Bridging Physics and Biology Teaching Through Modeling

In this blog, I often stress the value of toy models. I’m not the only one who feels this way. Anne-Marie Hoskinson and her colleagues suggest that modeling is an important tool for teaching at the interface of physics and biology (“Bridging Physics and Biology Teaching Through Modeling,” American Journal of Physics, Volume 82, Pages 434–441, 2014). They write

While biology and physics might appear quite distinct to students, as scientific disciplines they both rely on observations and measurements to explain or to make predictions about the natural world. As a shared scientific practice, modeling is fundamental to both biology and physics. Models in these two disciplines serve to explain phenomena of the natural world; they make predictions that drive hypothesis generation and data collection, or they explain the function of an entity. While each discipline may prioritize different types of representations (e.g., diagrams vs mathematical equations) for building and depicting their underlying models, these differences reflect merely alternative uses of a common modeling process. Building on this foundational link between the disciplines, we propose that teaching science courses with an overarching emphasis on scientific practices, particularly modeling, will help students achieve an integrated and coherent understanding that will allow them to drive discovery in the interdisciplinary sciences.

One of their examples is the cardiac cycle, which they compare and contrast with the thermodynamic Carnot cycle. The cardiac cycle is best described graphically, using a pressure-volume diagram. Russ Hobbie and I present a PV plot of the left ventricle in Figure 1.34 of Intermediate Physics for Medicine and Biology. Below, I modify this plot, trying to capture its essence while simplifying it for easier analysis. As is my wont, I present this toy model as a new homework problem.

Sec. 1.19

Problem 38 ½. Consider a toy model for the behavior of the heart’s left ventricle, as expressed in the pressure-volume diagram

(a) Which sections of the cycle (AB, BC, CD, DA) correspond to relaxation, contraction, ejection, and filling?

(b) Which points during the cycle (A, B, C, D) correspond to the aortic value opening, the aortic value closing, the mitral value opening, and the mitral valve closing?

(c) Plot the pressure versus time and the volume versus time (use a common horizontal time axis, but individual vertical pressure and volume axes).

(d) What is the systolic pressure (in mm Hg)?

(e) Calculate the stroke volume (in ml). 

(f) If the heart rate is 70 beats per minute, calculate the cardiac output (in m³ s^–1).

(g) Calculate the work done per beat (in joules).

(h) If the heart rate is 70 beats per minute, calculate the average power output (in watts).

(i) Describe in words the four phases of the cardiac cycle.

(j) What are some limitations of this toy model?

The last two parts of the problem are crucial. Many students can analyze equations or plots, but have difficulty relating them to physical events and processes. Translation between words, pictures, and equations is an essential skill. 

All toy models are simplifications; one of their primary uses is to point the way toward more realistic—albeit more complex—descriptions. Many scientific papers contain a paragraph in the discussion section describing the approximations and assumptions underlying the research.

Below is a Wiggers diagram from Wikipedia, which illustrates just how complex cardiac physiology can be. Yet, our toy model captures many general features of the diagram.

A Wiggers diagram summarizing cardiac physiology.
Source: adh30 revised work by DanielChangMD who revised original work of DestinyQx;
Redrawn as SVG by xavax, CC BY-SA 4.0, via Wikimedia Commons

I’ll give Hoskinson and her coworkers the last word.

“We have provided a complementary view to transforming undergraduate science courses by illustrating how physics and biology are united in their underlying use of scientific models and by describing how this practice can be leveraged to bridge the teaching of physics and biology.”

The Wiggers diagram explained in three minutes!
https://www.youtube.com/watch?v=0sogXvxxV0E

Magnetic Coil Stimulation of Straight and Bent Amphibian and Mammalian Peripheral Nerve in Vitro: Locus of Excitation

In this blog, I like to highlight important journal articles. One of my favorites is “Magnetic Coil Stimulation of Straight and Bent Amphibian and Mammalian Peripheral Nerve in Vitro: Locus of Excitation” by Paul Maccabee and his colleagues (Journal of Physiology, Volume 460, Pages 201–219, 1993). This paper isn’t cited in Intermediate Physics for Medicine and Biology, but it should be. It’s related to Homework Problem 32 in Chapter 8, about magnetic stimulation of a peripheral nerve.

The best part of Maccabee’s article is the pictures. I reproduce three of them below, somewhat modified from the originals. 

Fig. 1. The electric field and its derivative produced by magnetic stimulation using a figure-of-eight coil. Based on an illustration in Maccabee et al. (1993).

The main topic of the paper was how an electric field induced in tissue during magnetic stimulation could excite a nerve. The first order of business was to map the induced electric field. Figure 1 shows the measured y-component of the electric field, E_y, and its derivative dE_y/dy, in a plane below a figure-of-eight coil. The electric field was strongest under the center of the coil, while the derivative had a large positive peak about 2 cm from the center, with a large negative peak roughly 2 cm in the other direction. Maccabee et al. included the derivative of the electric field in their figure because cable theory predicted that if you placed a nerve below the coil parallel to the y axis, the nerve would be excited where −dE_y/dy was largest. 

Fig. 2. An experiment to show how the stimulus location changes with the stimulus polarity. Based on an illustration in Maccabee et al. (1993).

The most important experiment is shown in Figure 2. The goal was to test the prediction that the nerve was excited where −dE_y/dy is maximum. The method was to simulate the nerve using one polarity and then the other, and determine if the location where the nerve is stimulated shifted by about 4 cm, as Figure 1 suggests.

A bullfrog sciatic nerve (green) was dissected out of the animal and placed in a bath containing saline (light blue). An electrode (dark blue dot) recorded the action potential as it reached the end of the nerve. A figure-of-eight coil (red) was placed under the bath. First Maccabee et al. stimulated with one polarity so the stimulation site was to the right of the coil center, relatively close to the recording electrode. The recorded signal (yellow) consisted of a large, brief stimulus artifact followed by an action potential that propagated down the nerve with a speed of 40.5 m/s. Then, they reversed the stimulus polarity. As we saw in Fig. 1, this shifted the location of excitation to another point to the left of the coil center. The recorded signal (purple) again consisted of a stimulus artifact followed by an action potential. The action potential, however, arrived 0.9 ms later because it started from the left side of the coil and therefore had to travel farther to reach the recording electrode. They could determine the distance between the stimulation sites by dividing the speed by the latency shift; (40.5 m/s)/(0.9 ms) = 4.5 cm. This was almost the same as the distance between the two peaks in the plot of dE_y/dy in Figure 1. The cable theory prediction was confirmed. 

Fig. 3. The effect of insulating obstacles on the site of magnetic stimulation. Based on an illustration in Maccabee et al. (1993).

In another experiment, Maccabee and his coworkers further tested the theory (Fig. 3). The electric field induced during magnetic stimulation was perturbed by an obstruction. They placed two insulating lucite cylinders (yellow) on either side of the nerve, which forced the induced current to pass through the narrow opening between them. This increased the strength of the electric field (green), and caused the negative and positive peaks of the derivative of the electric field (dark blue) to move closer together. Cable theory predicted that if the cylinders were not present the latency shift upon change in polarity would be relatively long, while with the cylinders the latency shift would be relatively short. The experiment found a long latency (1.2 ms) without the cylinders and a short latency (0.3 ms) with them, confirming the prediction. This behavior might be important when stimulating, say, the median nerve as it passes between two bones in the arm.

In addition, Maccabee examined nerves containing bends, which created “hot spots” where excitation preferentially occurred. They also examined polyphasic stimuli, which caused excitation at both the negative and positive peaks of dE_y/dy nearly simultaneously. I won’t reproduce all their figures, but I recommend you download a copy of the paper and see them for yourself.

Why do I like this paper so much? For several reasons.

• It’s an elegant example of how theory suggests an experiment, which once confirmed leads to additional predictions, resulting in even more experiments, and so on; a virtuous cycle. 
• Their illustrations are informative and clear (although I do like the color in my versions). You should be able to get the main point of a scientific paper by merely looking through the figures, and you can do that with Maccabee et al.’s article.
  
• In vitro experiments (nerve in a dish) are nice because they strip away all the confounding details of in vivo (nerve in an arm) experiments. You can manipulate the system (say, by adding a couple lucite cylinders) and determine how the nerve responds. Of course, some would say in vivo experiments are better because they include all the complexities of an actual arm. As you might guess, I prefer the simplicity and elegance of in vitro experiments. 
• If you want a coil that stimulates a peripheral nerve below its center, as opposed to off to one side, you can use a four-leaf-coil.
  
• Finally, I like this article because Peter Basser and I were the ones who made the theoretical prediction that magnetic stimulation should occur where −dE_y/dy, not E_y, is maximum (Roth and Basser, “Model of the Stimulation of a Nerve Fiber by Electromagnetic Induction,” IEEE Transactions on Biomedical Engineering, Volume 37, Pages 588-597, 1990). I always love to see my own predictions verified. 

I’ve lost track of my friend Paul Maccabee, but I can assure you that he did good work studying magnetic stimulation of nerves. His article is well worth reading.

A Mechanism for the Dip in the Strength-Interval Curve During Anodal Stimulation of Cardiac Tissue

Scientific articles aren’t published until they’ve undergone peer review. When a manuscript is submitted to a scientific journal, the editor asks several experts to read it and provide their recommendation. All my papers were reviewed and most were accepted and published, although usually after a revision. Today, I’ll tell you about one of my manuscripts that did not survive peer review. I’m glad it didn’t.

In the early 1990s, I was browsing in the library at the National Institutes of Health—where I worked—and stumbled upon an article by Egbert Dekker about the dip in the anodal strength-interval curve.

Dekker, E. (1970)  “Direct Current Make and Break Thresholds for Pacemaker Electrodes on the Canine Ventricle,” Circulation Research, Volume 27, Pages 811–823.

In Dekker’s experiment, he stimulated a dog heart twice: first (S1) to excite an action potential, and then again (S2) during or after the refractory period. You expect that for a short interval between S1 and S2 the tissue is still refractory, or unexcitable, and you’ll get no response to S2. Wait a little longer and the tissue is partially refractory; you’ll excite a second action potential if S2 is strong enough. Wait longer still and the tissue will have returned to rest; a weak S2 will excite it. So, a plot of S2 threshold strength versus S1-S2 interval (the strength-interval curve) ought to decrease.

Dekker observed that the strength-interval curve behaved as expected when S2 was provided by a cathode (an electrode having a negative voltage). A positive anode, however, produced a strength-interval curve containing a dip. In other words, there was an oddball section of the anodal curve that increased with the interval. 

The cathodal and anodal strength-interval curves.

Moreover, Dekker observed two types of excitation: make and break. Make occurred after a stimulus pulse began, and break after it ended. Both anodal and cathodal stimuli could cause make and break excitation. (For more about make and break, see my previous post.)

I decided to examine make and break excitation and the dip in the anodal strength-interval curve using a computer simulation. The bidomain model (see Section 7.9 in Intermediate Physics for Medicine and Biology) represented the anisotropic electrical properties of cardiac tissue. The introduction of the resulting paper stated

In this study, my primary goal is to present a hypothesis for the mechanism of the dip in the anodal strength-interval curve: The dip arises from a complex interaction between anode-break and anode-make excitation. This hypothesis is explored in detail and supported by numerical calculations using the bidomain model. The same mechanism may explain the no-response phenomenon. I also consider the induction of periodic responses [a cardiac arrhythmia] from a premature anodal stimulus. The bidomain model was used previously to investigate the cathodal strength-interval curve; in this study, these calculations are extended to investigate anodal stimulation.

When I submitted this manuscript to a journal, it was rejected! Why? It contained a fatal flaw. To represent how the membrane ion channels opened and closed, I had used the Hodgkin and Huxley model, appropriate for a nerve axon. Yet, the nerve and cardiac action potentials are different. For example, the action potential in the heart lasts a hundred times longer than in a nerve.

After swearing and pouting, I calmed down and redid the calculation using an ion channel model more appropriate for cardiac tissue, and then published a series of papers that are among my best.

Roth, B. J. (1995) “A Mathematical Model of Make and Break Electrical Stimulation of Cardiac Tissue by a Unipolar Anode or Cathode,” IEEE Transactions on Biomedical Engineering, Volume 42, Pages 1174-1184.

Roth, B. J. (1996) “Strength-Interval Curves for Cardiac Tissue Predicted Using the Bidomain Model,” Journal of Cardiovascular Electrophysiology, Volume 7, Pages 722-737.

Roth, B. J. (1997) “Nonsustained Reentry Following Successive Stimulation of Cardiac Tissue Through a Unipolar Electrode,” Journal of Cardiovascular Electrophysiology, Volume 8, Pages 768-778.

I kept a copy of the rejected paper (you can download it here). It’s interesting for what it got right, and what it got wrong.
 

The response of cardiac tissue to S1/S2 stimulation, for a cathode (top) and anode (bottom).
"Strength" is the S2 strength, and "Interval" is the S1-S2 interval.
"No Response" (N) means S2 did not excite an action potential,
"Make" means an action potential was excited after S2 turned on,
"Break" means an action potential was excited after S2 turned off, and
"E" means one (gray) or more (black) extra action potentials were triggered by S2 (reentry).
Beware! These calculations were from my rejected paper.

What it got right: The paper identified make and break regions of the strength-interval curve, predicted a dip in the anodal curve but not the cathodal curve, and produced reentry for strong stimuli near the make/break transition. It even reproduced the no-response phenomenon, in which a strong stimulus excites an action potential but an even stronger stimulus does not.

What it got wrong: Cathode-break excitation was missing. The mechanism for anode-break excitation was incorrect. The Hodgkin-Huxley model predicts that anode-break excitation arises from the ion channel kinetics (for the cognoscenti, hyperpolarization removes sodium channel inactivation). This type of anode-break excitation doesn’t happen in the heart but did occur in my simulations, leading me astray. This wrong anode-break mechanism led to wrong explanations for the dip in the anodal strength-interval curve and the no-response phenomenon. (For the correct mechanism, look here.)

Below I reproduce the final paragraph of the manuscript, with the parts that were wrong in red.

“What useful conclusions result from these simulations?” is a fair question, given the limitations of the model. I believe the primary contribution is a hypothetical mechanism for the dip in the anodal strength-interval curve. The dip may arise from a complex interaction of anode-break and anode-make stimulation: A nonpropagating active response at the virtual cathode raises the threshold for anode-break stimulation under the anode. The same interaction could explain the no-response phenomenon. A second contribution is a hypothesis for the mechanism generating periodic responses to strong anodal stimuli: Anode-make stimulation cannot propagate back toward the anode because of the strong hyperpolarization, and the subsequent excitation of the tissue under the anode occurs with sufficient delay that a reentrant loop arises. This hypothesis is related to, but not the same as, the one presented by Saypol and Roth for cathodally induced periodic responses. These mechanisms are suggested by my numerical simulation using a simplified model; whether they play a role in the behavior of real cardiac tissue is unknown. Hopefully, my results will encourage more accurate simulations and, even more importantly, additional experimental measurements of the spatial-temporal distribution of transmembrane potential around the stimulating electrode during premature stimulation of cardiac tissue.

Even though this manuscript was flawed, it foreshadowed much of my research program for the mid 1990s; it was all there, in the rough. Moreover, in this case the reviewers were right and I was wrong. At the time, I was angry that anyone would reject my paper. Now, in retrospect, I realize they did me a favor; I benefited from their advice. For any young scientist who might be reading this post, don’t be too discouraged by critical reviews and rejection. Give yourself a day to whine and fuss, then fix the problems that need fixing and move on. That’s the way peer review works.

The Spectrum of Scattered X-Rays

Chapter 15 of Intermediate Physics for Medicine and Biology discusses Compton Scattering. In this process, an x-ray photon scatters off a free electron, creating a scattered photon and a recoiling electron. The wavelength shift between the incident and scattered photons, Δλ, is given by the Compton scattering formula (Eq. 15.11 in IPMB)

Δλ = h/mc (1 − cosθ) ,

where h is Planck’s constant, c is the speed of light, m is the mass of the electron, and θ is the scattering angle. The quantity h/mc is called the Compton wavelength of the electron.

I enjoy studying experiments that first measure fundamental quantities like the Compton wavelength. Such an experiment is described in Arthur Compton’s article

Compton, A. H. (1923) The Spectrum of Scattered X-Rays. Physical Review, Volume 22, Pages 409–413.

Compton’s x-ray source (emitting the K_α line from molybdenum) irradiated a scatterer (graphite). He performed his experiment for different scattering angles θ. For each angle, he first collimated the scattered beam (using small holes in lead sheets) and then reflected it from a crystal (calcite). The purpose of the crystal was to determine the wavelength of the scattered photon by x-ray diffraction. Russ Hobbie and I don’t analyze x-ray diffraction in IPMB. The wavelength λ of the diffracted photon is given by Bragg’s law

λ = 2 d sinϕ ,

where d is the spacing of atomic planes (for calcite, d = 3.036 Å) and ϕ is the angle between the x-ray beam and the crystal surface. For fixed θ, Compton would rotate the crystal, thereby scanning ϕ and analyzing the beam as a function of wavelength. The intensity of the beam would be recorded by a detector (an ionization chamber).

Let’s analyze Compton’s experiment in a new homework problem.

Section 15.4

Problem 7½. Use the data below to calculate the Compton wavelength of the electron (in nm). Estimate the uncertainty in your value. Compton’s experiment detected x-rays at both the incident wavelength (coherent scattering) and at a modified or shifted wavelength (Compton scattering).

I like this exercise because it requires the reader to do many things: 

• Decide which spectral line is coherent scattering and which is Compton scattering. 
• Choose which angle θ to analyze. 
• Estimate the angle ϕ of each spectral peak from the data. 
• Approximate the uncertainty in the estimation of ϕ. 
• Convert the values of ϕ from degrees/minutes to decimal degrees. 
• Determine the wavelength for each angle using Bragg’s law. 
• Calculate the wavelength shift. 
• Relate the wavelength shift to the Compton wavelength. 
• Compute the Compton wavelength. 
• Propagate the uncertainty. 
• Convert from Ångstroms to nanometers.
  

If you can do all that, you know what you’re doing.

Compton’s experiment played a key role in establishing the quantum theory of light and wave-particle duality. He was awarded the 1927 Nobel Prize in Physics for this research. Let’s give him the last word. Here are the final two sentences of his paper.

This satisfactory agreement between the experiments and the theory gives confidence in the quantum formula for the change in wave-length due to scattering. There is, indeed, no indication of any discrepancy whatever, for the range of wave-length investigated, when this formula is applied to the wave-length of the modified ray.