The Pitfalls of Using Handbooks and Formulae

Structures: Or Why Things Don't Fall Down, by J. E. Gordon.

Last week I discussed James Gordon’s book Structures: Or Why Things Don’t Fall Down. The book contains several appendices. The first appendix is ostensibly about using handbooks and formulas to make structural calculations.

Over the last 150 years the theoretical elasticians have analysed the stresses and deflections of structures of almost every conceivable shape when subjected to all sorts and conditions of loads…Fortunately a great deal of this information has been reduced to a set of standard cases or examples the answers to which can be expressed in the form of quite simple formulae.

Then, to my surprise, Gordon changes tack and warns about pitfalls when using these formulas. His counsel, however, applies to all calculations, not just mechanical ones. In fact, his advice is invaluable for any young scientist or engineer. Below, I quote parts of this appendix. Read carefully, and whenever you encounter a word specific to mechanics substitute a general one, or one related to your own field.

[Formulae] must be used with caution.

Appendix 1 of Structures.

1. Make sure that you really understand what the formula is about.
2. Make sure that it really does apply to your particular case.
3. Remember, remember, remember, that these formulae take no account of stress concentrations or other special local conditions.

After this, plug the appropriate loads and dimensions into the formula—making sure that the units are consistent and that the noughts are right. [I’m not sure what “noughts” are, but I think the Englishman Gordon is saying to make sure the decimal point is in the right place.] Then do a little elementary arithmetic and out will drop a figure representing a stress or a deflection.

Now look at this figure with a nasty suspicious eye and think if it looks and feels right. In any case you had better check your arithmetic; are you sure that you haven’t dropped a two?...

If the structure you propose to have made is an important one, the next thing to do, and a very right and proper thing, is to worry about it like blazes. When I was concerned with the introduction of plastic components into aircraft I used to lie awake night after night worrying about them, and I attribute the fact that none of these components ever gave trouble almost entirely to the beneficent effects of worry. It is confidence that causes accidents and worry which prevents them. So go over your sums not once or twice but again and again and again.

Structures: Or Why Things Don't Fall Down.

This is the attitude I try to instill in my students when teaching from Intermediate Physics for Medicine and Biology. I implore them to think before they calculate, and then think again to judge if their answer makes sense. Students sometimes submit an answer to a homework problem (almost always given to five or six significant figures) that is absurd because they didn't look at their answer with a “nasty suspicious eye.” I insist they "remember, remember, remember" the assumptions and limitations of a mathematical model and its resulting formulas. Maybe Gordon goes a little overboard with his “night after night” of lost sleep, but at least he cares enough about his calculation to wonder “again and again and again” if it is correct. A little worry is indeed a “right and proper thing.”

Who would of expected such wisdom tucked away in an appendix about handbooks and formulae?

Structures: Or Why Things Don't Fall Down

Structures: Or Why Things Don't Fall Down,
by James Gordon.

When I was in graduate school, I read a fascinating book by James Gordon titled Structures: Or Why Things Don’t Fall Down. It showed me to how engineers think about mechanics. Recently, I reread Structures and read for the first time its sequel The New Science of Strong Materials: Or Why You Don’t Fall Through the Floor. I enjoyed both books thoroughly.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss two mechanical properties of a material: stiffness and strength. Stiffness describes how much a material lengthens when pulled (that is, strains when stressed), and is quantified by its Young’s modulus. Strength measures how much stress a material can withstand before failing. Gordon summarizes these ideas succinctly.

A biscuit is stiff but weak, steel is stiff and strong, nylon is flexible and strong, raspberry jelly is flexible and weak. The two properties together describe a solid about as well as you can reasonably expect two figures to do.

Just two figures, however, are not sufficient to characterize a material, especially when it's used to build a structure.

The worst sin in an engineering material is not lack of strength or lack of stiffness, desirable as these properties are, but lack of toughness, that is to say, lack of resistance to the propagation of cracks.

Toughness is opposite to brittleness, and is related to but not identical to ductility. It is quantified by the work of fracture—the energy needed to produce a new surface by propagation of a crack through the material—a concept introduced by Alan Griffith during his research on fracture mechanics.

A strained material contains strain energy which would like to be released just as a raised weight contains potential energy and would like to fall…The relief of strain energy …. [is] proportional to the square of the crack length…On the other side of the account book is the surface energy…needed to form the new surfaces and clearly increases as only the first power of the depth of the crack…When the crack is shallow it is consuming more energy as surface energy than it is releasing as relaxed strain energy and therefore conditions are unfavorable for it to propagate. As the crack gets longer however these conditions are reversed and beyond the ‘critical Griffith length’ l_g the crack is producing more energy than it is consuming, so it may start to run away in an explosive manner.

In heterogeneous materials, internal interfaces act as crack stoppers. This makes wood exceptionally tough; Its cellular, fibrous structure prevents a crack from propagating. Toughness is important in biological materials that must undergo large strains without breaking. Wood is not dense (compared to, say, steel), so you get lots of toughness for little weight, which is one reason wood is so popular as a building material. On the other hand, wood isn’t very stiff, and it swells, burns, and rots.

Gordon provides deep insight into the behavior of structures and materials. Consider the stress in the wall of a cylindrical pressure vessel (a long cylinder with spherical end caps). The circumferential stress in the cylinder's wall is given by the Law of Laplace (see IPMB, Chapter 1, Problem 18). The longitudinal stress is equal to the stress in the end caps (the stress in a sphere is two times that in a cylinder, see Problem 19). Thus

the circumferential stress in the wall of a cylindrical pressure vessel is twice the longitudinal stress...One consequence of this must have been observed by everyone who has ever fried a sausage. When the filling inside the sausage swells and the skin bursts, the split is almost always longitudinal.

Then Gordon develops this theme.

Figure 5 from Structures.

If we make a tube or cylinder from such a material [as rubber] and then inflate it, by means of an internal pressure, so as to involve a circumferential strain of 50 per cent or more, then the inflation or swelling process will become unstable, and the tube will bulge out...into a spherical protrusion which a doctor would describe as an “aneurism”....Since veins and arteries do, in fact, generally operate at strains around 50 per cent, and since, as any doctor will tell you, one of the conditions it is most desirable to avoid in blood-vessels is the production of aneurisms, any sort of rubbery elasticity is quite unsuitable....The only sort of elasticity which is completely stable under fluid pressures at high strains is that which is represented by Figure 5 [showing the stress increasing exponentially with the strain]. With minor variations, this shape of stress-strain curve is very common indeed for animal tissue....Materials with this [exponential] type of stress-strain curve are extremely difficult to tear. One reason is, perhaps, that the strain energy stored under such a curve--and therefore available to propagate fracture...is minimized."

He continues

Perhaps partly for these reasons the molecular structure of animal tissue does not often resemble that of rubber or artificial plastics. Most of these natural materials are highly complex, and in many cases they are of a composite nature, with at least two components; that is to say, they have a continuous phase or matrix which is reinforced by means of strong fibres of filaments of another substance. In a good many animals this continuous phase or matrix contains a material called 'elastin', which has a very low modulus and a [flat] stress-strain curve...The elastin is, however, reinforced by an arrangement of bent and zig-zagged fibres of collagen...a protein, very much the same as tendon, which has a high modulus...Because the reinforcing fibres are so much convoluted, when the material is in its resting or low-strain condition they contribute very little to its resistance to extension, and the initial elastic behavior is pretty well that of elastin. However, as the composite tissue stretches the collagen fibres begin to come taut; thus in the extended state the modulus of the material is that of the collagen, which more or less accounts for Figure 5.

As you probably can tell, Gordon writes wonderfully and explains mechanics so it's understandable to a layman. His writing is a model of clarity.

Structures: Or Why Things Don't Fall Down.

Structures was my first exposure to continuum mechanics, but certainly not my last. I was a member of the Mechanical Engineering Section when I worked at the National Institutes of Health, so I was surrounded by outstanding mechanical engineers. My friend Peter Basser—himself a mechanical engineer—would lend me his books, and I recall reading classics such as Love’s A Treatise on the Mathematical Theory of Elasticity and Schlichting’s Boundary Layer Theory. I was impressed by Basser’s model of infusion-induced swelling in the brain and Richard Chadwick’s studies of cardiac biomechanics (Richard was another member of our Mechanical Engineering Section). In many ways, NIH provided a liberal education in physics applied to biology and medicine.

Throughout my career, most of my research has focused on bioelectricity and biomagnetism. Recently, however, I have been working on problems in biomechanics. But that is another story.

Computerized Transverse Axial Scanning (Tomography)

“Computerized Transverse Axial
Scanning (Tomography).”

Section 16.8 of Intermediate Physics for Medicine and Biology discusses computed tomography. Russ Hobbie and I describe the history of this technique.

The Nobel Prize in Physiology or Medicine was shared in 1979 by a physicist, Allan Cormack, and an engineer, Godfrey Hounsfield…. Hounsfield, working independently [of Cormack], built the first clinical [computed tomography] machine, which was installed in 1971. It was described in 1973 in the British Journal of Radiology. The Nobel Prize acceptance speeches (Cormack 1980; Hounsfield 1980) are interesting to read. A neurologist, William Oldendorf, had been working independently on the problem but did not share in the Nobel Prize…

Oddly, Russ and I did not include Hounsfield’s 1973 paper in our list of references. I decided to dig it up and have a look. The reference and abstract is:

Hounsfield GN (1973) Computerized transverse axial scanning (tomography): Part I. Description of the system. Br J Radiol 46:1016-1022

This article describes a technique in which X-ray transmission readings are taken through the head at a multitude of angles: from these data, absorption values of the material contained within the head are calculated on a computer and presented as a series of pictures of slices of the cranium. The system is approximately 100 times more sensitive than conventional X-ray systems to such an extent that variations in soft tissues of nearly similar density can be displayed.

A dozen comments:

1. This is Hounsfield’s most highly cited paper, with 4667 citations according to Google Scholar. That's a respectable number (ten times more than any of my papers have), yet seems curiously small for a Nobel Prize-winning advance.
2. Hounsfield’s paper is the first of a trilogy. Hounsfield is not a coauthor on the other two; they report clinical studies using the new technique.
3. Hounsfield lists his institution as “Central Research Laboratories of EMI Limited.” EMI is famous in the music industry; it is the recording label responsible for the early hits of the Beatles.
4. Hounsfield’s paper has only three references: two to his own preliminary reports and one to an article by Oldendorf. He didn’t cite Cormack’s papers.
5. Hounsfield sounds most impressed not by recreating three-dimensional images from two-dimensional projections (which to me is the big advance) but instead by the increased sensitivity of the technique to small differences in x-ray absorption coefficient.
6. Figure 3, illustrating the scanning device and sequence, is similar to Fig. 16.25 in IPMB.

   

   Fig. 16.25 of IPMB.

7. Hounsfield measured 160 points in each translation and performed 180 rotations. Each two-dimensional image was represented by an 80 × 80 grid of pixels.
8. The reconstruction method was different from the two Russ and I analyze in Chapter 12 of IPMB: i) Fourier transform reconstruction and ii) filtered back-projection. Instead, Hounsfield just fit his data using the least squares method (see Section 11.1 of IPMB). Hounsfield writes “Each beam path [in the CT scan], therefore, forms one of a series of 28,800 simultaneous equations, in which there are 6,400 variables and, providing that there are more equations than variables, the values of each [pixel] …. can be solved.”
9. The Hounsfield unit was introduced in Fig. 9, but he did not, of course, call it that. Interestingly, his definition is different than what is used today. Equation 16.25 in IPMB gives the Hounsfield unit as
   
   
   where μ_tissue and μ_water are x-ray attenuation coefficients. In his paper, Hounsfield defines the unit the same way, except he replaces 1000 by 500.
10. The article describes preliminary experiments using an iodine-containing contrast agent and digital subtraction, analogous to Fig. 16.23 in IPMB.
11. The computer equipment pictured in Hounsfield’s paper look big and clunky today. I can only guess what paltry computer power he had available for these first reconstructions.
12. I love the British Journal of Radiology, also known as BJR. (What journal did you think that Bradley John Roth would like?)

I’ll conclude with Hounsfield’s final paragraph. To my ear, it sounds like classic British understatement.

It is possible that this technique may open up a new chapter in X-ray diagnosis. Previously, various tissues could only be distinguished from one another if they differed appreciably in density. In this procedure absolute values of the absorption coefficient of the tissues are obtained. The increased sensitivity of computerized X-ray section scanning thus enables tissues of similar density to be separated and a picture of the soft tissue structure within the cranium to be built up.

Imaging and Velocimetry of the Human Retinal Circulation with Color Doppler Optical Coherence Tomography

Page 394 of IPMB.

Section 14.7 of Intermediate Physics for Medicine and Biology discuses optical coherence tomography (OCT). Russ Hobbie and I write

Optical range measurements using the time delay of reflected or backscattered light from pulses of a few femtosecond (10^-15 s) duration can be used to produce images similar to those of ultrasound…Since it is difficult to measure time intervals that short, most measurements are done using interference properties of the light. Optical coherence tomography is conceptually similar to range measurements but uses interference measurements…It is widely used in ophthalmology….

Fig. 14.15 of IPMB.

The basic apparatus [for OCT] is shown in Fig. 14.15….The light pulse travels over an optical fiber to a 50/50 beam splitter. Part travels to the sample, where it is reflected back to the 50/50 coupler and then to the detector. The other half of the light goes to the reference mirror, where it is also reflected back to the detector. Changing the position of the reference mirror changes the depth of the image plane in the sample….

Fig. 14.17 of IPMB.

It is possible to make many kinds of images. Fig. 14.17 shows the parabolic velocity profile of blood flowing in a retinal blood vessel 176 μm diameter. It was obtained by measuring the Doppler shift in light scattered from moving blood cells.

Yazdanfar et al. (2000).

Figure 14.17 is from the paper “Imaging and Velocimetry of the Human Retinal Circulation with Color Doppler Optical Coherence Tomography,” by Siavash Yazdanfar, Andrew Rollins, and Joseph Izatt (Optical Letters, Volume 25, Pages 1448–1450, 2000). [For some reason Russ and I did not include the year in our citation—another item for the errata].

Abstract: Noninvasive monitoring of blood flow in retinal microcirculation may elucidate the progression and treatment of ocular disorders, including diabetic retinopathy, age-related macular degeneration, and glaucoma. Color Doppler optical coherence tomography (CDOCT) is a technique that allows simultaneous micrometer-scale resolution cross-sectional imaging of tissue microstructure and blood flow in living tissues. CDOCT is demonstrated for the first time in living human subjects for bidirectional blood-flow mapping of retinal vasculature.

I like Fig 14.17 because it combines ideas from Chapters 1, 11, 13, and 14 of IPMB. It also highlights the excellent spatial resolution you can obtain with OCT.

The illustration below shows the geometry associated with Fig. 14.17. The light is reflected by blood cells moving at speed v, causing a Doppler shift in its frequency. By adjusting the reference mirror, different depths are selected. The vessel makes an angle θ relative to the incident light. As the depth is scanned across the vessel, the Doppler shift determines the blood flow profile.

The geometry associated with IPMB Fig. 14.17.

To help the reader learn more about the physics of OCT and Fig. 14.17, I have written two new homework problems. The solutions are included at the bottom of the post (upside down, to encourage readers to solve the problems themselves first). Enjoy!

Section 14.7

Problem 24 ⅓. This problem and the next explore the physics behind Fig. 14.17, which shows the velocity profile in a blood vessel measured using color Doppler optical coherence tomography. The data is based on an article by Yazdanfar et al. (2000). For this problem ignore the index of refraction of the tissue and assume θ = 60°.

(a) If the wavelength, λ, of the incident light is 832 nm and the wavelength bandwidth, Δλ, is 15 nm, determine the frequency, f, and the frequency bandwidth, Δf, in THz.

(b) The coherence time, τ_coh, is equal to 1/(πΔf). Calculate τ_coh in fs, and the coherence length, x₂ – x₁, in microns. The coherence length determines the spatial resolution of the measurement.

(c) Use Eq. 13.42 to derive an expression for the speed of blood flow in the direction of the light, v', in terms of the Doppler frequency shift, df. Assume that the speed of light, c, is much greater than v'. Calculate v' if df = 4 kHz. The Doppler technique measures the component of motion in the direction of the light. Determine the speed v along the vessel.

Problem 24 ⅔. The Doppler shift, df, of OCT data as a function of depth z across a blood vessel is given below. For viscous flow in a tube (Sec. 1.17), the blood speed varies parabolically across the vessel cross section (Eq. 1.37). Fit a parabola to this data of the form df = Az² + Bz + C, and determine constants A, B, and C. Use these constants to find the peak value of df in this vessel, the location of the center of the vessel, and the vessel diameter (the width of the parabola when df = 0). The measured diameter corresponds to an oblique section at θ = 60°. Correct this result to get the true diameter.

z (mm)	df (kHz)
0.15	3.26
0.20	5.20
0.25	6.12
0.30	6.02
0.35	4.89
0.40	2.75

I will give the final word to Yazdanfar, Rollins, and Izatt, who conclude

In summary, CDOCT has been applied for what is believed to be the first time to retinal blood-flow mapping in the human eye. Depth-resolved quantification of retinal hemodynamics may prove helpful in understanding the pathogenesis of several ocular diseases. Unlike fluorescein angiography, CDOCT is entirely noninvasive and does not require dilation of the pupil. Furthermore, CDOCT operates at longer wavelengths than does laser Doppler velocimetry, so light exposure times can be safely increased. CDOCT is believed to be the first technique for determining, with micrometer-scale resolution, the depth, diameter, and flow rate of blood vessels within the living retina.

Solution to new Homework Problem 24 ⅓

Solution to new Homework Problem 24 ⅔.

Venkataranan Ramakrishna, Biological Physicist

Failed physicist?

I was reading the November issue of APS News (A publication of the American Physical Society) when I noticed an article titled “Failed Physicist? From Biologist Turned Nobel Laureate to Author.” The article was about Venkataraman Ramakrishnan, winner of the 2009 Nobel Prize in Chemistry for “studies of the structure and function of the ribosome.” He received a PhD in physics before switching to biology. In the article, he calls himself a “failed physicist.”

Many readers of Intermediate Physics for Medicine and Biology may be in a similar position of having been trained in physics but now learning biology. Ramakrishnan provides an interesting case study in how to make such a transition. I looked up his biographical statement on the Nobel Prize website, and I reproduce excerpts below. Changing fields is not easy, but it is possible, and can ultimately lead to groundbreaking research.

Choosing Basic Science

[When Ramakrishna was growing up in India, he was looking for a university for his undergraduate studies.] A faculty member in the physics department in [the University of] Baroda, S.K. Shah, told me about a brand new curriculum they were introducing for their undergraduate course. It began with the Berkeley Physics Course, and was supplemented by the Feynman Lectures on Physics before moving on to more specialized areas. I therefore decided to enroll in the B.Sc. course in physics in Baroda, my hometown. Since I was only 16 when I began this course, I sensed that my parents, especially my father, were relieved that I was not leaving home….

I found myself tremendously interested by the articles in biology in Scientific American, to which I have subscribed off and on through the years. It appeared that hardly a month went by without a major breakthrough in the life sciences, whereas physics was having a hard time making any fundamental progress. Certainly I felt that if I continued in physics, I would be doing boring and tedious calculations rather than making really interesting advances. The result was that I felt so frustrated that I withdrew from my thesis work and spent an inordinate amount of time on extracurricular activities….

[Ramakrishna eventually finished his BS in physics, and then obtained a PhD in Physics at Ohio University]…. By that time I had already decided I was going to switch to biology.

Transition to Biology

Since I hardly knew any biology, I felt I needed formal training of some sort. I could go to graduate school again, with the option of getting a second Ph.D. or go to medical school, which was ironic since I had turned down the opportunity to do precisely that when I was younger. I took the MCAT …. but despite scoring in the 99th percentile in all the subjects, I only got one interview (at Yale) because I was not a U.S. citizen or even a permanent resident at that point…. However, I had also written to a number of graduate programs. Many of them said that they would not accept someone who already had a Ph.D. The chairman of the Molecular Biophysics and Biochemistry ... department at Yale, Franklin Hutchinson, wrote to me saying that while they could not take me as a graduate student, he would circulate my CV to faculty members for a potential postdoctoral position. Two of them responded: One was Don Engelman, and the other, ironically, was Tom Steitz, with whom I shared the Nobel Prize. Although I found their work very interesting, I thought doing a postdoc directly from a degree in physics would leave me with too narrow a background in biology to be an effective scientist. So when three schools accepted me into their graduate program, I chose to go to the University of California, San Diego (UCSD)…. During the first year, I did several lab rotations in biology and took as many undergraduate courses as I could possibly manage, including genetics, taught by Dan Lindsley, a well-known Drosophila geneticist, and biochemistry, where I was inspired by the brilliant and enthusiastic lectures of Paul Price.

In my second year [at UCSD], I settled down to do research in Mauricio Montal's lab. Mauricio had developed an ingenious method of incorporating conducting channels into lipid bilayers formed by bringing together two defined monolayers, and was thus doing single molecule biophysics at a time when nobody called it that. Around this time, however, I read an article in Scientific American by Don Engelman and Peter Moore about their ribosome work, and became interested in it. It also struck me that there was no longer any reason to continue on to obtain a second Ph.D. because I felt I had acquired the background I needed. I therefore wrote to Don Engelman, one of the two people from Yale who had responded to me earlier. Don was interested in membrane proteins, a subject I was already working on in Mauricio's lab. Don wrote back and said that he and Peter had a position open on their ribosome project, and I could always work on membrane-related projects once I got there. Peter arranged to meet me in San Diego in early 1978 and offered me a postdoctoral position soon afterwards. Thus began my lifelong interest in ribosomes….

In the APS News article, Ramakrishnan said something that could be the motto for IPMB.

Physics is a great training for every science because it teaches you quantitative and mathematical thinking, and that way of approaching problems is becoming increasingly important in every field, including biology.

Want to learn more? Ramakrishna has a new book out: Gene Machine: The Race to Dicipher the Secrets of the Ribosome. It's on my list of books to read. Below are a couple videos in which you can hear from Ramakrishna himself. Enjoy!

Write Mind

Write Mind, by Eric Maisel.

Every Saturday morning my wife and I visit the Rochester Hills Public Library. I like to browse the stacks, and sometimes I find a gem. Recently, I checked out Write Mind: 299 Things Writers Should Never Say to Themselves (and What They Should Say Instead) by Eric Maisel.

Write Mind contains some cognitive therapy jargon that I don’t care for, but its second sentence quotes Epictetus so it can’t be all bad. Seriously, at its core this delightful book is about attitude. Not all problems can be solved by a positive attitude but some can, whether you are an aspiring writer or a struggling physics student.

Write Mind is aimed at writers. Two hundred and ninety nine times it first states an incorrect, negative attitude (WRONG MIND) and then a better, positive attitude (RIGHT MIND).

To help students of Intermediate Physics for Medicine and Biology I have paraphrased Write Mind, providing 29 pairs of statements about physics applied to medicine and biology. Forgive me if sometimes they are corny; I hope you find them useful.

1. 

WRONG MIND: Mathematics and medicine are so different; I can’t learn both so I’ll settle for one or the other.

RIGHT MIND: I can master both mathematics and medicine.

2. 

WRONG MIND: Physiology requires so much memorization! I will give up and stick to physics.

RIGHT MIND: I can learn both physics and physiology.

3. 

WRONG MIND: I hate toy models because they oversimplify biology.

RIGHT MIND: I will gain insight from a toy model, and then analyze its strengths and weaknesses.

4. 

WRONG MIND: I have difficulty understanding what some homework problems are asking; I skip those.

RIGHT MIND: Research problems are often ill-defined. I will try my best to understand the question and then answer it.

5. 

WRONG MIND: Robert Plonsey, Art Winfree, and John Wikswo have contributed so much; I can never accomplish that much in my career.

RIGHT MIND: I intend to work hard, and take Plonsey, Winfree, and Wikswo as role models.

6. 

WRONG MIND: I’m good at math and I love medicine, but I have trouble connecting the two.

RIGHT MIND: Homework problems let me practice connecting math to medicine. Many students struggle with this difficulty. I am not alone.

7. 

WRONG MIND: IPMB and its blog recommend so many books; I don’t have time to read them all, so I won’t read any.

RIGHT MIND: I will find time to read one of the books recommended in IPMB or its blog. Once I have finished it, I will try to find time for another.

8. 

WRONG MIND: I like IPMB but I don’t have time to do the homework problems.

RIGHT MIND: Today I’ll make time to solve four homework problems. Tomorrow, four more.

9. 

WRONG MIND: You have to be a genius to apply physics and mathematics to biology and medicine; I have no chance.

RIGHT MIND: I can learn to apply physics and math to biology and medicine.

10. 

WRONG MIND: I am a biologist, and biologists can’t do math.

RIGHT MIND: I intend to learn math.

11. 

WRONG MIND: I love math, but my premed advisor says I don’t need math to become a medical doctor.

RIGHT MIND: I choose to learn math and to become a medical doctor.

12. 

WRONG MIND: Some students learn the topics in IPMB easily, but for me they are difficult. I am not meant to understand this subject.

RIGHT MIND: I can understand the topics in IPMB if I work hard.

13. 

WRONG MIND: Applying physics and mathematics to medicine and biology is difficult; I need so many skills. It isn’t worth it. I give up.

RIGHT MIND: I am learning how to apply physics and math to medicine and biology. I’m seeing how it all fits together. It’s so cool!

14. 

WRONG MIND: I got my homework back and it was covered with red ink. My instructor is an ass.

RIGHT MIND: I got my homework back and my instructor made many corrections. Such valuable feedback!

15. 

WRONG MIND: I spent 30 minutes solving a differential equation. After all that effort, I doubt my solution is correct.

RIGHT MIND: I spent 30 minutes solving a differential equation. Now I will spend 3 minutes plugging my solution back into the differential equation to check that it really works.

16. 

WRONG MIND: I solved the differential equation. The solution is complicated and I don’t understand what it means physically.

RIGHT MIND: I solved the differential equation. Now I will examine limiting cases to understand what it means physically.

17. 

WRONG MIND: My homework is due Friday. I don’t have to start working on it until Thursday night.

RIGHT MIND: My homework is due Friday. I will start working on it on Monday, leaving time to ask questions if I get stuck.

18. 

WRONG MIND: I need to read books by Steven Vogel, Mark Denny, and Knut Schmidt-Nielsen before I am ready to begin my homework.

RIGHT MIND: I would love to read books by Vogel, Denny, and Schmidt-Nielsen, but first I really need to start my homework.

19.

WRONG MIND: I know how to compute an answer to the homework, but it doesn’t mean anything.

RIGHT MIND: The purpose of computation is insight. I will think about my answer until I understand it physically.

20.

WRONG MIND: I have taken a calculus course, but I didn't really master the subject. IPMB uses calculus a lot; I shouldn’t take a course based on it.

RIGHT MIND: I will take a course based on IPMB, and use the experience to improve my math skills.

21.

WRONG MIND: Real-world problems are so complicated. The toy models presented in IPMB won’t prepare me for complex real-world problems.

RIGHT MIND: Solving toy models will help me build the skills and intuition I need to successfully attack more complicated real-world problems.

22.

WRONG MIND: To solve a homework problem, I search for an equation to put numbers into: “plug-and-chug.”

RIGHT MIND: I will think before I calculate. After I calculate, I will think if my answer makes sense. I will always think.

23.

WRONG MIND: Some homework problems ask me to “estimate” something, but they don’t give me all the data I need. What a bunch of BS.

RIGHT MIND: Part of learning to estimate is to make reasonable assumptions about data I do not have. I will develop this skill.

24.

WRONG MIND: I derived a complicated equation. I have no idea if it is correct.

RIGHT MIND: I will check my equation by verifying that it has the correct units. This doesn’t prove it’s right, but it could prove it’s wrong. I will practice this skill.

25.

WRONG MIND: Why does IPMB derive so many equations? I don’t need the derivation; I just want to use the equation to calculate numbers.

RIGHT MIND: A derivation is like a story. The derivation explains what is happening physically, and reminds me what assumptions were made.

26.

WRONG MIND: IPMB is always using math to model biological phenomena. This bugs me, and I dislike using “model” as a verb.

RIGHT MIND: I need to be able to build simple mathematical models of biological phenomena. I must learn to model.

27.

WRONG MIND: The computed tomography algorithms that create an image from projections are beautiful. I could never discover something that profound.

RIGHT MIND: With much hard work, I intend to discover something new. I will use those beautiful computed tomography algorithms to motivate me.

28.

WRONG MIND: I received a C- on my first exam, and a D+ on the second. I quit.

RIGHT MIND: I have learned so much from my mistakes on the first two exams. Had I gotten A’s on those exams, I wouldn’t be pushing myself hard enough.

29.

WRONG MIND: The homework is difficult for me, and my exam average is a C+. I will never achieve my goal of making new and valuable contributions to biomedical engineering.

RIGHT MIND: The skills needed in research are not identical to those needed in the classroom. I have as much to contribute on the job as the A student.

WRONG MIND: I would love to write. RIGHT MIND: I intend to write.

Mathematics is Biology’s Next Microscope, Only Better; Biology is Mathematics’ Next Physics, Only Better

Intermediate Physics for Medicine and Biology is full of equations. Equations are on almost every page, and often lots of them. Russ Hobbie and I use calculus without apology, and we discuss differential equations, Fourier analysis, and vector calculus. To understand biology, must we use all this mathematics?

“Mathematics is Biology’s Next Microscope, Only Better;
Biology is Mathematics’ Next Physics, Only Better”

The answer given by Joel Cohen of Rockefeller University is Yes! In his 2004 article “Mathematics is Biology’s Next Microscope, Only Better; Biology is Mathematics’ Next Physics, Only Better” (PLoS Biol 2:e439), Cohen argues that math skills are crucial for modern biologists. He writes

Although mathematics has long been intertwined with the biological sciences, an explosive synergy between biology and mathematics seems poised to enrich and extend both fields greatly in the coming decades.

The first half of his argument I believe enthusiastically: math has much to offer biology.

Mathematics broadly interpreted is a more general microscope. It can reveal otherwise invisible worlds in all kinds of data... For example, computed tomography can reveal a cross-section of a human head from the density of X-ray beams without ever opening the head, by using the Radon transform [see Chapter 12 of IPMB] ... Charles Darwin was right when he wrote that people with an understanding “of the great leading principles of mathematics… seem to have an extra sense”... Today’s biologists increasingly recognize that appropriate mathematics can help interpret any kind of data. In this sense, mathematics is biology’s next microscope, only better.

In IPMB, Russ and I illustrate how mathematical models can describe biological and medical systems. We don’t use sophisticated or complicated math, but instead focus on toy models that train students to analyze biological problems quantitatively. On the first day of my Biological Physics class, I tell the students that the course is a workshop on applying simple mathematical models to biological phenomena. Mathematics really is biology’s next microscope.

The second half of Cohen’s argument is not as obvious. Will biology lead to new advances in mathematics?

In the coming century, biology will stimulate the creation of entirely new realms of mathematics. In this sense, biology is mathematics’ next physics, only better. Biology will stimulate fundamentally new mathematics because living nature is qualitatively more heterogeneous than non-living nature.

Well, maybe, but I am skeptical. Cohen claims that biology generates large amounts of data, and biological systems are diverse and heterogeneous, which will lead to new math concepts that deal with what we now call Big Data. I hope this is true, but I expect much of the math already exists. Perhaps my skepticism arises because I love simple models, and the new math will certainly be elaborate and abstruse. We will see.

In his article, Cohen does more than make general claims; he gives specific examples. For instance, he tells a lovely story about how simple mathematical reasoning led William Harvey to predict the existence of capillaries

[Harvey’s] theoretical prediction, based on his meticulous anatomical observations and his mathematical calculations, was spectacularly confirmed more than half a century later when Marcello Malpighi (1628–1694) saw the capillaries under a microscope. Harvey’s discovery illustrates the enormous power of simple, off-the-shelf mathematics combined with careful observation and clear reasoning. It set a high standard for all later uses of mathematics in biology.

I encourage you all to read Cohen’s article. It makes a persuasive case that books such as Intermediate Physics for Medicine and Biology are necessary and even essential. Enjoy!

Marie Curie and her X-ray Vehicles’ Contribution to World War I Battlefield Medicine

Sunday is Veterans Day. This year the holiday is particularly significant because it marks the 100th anniversary of the armistice ending World War I.

Marie Curie in a Mobile Military Hospital X-Ray Unit.

Readers of Intermediate Physics for Medicine and Biology might wonder how the Great War influenced the application of physics to medicine. Timothy Jorgensen published a fascinating article on the website The Conversation discussing Nobel Prize-winner Marie Curie’s use of medical x-rays on the battlefield. Below are some annotated excerpts.

[In addition to her discovery of radium and polonium, Curie (1867-1934)] was also a major hero of World War I. In fact, a visitor to her Paris laboratory 100 years ago would not have found either her or her radium on the premises. Her radium was in hiding and she was at war.

The Guns of August,by Barbara Tuchman.

For Curie, the war started in early 1914, as German troops headed toward her hometown of Paris [“early” 1914? Germany declared war against France on August 3; see Guns of August by Barbara Tuchman]…[Curie] gathered her entire stock of radium, put it in a lead-lined container, transported it by train to Bordeaux…and left it in a safety deposit box at a local bank…

With the subject of her life’s work hidden far away…she decided to redirect her scientific skills toward the war effort; not to make weapons, but to save lives.

X-rays...had been discovered in 1895 by Curie’s fellow Nobel laureate, Wilhelm Roentgen.… Almost immediately after their discovery, physicians began using X-rays to image patients’ bones and find foreign objects – like bullets. But at the start of the war, X-ray machines were still found only in city hospitals, far from the battlefields where wounded troops were being treated. Curie’s solution was to invent the first “radiological car” – a vehicle containing an X-ray machine and photographic darkroom equipment – which could be driven right up to the battlefield where army surgeons could use X-rays to guide their surgeries....

[As the war progressed,] more radiological cars were needed. So Curie exploited her scientific clout to ask wealthy Parisian women to donate vehicles. Soon she had 20, which she outfitted with X-ray equipment. But the cars were useless without trained X-ray operators, so Curie started to train women volunteers. She recruited 20 women for the first training course, which she taught along with her daughter Irene [1897-1956, making her a teenager during much of the war], a future Nobel Prize winner herself….

Not content just to send out her trainees to the battlefront, Curie herself had her own “little Curie” [Petites Curies] – as the radiological cars were nicknamed – that she took to the front. This required her to learn to drive, change flat tires and even master some rudimentary auto mechanics, like cleaning carburetors. And she also had to deal with car accidents. When her driver careened into a ditch and overturned the vehicle, they righted the car, fixed the damaged equipment as best they could and got back to work [don’t you just love her?]....

Curie survived the war but was concerned that her intense X-ray work would ultimately cause her demise. Years later, she did contract aplastic anemia, a blood disorder sometimes produced by high radiation exposure. Many assumed that her illness was the result of her decades of radium work – it’s well-established that internalized radium is lethal [see The Radium Girls by Kate Moore]. But Curie was dismissive of that idea. She had always protected herself from ingesting any radium. Rather, she attributed her illness to the high X-ray exposures she had received during the war. (We will likely never know whether the wartime X-rays contributed to her death in 1934, but a sampling of her remains in 1995 showed her body was indeed free of radium.)

To learn more about Marie Curie, I recommend Jorgensen’s fine book Strange Glow: The Story of Radiation, or the article about Marie and her husband Pierre Curie and the discovery of polonium and radium, published by the Nobel Prize website. If you are a child at heart and enjoy Animated Hero Classic videos, watch this tearjerker.

To learn more about x-ray imaging, see Intermediate Physics for Medicine and Biology:

Chapter 16 describes the use of x rays for medical diagnosis and treatment. It moves from production to detection, to the diagnostic radiograph. We discuss image quality and noise, followed by angiography, mammography, fluoroscopy, and computed tomography. After briefly reviewing radiobiology, we discuss therapy and dose measurement. The chapter closes with a section on the risks from radiation.

To learn more about the First World War, visit the National World War I Museum and Memorial in Kansas City (know locally as Liberty Memorial).

National World War I Museum and Memorial in Kansas City.

Happy Veterans Day all who have defended our country in the military (including my dad and my brother-in-law). Thank you for your service.

Roderick MacKinnon's Nobel Lecture

In Chapter 9 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

Roderick MacKinnon and his colleagues determined the three-dimensional structure of a potassium channel using X-ray diffraction (Doyle et al. 1998; Jiang et al. 2003). MacKinnon received the 2003 Nobel Prize in Chemistry for his work on the potassium channel.

When I teach my graduate class on bioelectricity, we read the Doyle et al. article (“The Structure of the Potassium Channel: Molecular Basis of K⁺ Conduction and Selectivity,” Science, Volume 280, Pages 69–77, 1998). In my class, usually either my students and I discuss a paper or I explain some aspect of it. However, I’ve not found a better way to describe potassium channels than to watch MacKinnon’s brilliant Nobel lecture. I suggest you watch it too, using the embedded Youtube link below. It's 45 minutes long, but well worth the time.

If you have no time to spare, listen to the much shorter (less than two minute) interview where MacKinnon explains how being a scientist is like being an explorer.

Enjoy!

 

Roderick MacKinnon’s Nobel lecture.

 

“Being a scientist is like being an explorer.”

 

Earl Bakken (1924-2018)

Earl Bakken. From the Bakken Museum, via wikipedia.

Earl Bakken, cofounder of the medical device company Medtronic, died Sunday at the age of 94. In 1957 he developed the first external, battery-operated artificial pacemaker. Russ Hobbie and I don’t mention Bakken by name in Intermediate Physics for Medicine and Biology, but we do discuss pacemakers.

Cardiac pacemakers are a useful treatment for certain heart diseases. The most frequent are an abnormally slow pulse rate (bradycardia) associated with symptoms such as dizziness, fainting (syncope), or heart failure. These may arise from a problem with the SA node (sick sinus syndrome) or with the conduction system (heart block). One of the first uses of pacemakers was to treat complete or third degree heart block. The SA node and the atria fire at a normal rate but the wave front cannot pass through the conduction system. The AV node or some other part of the conduction system then begins firing and driving the ventricles at its own, pathologically slower rate…. A pacemaker stimulating the ventricles can be used to restore a normal ventricular rate.

In the 1950s, famed cardiac surgeon Dr. Walt Lillehei was at the University of Minnesota, where he met Bakken. Kirk Jeffrey tells the story of their collaboration in Machines In Our Hearts.

Machines In Our Hearts, by Kirk Jeffrey

The myocardial pacing wire was the first electrical device ever to be implanted in the human body and left there for a period of time. Surgeons at Minnesota were now able to pace children for days or weeks after heart surgery. By October 1957, they had used the technique with 18 patients. But Lillehei now grew uneasy about the Grass stimulator because it was bulky and plugged into the electrical system. The surgeon wanted to get his heart patients out of bed and moving around, but the stimulator had to accompany them on a wheeled cart. The electrical cord was a further nuisance. ‘‘Many of these [patients] were kids. They wanted to wander around and get active. Well, they were active. They couldn’t go any further than the cord. We had to string wires down the hall. . . . And then, if they needed an X ray or something that couldn’t be done in the room, you couldn’t get on the elevator so you had to string them down the stairwells. It seemed that almost everything you wanted was on a different floor. We needed something battery-operated.’’

From Machines In Our Hearts.

The plug-in stimulator was more than an inconvenience, for by introducing the myocardial pacing wire, Lillehei and his associates had connected the hearts of their surgical patients to the 110-volt electrical system of the hospital. Everyone in the program knew that an electrical surge might send patients into ventricular fibrillation or that a power outage could leave them without pacemaker support. On October 31, 1957, an equipment failure at a large Twin Cities power plant caused an outage lasting nearly three hours in Minneapolis. The University hospital had auxiliary power in its surgical suites and recovery area, but not in patients’ rooms. None of his heart patients died—but Lillehei viewed the event as a warning. Lillehei…turned to Earl Bakken…a young engineer who owned a small medical electronics business called Medtronic and repaired and serviced equipment for the Department of Surgery.

Bakken…realized that he could simply build a stimulator that used transistors and small batteries. ‘‘It was kind of an interesting point in history,’’ he recalled—‘‘a joining of several technologies.’’ In constructing the external pulse generator, Bakken borrowed a circuit design for a metronome that he had noticed a few months earlier in an electronics magazine for hobbyists. It included two transistors. Invented a decade earlier, the transistor was just beginning to spread into general use in the mid-1950s. Hardly anyone had explored its applications in medical devices. Bakken used a nine-volt battery, housed the assemblage in an aluminum circuit box, and provided an on-off switch and control knobs for stimulus rate and amplitude.

At the electronics repair shop that he had founded with his brother-in-law in 1949, Bakken had customized many instruments for researchers at the University of Minnesota Medical School…When Bakken delivered the battery-powered external pulse generator to Walt Lillehei in January 1958, it seemed to the inventor another special order, nothing more. The pulse generator was hardly an aesthetic triumph, but it was small enough to hold in the hand and severed all connection between the patient’s heart and the hospital power system. Bakken’s business had no animal-testing facility, so he assumed that the surgeons would test the device by pacing laboratory dogs. They did ‘‘a few dogs,’’ then Lillehei put the pacemaker into clinical use. When Bakken next visited the university, he was surprised to find that his crude prototype was managing the heartbeat of a child recovering from open-heart surgery.

The Bakken Museum. From wikipedia.

Russ Hobbie is retired from the University of Minnesota, and still lives in the area, so he is particularly familiar with Earl Bakken. He served on the board of the Bakken Museum, which is devoted to bioelectricity (my kind of museum). Russ say he ‘‘was impressed by Bakken's vision, energy, and desire to help people. The Bakken Museum has an extensive outreach program which does a lot of good things.’’ They recently posted a statement honoring their founder.

Medtronic is one of the largest medical device companies. I had a job interview there years ago, but I didn’t get the position. I came away impressed by the company, and wish I had bought stock.

Bakken is a member of the dwindling greatest generation; he was an airborne radar maintenance instructor during World War II. He had a long, full life, and we will miss him.

Below are a couple of videos about Earl Bakken. Enjoy!