

the spatial derivative of the electric field along the cellular fiber

(axon, or muscular long cylindrical cell)

 

TECHNICAL PROCEDURE

 

The electric field is produced through electromagnetic induction in the area of the TARGET CELLULAR TISSUE.

 

The inductor variable magnetic field is generated at the skin surface, by a current-carrying winding, the STIMULATING COIL.

 

 

SCOPE OF THE STUDY

 

Different forms of the stimulating coils are compared with regard to several efficiency criteria in order to analyze the magnitude and spatial distribution of AF and to find the best geometric configuration of the coil.

 

Efficiency criteria:

q       The AF peak value is maximized and focused on the target area

q       The induced electric field strength is maximized at a specific target area inside the body and minimized at the skin level

 

MATHEMATICAL MODEL

 

DOMAIN

 

COORDINATE SYSTEM: (x,y,z) Cartesian coordinate system

 

SIMPLIFIED MODEL OF ANATOMICAL TISSUE:

·        a straight nerve bundle embedded in a homogeneous conductive material;

·        the characteristic dimensions of anatomical domains are larger than the dimensions of the coils currently used.

·        (xOy, z=0) plane is considered the skin surface,

·        the TARGET CELLULAR TISUUE is a long fiber, parallel to (Ox) axis, embedded in (xOy z<0) half-space.

 

STIMULATING COIL: has no restriction to the shape and is placed above the

(xOy, z=0) plane

 

MEDIUM PROPERTIES:

(xOy, z<0) half-space is a homogeneous conductive medium (s @ 1 S/m)

(xOy, z>0) half-space is air

 

 

REGIME

 

Quasistatic regime – the frequency of the excitation current (f < 1kHz) and the usual tissues conductivity lead to a penetration depth much larger than the characteristic dimensions of anatomical domains.

 

ELECTRIC FIELD SOLUTION

 

Analytical solution originally derived by Esselle and Stuchly [Esselle K., Stuchly Maria, Neural Stimulation with Magnetic Fields: Analysis of Induced Electric Fields, IEEE Trans. on BME, vol. 39, no. 7, 1992, p. 693-700] and cited also in other works.

 

Electric field components produced at a specific location P(x,y,z), by the current element (idl) of the winding, which is situated at (x0,y0,z0):

 

,  ,  dEz = 0.

 

and the expression of the elementary AF,

 

,

 

 is the distance between (x0,y0,z0) and (x,y,z);

 is its projection on the (xOy, z = 0) plane;

m0 is the magnetic permeability of air;

 is the time derivative of the stimulating current.

 

INTEGRATION TECHNIQUE – CHANGE OF VARIABLES

 

The contribution of the total amperturns of a certain winding to the electric field solution results by integrating the equations above.

 

Change of variables, convenient for coils with circular turns

 

The elementary turn   

* is circular (of radius r)

* can rotate by an angle a

 (maintaining the fix point A)

 

 

 

APPLICATION 1     FOCALIZATION CRITERIUM

 

The basic STIMULATING COIL: 

4 circular turns, of radius r = 0.02 m, and

TARGET CELLULAR TISSUE:

the stimulated fiber is embedded within the tissue, at z = - 0.01 m

RESULTS: negative peaks of AF produce stimulation (cellular depolarization),

while positive peaks of AF produce inhibition (cellular hiperpolarization)

 

 

 

1 ´ 4 turns

F @ 12.5×10-6 m2

ER = 1

(a)     concentrated coil

 

 

 

2 ´ 2 turns

F @ 6.25×10-6 m2

ER = 1

(b) double coil (figure of 8 shape)

 

 

 

4 ´ 1 turns

F @ 6.25×10-6 m2

ER = 2.08

(c) quadruple coil (flower like shape)

 

geometry of the coils

repartition of AF in the

(xOy, z = -0.01m) plane

repartition of AF along the fiber (on the (Ox) direction, at y = 0 and z = -0.01 m)

 

 

CONCLUSION: The quadruple coil has the best performances and is the easiest in use because the AF peak corresponds to its center of symmetry.

 

 

APPLICATION 2   ELECTRIC FIELD CONTROL IN DEPTH OF TISSUE

 

Distribution of the AF in depth of tissue for three types of the stimulating coils

 

 

 

 

 (a) – quadruple horizontal coil

AF1

 

 

 (b) – combination of a quadruple horizontal coil and a quadruple coil with  inclined turns

(a = p/3)

AF2 

 

 

 (c) – combination of a quadruple horizontal coil and a quadruple coil with  inclined turns; the turns have double radius compared with case (a) and (b)

AF3 

 

 

 

 

CONCLUSION: The inclination and radius of turns could provide a good control for the localization of the AF values in depth of the tissue.

 

AF at z = 0 rated by AF at z = -0.01 m (the considered target area) is aprox.

2.5 in case (a)

1.8 in case (b)

1.15 in case (c)

CONCLUSIONS

q       Several forms of stimulating coils were used in MAGNETIC STIMULATION of the excitable tissue (nerve and muscle long fibers), in order to improve the distribution of the ACTIVATING FUNCTION (AF).

q       The quadruple coils produce the best concentration of the AF at the TARGET CELLULAR TISSUE, while minimizing the inhibitory effects on surrounding regions.

q       The combination of quadruple coils with horizontal and inclined turns may result into the control of the electric field strength magnitude and of the AF in depth of the stimulated tissue.  It may also concentrate the peak values of the AF at the fiber depth level, rather than at the skin surface.

 

 

FURTHER STUDY

q       The refinement of the coils forms by considering quadruple coils with eccentric turns. This geometry requires a slight modification in the geometric description of the circular turns.

q       The computation of the inductance for each specific coil shape; the spatial distribution of turns influences the inductance and, consequently, the waveform of the stimulating current.