Friday, June 29, 2012

Rosalyn Yalow and the Radioimmunoassay

The radioimmunoassay is a sensitive technique for measuring tiny amounts of biologically important molecules, such as the hormone insulin in the blood. The basic idea is to tag insulin with a radioisotope such as I-125, and mix it with antibodies for insulin. Then, add to this mix the patient’s blood. The insulin in the blood competes with the tagged insulin for binding to the antibodies. Next, remove the antibodies and their bound insulin, leaving just the free insulin in the supernatant. The radioactivity of the supernatant provides a way to determine the concentration of insulin in the blood.

Russ Hobbie and I describe the basics of a radioimmunoassay in Chapter 17 of the 4th edition of Intermediate Physics for Medicine and Biology.
“Four kinds of radioactivity measurements have proven useful in medicine. The first involves no administration of a radioactive substance to the patient. Rather, a sample from the patient (usually blood) is mixed with a radioactive substance in the laboratory, and the resulting chemical compounds are separated and counted. This is the basis of various competitive binding assays, such as those for measuring thyroid hormone and the availability of iron-binding sites. The most common competitive binding technique is called radioimmunoassay. A wide range of proteins are measured in this manner.”
The radioimmunoassay was developed by Rosalyn Yalow and Solomon Berson. Yalow received the Nobel Prize in Physiology or Medicine for this work in 1977 (Berson had died by then, and the Nobel committee never gives a prize posthumously). For readers of Intermediate Physics for Medicine and Biology, Yalow is interesting because she started out as a physics major, getting her bachelor’s degree in physics from Hunter College, part of the City University of New York (CUNY) system. In 1945 she obtained a PhD in nuclear physics from the University of Illinois at Urbana-Champaign. Building on this physics background, and collaborating with Berson, in the 1950s she developed the radioimmunoassay. Interestingly she and Berson refused to patent the method, wanting it to be freely available for use in medicine. Yalow died just over one year ago, at age 89.

You can learn more about Rosalyn Yalow and her inspiring life from her Nobel autobiography, her Physics Today obituary, her New York Times obituary, and the Jewish Woman’s Archive. For those who prefer a video, click here. I have not read Eugene Straus’s book Rosalyn Yalow: Nobel Laureate: Her Life and Work in Science, but I am putting it on my list of things to do.

I often like to finish a blog entry about a noteworthy scientist with their own words. Below are the opening paragraphs of Yalow’s Nobel Prize Lecture.
“To primitive man the sky was wonderful, mysterious and awesome but he could not even dream of what was within the golden disk or silver points of light so far beyond his reach. The telescope, the spectroscope, the radiotelescope - all the tools and paraphernalia of modern science have acted as detailed probes to enable man to discover, to analyze and hence better to understand the inner contents and fine structure of these celestial objects.

Man himself is a mysterious object and the tools to probe his physiologic nature and function have developed only slowly through the millenia. Becquerel, the Curies and the Joliot-Curies with their discovery of natural and artificial radioactivity and Hevesy, who pioneered in the application of radioisotopes to the study of chemical processes, were the scientific progenitors of my career. For the past 30 years I have been committed to the development and application of radioisotopic methodology to analyze the fine structure of biologic systems.

From 1950 until his untimely death in 1972, Dr. Solomon Berson was joined with me in this scientific adventure and together we gave birth to and nurtured through its infancy radioimmunoassay, a powerful tool for determination of virtually any substance of biologic interest. Would that he were here to share this moment.”

Friday, June 22, 2012

Mannitol

Elevated intracranial pressure often follows a traumatic brain injury. One way to lower pressure in the brain is to administer mannitol intravenously. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss mannitol.
"The converse of this effect [removal of urea during renal dialysis] is to inject into the blood urea or mannitol, another molecule that does not readily cross the blood–brain barrier. This lowers the driving pressure of water within the blood, and water flows from the brain into the blood. Although the effects do not last long, this technique is sometimes used as an emergency treatment for cerebral edema."
Mannitol (C6H14O6) has a similar size, structure, and chemical formula as glucose (C6H12O6). It is metabolically inert in humans. A 10% solution consists of 100 g of mannitol per liter (1000 g) of water. Mannitol has a molecular weight of 182 g/mole, implying an osmolarity of (100 g/liter)/(182 g/mole) = 0.55 moles/liter, or 550 mosmole. Blood has an osmolarity of about 300 mosmole, so 10% mannitol is significantly hypertonic. Problem 3 in Chapter 5 asks you to calculate the osmotic pressure produced by mannitol.
Problem 3 Sometimes after trauma the brain becomes very swollen and distended with fluid, a condition known as cerebral edema. To reduce swelling, mannitol may be injected into the bloodstream. This reduces the driving force of water in the blood, and fluid flows from the brain into the blood. If 0.01 mol l−1 of mannitol is used, what will be the approximate osmotic pressure?
Mannitol works best for short-term reduction of intracranial pressure. If it is administered continuously, eventually some of the mannitol may cross the blood-brain barrier, reducing its osmotic effect (Wakai et al., 2008). Then, when mannitol administration is discontinued, the mannitol that crossed into the brain can actually have the opposite effect of osmotically drawing water from the blood into the brain.

Interestingly, if you give a high enough concentration of mannitol, the osmotic shrinking of endothelia cells can disrupt the blood brain barrier. Sometimes mannitol is used to ensure that certain drugs are able to pass from the blood into the brain.

Friday, June 15, 2012

The Heating of Metal Electrodes

Twenty years ago, I published a paper titled “The heating of metal electrodes during rapid-rate magnetic stimulation: A possible safety hazard” (Electroencephalography and Clinical Neurophysiology, Volume 85, Pages 116-123, 1992). My coauthors were Alvaro Pascual-Leone, Leonardo Cohen, and Mark Hallett, all working at the National Institutes of Health at that time. The paper motivated two new homework problems in Chapter 8 of the 4th edition of Intermediate Physics for Medicine and Biology.
Problem 24 Suppose 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 × 106 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 25 Suppose 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 + x2 + ... + xn−1 = (1 − xn)/(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).]
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 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 + x2 + … + xN-1. If you multiply this expression by x, you get xS = x + x2 + … + xN. Now (and this is the clever trick), subtract xS from S, which gives (1 – x) S = 1 – xN. Note that all the terms x + x2 + … + xN-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.

Saturday, June 9, 2012

Law of Laplace

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I often introduce topics in the homework problems that we don’t have room to discuss fully in the text. For instance, Problem 18 in Chapter 1 asks the reader to derive the Law of Laplace, f = p R, a relationship between the pressure p inside a cylinder, its radius R, and its wall tension f.

In his book Vital Circuits, Steven Vogel explains the physiological significance of the Law of Laplace, particularly for blood vessels.
“The wall of the aorta of a dog is about 0.6 millimeters thick, while the wall of the artery leading to the head is only half that. Pressure differences across the wall are the same…, but the aorta has an internal diameter three times as great. The bigger vessel simply needs a thicker wall. An arteriole is 100 times skinnier yet; it wall is fifteen times thinner than that of the artery going to the head. A capillary is eight times still skinnier, with walls another twenty times thinner… A general rule that wall thickness is proportional to vessel diameter is clearly evident, just the relationship expected from Laplace’s law.”
In our homework problem 18, Russ and I write that “Sometimes a patient will have an aneurysm in which a portion of an artery will balloon out and possibly rupture. Comment on this phenomenon in light of the R dependence of the force per unit length.” The answer (spoiler alert) is explained by Vogel. He first examines what happens when inflating a balloon.
“About the same pressure is needed throughout the inflation, except for an extra bit to get started and (if you persist) another extra bit just before the final explosion… Pressure gets more effective in generating tension—stretch—as the balloon gets bigger [p = f/R], automatically providing the extra force needed as the rubber is expanded.”
What is the implication for an aneurysm?
“Pressure, we noted, is more effective in generating tension in the walls of a big cylinder than in those of a small cylinder. Blow into a cylindrical balloon, and one part of the balloon will inflate almost fully before the remainder expands. Pressure inside at any instant is the same everywhere, but the responsive stretching is curiously irregular…any part partially inflated is easier to inflate further than any part not yet inflated at all.”
Thus, the law of Laplace provides insight into aneurysms: just think of a bulge that develops when inflating a cylindrical balloon. Now the question can be turned on its head: why don’t all cylindrical vessels immediately develop aneurysms as soon as pressure is applied? In other words, why are we not all dead, killed by the law of Laplace? Vogel addresses this point too. “The primary question isn’t why aneurysms sometimes occur, but why they don’t normally happen…Why an arterial wall...behaves in a much friendlier manner.”

Vogel’s answer is that arteries “should surely stretch strangely” (I love the alliteration). The walls of an artery as designed such that
“a disproportionate force is needed for each incremental bit of stretch—the thing gets stiffer as it stretches further…As the vessels expand, pressure inside is increasingly effective at generating tension in their walls—that’s the unavoidable consequence of Laplace’s law. But that tension, the stress in the walls, is decreasingly effective in causing the walls to stretch. It all comes down to that curved, J-shaped line on the stress-strain graph, which means no aneurysm.”
Thank goodness nature found a way to avoid the aneurysms predicted by the law of Laplace!

There are many more biomedical applications of the Law of Laplace. In a review article, Jeffrey Basford describes the Law of Laplace and Its Relevance to Contemporary Medicine and Rehabilitation (Archives of Physical Medidine and Rehabilitation, 83:1165-1170, 2002). Basford considers many examples, including bladder function, compressive wraps to treat peripheral edema, the choice of where in the uterus to perform a Cesarean section. That is a lot of insight from one simple law relating pressure, radius and wall tension.

Friday, June 1, 2012

Andrew Huxley (1917-2012)

Andrew Huxley, the greatest mathematical biologist of the 20th century, died on Wednesday, May 30. Huxley won the Nobel Prize for his groundbreaking work with Alan Hodgkin that explained electrical transmission in nerves.

In Chapter 6 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the Hodgkin-Huxley model of membrane current in a nerve axon.
“Considerable work was done on nerve conduction in the late 1940s, culminating in a model that relates the propagation of the action potential to the changes in membrane permeability that accompany a change in voltage. The model [Hodgkin and Huxley (1952)] does not explain why the membrane permeability changes; it relates the shape and conduction speed of the impulse to the observed changes in membrane permeability. Nor does it explain all the changes in current…Nonetheless, the work was a triumph that led to the Nobel Prize for Alan Hodgkin and Andrew Huxley.”
The paper we cite (A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve, Journal of Physiology, Volume 117, Pages 500-544) is one of my favorites. Whenever I teach biological physics, I assign this paper to my students as an example of mathematical modeling in biology at its best. In 1981 Hodgkin and Huxley wrote a “citation classic” article about their paper, which has now been cited over 9300 times. They concluded
"Another reason why our paper has been widely read may be that it shows how a wide range of well-known, complicated, and variable phenomena in many excitable tissues can be explained quantitatively by a few fairly simple relations between membrane potential and changes of ion permeability—processes that are several steps away from the phenomena that are usually observed, so that the connections between them are too complex to be appreciated, intuitively. There now seems little doubt that the main outlines of our explanation are correct, but we have always felt that our equations should be regarded only as a first approximation that needs to be refined and extended in many ways in the search for the actual mechanism of the permeability change's on the molecular scale."
As one who does mathematical modeling of bioelectric phenomena for a living, I can think of no better way to honor Huxley than to show you his equations:


This set of four nonlinear ordinary differential equations, plus six expressions relating how the ion channel rate constants depend on voltage, not only describes the membrane of the squid giant nerve axon, but is the starting point for models of all electrically active tissue. Russ and I consider this model to be so important that we dedicate six pages to exploring it, and present in our Fig. 6.38 a computer program to solve the equations. For anyone interested in electrophysiology, becoming familiar with the Hodgkin-Huxley model is job one, just as analyzing the Bohr model for hydrogen is the starting point for someone interested in atomic structure. Remarkably, 60 years ago Huxley solved these differential equations numerically using only a hand-crank adding machine.

How can your learn more about this great man? First, the Nobel Prize website contains his biography, a transcript of his Nobel lecture, and a video of an interview. Another recent, more detailed interview is available on youtube in two parts, part1 and part 2. Huxley wrote a fascinating description of the many false leads during their nerve studies in a commemorative article celebrating the 50th anniversary of his famous paper. Finally, the Guardian published an obituary of Huxley yesterday.

I will conclude by quoting the summary at the end of Hodgkin and Huxley’s 1952 paper, which was the last of a series of five articles describing their voltage clamp experiments on a squid axon.

SUMMARY
1. The voltage clamp data obtained previously are used to find equations which describe the changes in sodium and potassium conductance associated with an alteration of membrane potential. The parameters in these equations were determined by fitting solutions to the experimental curves relating sodium or potassium conductance to time at various membrane potentials.
2. The equations, given on pp. 518-19, were used to predict the quantitative behaviour of a model nerve under a variety of conditions which corresponded to those in actual experiments. Good agreement was obtained in the cases:
(a) The form, amplitude and threshold of an action potential under zero membrane current at two temperatures.
(b) The form, amplitude and velocity of a propagated action potential.
(c) The form and amplitude of the impedance changes associated with an action potential.
(d) The total inward movement of sodium ions and the total outward movement of potassium ions associated with an impulse.
(e) The threshold and response during the refractory period.
(f) The existence and form of subthreshold responses.
(g) The existence and form of an anode break response.
(h) The properties of the subthreshold oscillations seen in cephalopod axons.
3. The theory also predicts that a direct current will not excite if it rises sufficiently slowly.
4. Of the minor defects the only one for which there is no fairly simple explanation is that the calculated exchange of potassium ions is higher than that found in Sepia axons.
5. It is concluded that the responses of an isolated giant axon of Loligo to electrical stimuli are due to reversible alterations in sodium and potassium permeability arising from changes in membrane potential.