Friday, February 24, 2012

The Hodgkin and Huxley Macarena

Last week, Oakland University had the honor of hosting James Keener, Distinguished Professor of Mathematics at the University of Utah. He gave a delightful talk as part of our Quantitative Biology lecture series. His book Mathematical Physiology won the 1998 Association of American Publishers award for the Best New Title in Mathematics. Somehow, Russ Hobbie and I failed to cite this book in the 4th edition of Intermediate Physics for Medicine and Biology. We did, however, cite Keener’s work with Sasha Panfilov on the three-dimensional propagation of electrical activity in the heart. You can learn more about Keener's career and research in a Society of Mathematical Biology newsletter.

One of my favorite features of Keener’s website is his instructions on how to do the Hodgkin-Huxley Macarena. A photograph shows a large group of researchers doing this dance at the Cold Spring Harbor Laboratory last summer. To make sense of the H&H Macarena, image that the left arm is the sodium channel “m” gate, and the right arm is the “h” gate, as discussed in chapter 6 of Intermediate Physics for Medicine and Biology. (Note: I assume the picture on Keener's website shows a person facing us, so that her left arm is on my right side). Initially h is open (right arm vertical) and m is closed (left arm horizontal). During an action potential, m opens (step 2 and 3) and then h closes (step 4 and 5) and the "nerve" becomes refractory. Since the h gate is slower than the m gate, perhaps you should imagine having a lead weight wrapped around your right wrist as you do the H&H Macarena. Unfortunately, Keener does not yet have a video posted (with music), but perhaps we can encourage him to make one. If readers of Introductory Physics for Medicine and Biology know only one dance, it should be the Hodgkin-Huxley Macarena (although the ECG dance is a close second).

Friday, February 17, 2012

Measurement of Blood Pressure

Last week I was in the hospital with pneumonia. I’m fine now, thank you, but I was there three days, and in fact last week’s blog entry was posted from my hospital bed (doesn’t everyone bring their laptop with them to the hospital?).

A hospital is a rich environment for a lover of physics applied to medicine. One thing that particularly caught my eye is their way of measuring blood pressure. I got interested when, after a cuff was inflated around my arm, instead of feeling the familiar slow steady release of pressure as the nurse listened to my arm (that’s the way they still do it at the blood drive), this cuff started gripping and ungripping my arm in a strange and almost belligerent way. I had several opportunities to observe the measurement of blood pressure, and I decided that it would be a good topic for this blog.

First, a bit about the basic physics and physiology. In Chapter 1 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write
“As the heart beats, the pressure in the blood leaving the heart rises and falls. The maximum pressure during the cardiac cycle is the systolic pressure. The minimum is the diastolic pressure. (A blood pressure reading is in the form systolic/diastolic, measured in torr.)”
To explain how blood pressure is measured traditionally, I will turn to my hero Isaac Asimov’s book The Human Body (I quote from my 1963 paperback copy).
“When blood is forced into the aorta, it exerts a pressure against the walls that is referred to as blood pressure. This pressure is measured by a device called a sphygmomanometer (sfig’ moh-ma-nom’ i-ter; “to measure the pressure of the pulse” [Greek]), an instrument which, next to the stethoscope, is surely the darling of the general practitioner. The sphygmomanometer consists of a flat rubber bag some 5 inches wide and 8 inches long. This is kept in a cloth bag that can be wrapped snugly about the upper arm, just over the elbow. The interior of the rubber bag is pumped up with air by means of a little rubber bulb fitted with a one-way valve. As the bag is pumped up, the pressure within it increases and that pressure is measured by a small mercury manometer to which the interior of the bag is connected by a second tube.

As the bag is pumped up, the arm is compressed until, at a certain point, the pressure of the bag against the arm equals the blood pressure. At that point, the main artery of the arm is pinched closed and the pulse in the lower arm (where the physician is listening with a stethoscope) ceases.

Now air is allowed to escape from the bag and, as it does so, the level of mercury in the manometer begins to fall and blood begins to make its way through the gradually opening artery. The person measuring the blood pressure can hear the first weak beats and the reading of the manometer at that point is the systolic pressure, for those first beats can be heard only during systole, when the blood pressure is highest. As the air continues to escape and the mercury level to fall, there comes a characteristic quality of the beat that indicates the diastolic pressure; the pressure when the heart is relaxed.”
What I experienced in the hospital was different than Asimov's explanation, and was more automated. I'm having a difficult time finding good technical literature about automated blood pressure monitors, but I am going out on a limb here and guess how they work. In the hospital there was an inflatable cuff around my forearm, but there was no one listening with a stethoscope. That person is replaced by an optical device (similar to a pulse oximeter, see Section 14.6, Biological Applications of Infrared Scattering, in Intermediate Physics for Medicine and Biology) clipped to my finger, which presumably can detect flow. The cuff then inflates and flow is measured, the cuff changes the pressure to a new level and flow is measured again, etc. This all happens rapidly; in fact, each new cuff pressure level was maintained for at most one second, implying that only a single heart beat sufficed to make the flow measurement. The cuff and optical device are attached to a computer, and the computer made the decision about when to increase or decrease pressure, and what values to use. It seemed to be doing some sort of binary search, first going above and then below the level that allows flow. The algorithm slowed as the threshold level was approached, and I suspect that in such cases several heartbeats were required to accurately determine if flow occurred. The device also output the pulse rate and, if my memory serves me well, blood oxygenation level. Both were recorded after the cuff had completed its measurement of blood pressure.

I like this method. It does not depend on someone carefully listening for delicate blood flow noises (also known as Korotkoff sounds). In fact, while the blood pressure was being measured, the nurse was usually checking my IV line or doing some other task; the method is truly automated. One time, I did a little experiment and fidgeted with the finger clip during the measurement. The nurse got a fright when she saw my blood pressure up around 220/150. But a quick repeat measurement (during which I behaved myself) revealed that my blood pressure was actually normal (about 110/70). I suspect that the use of the binary search and pulse oximeter provides a more accurate measurement than does the traditional method, although I have no evidence to support that opinion. Automated blood pressure recording is an excellent example of how physics and engineering can contribute to medicine and biology.

Friday, February 10, 2012

Decay Plus Input at a Constant Rate

Section 2.7 of the 4th edition of Intermediate Physics for Medicine and Biology is titled Decay Plus Input at a Constant Rate. When I taught Biological Physics last fall (using for my textbook--you guessed it--Intermediate Physics for Medicine and Biology), I found that we kept coming back to this section over and over. Russ Hobbie and I write
“Suppose that in addition to the removal of y from the system at a rate –by, y enters the system at a constant rate a, independent of y and t. The net rate of change of y is given by

dy/dt = a – by….. (2.24)

The solution is

y=a/b ( 1 – exp(-bt)).”
One of the first applications of this equation is to the speed of an animal falling under the force of gravity and air friction (Chapter 2, Problem 28). One can show that the terminal speed of the animal is a/b. If further one proves that the gravitational force (a) is proportional to volume, and the frictional force (-by) is proportional to surface area, then the implication is that larger animals fall faster than smaller ones. This led to Haldane’s famous quote “You can drop a mouse down a thousand-yard mine shaft: and arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes.”

We see this equation again in Chapter 3 when analyzing Newton’s law of cooling (Problem 45). The surrounding temperature plays the role of a, and the exponential cooling from convection is represented by a term like –by. The solution to the resulting differential equation is just the exponential solution presented in Sec. 2.7.

In Section 5.7 about the artificial kidney, the equation arises again in governing the concentration of solute in the blood when the concentration of the solute in the dialysis fluid is a constant. As with the animal falling, the ratio of blood volume V to membrane area S is a key parameter.

The equation appears twice in Chapter 6 (Impulses in Nerve and Muscle Cells). First, the gate variables m, h, and n in the Hodgkin and Huxley model obey this same differential equation. In the voltage-clamp case, the gates approach their steady-state values exponentially. Then, Problem 35 analyzes electrical stimulation of a space-clamped passive axon using a constant current, and finds that the transmembrane potential approaches its steady-state value exponentially also. This result is used to derive two quantities with colorful names—rheobase and chronaxie—that are important in neural stimulation.

By Chapter 10, some of the students have almost forgotten the equation when it appears again in the study of feedback loops (particularly Section 10.4). I am sure that the equation would appear even more times, except my one-semester class ended with Chapter 10.

Some might wonder why Intermediate Physics for Medicine and Biology contains an entire chapter (Chapter 2) about exponential growth and decay. I believe that the way we are constantly returning to the concepts introduced in Chapter 2 justifies why we organize the material the way we do. In fact, Chapter 2 has always been one of my favorite chapters in Intermediate Physics for Medicine and Biology.

Friday, February 3, 2012

Charles Dickens: Medical Physicist?

The 200th anniversary of the birth of Charles Dickens occurs this week (he was born February 7, 1812). I am a big Dickens fan, so I had to fit him into this week’s blog entry somehow. It is not easy, since there is little overlap between Dickens’ novels and the 4th edition of Intermediate Physics for Medicine and Biology. But let us try.

Dickens’ life spanned an incredibly productive era of Victorian science in England. He was born the same year as English Chemist Humphry Davy published his Elements of Chemical Philosophy. Dickens’ birth fell almost exactly halfway between the births of the two greatest of 19th century British physicists: Michael Faraday (September 1791) and James Maxwell (November 1831). Just as Maxwell was publishing his eponymous Maxwell’s equations, Dickens was publishing Great Expectations. The physician John Snow was born one year after Dickens. It was Snow who famously traced the source of the 1854 cholera epidemic to the Broad Street pump in London (read more about this story in The Ghost Map by Steven Johnson). Dickens was born just two years after the birth of Charles Darwin, and On The Origin of Species appeared almost simultaneously with Dickens' masterpiece A Tale of Two Cities. William Thomson (Lord Kelvin) was 12 years younger than Dickens. He formulated his version of the second law of Thermodynamics in 1851, soon after Dickens published David Copperfield.

A young Charles was working long hours at Warren’s Blacking Warehouse when the first issue of the British medical journal The Lancet appeared in 1823. Dickens published his first story the year after Faraday proposed his vision of electric and magnetic fields, and he got married and published his first novel (The Pickwick Papers) in 1836, the year Darwin returned from his voyage on the Beagle. Martin Chuzzlewit came out in 1843, the same year James Joule determined the mechanical equivalent of heat. Dickens traveled to France and Italy the year that the prominent English chemist John Dalton died in England. His last complete novel, Our Mutual Friend, appeared in 1865, the year before the first transatlantic telegraph cable was laid (for more about this fascinating story, read A Thread Across the Ocean by John Gordon). Kelvin developed the cable equation to govern the transmission of signals over this telegraph line, and the same equation is used nowadays to describe nerve axons. An elderly Dickens came to the United States for a reading tour in 1867, the year English surgeon Joseph Lister pioneered the use of antiseptic to sterilize surgical instruments. Charles Dickens died of a stroke in 1870—leaving the Mystery of Edwin Drood unfinished—just a few months before the birth of the greatest experimental physicist since Faraday, Ernest Rutherford.

The medical literature contains several studies of how medicine was portrayed in Dickens’ books. Howard Markel writes about Charles Dickens and the Art of Medicine in the Annals of Internal Medicine (Volume 101, Pages 408-411, 1984).
“Charles Dickens, the novelist, humanist, and social reformer, was a keen observer of all the characteristics of the people in his novels. Dickens observed physicians and visited hospitals so that he could record various illnesses and diseases of people he met during his life. Dickens also worked for many public health reforms in Victorian England. The author used his observations of sick people in many of his novels and produced several accurate descriptions of disease, including Ménière's disorder and acute leukemia.”
The article analyzes how, in Bleak House, Phil Squod (who was always “shoulding his way along walls”) demonstrated symptoms consistent with “dysfunction of the vestibular nerve…most likely Meniere’s disorder”. In Dombey and Son, the symptoms of young Paul Dombey “resemble those of a child with an acute form of leukemia.”

Kerrie Schoffer and John O’Sullivan focus on movement disorders in their study of Charles Dickens in the Journal of Clinical Neuroscience (Volume 13, Pages 898-901, 2006)
“Nineteenth-century Victorian novelists played an important role in developing our understanding of medicine and illness. With the eye of an expert clinician, Charles Dickens provided several detailed accounts of movement disorders in his literary works, many of which predated medical descriptions. His gift for eloquence, imagery, and precision attest not only to the importance of careful clinical observation, but also provide an insightful and entertaining perspective on movement disorders for modern students of neuroscience.“
So is Dickens a medical physicist? I guess not. But he was a great writer. My favorite Dickens book is A Christmas Carol; I read it every Christmas. I reread A Tale of Two Cities during my Paris trip two years ago (“…It is a far, far better thing that I do, than I have ever done; it is a far, far better rest that I go to than I have ever known.”). I enjoyed Bleak House a few years ago, although it took me a long time to plow through that 818 page tome. I love Dickens’ characters, like the Artful Dodger in Oliver Twist, and Wilkins Micawber in David Copperfield. What will be my next Dickens book? I haven’t read Nicholas Nickleby yet; I think that will be it.