Model and Computer Simulation for Precision Contactless Analgesia

 

 

Authors: J. Manne, S. Zou

 

Introduction

The object of this work was to develop a mathematical model and subsequent computer simulation for a technology that can provide neural blockade in a contactless form. It is known from earlier work that providing an electrode proximate to a neuron will create an electrical field that can either hyperpolarize or depolarize a neuron.¹ The object of this study was to demonstrate the creation of a similar external electrical field with a time varying magnetic field to block action potentials transmitting pain signals to the brain. In order to block the action potential from propagating, the interior of the cell must be sufficiently hyperpolarized so that the action potential with positive 40 mv depolarization cannot bring the cell to -60mv. Thus the interior should be hyperpolarized to greater than -100mv. Therefore the hyperpolarization produced by the external magnetic flux should be -30 mv or more. This then would cause a nerve block.

 

 

 

Materials and Methods

 

The model for electrical conduction in the neuron which

has been widely accepted is the modified cable equation:

 

 

 

where:

V represents voltage,

The gradient of the electric field on the right side of the equation is referred to as the

activating function by Rattay. It activated neuron using external electrodes.In order to

calculate the the total transmembrane voltage the Hodgkin-Huxley equation can be used.

 

 

The mathematical model was constructed to model an electrical coil connected into an RLC circuit with a time varying current in the coil. The coil is placed in a xy plane parallel to a plane localizing a peripheral nerve. The time varying current leads toeither an expanding or contracting magnetic field creating a secondary electrical field.

eqn.1

E = electrical field

A= magnetic vector potential leading to magnetic field

 

 

For the case of the coil with a time varying current, such as a coil in the present study, the equation for the induced magnetic vector potential has been described: ²

 

 

 

eqn.3

 

 

the equation simplifies to :

 

eqn.4

 

 

where k is defined by:

 

 

 

E and K are the elliptic integrals of this variable k

 

then the expression for the magnetic vector potential is used for the electric field along the

axis of the axon 12 (y component Figure 1)

 

 

The axial electric field gradient is the source term in the modified cable equation (Equation 1)

for nerve conduction.

 

 

There have been many publications describing the solution of the modified cable equation ³

 

In this method the nerve is divided into N compartments and modeled as a lumped circuit.⁴

The axial current in each compartment is secondary to two factors. The first is the voltage

gradient along the axon. The second is the extrinsically induced electric field from the

externally fluxing magnetic field. The current can be related to these electric potential terms

by Ohm's Law is applied to all nodes such that each node is successively treated as the

center node with the exception of the two terminal nodes:

 

 

 

 

Where :n= 2,3,4,...L-1

dl= the internodal distance

L = the total number of nodes in the nerve segment of interest

dx= length increment along axon

Gm= membrane conductance

Ex is the axial component of the magnetically induced electrical field

 

 

Thus there are L equations in L unknowns. The unknowns are the externally induced

transmembrane potentials located in the vector V =(V1,V2,V3,..Vl). The internodal potential

differences are due to the external electric field: E={E1,E2,E3,E4....EL). The equations are

expressed as the matrix product of the matrix: A which contains the coefficients for V1,V2...

and the product of the matrix: B which contains the coefficients of E1,E2,E3....leading to the

following equation:

 

 

 

Then V can be solved for:

 

 

V= transmembrane voltage

 

 

A computer program was written to run in the Octave environment(Linux analogue to

MATLAB)on a Intel corei5 motherboard running LINUX OS UBUNTU 20.04. to calculate the

externally induced transmembrane potentials in the subthreshold condition.

 

A computer program was written to calculate the desired externally induced transmembrane

potential when the coil and circuit characteristics are provided.

 

 

Results

 

Coil and Circuit Specifications

rc=radius of coil = 1 cm

rw=radius of wire = 1mm

Resistance R=0.21 ohms

Capacitance C= 0.0072 Farad

Inductance of coil = L= 9e-5 Henry

Voltage=Vo=700 volts

z0= height is coil above axon = 3 cm

time = 60 microseconds

 

FIGURE 1 and Figure 2. show the results of the calculation of induced electric field and

externally induced transmembrane potential, respectively, along the length and cross section

of the axon using the preceding coil and circuit values.

 

The time is 60 microseconds from the application of 100 volts dc across the circuit. The

surface plot of the induced electrical field in neuronal tissue in a plane 3 cm below the coil.

The x and y axes represent all points in that plane.

 

 

 

Discussion

From examination of the graph in Figure 2 it can be seen that there is a maxima and a

minima in the electric field. The minima correspond to hyperpolarized points. To block

propagation of neural impulses these points have to have an induced transmembrane

potential more negative than -30 mv. In that case the net transmembrane potential at these

points will be greater than -100 mv and thus cannot be depolarized by an advancing action

potential.

 

As the minima get closer and closer to

-30 mv the range of x values(along the cross section of the nerve),for which the induced

transmembrane potential is sufficient to cause neural blockade can be made as small as

desired providing a focussed target for analgesia.

 

 

 

 

 

 

 

 

 

 

 

 

 

References:

 

 

1. Rattay F, J Theor. Biol (1987) 125,339-349

 

 

2. Jackson JD. Classical Electrodynamics. 1962, New York

 

3. Nagarajan SS, Durand DM and Warman EN. Effects of Induced Electric Fields on Finite Neuronal

Structures: A Simulation Study. Transactions on Biomedical Engineering 40(11),pgs;1175-1187),1993

 

4. Segev I Fleshman W and Burke RE, "Compartmental Models of Complex Neurons" Methods of

Neuronal Modeling: From Synapses to

Networks, Koch C and Segev I, Eds. Cambridge, MA; MIT Press, 1989, pgs:63-97