The Electric Field Induced During Magnetic Stimulation

Chapter 8 of Intermediate Physics for Medicine and Biology discusses electromagnetic induction and magnetic stimulation of nerves. It doesn't, however, explain how to calculate the electric field. You can learn how to do this from my article “The Electric Field Induced During Magnetic Stimulation” (Electroencephalography and Clinical Neurophysiology, Supplement 43, Pages 268-278, 1991). It begins:

“The Electric Field Induced
During Magnetic Stimulation.”

Magnetic stimulation has been studied widely since its use in 1982 for stimulation of peripheral nerves (Polson et al. 1982), and in 1985 for stimulation of the cortex (Barker et al. 1985). The technique consists of inducing current in the body by Faraday’s law of induction: a time-dependent magnetic field produces an electric field. The transient magnetic field is created by discharging a capacitor through a coil held near the target neuron. Magnetic stimulation has potential clinical applications for the diagnosis of central nervous system disorders such as multiple sclerosis, and for monitoring the corticospinal tract during spinal cord surgery (for review, see Hallett and Cohen 1989). When activating the cortex transcranially, magnetic stimulation is less painful than electrical stimulation.

Appendix 1.

Although there have been many clinical studies of magnetic stimulation, until recently there have been few attempts to measure or calculate the electric field distribution induced in tissue. However, knowledge of the electric field is important for determining where stimulation occurs, how localized the stimulated region is, and what the relative efficacy of different coil designs is. In this paper, the electric field induced in tissue during magnetic stimulation is calculated, and results are presented for stimulation of both the peripheral and central nervous systems.

In Appendix 1 of this article, I derived an expression for the electric field E at position r, starting from

where N is the number of turns in the coil, μ₀ is the permeability of free space (4π × 10^-7 H/m), I is the coil current, r' is the position along the coil, and the integral of dl' is over the coil path. For all but the simplest of coil shapes this integral can't be evaluated analytically, so I used a trick: approximate the coil as a polygon. A twelve-sided polygon looks a lot like a circular coil. You can make the approximation even better by using more sides.

A circular coil (black) approximated by
a 12-sided polygon (red).

With this method I needed to calculate the electric field only from line segments. The calculation for one line segment is summarized in Figure 6 of the paper.

Figure 6 from “The Electric Field
Induced During Magnetic Stimulation.”

I will present the calculation as a new homework problem for IPMB. (Warning: t has two meanings in this problem: it denotes time and is also a dimensionless parameter specifying location along the line segment.)

Section 8.7

Problem 32 ½. Calculate the integral

for a line segment extending from x₂ to x₁. Define δ = x₁ - x₂ and R = r - ½(x₁ + x₂).

(a) Interpret δ and R physically.

(b) Define t as a dimensionless parameter ranging from -½ to ½. Show that r' equals r – R – tδ.

(c) Show that the integral becomes

(d) Evaluate this integral. You may need a table of integrals.

(e) Express the integral in terms of δ, R, and φ (the angle between R and δ).

The resulting expression for the electric field is Equation 15 in the article

Equation (15) in “The Electric Field Induced During Magnetic Stimulation.”

The photograph below shows the preliminary result in my research notebook from when I worked at the National Institutes of Health. I didn't save the reams of scrap paper needed to derive this result.

The November 10, 1988 entry
in my research notebook.

To determine the ends of the line segments, I took an x-ray of a coil and digitized points on it. Below are coordinates for a figure-of-eight coil, often used during magnetic stimulation. The method was low-tech and imprecise, but it worked.

The November 17, 1988 entry
in my research notebook.

Ten comments:

• My coauthors were Leo Cohen and Mark Hallett, two neurologists at NIH. I recommend their four-page paper “Magnetism: A New Method for Stimulation of Nerve and Brain.”
• The calculation above gives the electric field in an unbounded, homogeneous tissue. The article also analyzes the effect of tissue boundaries on the electric field.
• The integral is dimensionless. “For distances from the coil that are similar to the coil size, this integral is approximately equal to one, so a rule of thumb for determining the order of magnitude of E is 0.1 N dI/dt, where dI/dt has units of A/μsec and E is in V/m.”
• The inverse hyperbolic sine can be expressed in terms of logarithms: sinh^-1z = ln[z + √(z² + 1)]. If you're uncomfortable with hyperbolic functions, perhaps logarithms are more to your taste. 
  
• This supplement to Electroencephalography and Clinical Neurophysiology contained papers from the International Motor Evoked Potential Symposium, held in Chicago in August 1989. This excellent meeting guided my subsequent research into magnetic stimulation. The supplement was published as a book: Magnetic Motor Stimulation: Principles and Clinical Experience, edited by Walter Levy, Roger Cracco, Tony Barker, and John Rothwell. 
• Leo Cohen was first author on a clinical paper published in the same supplement: Cohen, Bandinelli, Topka, Fuhr, Roth, and Hallett (1991) “Topographic Maps of Human Motor Cortex in Normal and Pathological Conditions: Mirror Movements, Amputations and Spinal Cord Injuries.”
• To be successful in science you must be in the right place at the right time. I was lucky to arrive at NIH as a young physicist in 1988—soon after magnetic stimulation was invented—and to have two neurologists using the new technique on their patients and looking for a collaborator to calculate electric fields.
• A week after deriving the expression for the electric field, I found a similar expression for the magnetic field. It was never published. Let me know if you need it.
• If you look up my article, please forgive the incorrect units for μ₀ given in the Appendix. They should be Henry/meter, not Farad/meter. In my defense, I had it correct in the body of the article. 
• Correspondence about the article was to be sent to “Bradley J. Roth, Building 13, Room 3W13, National Institutes of Health, Bethesda, MD 20892.” This was my office when I worked at the NIH intramural program between 1988 and 1995. I loved working at NIH as part of the Biomedical Engineering and Instrumentation Program, which consisted of physicists, mathematicians and engineers who collaborated with the medical doctors and biologists. Cohen and Hallett had their laboratory in the NIH Clinical Center (Building 10), and were part of the National Institute of Neurological Disorders and Stroke. Hallett once told me he began his undergraduate education as a physics major, but switched to medicine after one of his professors tried to explain how magnetic fields are related to electric fields in special relativity.

A map of the National Institutes of Health campus
in Bethesda, Maryland.