Effect of Contour Shape of Nervous System Electromagnetic Stimulation Coils on the Induced Electrical Field Distribution

2.1. Theoretical background

A natural criterion for optimal coil design would be the electrical field induced in the region of interest. It is defined by:

where

is the induced field and is the magnetic vector potential obtained from the coil current pulses.

Electromagnetic stimulation processes could be studied using a more general relation:

where the electric potential component is added. However, it is usually neglected due to its relatively low value and hence weak influence. For a given coil current i(t) and contour shape (S), the magnetic vector-potential is defined by the known relation:

For the induced field and a plane contour placed in the plane XOY (Fig. 1),

The necessary and sufficient conditions for finding the extremum of Eq. (4) should be defined, for a known time-course of the current i(t). Thus, the induced field at point M (Fig. 1) will be at its maximum [16–18].

The function y(x) has to be found by solving the relation:

There are reasons to assume that a real solution of the above problem exists, and that it respects the condition of sufficiency.

The conditions for existence of extremum of Eq.(4), concerning y(x), can be defined by the function:

The conversion of Eq. (6) to a standard form is shown in Appendix A, which contains the succession of transformations leading to the necessary conditions for existence of extremum, respectively, to the Euler equations:

This system yields a stable numeric solution as a Cauchy problem, for a Lagrange constant value of λ = 1.5 and boundary conditions regarding the quantities x ₁ (t)_{t = 0} = 0.0025; x ₂ (t)_{t = 0} = 0.1; x'₁ (t)_{t = 0} = 4; x'₂ (t)_{t = 0} = 400. The extremal appears as shown in Fig. 2.

Analyzing the obtained solution, it should be taken into account that the axes scaling in Fig. 2 is influenced by the canceling of the multipliers to the functions of Eqs. (3) and (4).

A solution could be attempted not only as a Cauchy type, but also as a Sturm-Liouville problem, with the corresponding fundamental difficulties.

The extremal y(x) of Fig. 2 is considered in the class of symmetric curves with respect to OX, which is natural for this type of problem. The solution of the Cauchy problem in the class of closed curves creates difficulties with the use of numerical procedures. This is due do reaching, generally, points of unstable solution, where the first derivative is interrupted. Thus, it becomes necessary to look for the extremal in parametric form.

In addition, it can be shown that the necessary conditions considered are also sufficient (see Appendix B for details).

A. Necessary conditions for extremum

Conversion of Eq. (6) to a standard form can be accomplished as follows. The extremum should be investigated with the squared vector function

Then the condition for the existence of an extremum should be defined on the basis of :

It is not difficult to see that the extremum of (A1) is the sum of the extrema of its two squared components.

The theory of the calculus of variations makes clear that the extremum of a nonstandard function of the type F(y(x)) = F ₁(y(x))F ₂ (y(x)) is to be found on the basis of the linear combination:

G(y(x)) = F ₁ (y(x)) + λ₀ F ₂ (y(x)),     (A2)

where λ₀ is to be obtained from the ratio:

and y _extr (x) is defined using the conditions for extremum.

The functionals in (A1):

can be represented as a product of two equal functions. Then, in the two cases of U ₂ (y(x)) and U ₃ (y(x)) a trivial value for λ₀; λ₀ = 1 renders the problem for extremum of the sum of the squares of the functions to an extremum of the sum of their power of one and in a sense – to the sum of the extrema of their moduluses.

The integrals in (A1) should be transformed in parametric form with arguments t:

x = x ₁ (t); y = x ₂ (t)

Introducing also the condition for a constant perimeter, the calculus of variations problem is presented in a near standard form – the extremum of :

is to be found, having in view the limitation:

The necessary conditions for the existence of an extremum are defined using the Lagrange function:

where the Lagrange coefficient λ in this case is a constant, to be found according to the necessary conditions for extremum and the limitation (A5).

The necessary conditions for extremum can be defined by the Euler system of equations:

with boundary conditions:

It is known that the equations in this system are in linear relation. In this case, a calibration equation is recommended in the following form:

For this specific problem, the limitation condition (A5) can be taken as calibration equation.

Lagrange function (A6) leads to a system of differential equations in parametric form:

with boundary conditions (A9).

B. Sufficiency of the necessary conditions

For the second partial derivatives

and , taking into account Eq (9), we obtain:

and

It is obvious that (B-1) and (B-2) can have positive and negative values, which allows the conditions of Legendre to be met:

; < 0

• a condition for maximum, specifically a weak maximum, which is achievable with curves having zero order proximity (proximity to ordinates):

; > 0 - a condition for minimum

1. 4.

   It can be verified, based on Eq. (9), that the third partial derivative:

    

exists, which is also a part of the sufficiency conditions.

C. Field induced by a system of excitation coils

A relatively arbitrarily shaped contour L is considered (Fig. 1) with a current i(t) that excites an induced electrical field (x, y, z). The field is obtained from Eq. (4), or:

where

is a vector linear element of the integration contour (L) and r is the distance from the element to the point where (x, y, z) is computed.

Difficulties also appear for cases of contours in a relatively general position with respect to the coordinate axes [13]. Such difficulties are not insurmountable, but should be dealt with depending on the specific problem in view.

The present investigation relates to fields induced by flat contours approximated by n linear segments (Fig. 3) located in the XOY and YOZ planes. Applying the calculus of variation approach, such an approximation is justified. The numerical solution (Fig. 2) suggests that any other type of approximation would involve computational difficulties without major impact on the practical results. The solution accuracy depends, of course, on the value of n. For the triangular and square shapes, n is defined and does not affect the accuracy.

For a system of k contours, the resulting induced field is obtained by superposition, based on the linear relation between (x, y, z) and di / dt, as evident from Eq. C1. In this case, three coils (L ₁),(L ₂) and (L ₃) are involved, where (L ₁) and (L ₂) are symmetrically located in the XOY plane, and (L ₃) is in the YOZ plane, symmetrically oriented with respect to the negative Z-axis. The region to be stimulated is located in the positive half-space defined by XOY and the positive Z-axis.

(x, y, z) is to be obtained by solving the contour integrals along (L ₁), (L ₂) and (L ₃):

After decomposition of

over the respective axes, the following relation is obtained:

For the contours approximated by n segments (Fig. 3), sums of integrals are obtained for (L ₁) and (L ₂) respectively:

and for (L ₃):

The segments l _j are introduced in the contour integrals by the corresponding linear relations y = m _jk x + n _jk , k = 1, 2 ; j = 1,2,...,n in the XOY plane and z = m _j y + n _j in the YOZ plane.

The segments l _j are introduced in the contour integrals by the corresponding linear relations y = m _jk x + n _jk , k = 1, 2 ; j = 1,2,...,n in the XOY plane and z = m _j y + n _j in the YOZ plane.