Toy Models

Using Math in Physics:
4. Toy Models, by Joe Redish. 

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce many “toy models.” These are simplified models that strip away detail to expose fundamental processes. Why use toy models in biology, which is notoriously complex? To explore this question, I want to focus on an article by the late Joe Redish published in The Physics Teacher: “Using Math in Physics: 4. Toy Models” (Volume 59, Pages 683–688, 2021). The paper is one in a series of articles that Redish wrote about using math in physics, several of which will be cited in the 6th edition of IPMB.

In his introduction, Redish writes

As physicists, we consider our highly simplified models an obvious and natural way to approach physics. Mathematical models of complicated systems can be tricky, so the best way to understand the math is to take the simplest possible example that illustrates a phenomenon, then take it apart and put it back together again, matching the math with physical intuitions and building a mental blend of what the math means physically.

He goes on to say

Simple systems help build understanding: Learning to use this resource effectively to build new understanding is an important step in learning to be an effective scientist.

Toy models help students to learn

to blend physical concepts, knowledge, and intuition with mathematical symbols and processing.

I couldn’t agree more.

One important skill when using a toy model is deciding what to include and what to ignore. Redish addresses this issue:

In choosing a model, we have to decide what phenomena we are trying to describe, how to quantify the quantities involved, and, perhaps most important, what matters and what doesn’t. The world is too complex for us to include everything that’s going on. Deciding what matters and what can be ignored (at least at first) is an essential scientific skill, one that is, unfortunately, rarely taught explicitly even to our physics majors.

Toy models are useful for teaching students how to go back and forth from physics (and biology and medicine) to mathematics. When I was teaching, I noticed that many students understood the physics qualitatively and had good math skills, but had trouble translating between the two. They tend to think of these skill as being separate. Redish says

Once we’ve mapped our physical quantities onto math, we inherit processing tools from mathematics that let us solve problems that we might have difficulty solving. But once we have completed our calculation, we have to interpret the result back in the physics. What did the solution tell us about the physical world? Finally, we have to evaluate that interpretation. Is our model good enough for what we needed to do? Or are there refinements that we have to make, additional factors or effects that we really need to include?

Redish is explicit about why toy models are useful and important. 

We use toy models widely in introductory physics because they support multiple pedagogically valuable developments. 

• Toy models help students build the blend by focusing on the math-physics connection. 
• Toy models are built into most of our problems and can help build physical intuition. 
• Some toy models work way better than we might expect.

I consider the second bullet point to be particularly critical. Students need to gain intuition into how systems behave. They need insight. If they use no math, any insight is totally qualitative. If they use math, they risk missing the insight because they are focused entirely on manipulating the mathematical symbols. In graduate school, I took a course on general relativity. I learned how to do the math well enough to get an A, but I never felt I understood what was happening physically. I would have benefited from some toy models.

Some biologists and medical doctors like to put all possible details into a complicated “black box” computer model. While such an approach has its uses, such as for making numerical predictions to compare to experiments, it provides no insight. (Perhaps the researcher who writes the computer program gains some insight, but the user does not.) Redish says

Many real-world phenomena include lots of competing effects. Making sense of them, figuring out what matters most, and how to approach them can be challenging. Toy models are not just a way of learning to build the blend; they are an analytical tool for approaching a complicated system.

The sixth edition of Intermediate Physics for Medicine and Biology relies even more heavily on toy models than previous editions. If students can gain the intuition from these toy models and can practice building models and analyzing them mathematically, they will be ready to examine even more complicated and diverse biological and medical systems quantitatively.

Elective MRI Screening of the General Public—Buyer Beware

Elective MRI Screening of the 
General Public—Buyer Beware.

I saw an article by Matthew Davenport and Scott Reeder published recently in JAMA (The Journal of the American Medical Association) titled “Elective MRI Screening of the General Public—Buyer Beware.” My initial reaction was “oh no, not more nonsense about the health risks of static magnetic fields!” Fortunately, that’s not what the article is about. No risks from exposure to magnetic fields are mentioned. So, what’s the problem?

Apparently many people are paying for elective whole body magnetic resonance imaging scans, even when not recommended by standard medical practice. In these images, benign growths can look just like small, early-stage tumors; you can’t distinguish them. This leads to additional procedures—such as biopsies, endoscopies, or surgeries—which each carry their own risk. Also, a false positive result can cause anxiety, sleeplessness, and financial strain. When you put all these issues together, elective whole body MRI scans may cause more harm than good, even if the direct risks of having a MRI are nonexistent (take “direct risk” here to mean the risk you are exposed to if you have an MRI scan for free but then the image is accidentally deleted before anyone can look at it).

This may seem an odd topic for a blog about Intermediate Physics for Medicine and Biology. But it reminds us that while physics is important in medicine, other non-technical issues are also critical. You might think that having an image of the inside of the body is always better than not having an image. But apparently it’s not.

Does this mean we should stop striving to develop even better MRI scanners? No! Better scanners should lead to better images and therefore better diagnoses. But we must be careful when interpreting the images (even improved images) especially when screening, where false positives will always be a challenge. As Russ Hobbie and I say in Chapter 5 of IPMB when analyzing the artificial kidney, we must recognize “the distinction between a high-technology treatment [or, in this case, image] and a real conquest [or identification] of a disease.”

IPMB Machete Order

What’s the best order to watch the Star War movies? Aficionados have been arguing about this for years. (In fact, this debate raged before the sequel triology was released. For this post, we’ll pretend the Episodes VII, VIII, and IX don’t exist). Two options are to watch them in the order of their release (starting with Episode IV released in the summer of 1977, between my junior and senior years of high school), or to watch them in order of episode number (starting with Episode I). A third suggestion is to watch them in “Machete Order”:

1. Episode IV, A New Hope
   
2. Episode V, The Empire Strikes Back
3. Episode II, Attack of the Clones 
4. Episode III, Revenge of the Sith 
5. Episode VI, Return of the Jedi
   

 Machete Order on the The Big Bang Theory. 

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

Episode I, The Phantom Menace, is not included; Rod Hilton, the inventor of Machete Order, believed the series was better off without it. Unfortunately, by eliminating Episode I we miss out on most scenes with my favorite character: Jar Jar Binks. 

Jar Jar Binks scenes from Episode I. 

What does all this Star Wars talk have to do with Intermediate Physics for Medicine and Biology? It’s fun to invent a new order of the chapters and sections when studying from IPMB. Below I present my “IPMB Machete Order.” Enjoy. 

First Semester

Chapter 1a: Mechanics. Sections 1.1 and 1.2 are where everyone should start: talking about distance scales and models.

Chapter 2: Exponential Growth and Decay. Chapter 2 introduces some necessary mathematics before diving into the physics.

Chapter 10a: Feedback and Control. Much of Sections 10.1–10.7 and 10.11–10.12 can be considered mathematical biology and follows naturally after Chapter 2.

Chapter 1b, Mechanics. Physics almost always starts with mechanics. This ordering respects that tradition. Sections 1.3–1.20 discuss biomechanics and fluid dynamics.

Chapter 13: Sound and Ultrasound. Acoustics can be thought of as part of mechanics. The wave equation is introduced. Ultrasound is used for imaging, but Fourier analysis is not used as extensively as in other imaging techniques, so ultrasonic imaging can be analyzed before the power of Fourier methods are introduced.

Chapter 4: Transport in an Infinite Medium. Diffusion follows naturally from mechanics. The diffusion equation can be compared to and contrasted with the wave equation that will have been studied already.

Chapter 14a: Atoms and Light. Section 14.6 discusses the diffusion approximation of photon transport in a turbid medium. It might be best to study this section right after studying diffusion in general.

Chapter 5: Transport through Neutral Membranes. This chapter applies diffusion to transport through membranes. It also introduces osmosis.

Chapter 14b: Atoms and Light. Russ Hobbie and I don’t develop light as a consequence of Maxwell’s equations, so there’s no reason to delay Sections 14.1-14.5 and 14.7-14.15 and wait to discuss optics after the electricity and magnetism chapters.

Chapter 15: Interactions of Photons and Charged Particles with Matter. After describing infrared, visible, and ultraviolet light in Chapter 14, it is natural to analyze x-rays in Chapter 15.

Chapter 16: Medical Use of X-Rays. Chapter 16 follows naturally after Chapter 15 (the two are almost one long chapter about x-rays).

Chapter 3: Systems of Many Particles. Usually, an introductory physics sequence covers mechanics, heat, and sound in the first semester, and electricity, magnetism, and modern physics in the second. Therefore, Chapter 3 about thermodynamics fits naturally at the end of the first semester. One advantage of this ordering is that diffusion is already familiar, so the heat equation (another name for the diffusion equation) is easier to understand. It’s true that the concept of the Boltzmann factor and absolute temperature are used in earlier chapters. No ordering is perfect. 

 

Second Semester

Chapter 6: Impulses in Nerve and Muscle Cells. Chapter 6 begins a sequence of chapters about electricity and magnetism, the traditional starting point for a second semester of physics.

Chapter 7: The Exterior Potential and the Electrocardiogram. More bioelectricity.

Chapter 10b: Feedback and Control. Sections 10.8–10.9 apply mathematics to the heart. They fit naturally after Chapter 7.

Chapter 9a: Electricity and Magnetism at the Cellular Level. Sections 9.1–9.9 are about electricity, not magnetism, and might best follow the two chapters on bioelectricity.

Chapter 8: Biomagnetism. Magnetism traditionally follows electricity, and we won’t change that order.

Chapter 9b: Electricity and Magnetism at the Cellular Level. The last section, Section 9.10, deals with the effects of weak electric and magnetic fields, and can be covered after the biomagnetism chapter.

Chapter 11: The Method of Least Squares and Signal Analysis. Chapter 11 contains some heavy duty mathematics, including Fourier analysis, so it is appropriate that it comes late in the semester. Yet, the reason we include these topics is because they are essential for several chapters on imaging.

Chapter 12: Images. Chapter 12 follows naturally after Chapter 11. A highlight of the chapter is the analysis of tomography. General properties of two-dimensional images are developed.

Chapter 18: Magnetic Resonance Imaging. Now that you have the full power of Fourier techniques, you can study MRI.

Chapter 17: Nuclear Physics and Nuclear Medicine. Modern physics typically appears near the end of a introductory physics sequence, and we preserve that order. Yet, several of the nuclear medicine imaging techniques, such as positron emission tomography (PET) and single photon emission computed tomography (SPECT), require an understanding of tomographic methods.

The good news is that you don’t have to adopt IPMB Machete Order; you can just follow the order presented in the book. Or, you can make up your own schedule. When I taught Biological Physics and Medical Physics at Oakland University, I followed the order of the chapters in the book, except I skipped Chapter 9 (not enough time) and I moved Chapters 11 and 12 to after Chapter 16 to avoid starting the second semester with a lot of mathematics. If you find a different ordering that works for you, let me know. 

Oh My! I Was Wrong

Recently, there has been a brouhaha about “Ghost Murmur,” a proposed way to detect the magnetic field of a human heart from miles away. I was interviewed about Ghost Murmur by Deni Béchard, a reporter for Scientific American. He wrote¹

Bradley Roth, a physicist at Oakland University and author of the 2023 review Biomagnetism: The First Sixty Years, agrees. “People have been measuring the magnetic field of the heart for 60 years, and usually it’s done in a lab with shielding, and it’s done just a few centimeters or a couple inches from the heart, and even then you can barely record it.” A helicopter-borne version, he says, “would be not just a small advance, but it’d be a revolutionary advance from the state of the art.”

I stand by that statement, as written. In fact, I’m probably even more skeptical of the reality of Ghost Murmur now than when I spoke to Béchard. I confess, however, that I was wrong about one thing. I based my thinking on the magnetic field falling off as from a current dipole, so with the inverse square of the distance. Current dipoles are the usual way to model biomagnetic fields. I had seen other people online saying that the magnetic field falls off as from a magnetic dipole, so inverse cube, but I presumptuously assumed they were confusing a current dipole and a magnetic dipole.

Then I got to wondering if I was right. I know that for a current dipole in an unbounded conductor, the fall off is indeed one over distance squared. But how about for a bounded conductor, like the human body? Does that change things?

Being a physicist, my first impulse was to model the human body as a sphere. Anyone familiar with biomagnetism knows that a radial dipole in a sphere produces no magnetic field outside it. But how does the magnetic field of a tangential dipole in a sphere fall off with distance?

Homework Problem 21 in Chapter 8 of Intermediate Physics for Medicine and Biology provides the answer. It is a closed form expression for the magnetic field of a dipole in a sphere (originally derived by Jukka Sarvas).² The expression is sort of strange-looking, but it’s exactly what we need.

 

Let’s assume p = p x̂, r = r ẑ, and r₀ = r₀ ẑ, implying that a tangential dipole lies a distance r₀ from the sphere center along the z axis, and the magnetic field is measured a distance r from the sphere center also along the z axis, where r > r₀ ( r₀ is inside the sphere and r is outside). The homework problem defines a = r – r₀. I will take the limit as r >>> r₀. So a = r. In that case F = r (r² + r² – r₀ r) which approaches 2r³. The gradient of F becomes 6r². So the expression for the magnetic field falls with distance as 1/F (first term) or as r∇F/F² (second term). In both cases, the falloff is proportional to the inverse cube.

Ah Ha! I was wrong to say the magnetic field of a current dipole in a conducting sphere falls off as the inverse square. It is the inverse cube. The effect of the sphere boundary changes things from 1/r² to 1/r³. In fact, Flavio Grynszpan and David Geselowitz³ define a magnetic dipole m that is related to the electric, or current, dipole p for the case of a spherical conductor. So the folks who modeled the heart as a magnetic dipole knew what they were talking about. Apparently I didn’t. But at least I learned something, which is always a good thing.

What does this mean for Ghost Murmur? It means I was wrong in saying the magnetic field measured 60 km from the heart is one trillion (10¹²) times smaller than the magnetic field measured 60 mm from the heart. The field would be more like one quintillion (10¹⁸) times smaller. Yikes! 

 

 References

¹https://www.scientificamerican.com/article/what-is-the-quantum-ghost-murmur-purportedly-used-in-iran-scientists/ 

²Sarvas J (1987) Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys. Med. Biol. 32:11–22. 

³Grynszpan F, Geselowitz DB (1973) Model studies of the magnetocardiogram. Biophys. J. 13:911–925.