Both problems walk you through parts of our paper. I like Problem 24 because it provides a nice example of Faraday’s law of Induction, one of the topics discussed in Chapter 8 (Biomagnetism). Problem 25 could easily have been placed in Chapter 3 (Systems of Many Particles) because of its emphasis on thermal heating and Newton’s law of cooling.Problem 24Suppose one is measuring the EEG when a time-dependent magnetic field is present (such as during magnetic stimulation). The EEG is measured using a disk electrode of radius a = 5 mm and thickness d = 1 mm, made of silver with conductivity σ = 63 × 10^{6}S m^{−1}. The magnetic field is uniform in space, is in a direction perpendicular to the plane of the electrode, and changes from zero to 1 T in 200 μs.

(a) Calculate the electric field and current density in the electrode due to Faraday induction.

(b) The rate of conversion of electrical energy to thermal energy per unit volume (Joule heating) is the product of the current density times the electric field. Calculate the rate of thermal energy production during the time the magnetic field is changing.

(c) Determine the total thermal energy change caused by the change of magnetic field.

(d) The specific heat of silver is 240 J kg^{−1}◦C^{−1}, and the density of silver is 10 500 kg m^{−3}. Determine the temperature increase of the electrode due to Joule heating. The heating of metal electrodes can be a safety hazard during rapid (20 Hz) magnetic stimulation [Roth et al. (1992)].

Problem 25Suppose that during rapid-rate magnetic stimulation, each stimulus pulse causes the temperature of a metal EEG electrode to increase by ΔT (see Prob. 24). The hot electrode then cools exponentially with a time constant τ (typically about 45 s). If N stimulation pulses are delivered starting at t = 0 m with successive pulses separated by a time Δt, then the temperature at the end of the pulse train is T(N,Δt) = ΔT Σ e^{−iΔt/τ}[the sum goes from 0 to N-1]. Find a closed-form expression for T(N,Δt) using the summation formula for the geometric series: 1 + x + x^{2}+ ... + x^{n−1}= (1 − x^{n})/(1 − x). Determine the limiting values of T(N,Δt)for NΔt [much less than] τ and NΔt [much greater than] τ . [See Roth et al. (1992).]

Problem 25 also provides a physical example illustrating the mathematical expression for the summation of a geometric series. If you are not familiar with this sum, it is one that you don’t have to memorize because it is easy to derive. Let S = 1 + x + x

^{2}+ … + x

^{N-1}. If you multiply this expression by x, you get xS = x + x

^{2}+ … + x

^{N}. Now (and this is the clever trick), subtract xS from S, which gives (1 – x) S = 1 – x

^{N}. Note that all the terms x + x

^{2}+ … + x

^{N-1}cancel out! Solving for S gives you the equation in Problem 25. If x is between -1 and 1, and N goes to infinity, you get for the infinite sum S = 1/(1-x).

I should say a few words about my coauthors, who are all leaders in the field of transcranial magnetic stimulation. The project started when Alvaro Pascual-Leone, who had just arrived at NIH, mentioned to me that one patient of his had suffered a burn during rapid-rate magnetic stimulation (see: Pascual-Leone, A., Gates, J.R. and Dhuna, A.K. Induction of Speech arrest and counting errors with rapid transcranial magnetic stimulation. Neurology, 1991a, 41: 697-702.). This motivated Alvaro and I to launch a study using a variety of metal disks made by the NIH machine shop. Alvaro is now the Director of the Berenson-Allen Center for Noninvasive Brain Stimulation. Leo Cohen and Mark Hallett were both studying magnetic stimulation when I arrived at NIH in 1988. I was lucky to start collaborating with them, providing physics expertise to augment their extensive clinical experience. Both continue at the Human Motor Control Section at NIH in Bethesda, Maryland.

And a related but important challenge to obtain simultaneous EEG with functional MR imaging.

ReplyDeleteIt is a very closely related challenge. Transcranial magnetic stimulation and rapidly changing magnetic field gradients result in similar effects. In fact, one factor limiting the size and rate-of-change of gradients in MRI is neural stimulation. I suggest using small stainless steel EEG electrodes.

ReplyDeleteDo you have any references which provide detailed discussion about limits of B-field magnitudes and time dependent gradients in MRI out of concern for neural stimulation? I would be delighted to read these!

ReplyDeleteTry Medical and Biological Engineering and Computing, 27:101-110, 1989, "Peripheral nerve stimulation by induced electric currents: Exposure to time-varying magnetic fields" by J. P. Reilly

ReplyDeletehttp://www.springerlink.com/content/5175726236n4n827/

Not very recent, but a good starting point.

Inverse and Forward Problems, what is lost?: Would you consider discussing these at some point please?

ReplyDeleteI am struggling with the following: Suppose (a nerve is firing at some point inside a cylinder and) you measure the potential at many points on a surface that encloses a volume which contains an oscillating EM field somewhere inside.

Now suppose you have the same surface and volume and you duplicate and set the potential on the surface to the prior measured potential. Would you be able to recreate (to within an integration constant) the E field that was inside the volume?

Thanks for helping me wrap my head around this conundrum.

If the oscillations are small enough to ignore EM waves, then I think I know just how to do this. It is too long to explain in this small comment box. Basically, Laplace's equation allows you to determine the potential (and therefore E) given values on the surface. It even gives you the integration constant.

DeleteI guess the problem with this approach practically is that you can't set the surface potential completely. E.g. if you have a cylinder enclosing the arm, the ends of the cylinder are open and you're not going to be able to set the potentials on the endplate surfaces, and also the granularity with which you might be sampling and then fixing the surface potential may be insufficiently precise to reproduce the original E field

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