Bursting

Last week in this blog I talked briefly about bursting in pancreatic beta cells. A bursting cell fires several action potential spikes consecutively, followed by an extended quiescent period, followed again by another burst of action potentials, and so on. One of the first and best-known models for bursting was developed by James Hindmarsh and Malcolm Rose (“A Model of Neuronal Bursting Using Three Coupled First Order Differential Equations,” Proceedings of the Royal Society of London, B, Volume 221, Pages 87–102, 1984). Their analysis was an extension of the FitzHugh-Nagumo model, with an additional variable governed by a very slow time constant. Their system of equations is

dx/dt = y – x³ + 3 x² – z + I

dy/dt = 1 – 5 x² – y

dz/dt = 0.001 [ 4(x + 1.6) – z]

where x is the membrane potential (appropriately made dimensionless), y is a recovery variable (like a sodium channel inactivation gate), z is the slow bursting variable, and I is an external stimulus current. For some values of I, this model predicts bursting behavior.

Bursting: The Genesis of
Rhythm in the Nervous System,
by Stephen Coombes and Paul Bressloff.

There is an entire book dedicated to this topic: Bursting--The Genesis of Rhythm in the Nervous System, by Stephen Coombes and Paul Bressloff (World Sci. Pub. Co., 2005). The first chapter, co-written by Hindmarsh, provides a little of the history behind the Hindmarsh-Rose model:

The collaboration that led to the Hindmarsh-Rose model began in 1979 shortly after Malcolm Rose joined Cardiff University. The particular project was to model the synchronization of firing of two snail neurons in a relatively simple way that did not use the full Hodgkin-Huxley equations... A natural choice at the time was to use equations of the FitzHugh [type]…

A problem with this choice was that these equations do not provide a very realistic description of the rapid firing of the neuron compared to the relatively long interval between firing. Attempts were made to achieve a more realistic description by making the time constants … voltage dependent. In particular so the rates of change of x and y were much smaller in the subthreshold or recovery phase. These were not convincing and it was not until Malcolm raised the question about whether the FitzHugh equations could account for “tail current reversal” that progress was made.

The modification of the FitzHugh equations to account for tail current reversal was crucial for the development of the Hindmarsh-Rose model.

For those not familiar with the FitzHugh-Nagumo model, see Problem 33 in Chapter 10 of the 4th edition of Intermediate Physics for Medicine and Biology, or see the Scholarpedia article by FitzHugh himself, written before he died in 2007. If you want to see some bursting patterns, check out this youtube video. It is not great, but you will get the drift of what the model predicts.

My friend Artie Sherman also had a chapter in the bursting book, titled “Beyond Synchronization: Modulatory and Emergent Effects of Coupling in Square-Wave Bursting.” He has been working on bursting in pancreatic beta cells for years, as a member (and now chief) of the Laboratory of Biological Modeling in the Mathematical Research Branch, the National Institute of Diabetes, Digestive and Kidney Diseases, part of the National Institutes of Health. His work is the best I am aware of for modeling bursting.

Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations

I enjoy taking a scientific paper and reducing it to a homework problem. For example, one of the new homework problems in the 4th edition of Intermediate Physics for Medicine and Biology is Problem 23 of Chapter 4 (Transport in an Infinite Medium), based on a paper by John Wagner and Joel Keizer.

Problem 23 Calcium ions diffuse inside cells. Their concentration is also controlled by a buffer:

Ca + B ⇐⇒ CaB.

The concentrations of free calcium, unbound buffer, and bound buffer ([Ca], [B], and [CaB]) are governed, assuming the buffer is immobile, by the differential equations

∂[Ca]/∂t= D∇²[Ca] − k⁺[Ca][B] + k⁻[CaB],
∂[B]/∂t= −k⁺[Ca][B] + k⁻[CaB],
∂[CaB]/∂t= k⁺[Ca][B] − k⁻[CaB].

(a) What are the dimensions (units) of k⁺ and k⁻ if the concentrations are measured in mole l⁻¹ and time in s?

(b) Derive differential equations governing the total calcium and buffer concentrations, [Ca]_T = [Ca]+[CaB] and [B]_T= [B] + [CaB] . Show that [B]_T is independent of time.

(c) Assume the calcium and buffer interact so rapidly that they are always in equilibrium:

[Ca][B]/[CaB]= K,

where K = k⁻/k⁺.Write [Ca]_T in terms of [Ca] , [B]_T , and K (eliminate [B] and [CaB]).

(d) Differentiate your expression in (c) with respect to time and use it in the differential equation for [Ca]_T found in (b). Show that [Ca] obeys a diffusion equation with an “effective” diffusion constant that depends on [Ca]:

D_eff = D/(1 + K [B]_T/(K+[Ca])²) .

(e) If [Ca] < < K and [B]_T = 100K (typical for the endoplasmic reticulum), determine D_eff/D.

For more about diffusion with buffers, see Wagner and Keizer (1994).

The reference and abstract of the paper is given below.

John Wagner and Joel Keizer (1994) “Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations,” Biophysical Journal, Volume 67, Pages 447–456.

Based on realistic mechanisms of Ca²⁺ buffering that include both stationary and mobile buffers, we derive and investigate models of Ca²⁺ diffusion in the presence of rapid buffers. We obtain a single transport equation for Ca²⁺ that contains the effects caused by both stationary and mobile buffers. For stationary buffers alone, we obtain an expression for the effective diffusion constant of Ca²⁺ that depends on local Ca²⁺ concentrations. Mobile buffers, such as fura^-2, BAPTA, or small endogenous proteins, give rise to a transport equation that is no longer strictly diffusive. Calculations are presented to show that these effects can modify greatly the manner and rate at which Ca²⁺ diffuses in cells, and we compare these results with recent measurements by Allbritton et al. (1992). As a prelude to work on Ca²⁺ waves, we use a simplified version of our model of the activation and inhibition of the IP3 receptor Ca²⁺ channel in the ER membrane to illustrate the way in which Ca²⁺ buffering can affect both the amplitude and existence of Ca²⁺ oscillations.

John Wagner is currently with the Functional Genomics and Systems Biology Group of the IBM T. J. Watson Research Center. In the mid 1990s he was a research assistant with Joel Keizer.

Joel Keizer was a long-time member of the University of California at Davis. A UC Davis website states

Joel’s scientific legacy encompassed several fields. Joel originally trained as a chemist at the University of Oregon under Terrell Hill, where he received his doctorate in theoretical physical chemistry, and did postdoctoral work in chemical physics at the Battelle Institute in Columbus, Ohio. He began his career in 1971 at the University of California, Davis, as an assistant professor of chemistry. He pioneered an approach to the thermodynamics of non-equilibrium steady states, which culminated in the monograph, Statistical Thermodynamics of Nonequilibrium Processes in 1987. By this time, he had over 60 journal publications to his credit.

In the 1980s, Joel gradually shifted his research program and focused his powerful intellect on problems within the biological sciences, first on mathematical models of insulin secretion, and later on intracellular calcium oscillations and diffusion. He subsequently transferred his appointment to the Division of Biological Sciences, where both theoreticians and empiricists respected and admired Joel for his strong modeling work and his insightful collaborations with experimental biologists.

I never met Joel Keizer, but I did know a couple of his collaborators, John Rinzel and Arthur Sherman, both at NIH when I was there in the early 1990s. They worked on bursting in pancreatic beta-cells, and published some influential papers with Keizer (for example, see: Sherman, Rinzel, and Keizer (1988) “Emergence of Organized Bursting in Clusters of Pencreatic Beta-Cells by Channel Sharing,” Biophysical Journal, Volume 54, Pages 411–425).

Finally, the paper by Allbritton et al. cited in the Wagner and Keizer paper is:

Allbritton, Meyer, Stryer (1992) “Range of Messenger Action of Calcium-Ion and Inositol 1,4,5-Trisphosphate,” Science, Volume 258, Pages 1812–1815.

Superconductivity

This month marks the hundredth anniversary of the discovery of superconductivity. An article in the magazine IEEE Spectrum states

On April 8, 1911, physicist Heike Kamerlingh Onnes of Leiden University used an intricate glass cryostat to cool mercury down to just a few degrees above absolute zero. Then he scribbled down three words that ultimately marked the discovery of an entirely new physical phenomenon.

The phrase, jotted more than halfway down the page of a messy lab notebook, didn’t really match the occasion. What Kamerlingh Onnes wrote was “Mercury practically zero,” or, according to a more literal translation, “Quick[silver] near-enough dull.” But what he saw was the first evidence of superconductivity, the ability of some substances to conduct electricity with no resistance at all.

You can learn more about this landmark event in a Physics Today September 2010 article “The Discovery of Superconductivity,” by Dirk van Delft and Peter Kes, and the article “Superconductivity’s Smorgasbord of Insights: A Movable Feast,” in the April 8, 2011 issue of Science by Adrian Cho. Also, see the biography of Onnes on Nobelprize.org.

The Quest for Absolute Zero,
by Kurt Mendelssohn.

One of my favorite books is The Quest for Absolute Zero, by Kurt Mendelssohn. He starts his tale in 1877 with the liquefaction of oxygen and then tells the subsequent history of low temperature physics, including the fascinating story of how Onnes liquefied helium and his early superconductivity studies. According to Mendelssohn, the reason mercury was used for the first experiment is because it could be purified:

There was one other metal which might be obtained in an even purer state than gold, and that was mercury. Being a liquid at room temperatures, it can be distilled and re-distilled again and again until an extreme degree of purity is reached. The results were communicated to the Netherlands Royal Academy on the 28th April 1911, when Onnes reported that mercury, as well as a sample of very pure gold, had, at helium temperature, reached resistivities so low that his instruments had failed to detect them. He was particularly intrigued with the behavior of the mercury sample because it had still a fairly high resistance at liquid hydrogen temperatures and could also be recorded at the boiling point of liquid helium but then vanished at lower temperatures.

Russ Hobbie and I discuss superconductivity in Section 8.9 (Detection of Weak Magnetic Fields) of the 4th edition of Intermediate Physics for Medicine and Biology.

The [magnetic] signals from the body are weaker, and their measurement requires higher sensitivity and often special techniques to reduce noise. Hämäläinen et al. (1993) present a detailed discussion of the instrumentation problems. Sensitive detectors are constructed from superconducting materials. Some compounds, when cooled below a certain critical temperature, undergo a sudden transition and their electrical resistance falls to zero. A current in a loop of superconducting wire persists for as long as the wire is maintained in the superconducting state. The reason there is a superconducting state is a well-understood quantum-mechanical effect that we cannot go into here. It is due to the cooperative motion of many electrons in the superconductor [Eisberg and Resnick (1985), Sec. 14.1; Clarke (1994)].

We then go on to discuss superconducting quantum interference device (SQUID) magnetometers, which are often used to measure the small magnetic fields produced by the brain or the heart. Although not discussed in our book, superconductivity is also used in many MRI machines to produce the strong static magnetic field without losses due to heating of a copper coil.

The citations in the quote from our book are to:

Clarke, J. (1994) “SQUIDS,” Scientific American, Volume 1994, Pages 46–53.

Eisberg, R., and R. Resnick (1985) Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd ed. New York, Wiley.

Hämäläinen, M., R. Harri, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa (1993) “Magnetoencephalography—Theory, Instrumentation, and Applications to Noninvasive Studies of the Working Human Brain.” Reviews of Modern Physics, Volume 65, Pages 413–497.

Point/Counterpoint Revisited

In one of my first entries in this blog, I introduced readers to the Point/Counterpoint articles in the journal Medical Physics. I enjoy these articles immensely, and they provide valuable insight into important and controversial questions in the medical physics field. Each issue of Medical Physics contains one Point/Counterpoint, in which a proposition is stated and two prominent medical physicists debate it, one for and one against. They each have “opening statements” and then provide a “rebuttle” to their opponent’s claims. The articles make for more lively reading than a typical scientific paper full of jargon and technical content.

The September 2010 issue of Medical Physics contains a Point/Counterpoint article titled “Ultrasonography is Soon Likely to Become a Viable Alternative to X-Ray Mammography for Breast Cancer Screening.” Arguing for the proposition is Carri Glide-Hurst, a Senior Associate Physicist at Henry Ford Hospital in Detroit. Her opponent is Andrew Maidment, Associate Professor of Radiology at the University of Pennsylvania in Philadelphia. I am in favor of the proposition, for the silly reason that I always root for the home team. (Oakland University, where I work, is about 20 miles north of Detroit, and our medical physics program has several adjunct faculty at Henry Ford Hospital.)

The 4th edition of Intermediate Physics for Medicine and Biology provides much of the scientific background needed to understand this debate. Russ Hobbie and I added a new chapter to the 4th edition that describes ultrasound and its applications to medical imaging (Chapter 13, Sound and Ultrasound). After introducing the wave equation, we describe the decibel intensity scale, attenuation, medical uses of ultrasound, and the Doppler effect. Two chapters are dedicated to understanding x rays and x-ray imaging. In Chapter 15 (Interaction of Photons and Charged Particles with Matter) we analyze the basic mechanisms by which an x-ray photon affects tissue, including the photoelectric effect, Compton scattering, and pair production. Chapter 16 (Medical Use of X Rays) focuses on applications, including a section dedicated entirely to mammography.

Magnetic resonance imaging is another modality that produces no ionizing radiation. It is described in Chapter 18 of Intermediate Physics for Medicine and Biology. However, MRI is expensive, takes a long time, and cannot be used in some patients, such as those with surgical clips. Therefore the American Cancer Society recommends MRI only for a small group of patients.

Which side wins the debate? It’s always hard to say. I’m sure the moderator, Colin Orton, chooses only those questions that do not have obvious answers. Glide-Hurst concludes her opening statement by arguing

Ultrasound poses a practical and affordable solution for screening younger women with dense breasts, pregnant females, and those who do not meet the risk level requirements of breast MRI screening. Overall, whole-breast ultrasound is advantageous because it is volumetric, noninvasive, and nonionizing, and the current literature supports the routine implementation for breast cancer screening, particularly for women with dense breasts.

Maidment ends his opening statement by stating

Since ultrasound can distinguish solid tumors from fluidfilled cysts, it has a clear clinical role as a diagnostic tool in breast imaging. However, ultrasound does not appear useful for routine screening because of lower sensitivity and specificity compared to mammography, the suboptimal imaging of microcalcifications with ultrasonography, and the projected costs.

All things considered, I do know who wins the debate. The winner is the reader, who witnesses two experts carefully weighing the evidence, analyzing the physics, and predicting future trends. I encourage any student reading Intermediate Physics for Medicine and Biology to also browse through recent issues of Medical Physics. If, like me, you’re often short on time, skip the articles and just read Point/Counterpoint. You won’t regret it.

Fukushima Nuclear Reactors

Because of the scary events at the Fukushima nuclear reactors in Japan, the health hazards of radiation is in the news a lot. One place I turn to for authoritative information is the Health Physics Society. Here is what their website says:

As you are well aware, the Japanese experienced the worst earthquake in their history, followed by a devastating tsunami. These natural disasters have had a serious impact on several Japanese nuclear reactors, principally those at the Fukushima Daiichi site. The Health Physics Society is concerned about radiation exposures associated with these reactor problems and desires to keep our members and the concerned public advised on current events associated with the Japanese nuclear plants.

For information on the potential for radiation from the Japanese Nuclear Plants reaching the United States, see this Health Physics Society Ask the Experts FAQ. For information on radiation particle effects on food, read this Bloomberg FAQ.

Details of the status of the reactors at Fukushima are available in a document issued by the Japan Atomic Industrial Forum that is provided here. We will be updating this news item periodically to provide current information.

The Health Physics Society links to an interesting youtube video: an interview with John Boice of Vanderbilt University. He says “the fear is out of proportion to the risk,” and claims this event is no where near the situation in the Chernobyl diasater. (Warning: The interview was on March 20, and events seem to change daily.)

The Health Physics Society website also links to the following statement:

RADIATION RISKS TO HEALTH
A Joint Statement from the American Association of Clinical Endocrinologists, the American Thyroid Association, The Endocrine Society, and the Society of Nuclear Medicine
March 18, 2011

The recent nuclear reactor accident in Japan due to the earthquake and tsunami has raised fears of radiation exposure to populations in North America from the potential plume of radioactivity crossing the Pacific Ocean. The principal radiation source of concern is radioactive iodine including iodine-131, a radioactive isotope that presents a special risk to health because iodine is concentrated in the thyroid gland and exposure of the thyroid to high levels of radioactive iodine may lead to development of thyroid nodules and thyroid cancer years later. During the Chernobyl nuclear plant accident in 1986, people in the surrounding region were exposed to radioactive iodine principally from intake of food and milk from contaminated farmlands. As demonstrated by the Chernobyl experience, pregnant women, fetuses, infants and children are at the highest risk for developing thyroid cancer whereas adults over age 20 are at negligible risk.

Radioiodine uptake by the thyroid can be blocked by taking potassium iodide (KI) pills or solution, most importantly in these sensitive populations. However, KI should not be taken in the absence of a clear risk of exposure to a potentially dangerous level of radioactive iodine because potassium iodide can cause allergic reactions, skin rashes, salivary gland inflammation, hyperthyroidism or hypothyroidism in a small percentage of people. Since radioactive iodine decays rapidly, current estimates indicate there will not be a hazardous level of radiation reaching the United States from this accident. When an exposure does warrant KI to be taken, it should be taken as directed by physicians or public health authorities until the risk for significant exposure to radioactive iodine dissipates, but probably for no more than 1-2 weeks. With radiation accidents, the greatest risk is to populations close to the radiation source. While some radiation may be detected in the United States and its territories in the Pacific as a result of this accident, current estimates indicate that radiation amounts will be little above baseline atmospheric levels and will not be harmful to the thyroid gland or general health.

We discourage individuals needlessly purchasing or hoarding of KI in the United States. Moreover, since there is not a radiation emergency in the United States or its territories, we do not support the ingestion of KI prophylaxis at this time. Our professional societies will continue to monitor potential risks to health from this accident and will issue amended advisories as warranted.

News sources have been reporting that higher-than-normal radiation levels were detected in the United States. These observations say more about our ability to detect small amounts of radiation than about any risk to Americans. People living in the United States are at no risk of health hazards from radiation exposure caused by the Fukushima reactors.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the risk of radiation in Section 13 of Chapter 16 (Medical Use of X Rays). We introduce the unit of the sievert (Sv), one of the most important units used when discussing radiation risk.

Both the sievert and the gray are J kg^-1. Different names are used to emphasize the fact that they are quite different quantities. One is physical, and the other includes biological effects. An older unit … is the rem. 100 rem = 1 Sv.

We then analyze the natural background dose, which is about 3 mSv per year, and which arises from several sources, including cosmic radiation, terrestrial rocks, and inhalation of radon gas.

If you prefer learning from a video, watch Understanding the Reactor Meltdown in Fukushima, Japan from a Physics Perspective on Youtube.

Time will tell if this event turns into a full-scale disaster. At the moment, it is a serious situation, but does not appear to be a serious health hazard, except perhaps for the workers trying to repair the power plants.