Friday, May 25, 2018

The Constituents of Blood

Intermediate Physics for Medicine and Biology: The Constituents of BloodI'm a big supporter of blood donation. This week I gave another pint to the Red Cross, which brings my total to 8 gallons. As I lay there with a needle stuck in my arm, I began to wonder "what's in this blood they're squeezing out of me?"

Table 3.1 in Intermediate Physics for Medicine and Biology lists some constituents of blood. I reproduce the table below, with revisions.

Constituent Density in mg/cm3 Number in 1 μm3
Water 1000 33,000,000,000
Sodium 3 83,000,000
Glucose 1 3,300,000
Cholesterol 2 3,100,000
Hemoglobin 150 1,400,000
Albumin 45 390,000

This version of the table highlights several points. Water molecules outnumber all others by a factor of four hundred. Sodium ions are sixty times more common than hemoglobin molecules, but the mass density of hemoglobin is over fifty times that of sodium. In other words, if judged by number of molecules (and therefore the osmotic effect) sodium is most important, but if judged by mass or volume fraction, hemoglobin dominates. Glucose and cholesterol are intermediate cases. Albumin has a surprisingly small number of molecules, given that I thought it was one of the main contributors to osmotic pressure. It is a big molecule, however, so by mass it contributes nearly a third as much as hemoglobin.

Are other molecules in blood important? You can find a comprehensive list of blood constituents beautifully illustrated here. When judged by number, sodium is the most important small ion, but the chloride ion contributes nearly as much. Carbon dioxide and bicarbonate are also significant, and potassium has about the same number of molecules as glucose. If you drive drunk, you may have twice as many ethyl alcohol molecules as potassium ions (if the number of ethanol molecules reaches the level of sodium or chloride ions, you die). Urea has a similar number of molecules as hemoglobin.

Judged by mass, you get an entirely different picture. Large protein molecules dominate. Hemoglobin is by far the largest contributor to blood by mass (after water, of course), followed by albumin and another group of proteins called globulins. Next are glycoproteins such as the clotting factor fibrinogen and iron-binding transferrin.

Many trace constituents hardly affect the osmotic pressure or density of blood, but are excellent biomarkers for diagnosing diseases.

If you're starting to think that blood is awfully crowded, you're right. The picture below is by David Goodsell. No scale bar is included, but each candy-apple-red hemoglobin molecule in the lower left has a diameter of about 6 nm. The water, ions, and other small molecules such as glucose are not shown; if they had been they would produce a fine granular appearance (water has diameter of about 0.3 nm) filling in the spaces between the larger macromolecules.

Blood. Illustration by David S. Goodsell, the Scripps Research Institute.
Blood. Serum is in the upper right and a red blood cell is in the lower left. In the serum, the Y-shaped molecules are antibodies (an immunoglobulin), the long thin light-red molecules are fibrinogen (a glycoprotein), and the numerous potato-like yellow proteins are albumin. The red blood cell is filled with red hemoglobin molecules. The cell membrane is in purple. The illustration is by David S. Goodsell of the Scripps Research Institute.

In another eight weeks I will get free juice and cookies be eligible to give blood again. It doesn't hurt (much) or take (too) long. If you want to donate, contact the American Red Cross. Give the gift of life.

Friday, May 18, 2018

A Biological Constant

Intermediate Physics for Medicine and Biology: A Biological Constant
Membranes, Ions and Impulses, by Kenneth Cole
Physics has many famous constants: Planck’s constant, the speed of light, and the gravitational constant, to name a few. Biology has few such constants. Life is so full of variety that almost any parameter can vary between species or tissues. In fact, physicists differ from biologists by their focus on the unity rather than the diversity of life.

There is, however, one parameter that comes close to being a biological constant. All cells are surrounded by a membrane whose thickness and composition varies little among species. Therefore, the capacitance per unit area, Cm, of a membrane is as close to being a biological constant as you are likely to find.

In Section 6.16 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I calculate the capacitance of a lipid bilayer, and find that Cm is about 0.01 farads per square meter. Many of the key papers during the “golden age” of classical biophysics didn’t use standard SI units. Instead of measuring distance in meters, they used centimeters. If you express Cm as per square centimeter, and if you use microfarads instead of farads, you get the easy-to-remember value of Cm = 1 μF/cm2. Kenneth Cole wrote in his book Membranes, Ions and Impulses:
“This figure of about 1 μF/cm2 has been so confirmed and refined, extended and approximated for membranes of red cells and almost all other living cells, as to become a biophysical constant.”
Are there other biological constants? I suppose some constants governing the structure of key biological molecules, such as the distance between adjacent base pairs of the DNA double helix (0.34 nm), are conserved throughout biology. But these parameters belong more to the realm of biochemistry than biophysics. If you restrict your selection to parameters discussed in IPMB, Cm is one of the few biological constants.

Membranes, Ions and Impuses: A Chapter of Classical Biophysics, by Kenneth Cole

Membrane Capacitance, as discussed in Membranes, Ions and Impulses


 
Table of Contents for Membranes, Ions and Impulses, by Kenneth Cole

Friday, May 11, 2018

The Curie Temperature

Intermediate Physics for Medicine and Biology: The Curie Temperature In Chapter 8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetic materials, including ferromagnets (permanent magnets in which electron spins are aligned even in the absence of an external magnetic field). We write
“If the temperature of the sample is raised above a critical temperature called the Curie temperature, the magnetism is destroyed.”
When seeing such a sentence, my first inclination is to write a homework problem that uses a toy model to illustrate the physics behind the concept. Unfortunately, analyzing the Curie temperature is difficult, so no new homework problem appears in this post (readers are encouraged to try their hand at writing one).

Solid State Physics by Ashcroft and Mermin
When faced with a difficult concept in material physics, I reach for a copy of Solid State Physics by Neil Ashcroft and N. David Mermin. Regular readers of this blog may recall that I am a big fan of Mermin (for instance, see here and here). Everything I know about solid state physics (which isn't much) I learned from Solid State Physics. When Ashcroft and Mermin describe magnetic ordering of spins, they explain that
“Quantitative theories of magnetic ordering have proved most difficult to construct near the critical temperature Tc at which ordering disappears. The difficulty is not peculiar to the problem of magnetism. The critical points of liquid-vapor transitions, superconducting transitions (Chapter 34), the superfluid transition in liquid He4, and order-disorder transitions in alloys, to name just a few, present quite strong analogies and give rise to quite similar theoretical difficulties.”
They settle for a phenomenological description of the Curie temperature.
“The critical temperature Tc above which magnetic ordering vanishes is known as the Curie temperature in ferromagnets… As the critical temperature is approached from below, the spontaneous magnetization…drops continuously to zero. The observed magnetization just below Tc is well described by a power law.

M(T) ∼ (TcT)β,

where β is typically between 0.33 and 0.37.”
Below I plot of the spontaneous magnetization M versus the absolute temperature T for β=1/3.

magnetization versus temperature

The Curie temperature is interesting for two reasons. First, it is not named after Marie Curie, who plays such a big role in medical physics for isolating some of the first radioactive elements including radium and polonium. Instead, it is named after her husband Pierre Curie, who did important research on magnetism. Second, the ferromagnetic material that Russ and I discuss most in IPMB is magnetite (Fe3O4), which is found in magnetosomes, small magnetic particles that cause magnetotactic bacteria to align with the earth's magnetic field. The Curie temperature for magnetite is 585 °C, or 858 K, which is too hot to support life. Perhaps other substances exist for which the Curie temperature plays a role in biology and medicine, but I don’t know what they are.

I conclude with a quote from Mermin’s delightful essay “Writing Physics” in which he talks about writing Solid State Physics with Ashcroft. Enjoy!
“The striking exception to my inability to write collaboratively is my eight-year collaboration with Neil Ashcroft on our 800 page book on solid state physics. We have very different prose styles. Yet the book has a clear and distinctive uniform tone, which can't be identified as belonging to either of us. I think the reason this worked was that Neil knows solid state physics much better than I do. So he would produce the first drafts. Characteristically, I would not understand them. So I would try to make sense of what he was saying, and then produce my typical kind of irritating second draft. Neil, however, would now have to correct all my mistakes in a massively rewritten third draft. I would then have to root out any new obscurities he had introduced in a fourth draft. By this kind of tennis-playing, we would go through five or six drafts, and emerge with something that was clear, correct, and sounded like a human voice. That voice, however, was neither of ours.”

Friday, May 4, 2018

Strange Glow

Intermediate Physics for Medicine and Biology: Strange Glow
Strange Glow by Timothy Jorgensen
I recently read Strange Glow: The Story of Radiation, by Timothy Jorgensen. This book overlaps many of the topics Russ Hobbie and I discuss in Intermediate Physics for Medicine and Biology, particularly in Section 16.12 (The Risk of Radiation). Jorgensen writes
"The common denominator of most radiation exposure scenarios is fear. Just mention the word radiation, and you instill fear—a perfectly understandable response given the images of mushroom clouds and cancerous tumors that immediately come to mind. Those images would justifiably cause anyone to be anxious. Nevertheless, some people have also become highly afraid of diagnostic x-rays, luggage scanners, cell phones, and microwave ovens. This extreme level of anxiety is unwarranted, and potentially dangerous.

When people are fearful, they tend to exaggerate risk. Research has shown that people’s perception of risk is tightly linked to their fear level. They tend to overestimate the risk of hazards that they fear, while underestimating the risk of hazards they identify as being less scary. Often their risk perception has little to do with the facts, and the facts might not even be of interest to them. For example, many Americans are terrified of black widow spiders, which are found throughout the United States. They are uninterested in the reality that fewer than two people die from black widow bites each year, while over 1,000 people suffer serious illness and death annually from mosquito bites. Mosquitoes are just too commonplace to worry about. Likewise, the risk of commercial airplane crashes is tiny compared to motorcycle crashes, but many a biker is afraid to fly.

The point is that risk perception drives our decision making, and these perceptions often do not correspond to the real risk levels, because irrational fear is taking our brains hostage. When irrational fear enters the picture, it is difficult to objectively weigh risks. Ironically, health decisions driven by fear may actually cause us to make choices that increase, rather than decrease, our risks."
Strange Glow is written for a general audience. Those who have studied from IPMB will already have a stronger quantitative background in math and physics than is needed for Strange Glow’s qualitative discussion. However, as Jorgensen writes in the preface, “These highly quantitative approaches have proved to be largely ineffective in communicating the essence of risk to the public.” I can’t argue with that. I recommend IPMB for a technical background, but Strange Glow for appreciating the broader impact of radiation on society.

Like Gaul, Strange Glow is divided into three parts. Part 1 describes how radiation was discovered, Part 2 discusses the effects of radiation on human health, and Part 3 focuses on risk assessment. I liked the third part best. Jorgensen emphasizes the human stories behind the science. For instance, he begins the chapter about radon by telling the tale of the Watras house.
“On December 2, 1984, Stanley J. Watras, an engineer working on construction of the new Limerick nuclear power plant near Portstown, Pennsylvania, arrived at work. The plant, just seven miles from his home in Boyertown, was scheduled to begin generating power within three weeks, and the construction crew had just installed radiation detectors at the plant doors—a standard safeguard to ensure that nuclear workers don’t exit the plant with any radioactive contamination on their bodies. When Watras arrived that day, he set off the alarms on the detectors as he walked into the plant. Over the following two weeks he would set off the alarms every morning. Further investigation revealed that his clothes were contaminated with radioactivity that he had picked up at his home!

When radiation safety personnel from the plant visited Watras’s home, they discovered what they didn’t think possible. There was more radon gas in the Watras split-level house than was found in a typical uranium mine . . . about 20 times as much! Surprised, the radiation safety technicians checked the radon levels in the neighboring houses. “Our house,” Watras remarked in consternation, “had perhaps the highest contamination level in the world, but our next door neighbors had none.” How could this be?”
Jorgensen then describes how the Environmental Protection Agency publicized--and perhaps exaggerated--the risks of radon. But by trying to err on the side of safety, their efforts became a case study in the challenges of risk assessment.
“This is one of the trade-offs of using multiple, highly conservative assumptions in risk assessment. It may seem prudent to inflate the risk in order not to underestimate it. Nevertheless, by adopting high-end estimates for every uncertain risk parameter, the cascade of high-end risk assumptions can compound to the point where the final predicted risk levels become incredulous and may even defy common sense.”
I find Jorgensen’s evidence-based, unemotional discussion of risk assessment to be a breath of fresh air. Far too often public fears are driven by emotions and ignorance, rather than a balancing of risks and benefits. I highly recommend Strange Glow to anyone wondering or worrying about the danger of radiation. Sometimes the danger is real and sometimes it is not, and you need to know which is which.

Below are some videos about the book and its author. Enjoy!



Friday, April 27, 2018

Frequency Encoding and Phase Encoding

Intermediate Physics for Medicine and Biology: Frequency Encoding and Phase Encoding


I'm always searching for ways to illustrate concepts using “simple” analytical examples (I’ll let you decide whether or not this example is simple). Today, I present analytical examples of frequency and phase encoding during magnetic resonance imaging. Russ Hobbie and I discuss MRI in Chapter 18 of Intermediate Physics for Medicine and Biology.

1. Introduction

Our goal is to understand how the measured MRI signal changes when magnetic field gradients are present. These gradients are essential for "encoding" information about the spatial distribution of spins in the frequency and phase of the signal. To simplify our discussion, we make several assumptions:
  • The radio-frequency π/2 and π pulses, used to rotate the spins into the x-y plane and then create an echo, are so brief that the spins rotate instantaneously compared to all other time scales. Similarly, any slice selection gradient Gz = dBz/dz exists only during the radio-frequency pulses. We won’t include Gz in our drawings of pulse sequences. 
  • We ignore relaxation, so the longitudinal and transverse time constants T1 and T2 are infinite.
  • Despite ignoring relaxation, the spins do dephase leading to a free induction decay with time constant T2*. Dephasing is caused by a distribution of spin frequencies, corresponding to small-scale static heterogeneities of the magnetic field. We assume that the spin frequencies ω have the distribution
    The peak frequency ωo is the Larmor frequency equal to γBo, where γ is the gyromagnetic ratio and Bo is the main magnetic field. The time constant τ indicates the width of the frequency distribution.

  • The spins are distributed uniformly along the x axis from -Δx to +Δx.

2. Spin-Echo

The spin-echo pulse sequence, with no gradients and no frequency or phase encoding, is similar to Fig. 18.24 in IPMB. Our pulse sequences consist of three functions of time. The radio-frequency (RF) pulses are shown on the first line; the time between the π/2 and π pulses is TE/2. The magnetic field gradient in the x direction, Gx = dBz/dx, is indicated in the second line; for this first example Gx is zero. The recorded signal, Mx, is in the third line.
MRI Spin-echo pulse sequence
Our goal is to calculate Mx(t). During the time between the two radio frequency pulses, we calculate the signal by integrating the precessing spins over x and ω


In this case the x integral is trivial: the integrand does not depend on x. We can solve the ω integral analytically using the u-substitution u=τ(ω-ωo), the cosine addition formula cos(A+B) = cosA cosB – sinA sinB, and the definite integral
The resulting free induction decay (FID) is

where τ corresponds to T2*. The exponential shape of the free induction decay arises from the particular form of our spin distribution. The wider the distribution of frequencies, the faster the decay.

The spins accumulate phase relative to those precessing at the Larmor frequency. Just before the π pulse the extra phase is (ω-ωo)TE/2. The π pulse changes the sign of this phase, or in other words adds an additional phase -(ω-ωo)TE. After the π pulse the signal is


The x integral is again trivial and the ω integral produces an echo


which peaks at t = TE and decays with time constant τ.

3. Phase Encoding

Phase encoding adds a gradient field Gx of duration T between the radio-frequency π/2 and π pulses. It shifts the phase of the spins by different amounts at different x locations (thus, position information is encoded in the phase of the signal). This phase shift is then reversed by the π pulse.
MRI phase encoding pulse sequence
The trickiest part of calculating Mx(t) is keeping track of the phase shifts: (ω-ωo)t is the phase shift up to time t because of the distribution of frequencies, -(ω-ωo)TE arises because the spins are flipped by the π pulse, γGxxT is caused by the phase-encoding gradient, and -2γGxxT is again from flipping by the π pulse. During the echo the signal simplifies to


We can solve both the x and ω integrals by repeatedly using the cosine addition formula (it is tedious but not difficult; I leave the details to you), and find


The amplitude of the echo depends on the factor sin(γGxΔxT)/ (γGxΔxT). For a Gx of zero this factor is one and the result is the same as for the spin-echo. If we repeat this pulse sequence with different values of Gx and measure the amplitude of each echo, we can trace out the function sin(γGxΔxT)/ (γGxΔxT), which is the Fourier transform of the spin distribution as a function of position.

4. Frequency Encoding

To do frequency encoding, we add a readout gradient Gx that is on during the echo and lasts a time T, like in Fig. 18.26 of IPMB. In addition, we include a prepulse of opposite polarity and half duration just before the readout, to cancel any extra phase shift accumulated during the echo. (Russ and I discuss this extra lobe of the Gx pulse when analyzing Fig. 18.29c, but we get its sign wrong).
MRI frequency encoding pulse sequence

The free induction decay and the phase reversal caused by the π-pulse are the same as in the spin-echo example. Once Gx begins the result differs. The frequency again depends on x. The phase shifts are: (ω-ωo)t because of the distribution of frequencies, -(ω-ωo)TE from the π pulse, -γGxxT/2 caused by the prepulse, and γGxx(t-(TE -T/2)) during readout. The recorded signal simplifies to


The echo during the readout gradient is (you really must fill in the missing steps yourself to benefit from this post)
The envelope of the echo is the product of two terms, which are both functions of time: An exponential e-|t-TE|/τ that has the shape of the echo with no gradient, and a factor sin(γGxΔx(t-TE))/ (γGxΔx(t-TE)). The amplitude of the echo at t=TE is the same as if Gx were zero, but the shape of the echo has changed because of the time-dependent factor containing the gradient. The function containing the sine is the Fourier transform of the spin distribution. Therefore, the extra time-dependent modulation of the echo by Gx contains information about the spatial distribution of spins.

5. Conclusion

What do we learn from this example? A phase-encoding gradient changes the amplitude of the echo but not its shape. A frequency-encoding gradient, on the other hand, changes the shape but not the amplitude. Both can be written as a modulated Larmor-frequency signal. In the pulse sequences shown above, the Larmor frequency is drawn too low in order to make the figure clearer. In fact, the Larmor frequencies in MRI are many megahertz, and thousands of oscillations occur during the free induction decay and echo.

I analyzed both phase encoding and frequency encoding in the x direction and considered each individually, because I wanted to compare and contrast their behavior. In practice, frequency encoding is performed using a Gx gradient in the x direction and phase encoding with a Gy gradient in the y direction, mapping out the two-dimensional Fourier transform of the spin distribution (see IPMB for more). 

Until I did this calculation I never completely understood what the shape of the echo looks like during readout. I hope it helps you as much as it helped me. Enjoy!

Friday, April 20, 2018

Listmania! IPMB

Intermediate Physics for Medicine and Biology: Listmania! IPMB

Amazon used to have a feature called Listmania! You could make a list of up to 40 books that was visible at Amazon's website. Ten years ago I created a Listmania! list related to Intermediate Physics for Medicine and Biology, reproduced below. Because the list is old, it does not include recent books (such as The Optics of Life) or books that I have discovered recently (such as The First Steps in Seeing). To learn about newer books, search this blog for posts labeled "book review." Amazon has discontinued Listmania!, but you can still find the lists if you look hard. I miss it.

If you are interested in what I read for pleasure, look here.

Enjoy!

**********************************************************

Intermediate Physics for Medicine and Biology

 

 


Bradley J. Roth
The list author says: "Books that are cited by the 4th Edition of Intermediate Physics for Medicine and Biology. These are some of the best biological and medical physics books I know of, and are books that have been useful to me during my career."
Intermediate Physics for Medicine and Biology, 4th Edition (Biological and Medical Physics, Biomedical Engineering)
Intermediate Physics for Medicine and Biology, 4th Edition (Biological and Medical Physics, Biomedical Engineering)
"All the books listed below are cited in the 4th Edition of Intermediate Physics for Medicine and Biology, written by Russ Hobbie and me." 
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Applied Mathematics Series 55]
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Applied Mathematics Series 55]
"A math handbook that has everything you'll ever need to know."
The 2nd Law: Energy, Chaos, and Form (Scientific American Library Paperback)
The 2nd Law: Energy, Chaos, and Form (Scientific American Library Paperback)
"I love this coffee table book about the second law of thermodynamics.  A painless way to introduce yourself to the subject."
Introduction to Radiological Physics and Radiation Dosimetry
Introduction to Radiological Physics and Radiation Dosimetry
"Classic in the Medical Physics field."
The Essential Exponential! (For the Future of Our Planet)
The Essential Exponential! (For the Future of Our Planet)
"This book explains why we devoted an entire chapter of Intermediate Physics for Medicine and Biology to the exponential function."
Physics With Illustrative Examples From Medicine and Biology: Mechanics (Biological and Medical Physics, Biomedical Engineering)
Physics With Illustrative Examples From Medicine and Biology: Mechanics (Biological and Medical Physics, Biomedical Engineering)
"A classic textbook."
Physics With Illustrative Examples From Medicine and Biology: Electricity and Magnetism (Biological and Medical Physics, Biomedical Engineering)
Physics With Illustrative Examples From Medicine and Biology: Electricity and Magnetism (Biological and Medical Physics, Biomedical Engineering)
"The second edition of the book has much the same content as the first, but the quality of the printing and illustrations is vastly improved."
Physics With Illustrative Examples From Medicine and Biology: Statistical Physics (Biological and Medical Physics, Biomedical Engineering)
Physics With Illustrative Examples From Medicine and Biology: Statistical Physics (Biological and Medical Physics, Biomedical Engineering)
"Benedek and Villars were pioneers in biological and medical physics textbooks."
Random Walks in Biology
Random Walks in Biology
"The best book about the role of diffusion in biology that I know of."
Foundations of Medical Imaging
Foundations of Medical Imaging
"Fine book to study imaging algorithms."
Introduction to Membrane Noise
Introduction to Membrane Noise
"Great book on a little-known topic."
Air and Water
Air and Water
"One of my favorites. Written by a physiologist with an interest in physics (as opposed to Hobbie and I, who are physicists interested in physiology)."
Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
"My favorite modern physics textbook."
The Feynman Lectures on Physics (3 Volume Set) (Set v)
The Feynman Lectures on Physics (3 Volume Set) (Set v)
"What physics list could be complete without Feynman?"
From Clocks to Chaos
From Clocks to Chaos
"Excellent book to learn the biological and medical applications of chaos."
The Machinery of Life
The Machinery of Life
"Wonderful picture book.  Great way to visualize the relative sizes of biological objects."
Bioelectricity and Biomagnetism
Bioelectricity and Biomagnetism
"Good, thick tome on bioelectricity."
Textbook of Medical Physiology
Textbook of Medical Physiology
"The classic physiology textbook."
Radiobiology for the Radiologist
Radiobiology for the Radiologist
"Great place to learn about the biological effects of radiation."
Medical Imaging Physics
Medical Imaging Physics
"Standard textbook in medical physics. Hendee is a pioneer in the field."
Ion Channels of Excitable Membranes, Third Edition
Ion Channels of Excitable Membranes, Third Edition
"The bible for information about ion channels."
Machines in Our Hearts: The Cardiac Pacemaker, the Implantable Defibrillator, and American Health Care
Machines in Our Hearts: The Cardiac Pacemaker, the Implantable Defibrillator, and American Health Care
"Learn about the history of pacemakers and defibrillators."
The Physics of Radiation Therapy
The Physics of Radiation Therapy
"The place to go to learn about radiation therapy."
Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields
Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields
"Fine textbook on bioelectricity."
Powers of Ten (Revised) (Scientific American Library Paperback)
Powers of Ten (Revised) (Scientific American Library Paperback)
"Classic work describing how the world looks at different length scales. Required reading by anyone interested in science."
Electric Fields of the Brain: The Neurophysics of EEG,  2nd Edition
Electric Fields of the Brain: The Neurophysics of EEG, 2nd Edition
"Great way to learn about the physics of the electroencephalogram."
Bioelectricity: A Quantitative Approach
Bioelectricity: A Quantitative Approach
"Standard textbook for a class in bioelectricity."
Numerical Recipes 3rd Edition: The Art of Scientific Computing
Numerical Recipes 3rd Edition: The Art of Scientific Computing
"My go-to book on numerical methods."
Electricity and Magnetism (Berkeley Physics Course, Vol. 2)
Electricity and Magnetism (Berkeley Physics Course, Vol. 2)
"Best introduction to electricity and magnetism I know. Part of the great Berkeley Physics Course."
Statistical Physics: Berkeley Physics Course, Vol. 5
Statistical Physics: Berkeley Physics Course, Vol. 5
"Great intuitive introduction to statistical mechanics.  Part of the Berkeley Physics Course."
Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Fourth Edition)
Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Fourth Edition)
"Need a little review of vector calculus? This is the place to find it."
Scaling: Why is Animal Size so Important?
Scaling: Why is Animal Size so Important?
"Great book on biological scaling."
How Animals Work
How Animals Work
"Great physiology book. Quirky, but fun."
Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity)
Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity)
"Best book for a first course in nonlinear dynamics."
Life in Moving Fluids: The Physical Biology of Flow (Princeton Paperbacks)
Life in Moving Fluids: The Physical Biology of Flow (Princeton Paperbacks)
"Best book I know of on biological fluid dynamics. Not too mathematical, but full of insight. I recommend all of Vogel's books."
Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems
Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems
"Great for understanding the fluid dynamics of the circulatory system."
Lady Luck: The Theory of Probability (Dover Books on Mathematics)
Lady Luck: The Theory of Probability (Dover Books on Mathematics)
"I often find probability theory boring, but not this book. An oldie but goodie."
The Geometry of Biological Time (Interdisciplinary Applied Mathematics)
The Geometry of Biological Time (Interdisciplinary Applied Mathematics)
"Classic by Art Winfree, who was a leading mathematical biologists.  Be sure to get the 2nd edition."
When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias
When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias
"Winfree's classic on the nonlinear dynamics of the heart."
Cardiac Electrophysiology: From Cell to Bedside, 4e
Cardiac Electrophysiology: From Cell to Bedside, 4e
"Comprehensive reference on cardiac electrophysiology."