Used Math

Used Math,
by Clifford Swartz.

How much mathematics is needed when taking a class based on Intermediate Physics for Medicine and Biology? Students come to me all the time and say “I am interested in your class, but I don’t know if I have enough math background.” I wish I had a small book that reviewed the math needed for a class based on IPMB. Guess what? Used Math by Clifford Swartz is just what I need. In the preface, Swartz writes:

In this book, which is part reference and part reminder, we are concerned with how to use math. We concentrate on those features that are most needed in the first two years of college science courses. That range is not rigorously defined, of course. A sophomore physics major at M.I.T. or Cal. Tech. must use differential equations routinely, while a general science major at some other place may still be troubled by logarithms. It is possible that even the Tech student has never really understood certain things about simple math. What, for instance, is natural about the natural logs? We have tried to cover a broad range to topics—all the things that a science student might want to know about math but has never dared ask.

Students and instructors might benefit if I went through Used Math chapter by chapter, assessing what math is needed, and what is math not needed, when studying from IPMB. Also, what math is needed but is not included in Used Math.

Chapter 1: Reporting and Analyzing Uncertainty

Russ Hobbie and I assume our readers know about scientific notation and significant figures. The best time to teach significant figures is during laboratory. (Wait! Is there is a lab that goes along with IPMB? No. At least not that I know of. But perhaps there should be.) In my Biological Physics class, students often answer homework using too many significant figures. I don’t take off points, but I write annoying notes in red ink.

Chapter 2: Units and Dimensions

Russ and I do not review how to convert between units. My students usually don’t have trouble with this. Often, however, they will do algebra and derive an equation that is dimensionally wrong (for example, containing “a + a²” where a has units of length). I take off extra points for such mistakes, and I harp about them in class.

Chapter 3: Graphs

We assume students can plot a simple graph of y(x) versus x. In class, when we derive a result such as y(x) = x/(x² + a²), I ask the students what a sketch of this function looks like. Often they have trouble drawing it. Our homework problems routinely ask students to plot their result. I deduct points if these plots are not qualitatively correct. IPMB discusses semilog and log-log plots in Chapter 2.

Chapter 4: The Simple Functions of Applied Math

Students should be familiar with powers, roots, trigonometric functions, and the exponential function before taking a class based on IPMB. Chapter 2 is devoted to the exponential, and Appendix C lists properties of exponents and logarithms. We define the hyperbolic functions sinh and cosh upon first use (Eq. 6.98). I don’t give placement quizzes at the first class meeting, but if I did I would have the students sketch plots of x², √x, sin(x), cos(x), tan(x), e^x, log(x), sinh(x), cosh(x), and tanh(x). If you can’t do that, you will never be able to translate mathematical results into physical insight.

Chapter 5: Statistics

I discussed the statistics used in IPMB before in this blog. We analyze probability distributions in Chapter 3 on thermodynamics and Chapter 4 on diffusion, and go into more detail in Appendix G (mean and standard deviation), Appendix H (the binomial distribution), Appendix I (the Gaussian distribution), and Appendix J (the Poisson distribution). We don’t discuss analyzing data, such as testing a hypothesis using a student t-test. One topic missing from Used Math is simple concepts from probability; for example, when you role two dice what is the probability that they add to five? When I taught quantum mechanics (a subject in which probability is central), I spent an entire class calculating the odds of winning at craps. You will understand probability by the time you finish that calculation.

Chapter 6: Quadratic and Higher Power Equations

Russ and I use the quadratic equation without review. We don’t solve any higher order equations in IPMB, and we never ask the student to factor a polynomial using a procedure similar to long division (yuk!).

Chapter 7: Simultaneous Equations

Students should know how to solve systems of linear equations. I often solve small systems (two or three equations) in class. Sometimes when teaching I derive the equations and then say “the rest is just math” and state the solution. This happens often when doing a least-squares fit at the start of Chapter 11. I don’t ask students to solve a system of many (say, five) equations.

Chapter 8: Determinants

IPMB does not stress linear algebra and we never require that students calculate the determinant of a matrix. However, we do occasionally require the student to calculate a cross product using a method similar to taking a determinant (Eq. 1.9), so students need to know the rules for evaluating 2 × 2 and 3 × 3 determinants.

Chapter 9: Geometry

Used Math goes into more detail about analytical geometry (conic sections, orbits, and special curves like the catenary) than is needed in IPMB. The words ellipse and hyperbola never appear in our book (parabola does.) We discuss cylindrical and spherical coordinates in Appendix L. Students should know how to find the surface area and volume of simple objects like a cube, cylinder, or sphere.

Chapter 10: Vectors

Russ and I use vectors throughout IPMB. They are reviewed in Appendix B. We define the dot and cross product of two vectors when they are first encountered.

Chapter 11: Complex Numbers

We avoid complex numbers. I hate them. One exception: we introduce complex exponentials when discussing Fourier methods, where we present them as an alternative to sines and cosines that is harder to understand intuitively but easier to handle algebraically. You could easily skip the sections using complex exponentials, thereby banishing complex numbers from the class.

Chapter 12: Calculus—Differentiation

Students must know the definition of a derivative. In class I derive a differential equation for pressure by adding the forces acting on a small cube of fluid and then taking the limit as the size of the cube shrinks to zero. If students don’t realize that this process is equivalent to taking a derivative, they will be lost. Also, they should know that a derivative gives the slope of a curve or a rate of change. What functions should students be able to differentiate? Certainly powers, sines and cosines, exponentials, and logarithms. Plus, students must know the chain rule and the product rule. They should be able to maximize a function by setting its derivative to zero, and they should realize that a partial derivative is just a derivative with respect to one variable while the other variables are held constant (Appendix N).

Chapter 13: Integration

Students must be able to integrate simple functions like powers, sines and cosines, and exponentials. They should know the difference between a definite and indefinite integral, and they should understand that an integral corresponds to the area under a curve. Complicated integrals are provided to the student (for example, Appendix K explains how to evaluate integrals of e^-x2) or a student must consult a table of integrals. In my class, I always use the “guess and check” method for solving a differential equation: guess a solution containing some unknown parameters, plug it into the differential equation, and determine what parameters satisfy the equation; no integration is needed. One calculation that some students have problems with is integrating a function over a circle. In class, I carefully explain in how the area element becomes rdrdθ. At first the students look bewildered, but most eventually master it. I avoid integration by parts (which I dislike), but it is needed when calculating the electrical potential of a dipole. Perhaps you can devise a way to eliminate integration by parts altogether?

Chapter 14: Series and Approximations

Appendix D of IPMB is about Taylor series. If you remember only that e^x is approximately 1+x, you will know 90% of what you need. The expansions of sin(x) and cos(x) are handy, but not essential. When deriving the dipole approximation, I use the Taylor series of 1/(1-x). (The day I discuss the dipole is one of the most mathematical of the semester.) Student’s never need to derive a Taylor series, and they rarely require more than the first two terms of the expansion. The geometric series (1+x+x²+…) appears in Homework Problem 28 of Chapter 8, but the sum of the series is given. In IPMB, we never worry about convergence of an infinite series. Fourier series is central to imaging. In Medical Physics (PHY 3260), I spend a couple weeks discussing Fourier series and Fourier transforms, the most mathematically intensive part of IPMB. If students can handle Chapters 11 and 12, they can handle any math in the book.

Chapter 15: Some Common Differential Equations

I always tell my class “if you can solve only one differential equation, let it be dy/dx = by” (in case you are wondering, the solution is y = e^bx). As I mentioned earlier, I preferred to solve differential equations by guess and check. In IPMB, you can get away with guesses that involve powers, trig functions, and exponentials. Some students claim that a course in differential equations is needed before taking a class using IPMB. I disagree. We don’t need advanced methods (e.g., exact differential equations) and we never analyze existence and uniqueness of solutions. We just guess and check. Appendix F discusses differential equations in general, but my students rarely need to consult it. I emphasize understanding differential equations from a physical point-of-view. I expect my students to be able to translate a physical statement of a problem into a differential equation. Yes, I put such questions on my exams. To me, that is a crucial skill.

Chapter 16: Differential Operators

What Used Math calls differential operators, I call vector calculus: divergence, gradient, and curl. Russ and I use vector calculus occasionally. I expect students to be able to do homework problems using it, but I don’t expect them to do such calculations on exams. Mostly, vector calculus appears when talking about electricity and magnetism in Chapters 6-8. I think an instructor could easily design the class to avoid vector calculus altogether. Whenever Russ and I use vector calculus, we typically cite Div, Grad, Curl and All That, which is my favorite introduction to these concepts.


That sums up of the topics in Used Math. Is there any other math in IPMB? Special functions sometimes pop up, such as Bessel functions, the error function, and Legendre polynomials. Usually these appear in homework problems that you don’t have to assign. We occasionally ask students to solve differential equations numerically (see Sec. 6.14), usually in the homework. I skip these problems when I teach from IPMB; there is not enough time for everything. In some feedback problems in Chapter 10 (for example, Problems 10.12 and 10.17) the operating point must be evaluated numerically. I do assign these problems, and I tell students to find the solution by trial and error. We don’t spend time developing fancy methods for solving nonlinear equations, but I want students to realize they can solve equations such as xe^x = 1 numerically (the solution is approximately x = 0.57).

In summary, Used Math contains almost all the mathematics you need when taking a class from IPMB. It would be an excellent supplementary reference for students. From now on, when students ask me how much math they need to know for my Biological Physics or Medical Physics class, I will tell them all they need is in Used Math.

Abramowitz and Stegun

Handbook of Mathematical Functions,
by Abramowtiz and Stegun

When Russ Hobbie and I discuss a special mathematical function in Intermediate Physics for Medicine and Biology, we often cite the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, by Milton Abramowitz and Irene Stegun. My copy is the paperback Dover edition, which is a unaltered replication of the original book published in 1964 by the National Bureau of Standards. According to Google Scholar, the Handbook is approaching its 100,000^th citation, which means it has been cited on average about once every five hours for over 50 years!

“Abromowitz and Stegun,” as the Handbook is universally known, has a fascinating history. According to David Alan Grier’s article “Irene Stegun, the ‘Handbook of Mathematical Functions’, and the Lingering Influence of the New Deal” (The American Mathematical Monthly, Volume 113, pages 585–597, 2006), the Handbook began as part of President Franklin Roosevelt's Work Projects Administration during the Great Depression. Grier describes the group gathered to calculate mathematical tables in the days before electronic computers.

The WPA organized the Mathematical Tables Project in the fall of 1937 and began its operations at the start of February 1938. The project was designed to employ 450 workers as human computers, individuals who did scientific calculations by hand. Using nothing but paper and pencil, these workers were instructed to create large, high precision tables of mathematical functions: the exponential function, natural and common logarithms, circular and hyperbolic trigonometric functions, probability functions, gamma, elliptical, and Bessel functions.

The Mathematical Tables Project operated in New York City and occupied the top floor of a decaying industrial building in the “Hell’s Kitchen” neighborhood of New York City. All of the project’s computers were drawn from the city’s welfare rolls and were desperately poor. Most had been unemployed for at least a year. Only a few had attended high school. Roughly 20 percent of them were polio victims, amputees, or handicapped in some way. Another 20 percent of the computers were African American. The project also had a large cohort of Jewish workers from the tenements of the lower East Side of Manhattan and a group, of roughly equal size, from the cramped Irish neighborhoods on the West Side. Approximately 45 percent of the computers were women who were supporting their families.

This motley crew performed the heroic work that led eventually to the accurate tables in the Handbook. Grier continues

The [Planning] committee [which included Abramowitz and Stegun] occupied a few desks in one corner of the project’s vast computing floor. From that corner, the committee oversaw the 450 human computers. Most of these computers had no mathematical training. Many of them did not know the steps of long division and at least a few did not understand negative numbers.

The Mathematical Tables Project eventually morphed into an effort to produce a definitive mathematical handbook.

The Handbook of Mathematical Functions…is an unusual artifact, for it is both an example of a large, collaborative research project, which is a rare activity in the mathematical sciences, and one of the very few scientific activities of the 1950s led by a woman. As was often the case with early scientific contributions by women, the gender of the leader is obscure. The title page lists two editors, one male and one female: Milton Abramowitz (1913-1958) and Irene Stegun (1919?)… Initially, these two mathematicians shared editorial duties, though Abramowitz clearly played the leading role. He prepared the outline for the book, drafted preliminary material, and recruited the first group of contributors.... His role ended on a hot summer's day in 1958, when he unwisely decided to mow the lawn of his home in suburban Washington. Succumbing to the heat, he collapsed and died, leaving Stegun as the sole editor.

When Computers Were Human,
by David Alan Grier.

Grier has written an entire book about large scale computations done before the invention of electronic computers: When Computers Were Human. I've gotten it interlibrary loan and plan to read it over the holiday break. If you prefer listening to reading, watch the Youtube video of a talk by Grier, shown below. Anyone who watched the movie Hidden Figures saw human computers working for NASA.

One of the best things about Abramowitz and Stegun is that you can access it for free online at http://numerical.recipes/aands. NIST (the National Institutes of Standards and Technology, formerly known as the National Bureau of Standards) also maintains an updated electronic math handbook at http://dlmf.nist.gov. 

Enjoy!

Gopalasamudram Narayanan Ramachandran, Biological Physicist

Many followers of the Intermediate Physics for Medicine and Biology Facebook page are from India, and I would like to somehow thank them for their interest in our book. The only way I can express my appreciation is by writing in this blog. So, today’s post is about the great Indian physicist Gopalasamudram Narayanan Ramachandran (1922-2001).

In an obituary published in the Biographical Memoirs of Fellows of the Royal Society (Volume 51, Pages 367–377, 2005), Vijayan and Johnson write

G. N. Ramachandran has been among the most outstanding crystallographers and structural biologists of our times. He is considered by many to be the best scientist to have worked in independent India. The model of collagen developed by him has stood the test of time and has contributed greatly to understanding the role of this important fibrous protein. His pioneering contributions in crystallography, particularly in relation to methods of structure analysis using Fourier techniques and anomalous dispersion, are well recognized. A somewhat less widely recognized contribution of his is concerned with three-dimensional image reconstruction. Much of the foundation of the currently thriving field of molecular modelling was laid by him. The Ramachandran plot remains the simplest and the most commonly used descriptor and tool for the validation of protein structures.

Ramachandran appears in Intermediate Physics for Medicine and Biology in Chapter 12, when Russ Hobbie and I discuss computed tomography. He and A. V. Lakshminarayanan developed one of the two man main tomographic techniques: filtered back projection. We write

Filtered back projection is more difficult to understand than the direct Fourier technique. It is easy to see that every point in the object contributes to some point in each projection. The converse is also true. In a back projection every point in each projection contributes to some point in the reconstructed image…A very simple procedure would be to construct an image by back-projecting every projection…We will now show that the image f_b(x,y) obtained by taking projections of the object F(θ,x') and then backprojecting them is equivalent to taking the convolution of the object with the function h.

h(x) is

Unfortunately, this function does not exist; the integral doesn’t converge. The factor |k| diverges as k goes to ±∞. But Ramachandran and Lakshminarayanan realized that you don’t need to integrate to infinity. In the above integral, k is the spatial frequency. He suggested there should be an upper limit on the spatial frequency, k_max. What should the upper limit be? The measured projection F(θ,x') is not a continuous function of position x'. The data is discrete, measured at a finite number of points. The largest spatial frequency is that given by the Nyquist sampling criterion: there should be at least two points per wavelength. Using this upper limit for k_max, Ramachandran and Lakshminarayanan were able to solve the integral for h(x) analytically, and found that

where i denotes the ith discrete value of x. This result looks slightly different than Eq. 12.44 in IPMB; Here I factored N²/16 out of each term, and I use i for the integer instead of k, because I don’t want to use k for both spatial frequency and an integer. Below is a plot of h(i).

Convolution with function h(i) corresponds to a passing the signal through a high pass filter (often called the "Ram-Lak filter"). Therefore, the convolution of a constant should vanish, implying that all the values of h(i) should add to zero. In fact, this is true. The infinite series
 

 is exactly what is needed to ensure this.

At the end of their obituary, Vijayan and Johnson discuss Ramachandran’s impact on science and India.

To more than a generation of scientists in India, and some abroad, Ramachandran was a source of scientific and personal inspiration. Many of his contributions were based on simple but striking ideas. He demonstrated how international science could be influenced, even from less well-endowed neighbourhoods, through ingenuity and imagination. It is remarkable that although Ramachandran left structural biology and mainstream research about a quarter of century ago, his presence in the field remains as vibrant as ever. Indeed, Ramachandran established a great scientific tradition. That tradition lives on and thrives in the world, in India, and in the two research schools he founded.

Thanks to all the Indian readers of IPMB. I’m glad you like the book.

Shattered Nerves: How Science is Solving Modern Medicine's Most Perplexing Problem

Shattered Nerves:
How Science is Solving Modern
Medicine's Most Perplexing Problem,
by Victor Chase.

In his book Shattered Nerves: How Science is Solving Modern Medicine’s Most Perplexing Problem, Victor Chase tells the story of neural prostheses. Russ Hobbie and I discuss neural stimulation in Section 7.10 of Intermediate Physics for Medicine and Biology.

The information that has been developed in this chapter can also be used to understand some of the features of stimulating electrodes. These may be used for electromyographic studies; for stimulating muscle to contract called functional electrical stimulation (Peckham and Knutson 2005); for a cochlear implant to partially restore hearing (Zeng et al. 2008); deep brain stimulation for Parkinson’s disease (Perlmutter and Mink 2006); for cardiac pacing (Moses and Mullin 2007); and even for defibrillation (Dosdall et al. 2009).

Chase begins by describing the cochlear implant. A common cause of deafness is the death of hair cells in the cochlea, leaving the auditory nerve intact but not activated by sound. A cochlear implant stimulates the auditory nerve using several electrodes, each corresponding to a different frequency. Chase often describes medical devices from the point of view of a patient, and in this case he tells the story of Michael Pierschalla, who not only benefited from this technology but contributed to its development.

I am fascinated by idiosyncratic inventors such as Giles Brindley. Chase writes

An often-told tale about Giles Brindley might reveal something about the person referred to as the grandfather of neural prostheses. In 1983, the inveterate innovator and self-experimenter stood before a scientific audience and removed his pants. The venue was Las Vegas, Nevada, and the audience that witnessed this occurrence was the membership of the American Urological Association. Brindley was demonstrating, quite graphically, the success of an injection of phenoxybenzamine, a treatment he had developed for erectile dysfunction.

Brindley developed one of the first visual prostheses that stimulated the brain. He also invented a musical instrument he called the “undilector,” which is something like a computer-controlled bassoon.

One hero of Chase’s story is F. Terry Hambrecht, who led the National Institutes of Health Neural Prosthesis Program. When I was working at the NIH intramural program in the early 1990s, I often attended Hambrecht’s annual Neural Prosthesis Workshop. Sometimes I would submit a poster about magnetic stimulation. It was close enough to the workshop’s theme to be worth a poster, but far enough from its main thrust to be a little off-topic. At these workshops, held on the NIH campus, I met many of the scientists highlighted by Chase.

Shattered Nerves focuses on research performed at Case Western Reserve University. J. Thomas Mortimer founded the Applied Neural Control Laboratory there. His student P. Hunter Peckham developed a prosthetic device to restore function to a patient's paralyzed hand. Another Case researcher, Ronald Triolo, invented a stimulator that allowed a wheelchair-bound patient to stand and move around. Quadriplegics often have difficulty controlling their urination and bowel movements. Mortimer and Graham Creasey developed a prosthesis to control the bladder and bowel muscles.

Rather than summarizing Shattered Nerves myself, I will let Chase do so in his own words.

Unfortunately, in some people, the circuitry that generates and conducts electrical signals goes bad, rendering them unable to fully partake of the miracle of the senses, as in the case of the blind, when the rod and cone photoreceptors inside the eye can no longer translate light into the electrical signals that send information to the brain. Or when the hair cells inside the cochlea of the inner ear, which process sound waves, die off, and a person loses the ability to hear. Failure of the body's electrical circuitry is also responsible for paralysis that occurs when spinal cord injuries damage the nerve cells that carry electrical signals from the brain's motor cortex to the muscles and from the skin's tactile receptors to the somatosensory portion of the brain. Until recently, these conditions were deemed irreversible. Now there is hope.

What did I gain from reading Shattered Nerves? First, I like to study the history of a field in order to better appreciate the current problems and future directions. Second, the researchers and patients that Chase describes are inspirational. Third, I was amazed at how these pioneers combined physics and engineering with medicine and biology, as Russ and I advocate in IPMB.

All books have advantages and disadvantages. One disadvantage is that Shattered Nerves was written in 2006. In a fast moving field like neural prostheses, I wish the book was up-to-date. An advantage is that you can read it for free through Project Muse.

Enjoy!

Suki Has Fleas

Suki

Suki has fleas. It’s her worst infestation ever. My wife and I have battled them for about a month, and are finally gaining the upper hand by constantly vacuuming the house, washing her bedding, and giving her baths.

While I am sure you empathize with our little puppy, you are probably asking “what do Suki’s fleas have to do with Intermediate Physics for Medicine and Biology?” A lot! In Problem 47 of Chapter 2, Russ Hobbie and I ask students to determine how jumping height scales with mass. I won’t give away the answer here, but when you are asked how something scales with mass, one possible answer is that it doesn’t. In other words, if the allometric relationship is Jumping Height = C Massⁿ, where C and n are constants, then one possible value for n is zero; jumping height is independent of mass.

Scaling: Why is Animal
Size So Important?
by Knut Schmidt-Nielsen.

The next homework exercise, Problem 48, analyzes one of the many scaling arguments made by Knut Schmidt-Nielsen in his marvelous book Scaling: Why is Animal Size so Important? The start of Problem 48 gives away the answer to Problem 47:

Problem 48. In Problem 47, you should have found that all animals can jump to about the same height (approximately 0.6 m), independent of their mass M.

Are you skeptical that, for instance, a tiny flea can jump 60 cm (about two feet)? I can tell you from first-hand experience that those little buggers can really jump. Each evening we inspect Suki with a flea comb, and sometimes a flea jumps away before we can kill it. 

Problem 48 requires that students calculate the flea's acceleration. Again, I won’t give you the answer, but those fleas sure undergo large accelerations! If you don’t believe me, do Problem 48, or read what Knut Schmidt-Nielsen writes.

For a flea, acceleration takes place over less than 1 mm, and takeoff time is less than 1 msec. The average acceleration during takeoff must therefore exceed 200 g. It is worth a moment's reflection to think of what such high acceleration means. It means that the force on the animal is 200 times its weight (any mammal would be totally crushed under such forces), and the insect must have a skeleton and internal organs able to resist such acceleration forces.

I wonder how fleas avoid concussions?

I’m glad that our little fleabag is cleaning up her act. Suki turned 15 a few months ago, and she is the old lady of the family. But she is still up for our walks, during which I listen to audiobooks and she snoots around (that’s probably how she got the fleas). And what is her favorite book? Intermediate Physics for Medicine and Biology, of course.