The Goldman-Hodgkin-Katz Equation Including Calcium

In Section 9.6 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I derive the Goldman-Hodgkin-Katz equation. It accounts for both diffusion and electrical forces acting on ions in the membrane (presumably passing through ion channels spanning the lipid bilayer). If only one ion were present, its concentration on each side of the membrane would determine the equilibrium, or reversal, potential. For instance more potassium is inside a cell than outside, so diffusion pushes the positively charged potassium ions out. As the outside becomes positive, the resulting electric field in the membrane pushes potassium back in. The reversal potential, v_rev, is the potential across the membrane when diffusion and electrical forces balance.

Mathematically, we can derive the reversal potential for any ion C by starting with an expression for its current density, J_C

where z is the valence, e is the elementary charge, v is the potential, ω_C is the permeability, N_A is Avogadro's number, k_B is the Boltzmann constant, T is the absolute temperature, and [C₁] and [C₂] are the concentrations outside and inside the membrane. (See IPMB for a derivation of this complicated equation.) To find the reversal potential, we set J_C to zero and solve for v.

When more than one ion can cross the membrane, the situation is more complicated. Russ and I examined a membrane that can pass three ions: sodium, potassium, and chloride. The resulting equation for the reversal potential—also known as the Goldman-Hodgkin-Katz equation—is

We then write

When ions have different valences, the GHK equation becomes more complicated. Lewis (1979) has derived an analogous equation for transport of sodium, potassium, and calcium.

The citation is to

Lewis CA (1979) “Ion-concentration dependence of the reversal potential and the single channel conductance of ion channels at the frog neuromuscular junction.” Journal of Physiology, Volume 286, Pages 417–445.

Below is a new homework problem, based on Appendix A of Lewis’s paper, analyzing a more complicated GHK equation that includes calcium along with sodium and potassium.

Section 9.6

Problem 20 ½. Derive an expression for the Goldman-Hodgkin-Katz equation when you have three ions that can pass through the membrane: sodium, potassium, and calcium.

(a) Write down an expression like Eq. 9.53 for the current density for each ion: J_Na, J_K, and J_Ca. Hint: be careful to include the valence z properly.

(b) Assume the amount of charge in the cell does not change with time, so J_Na + J_K + J_Ca = 0. Try to solve the resulting equation for the reversal potential, v_rev. You should find it difficult, because the expression for J_Ca has a different denominator than do J_Na and J_K.

(c) Define a new permeability for calcium,

Now derive an expression for v_rev. Your result should look similar to Eq. 9.55, except for some factors of four, and in the numerator the new calcium permeability will be multiplied by a voltage-dependent factor.

What’s the lesson to be learned from this homework problem? First, the GHK expression including calcium has the potential on the left side of the equation, but also on the right side, inside a logarithm. No simple way exists to calculate v_rev. My first thought is to use an iterative method, but I haven’t looked into this in detail. Second, notice how a small modification to the problem—changing chloride to calcium—made a major change in how difficult the problem is to solve. Adding the negative chloride ion to positive sodium and potassium resulted in a trivial change to the GHK equation (the inside chloride concentration appears in the numerator rather than the outside concentration). However, adding the divalent cation calcium totally messes up the equation, making it difficult to solve except with numerical methods.

I advocate for simple models. They provide tremendous insight. However, the moral of this story is if you push a toy model too hard, it can become complicated; it’s no longer a toy.

A Severe Shortage of Blood!

Blood donation bags
at a blood drive in
Rochester Michigan
on March 24, 2020.

Yesterday Michigan’s governor, Gretchen Whitmer, announced a shelter-in-place order. Now I need a legitimate reason to leave home. I have one: To donate blood! The Red Cross has a severe blood shortage. I encourage all readers of Intermediate Physics for Medicine and Biology to make an appointment.

What’s the purpose of blood, anyway? To carry oxygen. Let’s estimate the concentration of oxygen in blood (Fermi problem time). As a first step, Homework Problem 1 in Chapter 1 of IPMB asks the reader to estimate how much hemoglobin is in a red blood cell.

Problem 1. Estimate the number of hemoglobin molecules in a red blood cell. Red blood cells are little more than bags of hemoglobin, so it is reasonable to assume that the hemoglobin takes up all the volume of the cell.

My name tag at the blood drive.

Russ Hobbie and I don’t send IPMB’s solution manual to anyone except instructors, but because you all are going to make blood-donation appointments as soon as you’re done reading this post, I’ll share the solution to this problem.

1.1 An important skill for students to learn is order-of-magnitude estimation. The first four problems in this chapter require the students to estimate some quantity of biological interest.

Approximate the dimensions of a red blood cell as 8 μm × 8 μm × 2 μm. Approximate the dimensions of a hemoglobin molecule as 6 nm × 6 nm × 6 nm. The number N of hemoglobin molecules is equal to the volume of a red blood cell divided by the volume of a hemoglobin molecule: 

We do not expect a “back-of-the-envelope” estimate such as this one to be accurate to, say, a factor of 2 or π. But it should give a quick order of magnitude approximation.

I had a difficult time taking
this selfie: one hand holding my
phone, the book balanced on my
chest, and a needle in the other arm.

Each hemoglobin molecule can bind with four oxygen molecules, so a red blood cell can contain 2400 million oxygen molecules. I’ll assume the hemoglobin isn’t packed too tightly, so let’s round that down to 2000. The volume of a red blood cell is 128 cubic microns. Inside a red blood cell the oxygen concentration is therefore 2000 million molecules per 128 cubic microns, or about 16,000,000/μm³. A typical hematocrit (fraction of blood volume occupied by red blood cells) is 40%. Therefore blood has an oxygen concentration of around 6 million per cubic micron.

I admit, those are strange units. A cubic micron is 10^-15 liters, and 6 million molecules is 10^-17 moles. So, the concentration of oxygen in blood is about 0.01 molar, or 10 mM.

You can estimate the concentration of oxygen in air using the ideal gas law, pV = nRT. Air is about 20% oxygen, so using p = 0.2 atm, T = 310 K, and R = 0.082 liter atm/(mole K), you get n/V = 0.008, or 8 mM. Within the uncertainty of our rough estimate, this result implies that the concentration of oxygen in blood is nearly the same as the concentration of oxygen in air. As it should be! The whole point of blood is to get oxygen from the air into the tissues.

The best part of blood donation.

Thanks to all the phlebotomists and volunteers for collecting blood, despite the risk; they’re heroes. I won’t be able to give blood again for another eight weeks. By that time I hope the @#$%&! coronavirus is gone and life has returned back to normal.

After giving blood. My daughter Stephanie,
who also donated, took the photo.

Bob Park’s What’s New has been Restored!

Screenshot of the What's New website,
whatsnewbobpark.com.

From 1983–2006, physicist Bob Park was the director of public information in the Washington D. C. office of the American Physical Society. During this time, he wrote the delightful weekly column What’s New. When I was in graduate school, every Friday I’d look forward to a new post.

For years What’s New disappeared from the internet, but recently it’s been restored (at least partially) from internet archives. You can find it at http://whatsnewbobpark.com. [Note added October 9, 2020: the link no longer works properly. However, here is one that does: https://web.archive.org/web/20140124195058/http://bobpark.physics.umd.edu/archives.html] If you click on “About Bob” you’ll see:

Robert L. (Bob) Park is professor of physics and former chair of the Department of Physics at the University of Maryland. For twenty years, research into the properties of crystal surfaces occupied most of his waking hours, but in 1983 he was recruited by astrophysicist Willie Fowler (who was awarded the Nobel Prize in Physics later that year) to open a Washington Office of the American Physical Society. Bob initiated a weekly report of happenings in Washington that were important to science, and with the development of the internet, the weekly report evolved into the news/editorial column What’s New. For the next twenty years he divided his time between the University and the Washington Office. In 2003 he returned to the University full time. With the support of the Department of Physics of the University of Maryland, he continues to write the occasionally controversial What’s New, which has developed a following that extends beyond physics.

Dr. Park has also written two books based on his Washington experience:

Voodoo Science: The Road from Foolishness to Fraud (Oxford, 2000)

Superstition: Belief in the Age of Science (Princeton, 2008)

In What’s New, Park would return to certain topics again and again; for example, cell phones and cancer, creationism, climate change, and cold fusion. Often he would debunk alternative medicine, such as homeopathy, biomagnetic therapy, and therapeutic touch. Readers of Intermediate Physics for Medicine and Biology will enjoy how he applied physics reasoning to medicine. Below are excerpts from 2011, so you can sample Park’s writing style. 

Friday, June 10, 2011

1. ET TU TARA? “PIERCING THE FOG AROUND CELLPHONES AND CANCER.” The WELL blog by Tara Parker-Pope was the top story in Tuesdays NYT Health Section. Her story is not wrong, but its told in the wrong context. Science is a search for cause and effect, not an epidemiologic majority. To settle the question, WHO invited 31 experts to spend a week in Lyon, the culinary capital of France, strategically located between the two best wine regions. Meanwhile, much had been made of a study showing that the brain is “activated” by microwave radiation. Of course, it is. The effect of microwaves on the human brain, as on cold pizza, is to cause chemical bonds to vibrate, which we sense as heat. Unlike cold pizza, however, the human brain resists being heated. Deep within the brain, the hypothalamus, the thing below the thalamus, senses any increase in blood temperature. It calls on blood vessels in the heated area to expand, and increases the heart rate. The fresh blood is a coolant, but incidentally, also increases the rate of metabolism. “Microwaves have activated the brain,” the human observers shouted. The shout was heard in Lyon. Amidst the clinking of glasses, the vote of the expert panel tipped from “no effect” to “possibly carcinogenic to humans.” What could it matter? No one is going to stop using cell phones anyway. Does anyone care? One enormously powerful group cares, the tort industry.

Friday, August 19, 2011

1. HOMEOPATHY: THE DILUTION LIMIT AND THE CULTURE OF CREDULITY. Based in France, Boiron, a huge multinational maker of homeopathic-remedies, is suing an Italian blogger, Samuele Riva, for saying oscillococcinum, the companys featured flu medication, has no active ingredient. Congratulations Sam, I gave up trying to get Boiron to sue me, years ago but the Center for Inquiry, of which I’m a member, is pleading with Boiron to sue us. “Anas barbariae hepatis et cordis extractum,” is listed as the active ingredient by the company. Its prepared at a concentration of 200CK HPUS from the liver of the Barbary duck. The 200CK means the solution has been diluted 1 part in 100, shaken, and repeated sequentially 200 times. HPUS means the medication is listed in the Homeopathic Pharmacopeia of the United States, and prepared according to 1938 federal guidelines. Its a national disgrace that the antiquated law sanctioning homeopathy, introduced by Sen. Royal Copeland, himself a homeopathist, is still be on the books. The dilution claim is totally meaningless. Somewhere around the 30th of the 200 sequential dilutions, the dilution limit of Earth would be reached, with the entire Earth becoming the solute. That is, the possibility of even one molecule of the duck-liver extract remaining in the solution beyond that point would be negligible. Long before the 200th dilution, the dilution limit of the entire visible universe would have been reached. This is all quite meaningless. Astronomers put the number of atoms in the visible universe at about 10 to the 80th power. It would take many universes to get to a dilution of 200 C.

Friday, September 16, 2011

1. WI-FI REFUGE: UNITED STATES NATIONAL RADIO QUIET ZONE. A 34,000 km² rectangle of land straddling the border of Virginia and West Virginia surrounds The Robert C. Byrd Green Bank Telescope, the world’s largest fully steerable radio telescope. The site was chosen partly because the Allegheny Mountains block the horizontal propagation of radio signals, but mostly because Robert C Byrd (D-WV) was one powerful US Senator. Radio transmission in the zone is either limited or banned outright. In addition to radio astronomers, the quiet zone has also attracted a colony of people who say they suffer from Electromagnetic Hypersensitivity (EHS). They certainly suffer from something, but EHS is not medically recognized in the US. In a BBC News interview last week I suggested that the appropriate treatment for a non-ailment such as EHS would be homeopathic medicine.

Sunday, October 23, 2011

2. CELL PHONEYS: BRAIN CANCER LINK IS REJECTED AGAIN. Ten years ago, a brilliant Danish epidemiological study found no link between mobile phone use and brain cancer (JNCI 2001, 93: 203-7). A decadal reexamination by Denmarks Institute of Cancer Epidemiology, released last week, again found no link. The object of the new study was to look for any evidence of latent cancer that had not yet shown up in 2001; none was found. In a 2001 JNCI editorial I pointed out that none would be expected, since microwave radiation is non-ionizing, Park, Robert L, JNCI 2001, 93: 166-167. Can we now put the damned cell-phone/cancer scare behind us?

Wednesday, November 16, 2011

2. CANCER AND CAUSALITY: EINSTEIN DIDNT HAVE A CELL-PHONE. Of the worlds 7 billion people, an incredible 5 billion have cell phones (mobiles in most countries). The safe use of mobiles is therefore a global health concern. The response of the World Health Organization was to conduct a huge epidemiologic study aimed at demonstrating a link between cell-phone radiation and brain cancer. The effort was seriously misguided no such link exists. The study served only to raise widespread public alarm over a nonexistent hazard. Epidemiology, which is the study of health patterns in populations; is important, but its not a substitute for science. Science is the organization of knowledge into testable laws and theories. It has been known for more than 100 years that electromagnetic radiation at frequencies below the ultraviolet is non-ionizing, and thus cannot create the mutant strands of DNA that constitute incipient cancers. In 1905, Einsteins miracle year, he theorized that electromagnetic radiation consists of discrete units of energy, now called photons, which are equal in energy to the frequency multiplied by Planck’s constant. It marked the origin of wave-particle duality and earned Einstein his 1921 Physics Nobel Prize. His theory is verified every time a cell phone works.

I miss Bob Park. We still need him. His mantle has been taken up by people like the Skep Doc Harriet Hall. We must expose quackery and embrace evidence-based science and medicine.

What’s New was hosted by a University of Maryland website. At the bottom of the page was the disclaimer:

Opinions are the author’s and are not necessarily shared by the University, but they should be.

Bob Park is featured in the video 

“You Don't Have to Be a Scientist to Spot 

the Fraudulent Science that Swirls Around Us (2000)” 

https://www.youtube.com/watch?v=yAFUpB5cQOs

 Part 1 of Superstition: Belief in the Age of Science
featuring Bob Park (you can find the other six parts on YouTube).

https://www.youtube.com/watch?v=lgK9vNJFDj4&list=PL5C8D41EB69CB1096&index=2&t=197s

Practice Problems in Bioelectricity and Biomagnetism

It’s funny how your memory can deceive you. I thought my interest in writing homework problems began with my work on Intermediate Physics for Medicine and Biology. Recently, however, I was rummaging through some old papers and discovered that I’ve been writing homework problems for a lot longer. This habit traces back to my graduate school days at Vanderbilt University, when I worked for John Wikswo. Among my old documents, I found a brittle yellowed copy of “The Magnetic Field of a Single Axon: Practice Problems.” It begins

These problems are presented to help someone to become familiar with the analytic volume conduction models of electric potentials and magnetic fields produced by nerve axons or bundles of nerve or muscle fibers developed between 1982 and 1988 in the Living State Physics Group. The problems vary in difficulty, with the very difficult ones marked by a *.

The problems are drawn from eight publications I helped write back in the day. If you need copies of these articles so you can solve the problems, just email me: roth@oakland.edu. (Technically the journal owns the copyright, so I won’t link to the pdfs in this blog. 😞)

J. K. Woosley, B. J. Roth, and J. P. Wikswo, Jr. (1985) “The Magnetic Field of a Single Axon: A Volume Conductor Model.” Mathematical Biosciences, Volume 75, Pages 1-36.

B. J. Roth and J. P. Wikswo, Jr. (1985)  “The Magnetic Field of a Single Axon: A Comparison of Theory and Experiment.” Biophysical Journal, Volume 48, Pages 93-109.

B. J. Roth and J. P. Wikswo, Jr. (1985) “The Electrical Potential and Magnetic Field of an Axon in a Nerve Bundle.” Mathematical Biosciences, Volume 76, Pages 37-57.

B. J. Roth and J. P. Wikswo, Jr. (1986) “A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue.” IEEE Transactions on Biomedical Engineering, Volume 33, Pages 467-469.

B. J. Roth and J. P. Wikswo, Jr. (1986) “Electrically Silent Magnetic Fields.” Biophysical Journal, Volume 50, Pages 739-745.

B. J. Roth and F. L. H. Gielen (1987) “A Comparison of Two Models for Calculating the Electrical Potential in Skeletal Muscle.” Annals of Biomedical Engineering, Volume 15, Pages 591-602.

J. P. Wikswo, Jr. and B. J. Roth (1988) “Magnetic Determination of the Spatial Extent of a Single Cortical Current Source: A Theoretical Analysis.” Electroencephalography and Clinical Neurophysiology, Volume 69, Pages 266-276.

B. J. Roth (1987) “Longitudinal Resistance in Strands of Cardiac Muscle.” Ph.D. Thesis, Vanderbilt University, Nashville, Tennessee.

Many of these problems require analyzing Bessel functions and Fourier transforms; I was enamored by those mathematical methods in the 80’s. You can get a hint of what these old homework problems are like by looking at Problem 16 in Chapter 8 of IPMB, where the reader must use these techniques to calculate the magnetic field of a nerve axon.

Let me give you another example. Problem 10 in this ancient collection is based on the  paper by Woosley et al.:

Prove that Eq. (36) and (45) are equal.

Equation (36) is the magnetic field of a nerve axon derived using the law of Biot and Savart, and Equation (45) is the magnetic field derived using Ampere’s law. I’ve discussed before in this blog how I could not prove these two equations were equivalent until I found a Wronskian relating Bessel functions.

I’ve also analyzed the IEEE TBME paper in a blog post from 2018.

These practice problems are not for the faint of heart. Nevertheless, most of what you need to solve them is in IPMB. If you want to learn some advanced methods in theoretical bioelectricity and biomagnetism, download the practice problems and give them a try. If nothing else, they will provide insight into what I used to work on as a graduate student. Besides, with the coronavirus pandemic holding you in quarantine, what else do you have to do?

Traveling Waves and Standing Waves

Section 13.2 of Intermediate Physics for Medicine and Biology discusses waves. Russ Hobbie and I note that two solutions to the wave equation exist: traveling waves and standing waves.

Traveling Waves

We write that the pressure distribution p(x,t) = f(x − ct), where f is any function,

obeys the wave equation... It is called a traveling wave. A point on f(x − ct), for instance its maximum value, corresponds to a particular value of the argument x − ct. To travel with the maximum value of f(x − ct), as t increases, x must also increase in such a way as to keep x − ct constant. This means that the pressure distribution propagates to the right with speed c... Solutions p(x,t) = g(x + ct), where g is any function, also are solutions to the wave equation, corresponding to a wave propagating to the left.

Standing Waves

We then discuss standing waves.

Standing waves such as p(x,t) = p cos(ωt) sin(kx) are also solutions to the wave equation… [This] standing wave… has nodes fixed in space where sin(kx) is zero… A standing wave can also be written as the sum of two sinusoidal traveling waves, one to the left and one to the right. Conversely, two standing waves can be combined to give a traveling wave.

Converting Traveling Waves to Standing Waves

IPMB includes a homework problem asking the reader to show analytically that two traveling waves combine to make a standing wave, and vice versa.

Problem 8. Use the trigonometric identity sin(a ± b) = sin a cos b ± cos a sin b to show that a traveling wave can be written as the sum of two out-of phase standing waves, and that a standing wave can be written as the sum of two oppositely-propagating traveling waves.

Visualizations

Russ and I also include figures illustrating the difference between a traveling wave (our Fig. 13.4) and a standing wave (Fig. 13.5). To gain insight, however, nothing can replace a dynamic visualization. Fortunately, the internet is full of such visualizations. One appears in the Wikipedia article about standing waves. The Physics Hypertextbook also has traveling and standing wave animations.

This Youtube video shows trigonometry in action: the sum of two oppositely going traveling waves (blue wave propagating right, and green left) add to form a single standing wave (red).

Two traveling waves adding to form a standing wave.

https://www.youtube.com/watch?v=X8qZO6g_X5Q

I like the next video because it shows a traveling wave turning into a standing wave when it reflects off a boundary.

A traveling wave turning into a standing wave when it reflects off a boundary.

https://www.youtube.com/watch?v=2KBJp5ysS74

Here’s a nice video showing how standing waves can be created experimentally.

Standing waves created experimentally on a string fastened at both ends.

https://www.youtube.com/watch?v=-gr7KmTOrx0

Finally, here’s a Flipping Physics video comparing standing and traveling waves. It’s a little corny, but I like it that way.

A lecture about waves from Flipping Physics.

https://www.youtube.com/watch?v=DUPLRe5PHvE

Enjoy!

Physics Girl

Because of the coronavirus, I had to transform my introductory physics course from in-person to online (in two days!). I thought: If I’m going to teach remotely, I might as well use some of the excellent resources that are available on the internet. This led me to Physics Girl.

Dianna Cowern produces funny and informative videos about physics. Some even deal with medical and biological physics. Below I have embedded a few about biomechanics, sound perception, sun screen, color vision, magnetic resonant imaging, and bioelectricity.

If you’re studying from Intermediate Physics for Medicine and Biology, consider these videos as supplementary material. If you like them, plenty more are at the Physics Girl YouTube channel.

Happy Physicsing!

Testing what exercise actually does to your butt.

https://www.youtube.com/watch?v=Dh8Yd0mQnB8

What stretching actually does to your body.

https://www.youtube.com/watch?v=1JgBp7dX4AU

Can you guess this note? Perfect pitch and physics.

https://www.youtube.com/watch?v=-65FMJTKGLs

Sunscreen in the UV.

https://www.youtube.com/watch?v=GRD-xvlhGMc

Does this look like white to you?

https://www.youtube.com/watch?v=uNOKWoDtbSk&t=5s

The projector illusion.

https://www.youtube.com/watch?v=Xp6bxCh_p7c

Are MRIs safe?
https://www.youtube.com/watch?v=hlRpl_GMPPc

My dad was struck by lightening (TWICE!)
https://www.youtube.com/watch?v=9-E7koihWF4

Videos for PHY 3250, Biological Physics

Last fall, I recorded my lectures for my PHY 3250 (Biological Physics) class, and posted them on YouTube. The videos are not great; they are nowhere near professional quality, and often the chalkboard is difficult to read. I originally recorded them as a backup for my students, in case they missed a class or wanted to review something they heard me say in a lecture. Nevertheless, I think that students and instructors may find these videos useful.

My Biological Physics class covers the first ten chapters in Intermediate Physics for Medicine and Biology. Topics include biomechanics, fluid dynamics, the exponential function, biothermodynamics, diffusion, osmotic pressure, bioelectricity, biomagnetism, and feedback.

Some videos are missing: Monday, September 30 was Exam 1; Wednesday, October 30 was Exam 2; Wednesday, November 27 the class played Trivial Pursuit IPMB; and Friday, November 29 was the day after Thanksgiving.

If you are sitting at home self-quarantining with nothing to do, feel free to binge.

Enjoy!

Wednesday, September 4, 2019. Introduction.

https://www.youtube.com/watch?v=BpZO44XlKps&t=2915s

Friday, September 6, 2019. Biomechanics.

https://www.youtube.com/watch?v=z2-_ynK7mC4&t=28s

Monday, September 9, 2019. Hydrostatics.

https://www.youtube.com/watch?v=JIzzxaT8GUs&t=3420s

Wednesday, September 11, 2019. Fluid Dynamics.

https://www.youtube.com/watch?v=epBEt4IrAog

Friday, September 13, 2019.  The exponential function.

https://www.youtube.com/watch?v=PqoDQMIn2Z8

Monday, September 16, 2019. Scaling.

https://www.youtube.com/watch?v=8HapJ_irxb8

Wednesday, September 18, 2019. Boltzmann factor.

https://www.youtube.com/watch?v=4Yh9D4-X2cI

Friday, September 20, 2019. Heat capacity.

https://www.youtube.com/watch?v=7KNYTSKfqDE

Monday, September 23, 2019. Heat transfer.

https://www.youtube.com/watch?v=8zHLDM_rzK8

Wednesday, September 25, 2019. Review for Exam 1.

https://www.youtube.com/watch?v=vJ0mlt4zAJY

Friday, September 27, 2019. Review for Exam 1 (cont.).

https://www.youtube.com/watch?v=m-e04OyfExQ

Wednesday, October 2, 2019. Heat conduction.

https://www.youtube.com/watch?v=eOvRKLR5Uvk

Friday, October 4, 2019. Diffusion.

https://www.youtube.com/watch?v=qO1rNWI0l6I

Monday, October 7, 2019. Diffusion and convection.

https://www.youtube.com/watch?v=ChJs5D7stA8

Wednesday, October 9, 2019. Osmotic pressure.

https://www.youtube.com/watch?v=wdUli6XsSkY

Friday, October 11, 2019. Countercurrent exchange.

https://www.youtube.com/watch?v=ObwElQROX8s

Monday, October 14, 2019. Bioelectricity.

https://www.youtube.com/watch?v=x6Voj3puTks

Wednesday, October 16, 2019. Hodgkin & Huxley model.

https://www.youtube.com/watch?v=kZVN14rrzLg

Friday, October 18, 2019. Hodgkin & Huxley model (cont.).

https://www.youtube.com/watch?v=fg34n0ShzFc

Monday, October 21, 2019. The cable equation.

https://www.youtube.com/watch?v=alUujlzUPQ8

Wednesday, October 23, 2019. Action potential propagation.

https://www.youtube.com/watch?v=ZmmDQLvT0sI

Friday, October 25, 2019. Review for Exam 2.

https://www.youtube.com/watch?v=7_WNLgDPydY

Monday, October 28, 2019. Review for Exam 2 (cont.).

https://www.youtube.com/watch?v=onPXPYwedlg

Friday, November 1, 2019. Extracellular stimulation of nerves.

https://www.youtube.com/watch?v=fj6v7xvFwus

Monday, November 4, 2019. Extracellular potentials and the dipole.

https://www.youtube.com/watch?v=gcXAlrJPxCE

Wednesday, November 6, 2019. The heart.

https://www.youtube.com/watch?v=oalxSQ0YN-w

Friday, November 8, 2019. The electrocardiogram.

https://www.youtube.com/watch?v=lxYbFaNfVYQ

Monday, November 11, 2019. Pacemakers and defibrillators.

https://www.youtube.com/watch?v=49T0eNyKxHY

Wednesday, November 13, 2019. The electroencephalogram.

https://www.youtube.com/watch?v=UP-twf7MK3Y

Friday, November 15, 2019. Biomagnetism.

https://www.youtube.com/watch?v=5ghPyovl9VM

Monday, November 18, 2019. Transcranial magnetic stimulation.

https://www.youtube.com/watch?v=4XQD0viWpy4

Wednesday, November 20, 2019. Cardiac restitution.

https://www.youtube.com/watch?v=oef4la5psq4

Friday, November 22, 2019. Cellular automata.

https://www.youtube.com/watch?v=e0GiDcBlem0

Monday, November 25, 2019. Feedback.

https://www.youtube.com/watch?v=1n1DMCq-2TM

Monday, December 2, 2019. Feedback (cont.).

https://www.youtube.com/watch?v=r7RoUdMH7Ds

Monday, December 4, 2019. Review for Exam 3.

https://www.youtube.com/watch?v=QT2SVuZKOkM

 Wednesday, December 6, 2019. Review for Exam 3 (cont.).

https://www.youtube.com/watch?v=LgADeCoy_c8

The Ophthalmoscope

The First Steps in Seeing,
by Robert Rodieck.

In The First Steps in Seeing, Robert Rodieck describes the ophthalmoscope.

Light passes into the eye through the pupil, and continues through its mainly transparent interior to reach the retina. The portion of the light that is not caught by the photoreceptors is either absorbed or scattered in all directions by the underlying tissues. Some of the scattered light passes back through the pupil and out of the eye. But when we look into another person’s pupil, the back of the eye, or fundus, appears black. This is because the optical pathway of the light that enters the eye and falls on a given region of the fundus is the same as that of the light scattered from that region, which leaves the eye through the pupil. In effect, in order to see the interior of the eye under ordinary conditions, one has to place one’s head into this common pathway of the light.

A brilliant young clinician, Hermann von Helmholtz (1821-1894), grasped this issue, and realized that all he needed to do to see the interior of another person’s eye was to devise an optical device by which he could get both his head and the light into the pathway. He did so by placing a piece of glass between his eye and the patient’s and angling the glass so that it partially reflected the light from a lamp into the patient’s eye… The piece of glass and the lamp formed a device termed an ophthalmoscope (Greek opthalmos = eye + skopion, from skopein = to see). Modern ophthalmoscopes have a built-in light source, colored filters to emphasize some aspect of the view, and lenses to correct for any error in the optics of the clinician or patient (i.e., lenses of the same power that they might use in spectacles.)

The picture below shows a simple ophthalmoscope, which consists of just a light source, a semi-reflecting mirror, and two eyes.

An ophthalmoscope.

An image of the retina, as might be seen using an ophthalmoscope, is shown below. The dark patch in the center is the fovea, where the cone density is greatest. The light patch to its right is the optic disc where the optic nerve enters the blood vessels converge.

An image of the retina.
From Häggström, Mikael (2014). “Medical Gallery of Mikael Häggström 2014.”
WikiJournal of Medicine 1 (2). DOI:10.15347/wjm/2014.008

Learn more about the ophthalmoscope and its history from a website maintained by the College of Optometrists. Learn more about Helmholtz in one of my previous posts. Learn more about the physics of the eye in Chapter 14 of Intermediate Physics for Medicine and Biology.

The ophthalmoscope is yet one more example of how physics contributes of medicine and biology. 

Visual Acuity

The coronavirus has led to events being canceled, people being isolated, and classes being disrupted. What can I do to help? I plan to post to this blog more often, so students can learn how physics is applied to medicine and biology. I can’t promise daily posts, but I’ll do what I can.

Russ Hobbie and I discuss visual acuity—how sharp your vision is—in Chapter 14 of Intermediate Physics for Medicine and Biology.

The maximum photopic (bright-light) resolution of the eye is limited by four effects: diffraction of the light passing through the circular aperture of the pupil (5–8 μm), spacing of the receptors (≈ 3 μm), chromatic and spherical aberrations (10–20 μm), and noise in eyeball aim (a few micrometers)… The total standard deviation is (6²+3²+15²+5²)^1/2 = 17 μm in the image on the retina. Since the diameter of the eyeball is about 2 cm, this corresponds to an angular size... of (17 × 10^-6)/(2 × 10^-2) = 8.5 × 10^-4 rad = 0.048 ° = 2.9 min of arc.

Let’s examine the factors contributing to acuity, one by one.

Diffraction

The Rayleigh criterion specifies the minimum angular separation, θ_min, of two objects that can just be resolved. The criterion can be expressed as θ_min = 1.22 λ/D, where λ is the wavelength of light and D is the diameter of the pupil. If we use light from the center of the visible spectrum—say green light with wavelength 550 nm—and a pupil diameter of 2.5 mm, we get θ_min = 0.00027 radians, which is 0.015° or 0.93 minutes of arc. If we take the eyeball diameter to be 2 cm, that translates into a minimum separation on the retina of 5.4 μm.

The Spacing of Receptors

According to The First Steps in Seeing, by Robert Rodieck, in the fovea cones have a density of about 0.1 per square micron. That translates roughly into a 3 micron separation between cones. The cone density is down by a factor of ten in other parts of the retina.

Chromatic and Spherical Aberration

The First Steps in Seeing,
by Robert Rodieck

Chromatic aberration arises because the index of refraction of the eye, including the lens, depends on wavelength of the light. Therefore, different colors form images at different locations. Spherical aberration arises because a spherical lens is not ideal for forming images; off-axis rays have a different focal point than on-axis rays. The eye and its lens, however, are not truly spherical, so when we speak of spherical aberration in the context of vision, we mean heterogeneities in the imaging system that cause the image to be blurred. Rodieck says

At night the pupil is fully open, and the spread of photons is due mainly to the optical imperfections of the eye; the effects of these imperfections increase rapidly with pupil size. The other factor that contributes to the spread of photons is intrinsic to the nature of how photons go from place to place, and is termed diffraction. This factor is not significant here, but in daylight, when the pupil is small, the spread of photons in the retinal image is due mainly to diffraction.

Noise in Eyeball Aim

Rodieck explains how your gaze is always moving, even when staring at a stationary object.

Gazing at a stationary object also involves smooth eye movements. This is because your head is always in slight motion as the muscle of your body and neck attempt to maintain your posture. Thus when you look as steadily as possible at some small stationary object, such as a pebble on the ground, your slight head movements cause the image of the pebble to move on your retina.

This motion has some noise, which limits our visual acuity.

A Snellen chart.

A Snellen chart is the traditional way to measure visual acuity. You stand 6 meters (20 feet) from the chart and read the letters with one eye. If you plan to print out this chart, you need to make sure it is the correct size; the topmost “E” should be 87.3 mm tall. In that case, the 20/20 row corresponds to letters that subtend 5 minutes of arc.

The Big Dipper. The second star in the handle is a double.

Another test of visual acuity arises because the second star from the end of the handle of the Big Dipper is actually a double star: Mizar and Alcor. They are separated by 12 minutes of arc. George Bohigian published an article in Survey of Ophthalmology (Volume 53, Pages 536-539, 2008) about “An Ancient Eye Test—Using the Stars.” He begins

A common vision test in ancient Persia used the double star of the Big Dipper in the constellation Ursa Major or the Big Bear. This vision evaluation test was given to elite warriors in the ancient Persian army and was called “the test” or “the riddle.” The desert Arabs, especially the Bedouins, used the separation of Mizar and Alcor as a test of good vision. The separation of these two stars is known as the Arab Eye Test , and has been used in antiquity to test children's eyesight. This article explores the origin, history, and practicality of this eye test and how it correlates with the present-day Snellen visual acuity test.

He concludes

The Arab Eye Test using the double star of Mizar and Alcor remains a practical test of visual acuity and visual function as it was over 1000 years ago. This test is somewhat equivalent to the 20/20 in the Snellen visual acuity nomenclature. This is the first report that correlates the Mizar–Alcor naked eye test with the current Snellen visual acuity test. With the spread of Islam from Spain to Central Asia, the Arabs brought their knowledge of astronomy mixed with the traditions of Greece, India, Babylonia, and Persia to Western civilization.

Throughout our history the stars have been a constant guide to navigation, measure the seasons, to divine the future, and to measure eyesight. The Arab Eye test is an example of how a natural phenomenon has been used for a practical purpose.

Arguing With Zombies

Arguing With Zombies:
Economics, Politics, and the
Fight for a Better Future,
by Paul Krugman.

Recently I read Arguing With Zombies: Economics, Politics, and the Fight for a Better Future, by Paul Krugman. The book is a collection of editorials and blog posts Krugman wrote for the New York Times, plus a few other previously-published articles. I enjoy Krugman’s writings, but what do they have to do with biological physics or medical physics? Based on the first 390 pages of his book, the answer is: nothing. But near the end was a 1993 article that appeared in The American Economist titled “How I Work” that is relevant to Intermediate Physics for Medicine and Biology. One feature I like best about IPMB is its emphasis on deriving simple “toy models” that provide insight. Simple models aren’t in fashion in biomedical research, but I like them and so does Krugman.

“How I Work” lists Krugman’s four basic rules governing his research. You can read excerpts of his analysis below. Whenever he starts applying his rules specifically to economics, just replace all the financial talk with illustrations from physics applied to medicine and biology.

Listen to the Gentiles

What I mean by this rule is “Pay attention to what intelligent people are saying, even if they do not have your customs or speak you analytical language.”…

I am a strong believer in the importance of models, which are to our minds what spear-throwers were to stone age arms: they greatly extend the power and range of our insight. In particular, I have no sympathy for those people who criticize the unrealistic simplifications of model builders, and imagine that they achieve greater sophistication by avoiding stating their assumptions clearly. The point is to realize that economic models are metaphors, not truth.

For a physicist working in medicine and biology, the “gentiles” would be the biologists and medical doctors. They have much to tell us. For example, when I worked at the National Institutes of Health I learned a lot about magnetic stimulation of nerves from Mark Hallett and Leo Cohen, even if sometimes they mixed up their electricity and magnetism.

I like Krugman’s emphasis on using models to extend our insight. Models may not be as common in pure physics, where you can deduce things from fundamental principles, but biology is so complicated that you can rarely start from Schrödinger's equation and get anywhere. You build models to make sense of biological complexity.

Question the Question

In people in a field have bogged down on questions that seem very hard, it is a good idea to ask whether they are really working on the right questions. Often some other question is not only easier to answer but actually more interesting!

Organisms are so complex that often the right questions aren’t obvious. It’s hard to teach a student how to ask better questions, but we must try.

Dare to be Silly

If you want to publish a paper in economic theory, there is a safe approach: make a conceptually minor but mathematically difficult extension to some familiar model. Because the basic assumptions of the model are already familiar, people will not regard them as strange; because you have done something technically difficult, you will be respected for your demonstration of firepower. Unfortunately, you will not have added much to human knowledge.

What I found myself doing in the new trade theory was pretty much the opposite. I found myself using assumptions that were unfamiliar, and doing very simple things with them. Doing this requires a lot of self-confidence, because initially people (especially referees) are almost certain not simply to criticize your work but to ridicule it….

The age of creative silliness is not past. Virtue, as an economic theorist, does not consist in squeezing the last drop of blood out of assumptions that have come to seem natural because they have been used in a few hundred earlier papers. If a new set of assumptions seems to yield a valuable set of insights, then never mind if they seem strange.

Throughout my career, I have developed simple models. For example, one of my favorite publications is “How to Explain Why Unequal Anisotropy Ratios is Important Using Pictures but No Mathematics.” It consists of some almost silly hand-waving that is amazingly effective at explaining how electric fields interact with cardiac tissue. Another example is my paper “Virtual Electrodes Made Simple” in which I use a trivial little cellular automaton to explain how certain cardiac arrhythmias begin.

Biomedical engineers are doing some incredibly sophisticated calculations to simulate how our bodies work, and these studies are necessary and valuable. But I believe that for those of us who apply physics to medicine and biology, the age of creative silliness expressed through simple models is not yet over. That’s why Russ Hobbie and I stress building models in Intermediate Physics for Medicine and Biology.

Simplify, Simplify

The injunction to dare to be silly is not a license to be undisciplined. In fact, doing really innovative theory requires much more intellectual discipline than working in a well-established literature. What is really hard is to stay on course: since the terrain is unfamiliar, it is all too easy to find yourself going around in circles…. And it is also crucial to express your ideas in a way that other people, who have not spent the last few years wrestling with your problems and are not eager to spend the next few years wrestling with your answers, can understand without too much effort.

Fortunately, there is a strategy that does double duty: it both helps you keep control of your own insights, and makes those insights accessible to others. The strategy is: always try to express your ideas in the simplest possible model. The act of stripping down to this minimalist model will force you to get to the essence of what you are trying to say….

The downside of this strategy is, of course, that many of your colleagues will tend to assume that an insight that can be expressed in a cute little model must be trivial and obvious—it takes some sophistication to realize that simplicity may be the result of years of hard thinking…. There is a special delight in managing not only to boldly go where no economist has gone before, but to do so in a way that seems after the fact to be almost child’s play.

Physicists working in medicine share some of the frustrations that Krugman experiences. Reviewers of papers—and especially reviewers of grant proposals for the National Institutes of Health—often don’t appreciate simple models. My simulations of cardiac electrophysiology have always lacked the particular ion channel that the referee believed was critical, and my biomechanics models tend to use simplifications such as linear strains that trigger objections. (A referee for one of my National Science Foundation applications claimed “this proposal should never have been submitted.”😮)

I often discard biological realism in order to focus on the one or two new features of a model. I’m not asserting that the discarded behavior is unimportant. Rather, I want a simple model so I can highlight the new feature that I’m studying. I don’t want my message to be frittered away by detail. Like Thoreau, Krugman and I strive to simplify, simplify! I hope students using Intermediate Physics for Medicine and Biology learn to appreciate the value of a simple model.

Read “How I Work” online for free.

Listen to Paul Krugman explain how he revolutionized trade theory.

He and I are both big Asimov fans.

https://www.youtube.com/watch?v=pfte6Ynpe8w