The Development of Transcranial Magnetic Stimulation

When I worked at the National Institutes of Health, I studied transcranial magnetic stimulation. In Chapter 8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe this technique to activate neurons in the brain.

Since a changing magnetic field generates an induced electric field, it is possible to stimulate nerve or muscle cells without using electrodes. The advantage is that for a given induced current deep within the brain, the currents in the scalp that are induced by the magnetic field are far less than the currents that would be required for electrical stimulation. Therefore transcranial magnetic stimulation (TMS) is relatively painless.

The method was invented in 1985 and when I arrived at NIH in 1988 the field was new and ripe for analysis. I spent the next seven years calculating electric fields in the brain and determining how the electric field couples to a nerve.

Roth, B. J. (2022) The Development of
Transcranial Magnetic Stimulation,
BOHR International Journal of
Neurology and Neuroscience,
Volume 1, Pages 8–20.

Recently, I wrote a review article telling the story of how transcranial magnetic stimulation began. You can get a copy at https://journals.bohrpub.com/index.php/bijnn/article/view/28; it is an open access article so everyone is free to download it. The abstract states

This review describes the development of transcranial magnetic stimulation in 1985 and the research related to this technique over the following 10 years. It not only focuses on work done at the National Institutes of Health but provides a survey of other related research as well. Key topics are the calculation of the electric field produced during magnetic stimulation, the interaction of this electric field with a long nerve axon, coil design, the time course of the magnetic stimulation pulse, and the safety of magnetic stimulation.

Readers of this blog will recognize some of the topics from earlier posts, such as the calculation of the induced electric field, determining the site of stimulation along a peripheral nerve, Paul Maccabee’s wonderful article, the four-leaf coil, the heating of metal electrodes, implantable microcoils, and Tony Barker's online interview. You could almost say I pre-wrote much of the review using this blog as my test bed. 

I like magnetic stimulation because it's a classic example of how a fundamental concept from physics can have a major impact in biology and medicine. If you combine this review of transcranial magnetic stimulation together with my earlier review of the bidomain model of cardiac tissue, you get a pretty good summary of my most important research.

Enjoy!

Think Before You Calculate!

I encourage students to build their qualitative problem solving skills by recasting equations in dimensionless variables, analyzing the limiting behavior of mathematical expressions, and sketching plots showing how functions behave. “Think Before You Calculate!” is my mantra. But how, specifically, do you do this? Let me show you an example.

Fig. 2.16 from IPMB. A plot of the solution
of the logistic equation when y₀ = 0.1,
y_∞ = 1.0, b₀ = 0.0667. Exponential
growth with the same values of
y₀ and b₀ is also shown.

In Section 2.10 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the logistic model.

Sometimes a growing population will level off at some constant value. Other times the population will grow and then crash. One model that exhibits leveling off is the logistic model, described by the differential equation

dy/dt = b₀ y (1 – y/y_∞) ,                           (2.28)

where b₀ and y_∞ are constants….

If the initial value of y is y₀, the solution of Eq. 2.28 is

y(t) = 1 / [1/y_∞ + (1/y₀ – 1/y_∞) e^−b₀t] .    (2.29)

Below is a new homework problem, analyzing the logistic equation in a way to build insight. Consider it an early Christmas present. Santa won’t give you the answer, so you need to solve the problem yourself to gain anything from this post.

Section 2.10

Problem 36 ½. Consider the logistic model.

(a) Write Eq. 2.28 in terms of dimensionless variables Y and T, where Y = y/y_∞ and T = b₀t.

(b) Express the solution Eq. 2.29 in terms of Y, T, and Y₀ = y₀/y_∞.

(c) Verify that your solution in part (b) obeys the differential equation you derive in part (a).

(d) Verify that your solution in part (b) is equal to Y₀ at T = 0.

(e) In a plot of Y(T) versus T, which of the three constants (y_∞, y₀, and b₀) affect the qualitative shape of the solution, and which just scale the Y and T axes? 

(f) Verify that your solution in part (b) approaches 1 as T goes to infinity.

(g) Find an expression for the slope of the curve Y = Y(T). What is the slope at time T = 0? For what value of Y₀ is the initial slope largest? For what values of Y₀ is the slope small?

(h) The plot in Fig. 2.16 compares the solution of logistic equation with the exponential Y = Y₀ e^T. The figure gives the impression that the exponential is a good approximation to the logistic curve at small times. Do the two curves have the same value at T = 0? Do the two curves have the same slope at T = 0?

(i) Sketch plots of Y versus T for Y₀ = 0.0001, 0.001, 0.01, and 0.1.

(j) Rewrite the solution from part (b), Y = Y(T), using the constant T₀, where T₀ = ln[(1−Y₀)/Y₀]. Show that varying Y₀ is equivalent to shifting the solution along the T axis. What value of Y₀ corresponds to T₀ = 0?

(k) How does the logistic curve behave if Y₀ > 1? Sketch a plot of Y versus T for Y₀ =1.5.

(l) How does the logistic curve behave if Y₀ < 0? Sketch a plot of Y versus T for Y₀ = –0.5.

(m) Plot Y versus T for Y₀ = 0.1 on semilog graph paper.

If you solve this new homework problem and want to compare you solution to mine, email me at roth@oakland.edu and I’ll send you my solution. 

The Logistic Equation, MIT OpenCourseWare

https://www.youtube.com/watch?v=TCkLSYxx21c&t=69s

Mark Hallett Festschrift

Mark Hallett

Last Monday I attended (over the internet) a Festschrift to honor the retirement of Mark Hallett from the intramural program at the National Institutes of Health. Russ Hobbie and I cite Hallett in Chapter 8 of Intermediate Physics for Medicine and Biology.

Magnetic stimulation can be used to diagnose central nervous system diseases that slow the conduction velocity in motor nerves without changing the conduction velocity in sensory nerves (Hallett and Cohen 1989).

The reference is to a wonderful paper that Hallett and Leo Cohen wrote in the Journal of the American Medical Association.

Hallett M, Cohen LG (1989) Magnetism: A new method for stimulation of nerve and brain. JAMA 262:538–541.

Hallett came to NIH in 1984 and worked there for almost 40 years. I collaborated with him in the early 1990s, when I was working in NIH’s Biomedical Engineering and Instrumentation Program. My role was to calculate the electric field induced in the brain during transcranial magnetic simulation.

For a long time, Hallett was the clinical director for the National Institute of Neurological Disorders and Stroke intramural program. According to Google Scholar, his papers are cited about ten times every day, and his h-index is over 100, meaning he has published over 100 papers that have each been cited over 100 times. He has had an enormous impact on the field of neurophysiology. In particular, he trained an amazing number of young scholars who have gone on to be leaders in the field themselves, many of who spoke at the event.

Mark Hallett

Below is Hallett’s biography found on his NIH webpage.

Dr. Hallett obtained his A.B. and M.D. at Harvard University, had his internship in Medicine at the Peter Bent Brigham Hospital and his Neurology training at Massachusetts General Hospital. He had fellowships in neurophysiology at the NIH and in the Department of Neurology, Institute of Psychiatry in London, where he worked with C. David Marsden. Before coming to NIH in 1984, Dr. Hallett was the Chief of the Clinical Neurophysiology Laboratory at the Brigham and Women's Hospital in Boston and progressed to Associate Professor of Neurology at Harvard Medical School. He is currently Chief of the Medical Neurology Branch and Chief of its Human Motor Control Section. He is now Past-President of the International Federation of Clinical Neurophysiology. He has been President of the International Parkinson and Movement Disorder Society and Vice-President of the American Academy of Neurology. He served as Editor in Chief of Clinical Neurophysiology. Among many awards, in 2012 he became an Honorary Member of the American Neurological Association, and in 2014 won the Lifetime Achievement Award of the American Association of Neuromuscular and Electrodiagnostic Medicine. In 2017 he received the degree of Doctor of Medicine Honoris Causa from the University of Hamburg, and in 2018 was made an Honorary Member of the European Academy of Neurology. His research activities focus on the physiology of human voluntary movement and its pathophysiology in disordered voluntary movement and involuntary movement.

Once Hallett told me that he started out college studying physics, but when his instructor explained to his class that a magnetic field is just a relativistic effect of an electric field (see Problem 5 in Chapter 8 of IPMB) he switched to a premed program! 

At the end of his Festschrift, Hallett spoke and honored his many mentors. His final words were "I will be retiring, but not too much."

Enjoy your retirement, Mark Hallett, but not too much. Working with you was a delight.

https://www.youtube.com/watch?v=eR-D9bLWKhQ 

Oral History 2013: Stanley Fahn Interviews Mark Hallett

Surface Tension

Air and Water,
by Mark Denny.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I don’t talk much about surface tension. However, in Air and Water, Mark Denny devotes several chapters to it. He begins

We are now at the first of three chapters in which we explore the physics of the interface between water and air. As we will soon discover, the interface is a bizarre and fascinating place. To begin with, it is truly two-dimensional: it has neither outside nor inside. It is not contained in the water nor is it contained in the air; it is simply the place at which they meet. As such, its properties are not those of air or water alone, but of their mutual interaction, and these can be both surprising and nonintuitive…

We begin this exploration with an examination of the phenomenon known as surface tension. This is the force that keeps water droplets spherical, and it has a variety of biological consequences. For instance, we will see how surface tension allows trees to grow to majestic heights and flies to adhere to glass; how surface tension allows some insects to breathe under water and others to walk upon it.

Surface tension arises because a water molecule is usually hydrogen bonded to several other water molecules surrounding it. On the water surface, however, there is at least one hydrogen bond missing, so a higher surface energy is required to produce additional surface area. The surface energy of water in air is denoted γ and has a value of 0.07 J/m². This is a fairly high surface energy compared to most other liquids (for instance, everyone’s favorite, ethanol, has a surface energy of only 0.02 J/m²).

After explaining the origin of surface energy, Denny adds an important caveat.

In nature it is extremely rare to find an air-water interface that is not fouled to some extent with an ill-defined organic film. Most biological molecules (fatty acids in particular) can lower the surface energy to a fraction of that found in pure water. As a result, the surface of all but the cleanest bodies of water is likely to have a lower surface energy than reported here.

Denny then relates the concept of surface energy to that of surface tension.

To this point we have discussed the air-water interface in terms of its surface energy. Why, then, is this chapter entitled “Surface Tension”? It turns out that surface tension is just another way of expressing surface energy.

In the abstract, this is easily seen by comparing the units of the two expressions. Surface energy is expressed as J m⁻². But a joule is one newton meter, so J m⁻² is the same as N m⁻¹, that is, force per distance, a tension.

Surface tension is related to the concept of capillarity. Water tends to adhere to a clean glass surface. Therefore, water will rise in a hollow glass tube until it reaches a height at which the weight of the column is balanced by the adhesion force. This height is proportional to the surface tension and is inversely proportional to the radius of the tube. Denny observes that

Because the water in the tube is, in essence, hanging from the air-water interface, it is at a lower pressure than that of the surrounding air. As a result, if one were to poke a hole in the side of the tube, air would be drawn in rather than water being forced out.

Denny shows how capillarity is used to get water up a tree. Even more interesting is his discussion of insect tracheae.

Consider, for instance, the problems faced by insects. These animals rely on their tracheae and tracheoles to deliver oxygen to their muscles and to remove carbon dioxide… This system works only because the tracheae are tracheoles are filled with air. If these small tubes become filled with water, the rate at which they transport O₂ and CO₂ decreases 10,000-fold [because of the 100-fold difference of the diffusion constant in air and water]. How does the respiratory system of insects keep from filling up with water via capillarity?

The answer is likely to be that the inner surface of the respiratory system is coated with some substance that is not wetted by water… If the tracheae and tracheoles are coated with a waxy substance similar to that found on the external cuticle of many insects, water has no tendency to fill the system, and effective respiration is possible.

Denny then analyzes the law of Laplace, relating the air pressure inside a bubble to its radius and water’s surface tension. Russ and I analyze the case of a spherical bubble in Homework Problem 60 of Chapter 3 in IPMB. 

Denny concludes the chapter by examining animals that can walk on water.

The surface of lakes and streams provides a unique opportunity for terrestrial organisms. An animal that can walk on water has available to it a flat substratum from which to hunt aquatic prey and a refuge on which to escape from predators. There is just one problem: How does an animal manage to walk on water?

Surface tension, of course, provides the answer. If the animal contacts the water with a nonwettable structure of sufficient perimeter, the upward force of surface tension can support the organism’s weight. 

This only works for small animals.

The prospect of walking on water becomes less likely the larger the animal becomes. The crux of the problem is that an animal’s weight increases as the cube of a linear measure of its size, whereas the force due to surface tension increases in direct proportion to a linear measure (in this case, perimeter). For example, a mouse with a mass of 100 g weights 1 N. If the animal had feet coated with the same wax available to a water strider, it would need a perimeter of 40.3 m in contact with the water, roughly 10 m per foot. Feet of this size would clearly be impractical, and for this reason animals the size of mice do not walk on water.

The Neper

When discussing the attenuation of sound in Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

In acoustics, the attenuation is usually expressed in decibels per meter.

At the bottom of the page is this footnote:

Sometimes the attenuation coefficient is expressed in nepers m⁻¹, in which case the natural logarithm of the intensity or pressure ratio is used.

The neper? What’s that?

First, let’s review the decibel. If the pressure amplitude of two sound waves are p₁ and p₂, their relative pressure can be written as

            20 log₁₀(p₂/p₁).

When expressed in this way, the pressure difference is said to be in decibels (dB). If p₂ is ten times p₁, then the pressure difference is 20 log₁₀(10) or 20 dB.

Often we express sound in terms of intensity rather than pressure. The intensity is proportional to the pressure squared, so the relative intensity difference of I₁ and I₂ is

            10 log₁₀(I₂/I₁).

The leading factor of 20 in the expression containing the pressures is replaced by 10 in the expression containing intensities. If you don’t like that factor of ten out front, you could not use it, in which case your intensity difference is expressed in the rarely used unit of bels rather than decibels.

Notice that the decibel is defined using of a logarithm with base ten, also known as the common logarithm. Alternatively, we could use the natural logarithm, with base e = 2.718..., which leads to the neper. However—and this is the confusing part—instead of having the leading factor of 20 in the expression for decibels in terms of pressure, the expression for nepers has no leading factor at all. A factor of 2 is removed because the neper is defined in terms of the pressure and not intensity, and a factor of 10 is removed because nepers are like bels and not decibels. So,

            ln(p₂/p₁)

is the pressure difference in nepers. If you insist on using intensity rather than pressure, you must use the ugly-looking expression

            ½ ln(I₂/I₁) .

A ten-fold difference in intensity is 1.15 nepers (Np), so 10 dB is the same as 1.15 Np, or 1 Np = 8.7 dB. If a sound wave attenuates at a rate of 1 neper per meter that means for every meter traveled the pressure falls by a factor of e and the intensity falls by a factor of 7.4. In tissue, attenuation is usually proportional to frequency, so as a rule of thumb the attenuation is about 100 dB per meter per megahertz or roughly 12 neper per meter per megahertz.

Asimov's Biographical Encyclopedia
of Science & Technology,
by Isaac Asimov.

Where does the strange name “neper” come from? It honors the inventor logarithms, John Napier. Here is a excerpt about Napier from Asimov's Biographical Encyclopedia of Science & Technology.

NAPIER, John (nay’pee-ur) 

Scottish mathematician

Born: Merchiston Castle, near Edinburgh, 1550 

Died: Merchiston Castle, near Edinburgh, April 4, 1617

... Napier’s solid reputation rests upon a new method of calculation that first occurred to him in 1594… It occurred to Napier that all numbers could be expressed in exponential form. That is, 4 can be written as 2², while 8 can be written as 2³, and 5, 6, and 7 can be written as 2 to some fractional power between 2 and 3. Once numbers were written in such exponential form, multiplication could be carried out by adding exponents, and division by subtracting exponents. Multiplication and division would at once become no more complicated than addition and subtraction.

Napier spent twenty years working out rather complicated formulas for obtaining exponential expressions for various numbers. He was particularly interested in the exponential forms of the trigonometric functions, for these were used in astronomical calculations and it was these which Napier wanted to simplify. His process of computing the exponential expressions led him to call them logarithms (“proportionate numbers”) and that is the word still used.

Finally, in 1614, Napier published his tables of logarithms, which were not improved on for a century, and they were seized on with avidity. Their impact on the science of the day was something like that of computers on the science of our own time. Logarithms then, like the computers now, simplified routine calculations to an amazing extent and relieved working scientists of a large part of the noncreative mental drudgery to which they were subjected.