Data Reduction and Error Analysis for the Physical Sciences

Data Reduction and Error Analysis
for the Physical Sciences,
by Philip Bevington and Keith Robinson.

In Chapter 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I cite the book Data Reduction and Error Analysis for the Physical Sciences, by Philip Bevington and Keith Robinson.

The problem [of fitting a function to data] can be solved using the technique of nonlinear least squares…The most common [algorithm] is called the Levenberg-Marquardt method (see Bevington and Robinson 2003 or Press et al. 1992).

I have written about the excellent book Numerical Recipes by Press et al. previously in this blog. I was not too familiar with the book by Bevington and Robinson, so last week I checked out a copy from the Oakland University library (the second edition, 1992).

I like it. The book is a great resource for many of the topics Russ and I discuss in IPMB. I am not an experimentalist, but I did experiments in graduate school, and I have great respect for the challenges faced when working in the laboratory.

Their Chapter 1 begins by distinguishing between systematic and random errors. Bevington and Robinson illustrate the difference between accuracy and precision using a figure like this one:

a) Precise but inaccurate data. b) Accurate but imprecise data.

Next, they present a common sense discussion about significant figures, a topic that my students often don’t understand. (I assign them a homework problem with all the input data to two significant figures, and they turn in an answer--mindlessly copied from their calculator--containing 12 significant figures.)

In Chapter 2 of Data Reduction and Error Analysis, Bevington and Robinson introduce probability distributions.

Of the many probability distributions that are involved in the analysis of experimental data, three play a fundamental role: the binomial distribution [Appendix H in IPMB], the Poisson distribution [Appendix J], and the Gaussian distribution [Appendix I]. Of these, the Gaussian or normal error distribution is undoubtedly the most important in statistical analysis of data. Practically, it is useful because it seems to describe the distribution of random observations for many experiments, as well as describing the distributions obtained when we try to estimate the parameters of most other probability distributions.

Here is something I didn’t realize about the Poisson distribution:

The Poisson distribution, like the bidomial distribution, is a discrete distribution. That is, it is defined only at integral values of the variable x, although the parameter μ [the mean] is a positive, real number.

Figure J.1 of IPMB plots the Poisson distribution P(x) as a continuous function. I guess the plot should have been a histogram.

Chapter 3 addresses error analysis and propagation of error. Suppose you measure two quantities, x and y, each with an associated standard deviation σ_x and σ_y. Then you calculate a third quantity z(x,y). If x and y are uncorrelated, then the error propagation equation is

For instance, Eq. 1.40 in IPMB gives the flow of a fluid through a pipe, i, as a function of the viscosity of the fluid, η, and the radius of the pipe, R_p

The error propagation equation (and some algebra) gives the standard deviation of the flow in terms of the standard deviation of the viscosity and the standard deviation of the radius

When you have a variable raised to the fourth power, such as the pipe radius in the equation for flow, it contributes four times more to the flow’s percentage uncertainty than a variable such as the viscosity. A ten percent uncertainty in the radius contributes a forty percent uncertainty to the flow. This is a crucial concept to remember when performing experiments.

Bevington and Robinson derive the method of least squares in Chapter 4, covering much of the same ground as in Chapter 11 of IPMB. I particularly like the section titled A Warning About Statistics.

Equation (4.12) [relating the standard deviation of the mean to the standard deviation and the number of trails] might suggest that the error in the mean of a set of measurements x_i can be reduced indefinitely by repeated measurements of x_i. We should be aware of the limitations of this equation before assuming that an experimental result can be improved to any desired degree of accuracy if we are willing to do enough work. There are three main limitations to consider, those of available time and resources, those imposed by systematic errors, and those imposed by nonstatistical fluctuations.

Russ and I mention Monte Carlo techniques—the topic of Chapter 5 in Data Reduction and Error Analysis—a couple times in IPMB. Then Bevington and Robinson show how to use least squares to fit to various functions: a line (Chapter 6), a polynomial (Chapter 7), and an arbitrary function (Chapter 8). In Chapter 8 the Marquardt method is introduced. Deriving this algorithm is too involved for this blog post, but Bevington and Robinson explain all the gory details. They also provide much insight about the method, such as in the section Comments on the Fits:

Although the Marquardt method is the most complex of the four fitting routines, it is also the clear winner for finding fits most directly and efficiently. It has the strong advantage of being reasonably insensitive of the starting values of the parameters, although in the peak-over-background example in Chapter 9, it does have difficulty when the starting parameters of the function for the peak are outside reasonable ranges. The Marquardt method also has the advantage over the grid- and gradient-search methods of providing an estimate of the full error matrix and better calculation of the diagonal errors.

The rest of the book covers more technical issues that are not particularly relevant to IPMB. The appendix contains several computer programs written in Pascal. The OU library copy also contains a 5 1/2 inch floppy disk, which would have been useful 25 years ago but now is quaint.

Philip Bevington wrote the first edition of Data Reduction and Error Analysis in 1969, and it has become a classic. For many years he was a professor of physics at Case Western University, and died in 1980 at the young age of 47. A third edition was published in 2002. Download it here.

Rall's Equivalent Cylinder

Chapter 6 of Intermediate Physics for Medicine and Biology discusses nerve electrophysiology. In particular, Russ Hobbie and I derive the cable equation. This equation works great for a peripheral nerve with its single long cylindrical axon. In the brain, however, nerves end in branching networks of dendrites (see one of the famous drawings by Ramón y Cajal below). What equation describes the dendrites?

A drawing of a dendritic tree,
by Ramón y Cajal.

Wilfrid Rall answered this question by representing the dendrites as a branching network of fibers: the Rall model (Annals of the New York Academy of Sciences, Volume 96, Pages 1071–1092, 1962). Below I’
--> -->ll rederive the Rall model using the notation of IPMB. But—as I know some of you do not enjoy mathematics as much as I do—let me first describe his result qualitatively. Rall found that as you move along the dendritic tree, the fiber radius a gets smaller and smaller, but the number of fibers n gets larger and larger. Under one special condition, when na^3/2 is constant, the voltage along the dendrites obeys THE SAME cable equation that governs a single axon. This only works if distance is measured in length constants instead of millimeters, and time in time constants instead of milliseconds. Dendritic networks don't always have na^3/2 constant, but it is not a bad approximation, and provides valuable insight into how dendrites behave.
But instead of me explaining Rall’s goals, why not let Rall do so himself.

In this paper, I propose to focus attention upon the branching dendritic trees that are characteristic of many neurons, and to consider the contribution such dendritic trees can be expected to make to the physiological properties of a whole neuron. More specifically, I shall present a mathematical theory relevant to the question: How does a neuron integrate various distributions of synaptic excitation and inhibition delivered to its soma-dendritic surface. A mathematical theory of such integration is needed to help fill a gap that exists between the mathematical theory of nerve membrane properties, on the one hand, and the mathematical theory of nerve nets and of populations of interacting neurons, on the other hand.

I had the pleasure of knowing Rall when we both worked at the National Institutes of Health in the 1990s. He was trained as a physicist, and obtained his PhD from Yale. During World War II he worked on the Manhattan Project. He spent most of his career at NIH, and was a leader among scientists studying the theoretical electrophysiology of dendrites.

Rall receiving the Swartz Prize.

Now the math. First, let me review the cable model for a single axon, and then we will generalize the result to a network. The current i_i along an axon is related to the potential v and the resistance per unit length r_i by a form of Ohm's law

(Eq. 6.48 in IPMB). If the current changes along the axon, it must enter or leave through the membrane, resulting in an equation of continuity

(Eq. 6.49), where g_m is the membrane conductance per unit area and c_m is the membrane capacitance per unit area. Putting these two equations together and rearranging gives the cable equation

The axon length constant is defined as

and the time constant as

so the cable equation becomes

If we measure distance and time using the dimensionless variables X = x/λ and T = t/τ, the cable equation simplifies further to

Now, let’s see how Rall generalized this to a branching network. Instead of having one fiber, assume you have a variable number that depends on position along the network, n(x). Furthermore, assume the radius of each individual fiber varies, a(x). The cable equation can be derived as before, but because r_i now varies with position (r_i = 1/nπa²σ, where σ is the intracellular conductivity), we pick up an extra term

When I first looked at this equation, I thought “Aha! If r_i is independent of x, the new term disappears and you get the plain old cable equation.” It’s not quite that simple; λ also depends on position, so even without the extra term this is not the cable equation. Remember, we want to measure distance in the dimensionless variable X = x/λ, but λ depends on position, so the relationship between derivatives of x and derivatives of X is complicated

In terms of the dimensionless variables X and T, the cable equation becomes

If λr_i is constant along the axon, the ugly new term vanishes and you have the traditional cable equation. If you go back to the definition of r_i and λ in terms of a and n, you find that this condition is equivalent to saying that na^3/2 is constant along the network. If one fiber branches into two, the daughter fibers must each have a radius of 0.63 times the parent fiber radius. Dendritic trees that branch in this way act like a single fiber. This is Rall’s result: the Rall equivalent cylinder.

If you want to learn more about Rall’s work, read the book The Theoretical Foundation of Dendritic Function: Selected Papers of Wilfrid Rall with Commentaries, edited by Idan Segev, John Rinzel, and Gordon M. Shepherd. The foreword, by Terrence J. Sejnowski, says

The exploration of the electrical properties of dendrites by Wilfrid Rall provided many key insights into the computational resources of the neurons. Many of the papers in this collection are classics: dendrodendritic interactions in the olfactory bulb; nonlinear synaptic integration in motoneuron dendrites; active currents in pyramidal neuron apical dendrites. In each of these studies, insights arose from a conceptual leap, astute simplifying assumptions, and rigorous analysis. Looking back, one is impressed with the foresight shown by Rall in his choice of problems, with the elegance of his methods in attacking them, and with the impact that his conclusions have had for our current thinking. These papers deserve careful reading and rereading, for there are additional lessons in each of them that will reward the careful reader....It would be difficult to imagine the field of computational neuroscience today without the conceptual framework established over the last thirty years by Wil Rall, and for this we all owe him a great debt of gratitude.

Magneto-Aerotactic Bacteria Deliver Drug-Containing Nanoliposomes to Tumour Hypoxic Regions

In Chapter 8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe magnetotactic bacteria.

Several species of bacteria contain linear strings of up to 20 particles of magnetite, each about 50 nm on a side encased in a membrane (Frankelet al. 1979; Moskowitz 1995). Over a dozen different bacteria have been identified that synthesize these intracellular, membrane-bound particles or magnetosomes (Fig. 8.25). In the laboratory the bacteria align themselves with the local magnetic field. In the problems you will learn that there is sufficient magnetic material in each bacterium to align it with the earth’s field just like a compass needle. Because of the tilt of the earth’s field, bacteria in the wild can thereby distinguish up from down.

Other bacteria that live in oxygen-poor, sulfide-rich environments contain magnetosomes composed of greigite (Fe₃S₄), rather than magnetite (Fe₃O₄). In aquatic habitats, high concentrations of both kinds of magnetotactic bacteria are usually found near the oxic–anoxic transition zone (OATZ). In freshwater environments the OATZ is usually at the sediment–water interface. In marine environments it is displaced up into the water column. Since some bacteria prefer more oxygen and others prefer less, and they both have the same kind of propulsion and orientation mechanism, one wonders why one kind of bacterium is not swimming out of the environment favorable to it. Frankel and Bazylinski(1994) proposed that the magnetic field and the magnetosomes keep the organism aligned with the field, and that they change the direction in which their flagellum rotates to move in the direction that leads them to a more favorable concentration of some desired chemical.

I enjoy learning about the biology and physics of magnetotactic bacteria, but I never expected that they had anything to do with medicine. Then last month a paper published in Nature Nanotechnology discussed using these bacteria to treat cancer!

Oxygen-depleted hypoxic regions in the tumour are generally resistant to therapies. Although nanocarriers have been used to deliver drugs, the targeting ratios have been very low. Here, we show that the magneto-aerotactic migration behaviour of magnetotactic bacteria, Magnetococcus marinus strain MC-1 (ref. 4), can be used to transport drug-loaded nanoliposomes into hypoxic regions of the tumour. In their natural environment, MC-1 cells, each containing a chain of magnetic iron-oxide nanocrystals, tend to swim along local magnetic field lines and towards low oxygen concentrations based on a two-state aerotactic sensing system. We show that when MC-1 cells bearing covalently bound drug-containing nanoliposomes were injected near the tumour in severe combined immunodeficient beige mice and magnetically guided, up to 55% of MC-1 cells penetrated into hypoxic regions of HCT116 colorectal xenografts. Approximately 70 drug-loaded nanoliposomes were attached to each MC-1 cell. Our results suggest that harnessing swarms of microorganisms exhibiting magneto-aerotactic behaviour can significantly improve the therapeutic index of various nanocarriers in tumour hypoxic regions.

The IOP website physicsworld.com published an article by Belle Dumé describing this study. It begins

Bacteria that respond to magnetic fields and low oxygen levels may soon join the fight against cancer. Researchers in Canada have done experiments that show how magneto-aerotactic bacteria can be used to deliver drugs to hard-to-reach parts of tumours. With further development, the method could be used to treat a variety of solid tumours, which account for roughly 85% of all cancers.

A similar article, also by Dumé, can be found on medicalphysicsweb.com

As cancer cells proliferate, they consume large amounts of oxygen. This results in oxygen-poor regions in a tumour. It is notoriously difficult to treat these hypoxic regions using conventional pharmaceutical nanocarriers, such as liposomes, micelles and polymeric nanoparticles.

Now, a team led by Sylvain Martel of the NanoRobotics Laboratory at the Polytechnique Montréal has developed a method that exploits the magnetotactic bacteria Magnetoccus marinus (MC-1) to overcome this problem.

Pretty cool stuff.

Rutherford Scattering and the Differential Cross Section

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the differential cross section.

We may wish to know the probability that particles…are scattered in a certain direction. We have to consider the probability that they are scattered into a small solid angle dΩ. In this case, σ is called the differential scattering cross section and is often written as

The units of the differential scattering cross section are m² sr^-1. The differential cross section depends on θ, the angle between the directions of travel of the incident and scattered particles.

Perhaps the most famous differential cross section is the Rutherford scattering formula. Ernest Rutherford (who I have discussed before in this blog) derived this formula to explain the results of his alpha particle scattering experiments, in which he fired alpha particles at a thin metal foil and determined the angle of scattering by observing the light produced when a scattered particle hit a zinc sulfide screen. His formula assumes a nonrelativistic alpha particle scatters off a massive (no recoil), spinless, bare, positively charged target nucleus. Below is a new homework problem providing some practice with the Rutherford formula

Problem 16 ½. An example of a differential cross section is the Rutherford scattering formula

(a) Plot dσ/dΩ versus θ over the range 0 to π.

(b) Repeat part (a) using semilog graph paper.

(c) The constant A is equal to

where q and Q are the charges of the alpha particle and nucleus, and E is the alpha particle energy. Show that A has the units of m² sr^-1. Hint: steradians, like radians, are dimensionless (see Appendix A).

(d) Interpret what happens physically when θ is π. What is the value of the cosecant of π/2? Write A in terms of the distance of closest approach of an alpha particle to the nucleus. Hint: see Chapter 17, Problem 2.

(e) Note that dσ/dΩ goes to infinity as θ goes to zero. Interpret this result physically. What assumption did Rutherford make that may be responsible for this unphysical behavior?

(f) Integrate dσ/dΩ over θ from 0 to π. You may need to use a good table of integrals. Explain your result (which may surprise you) physically.

The Making of the Atomic Bomb,
by Richard Rhodes.

Here is the history of the Rutherford scattering experiment, as told by Richard Rhodes in The Making of the Atomic Bomb.

[Hans] Geiger [Rutherford’s assistant] went to work on alpha scattering, aided by Ernest Marsden, then an eighteen-year-old Manchester undergraduate. They observed alpha particles coming out of a firing tube and passing through foils of such metals as aluminum, silver, gold, and platinum. The results were generally consistent with expectation: alpha particles might very well accumulate as much as two degrees of total deflection bouncing around among atoms of the plum-pudding sort [an early model of atomic structure proposed by J. J. Thomson]. But the experiment was troubled with stray particles. Geiger and Marsden thought molecules in the walls of the firing tube might be scattering them. They tried eliminating the strays by narrowing and defining the end of the firing tube with a series of graduated metal washers. That proved no help.

Rutherford wandered into the room. The three men talked over the problem. Something about it alerted Rutherford’s intuition for promising side effects. Almost as an afterthought he turned to Marsden and said, “See if you can get some effect of alpha particles directly reflected from a metal surface.” Marsden knew that a negative result was expected—alpha particles shot through thin foils, they did not bounce back form them—but that missing a positive result would be an unforgivable sin. He took great care to prepare a strong alpha source. He aimed the pencil-narrow beam of alphas at a forty-five degree angle onto a sheet of gold foil. He positioned his scintillation screen on the same side of the foil, beside the alpha beam, so that a particle bouncing back would strike the screen and register as a scintillation. Between firing tube and screen he interposed a thick lead plate so no direct alpha particles could interfere.

Immediately, and to his surprise, he found what he was looking for. “I remember well reporting the result to Rutherford,” he wrote, “…when I met him on the steps leading to his private room, and the joy with which I told him…”

Rutherford had been genuinely astonished by Marsden’s results. “It was quite the most incredible event that has ever happened to me in my life,” he said later. “It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration I realized that this scattering backwards must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greatest part of the mass of the atom was concentrated in a minute nucleus.”

The Biomechanics of Solids and Fluids: The Physics of Life

“The Biomechanics of Solids and Fluids:
The Physics of Life,”
by David Alexander.

This summer a review article about biomechanics by David Alexander appeared in the European Journal of Physics: “The Biomechanics of Solids and Fluids: The Physics of Life” (Volume 37, Article 053011, 2016). It serves as an excellent supplement for much of the material in Chapter 1 (Mechanics) in Intermediate Physics for Medicine and Biology. It describes the biomechanics of solids (elasticity) and fluids (fluid mechanics).

Biomechanics borrows and extends engineering techniques to study the mechanical properties of organisms and their environments. Like physicists and engineers, biomechanics researchers tend to specialize on either fluids or solids (but some do both). For solid materials, the stress–strain curve reveals such useful information as various moduli, ultimate strength, extensibility, and work of fracture. Few biological materials are linearly elastic so modified elastic moduli are defined. Although biological materials tend to be less stiff than engineered materials, biomaterials tend to be tougher due to their anisotropy and high extensibility. Biological beams are usually hollow cylinders; particularly in plants, beams and columns tend to have high twist-to-bend ratios. Air and water are the dominant biological fluids. Fluids generate both viscous and pressure drag (normalized as drag coefficients) and the Reynolds number (Re) gives their relative importance. The no-slip conditions leads to velocity gradients (‘boundary layers’) on surfaces and parabolic flow profiles in tubes. Rather than rigidly resisting drag in external flows, many plants and sessile animals reconfigure to reduce drag as speed increases. Living in velocity gradients can be beneficial for attachment but challenging for capturing particulate food. Lift produced by airfoils and hydrofoils is used to produce thrust by all flying animals and many swimming ones, and is usually optimal at higher Re. At low Re, most swimmers use drag-based mechanisms. A few swimmers use jetting for rapid escape despite its energetic inefficiency. At low Re, suspension feeding depends on mechanisms other than direct sieving because thick boundary layers reduce effective porosity. Most biomaterials exhibit a combination of solid and fluid properties, i.e., viscoelasticity. Even rigid biomaterials exhibit creep over many days, whereas pliant biomaterials may exhibit creep over hours or minutes. Instead of rigid materials, many organisms use tensile fibers wound around pressurized cavities (hydrostats) for rigid support; the winding angle of helical fibers greatly affects hydrostat properties. Biomechanics researchers have gone beyond borrowing from engineers and adopted or developed a variety of new approaches—e.g., laser speckle interferometry, optical correlation, and computer-driven physical models—that are better-suited to biological situations.

One of my favorite parts of the review are the references. Alexander cites many of his own publications, including his book Nature’s Flyers: Birds, Insects, and the Biomechanics of Flight. For some reason, he didn’t cite his recent book On the Wing: Insects, Pterosaurs, Birds, Bats and the Evolution of Animal Flight. By the way, David Alexander is not the same as R. McNeill Alexander, who published Principles of Animal Locomotion, which is also cited in the review, and who died earlier this year. The review cites several works by Mark Denny, although not my favorite: Air and Water. Alexander cites over a dozen works by Steven Vogel, whose Life in Moving Fluids appears on my ideal bookshelf. Finally, he writes that “James Gordon’s book Structures, or Why Things Don’t Fall Down (Gordon 1978) is one of the most entertaining and readable introductions to a technical topic ever written.” I read Gordon’s book many years ago and had almost forgotten it. Alexander is right, it’s a gem.

In Figure 1.21, Russ Hobbie and I show a typical stress-strain curve. Alexander shows similar curves, and analyzes them in more detail. Like our book, he develops the concepts of Young’s modulus, shear modulus, strength, and Poisson’s ratio. Alexander introduces another concept: the strain energy density, which is the area under the stress-strain curve. Stress has units of N/m², and strain is dimensionless, so the strain energy density has units of N/m² = J/m³. Alexander writes “this key value measures how much work a material absorbs before breaking, and is sometimes referred to as ‘toughness’. Perhaps counterintuitively, some very hard, rigid materials are not very tough, whereas many floppy, easily extended materials are very tough.”

The section on fluid dynamics covers much of the same ground as analyzed in IPMB. It also discusses high Reynold’s number flow, including turbulence, flow separation, boundary layers, lift, and drag. These are fascinating topics, and are vital for understanding animal flight, but do not impact the low Reynold’s number flow that Russ and I focus on.

One topic that Russ and I give a brief mention is viscoelasticity. Alexander spends more time on this interesting subject.

Most biological materials do not fit perfectly into the solid or fluid categories as engineers and physicists have usually defined them. Many biological structures that we would ordinarily consider solid actually have a time-dependent response to loading that gives them a partly fluid character. A proper Hookean material behaves the same way whether it is loaded for a second or a week: remove the load and it returns to its original shape. A viscoelastic solid, however, displays a property called creep : apply a load briefly and the material will spring back just as if it were Hookean. Apply the same load for a prolonged period, however, and the material will continue to deform gradually. When the load is removed, the material may have acquired a permanent deformation, and if so, the longer it is loaded, the greater the permanent deformation.

Alexandar’s review is a great place to go for more about biomechanics after reading Chapter 1 of IPMB. I highly recommend it.

Whiplash

Last week, my wife Shirley and I were in an automobile accident. We suffered no serious injuries, thank you, but the car was totaled and we were sore for several days. After the obligatory reflections on the meaning of life, I began to think critically about the biomechanics of auto accident injuries.

Our car was at a complete stop, and the idiot in the other car hit us from behind. The driver’s side air bag deployed and the impact pushed us off to the right of the road (we hit the car in front of us in the process), while the idiot’s car ended up on the opposite shoulder. It looked a little like this; we were m₂ and the idiot was m₁:

The collision dynamics of our car accident.

The police came and our poor car was carried off on a wrecker to a junk yard. Shirley and I walked home; the accident occurred about a quarter mile from our house.

My neck is still stiff. Presumably I suffered a classic—but not too severe—whiplash. Although Intermediate Physics for Medicine and Biology does not discuss whiplash, it does cover most of the concepts needed to understand it: acceleration, shear forces, torques, and biomechanics. Paul Davidovits describes whiplash briefly in Physics in Biology and Medicine. From the second edition:

5.7  Whiplash Injury

Neck bones are rather delicate and can be fractured by even a moderate force. Fortunately the neck muscles are relatively strong and are capable of absorbing a considerable amount of energy. If, however, the impact is sudden, as in a rear-end collision, the body is accelerated in the forward direction by the back of the seat,  and the unsupported neck is then suddenly yanked back at full speed. Here the muscles do not respond fast enough and all the energy is absorbed by the neck bones, causing the well-known whiplash injury.

You can learn more about the physics of whiplash in the paper “Kinematics of a Head-Neck Model Simulating Whiplash” published in The Physics Teacher (Volume 46, Pages 88–91, 2008).

In a typical rear-end collision, the vehicle accelerates forward when struck and the torso is pushed forward by the seat. The structural response of the cervical spine is dependent upon the acceleration-time pulse applied to the thoracic spine and interaction of the head and spinal components. During the initial phases of the impact, it is obvious that the lower cervical vertebrae move horizontally faster than the upper ones. The shear force is transmitted from the lower cervical vertebrae to the upper ones through soft tissues between adjacent vertebrae one level at a time. This shearing motion contributes to the initial development of an S-shape curvature of the neck (the upper cervical spine undergoes flexion while the lower part undergoes extension), which progresses to a C-shape curvature. At the end of the loading phase, the entire head-neck complex is under the extension mode with a single curvature. This implies the stretching of the anterior and compression of the posterior parts of the cervical spine.

Here are links to videos showing what happens to the upper spine during whiplash:

http://www.injuryresources.com/brain-injury/injury44.html

http://www.injuryresources.com/brain-injury/injury45.html

http://www.injuryresources.com/brain-injury/injury46.html

Injury from whiplash depends on the acceleration. What sort of acceleration did my head undergo? I don’t know the speed of the idiot’s car, but I will guess it was 25 miles per hour, which is equal to about 11 meters per second. Most of the literature I have read suggests that the acceleration resulting from such impacts occurs in about a tenth of a second. Acceleration is change in speed divided by change in time (see Appendix B in IPMB), so (11 m/s)/(0.1 s) = 110 m/s², which is about 11 times the acceleration of gravity, or 11 g. Yikes! Honestly, I don’t know the idiot’s speed. He may have been slowing down before he hit me, but I don’t recall any skidding noises just before impact.

What lesson do I take from this close call with death? My hero Isaac Asimov—who wrote over 500 books in his life—was asked what he would do if told he had only six months to live. His answer was “type faster.” Sounds like good advice to me!

Our car, after the accident.

Everything's Up To Date in Kansas City

Union Station.

I posted the last two entries in this blog while on a trip to Kansas City to visit my parents. I didn’t grow up in the Kansas City area, but I did graduate from Shawnee Mission South High School in Overland Park Kansas, and I was a physics major at the University of Kansas in Lawrence. My dad is a native of Kansas City Missouri, while my mom moved to Kansas City Kansas when young and attended Wyandotte High School.

Kauffman Center
for the Performing Arts.

We had a great visit, including a ride on Kansas City’s new streetcar, which travels a route along Main Street from Union Station, past the Power and Light District, within a few blocks of the Kauffman Center, to the River Market. We also had a great pork sandwich at Pigwich in the East Bottoms. Kansas City is booming.

Liberty Memorial.

Is there any connection between Kansas City and medical physics? Yes, there is. Rockhurst University, a liberal arts college located a mile west of where my dad grew up on Swope Parkway, offers an undergraduate program in the physics of medicine, which is similar to the medical physics major we offer at Oakland University. I thought the readers of Intermediate Physics for Medicine and Biology might like to see how another school other than Oakland structures its undergraduate medical physics curriculum.

From their Physics of Medicine website:

POM Program Overview: To suit your interests and career goals the POM Program has three program choices:

• Medical Physics Major - Major Track designed for students wishing to enter graduate school in Medical Physics
• Physics of Medicine (POM) Pre-Professional Major - Major Track designed for students wishing to enter a Medical/Healthcare Graduate Program
• Physics of Medicine (POM) Minor— Minor designed to complement pre-healthcare or pre-medicine program

Advantages to students of the POM Program are:

1. Deeper understanding of physics principles and their applicability to a medical or health field career.
2. Stronger post-graduate application to competitive health field programs.
3. Undergraduate research opportunities – potential for capstone area or future graduate work. 
4. Value to students of interdisciplinary study, allowing them to tie together coursework in science/math with professional goals. 

All Physics of Medicine Coursework is designed to complement the Scientific Foundations for Future Physicians Report, Report of the AAMC-HHMI Committee (2009).

Some courses specific to the program are:

PH 3200 Physics of the Body I: This course expands on the physics principles developed in introductory physics courses through an in-depth study of mechanics, fluids and thermodynamics as they are applied to the human body. Areas of study include the following: biomechanics (torque, force, motion and lever systems of the body; application of vector analysis of human movement to video), thermodynamics and heat transfer (food intake and mechanical efficiency) and the pulmonary system (pressure, volume and compliance relationships). Guest speakers from the medical community will be invited. [This course appears to cover the material in Chapters 1-3 in IPMB]

PH 3210 Physics of the Body II: This course is a continuation of Physics of the Body I with a concentration on the cardiovascular system, electricity and wave motion. Areas of study include the following: cardiovascular system (heart as a force pump, blood flow and pressure), electricity in the body (action potentials, resistance-capacitance circuit of nerve impulse propagation, EEG, EKG, EMG), and sound (hearing, voice production, sound transfer and impedance, ultrasound – transmission and reflection). In addition, students complete a guided, in-depth, individual investigation on a topic pertinent to Physics of the Body. Guest speakers from the medical community will be invited. [Approximately Chapters 6, 7, and 13 in IPMB. PH 3200 and 3210 together are similar to Oakland University’s PHY 325, Biological Physics]

PH 3240 Physics of Medical Imaging: This course focuses on an introduction to areas of modern physics required for an understanding of contemporary medical diagnostic and treatment procedures. Topics include a focus on the physics underlying modern medical imaging instruments: the EM Spectrum, X-Ray, CT, Gamma Camera, SPECT, PET, MRI and hybrid instrumentation. In this course, students learn about the physics involved in how these diagnostic and therapeutic instruments work as well as the numerous physics and patient factors that contribute to the choice of instrument for diagnosis. There will be field trips to local hospitals and medical imaging facilities and invited guest speakers. [Chapters 15-18 in IPMB; similar to OU’s PHY 326, Medical Physics]

PH 4400 Optics: This course covers both the geometric and physical properties of optical principles including optics of the eye, lasers, fiber optics, and use of endoscopy in medicine. Students will complete a final optics research project in which they relate content learned to an area of optics research. [Chapter 14 in IPMB. We have no comparable course at OU. We offer a standard optics class, but with no biomedical emphasis. This class intrigues me.]

PH 4900 Statistics for the Health Sciences: This course introduces the basic principles and methods of health statistics. Emphasis is on fundamental concepts and techniques of descriptive and inferential statistics with applications in health care, medicine and public health. Core content includes research design, statistical reasoning and methods. Emphasis will be on basic descriptive and inferential methods and practical applications. Data analysis tools will include descriptive statistics and graphing, confidence intervals, basic rules of probability, hypothesis testing for means and proportions, and regression analysis. Students will use specialized statistical software to conduct data analysis of health related data sets. [Nothing exactly like this in IPMB. At OU, we require all medical physics majors to take a statistics class, taught by the Department of Mathematics and Statistics.]

PH 4900 Research in Physics of Medicine: Independent student research on coursework from Physics of Medicine Program. Students will choose topic from Physics of Medicine Program coursework to investigate further and prepare for presentation submission. This course will serve as a capstone course for Medical Physics and Physics of Medicine Pre-Professional Majors. [I am a big supporter of undergraduate research. At OU, medical physics majors can satisfy their capstone requirement by either research or our seminar class.]

MT 3260 Mathematical Modeling in Medicine: Students will build mathematical models and use these models to answer questions in various areas of medicine. Topics may include: Epidemic modeling, genetics, drug treatment, bacterial population modeling, and neural systems/networks. [IPMB is focused on mathematical modeling. I teach PHY 325 and 326 as workshops on mathematical modeling in biology and medicine.]

The Rockhurst physics of medicine minor looks like an idea I am tempted to steal. Their requirements are:

To complete the Physics of Medicine Minor:

Prerequisites: one year of introductory/general physics and Calculus I (complete in first two years)

Upper Division Courses: complete 4 upper-division POM courses (12 hrs.total)

Required:

• PH 3200: Physics of the Body I (3 Hours, Offered Fall Semester Odd years) 
• PH 3210: Physics of the Body II (3 Hours, Offered Spring Semester even years)

Choose 2 from the following:

• PH 3240: Physics of Medical Imaging (3 Hours, Offered Spring Semester Odd Years) 
• PH 4400: Optics (3 hours, Offered Fall Semester Even Years) 
• MT 3260: Mathematical Modeling in Medicine (3 Hours, Offered Fall Semester Even years)
• PH 4900: Statistics for the Health Sciences (3 Hours, Offered Spring semesters)

An OU version might be Biological Physics (PHY 325) and Medical Physics (PHY 326), plus their prerequisites: two semesters of introductory physics and two semesters of calculus.

The Nelson Art Gallery.

I enjoy my trips to Kansas City because there is a lot to do and see there, from the Nelson Art Gallery to Crown Center to the Liberty Memorial and the National World War I Museum. I remember in high school attending shows at the Starlight Theater in Swope Park, and watching many Kansas City Royals baseball games at Kauffman Stadium (where I saw George Brett play in the World Series!). The Truman Library is in nearby Independence Missouri.

The Country Club Plaza.

I didn’t expect to find a hub of medical physics education in Kansas City, but there it is. In addition to the Rockhurst program, the Kansas University Medical Center has a CAMPEP-accredited clinical medical physics residency (while driving on I-35, I could see cranes putting up a new KU Med Center building), and the Stowers Institute, less than a mile north of Rockhurst and just east of the Country Club Plaza, has a strong biomedical research program. As the song says, Everything's Up To Date in Kansas City.

Kansas City celebrating the 2015 Royals World Series Championship.

How to Explain Why Unequal Anisotropy Ratios is Important Using Pictures but No Mathematics

Ten years ago, at the IEEE Engineering in Medicine and Biology Society Annual Conference in New York City, I presented a paper titled “How to Explain Why Unequal Anisotropy Ratios is Important Using Pictures but No Mathematics.” Although it was only a four-page conference proceeding, it remains one of my favorite papers.

I. Introduction 

The bidomain model describes the electrical properties of cardiac tissue. The term “bidomain” arises because the model accounts for two (“bi”) spaces (“domains”): intracellular and extracellular. Both spaces are anisotropic; the electrical conductivity depends on the direction relative to the myocardial fibers. Moreover, the intracellular space is more anisotropic than the extracellular space, a condition referred to in the literature as “unequal anisotropy ratios.” This condition has important consequences for the electrical behavior of the heart.

Many papers describe the implications of unequal anisotropy ratios. The mathematical derivations and numerical calculations in these reports emphasize the consequences of unequal anisotropy ratios, but they often fail to explain physically why these consequences occur. For example, Sepulveda et al. discovered that during unipolar stimulation, depolarization occurs under the cathode but hyperpolarization exists adjacent to it along the fiber direction. The hyperpolarized regions affect the mechanism of excitation, the shape of the strength-interval curve, and the induction of reentry. Yet, when I am asked why the hyperpolarization appears, I find it difficult to give an intuitive, nonmathematical answer.

In this paper, I try to answer the “why” questions that arise from the bidomain model. I present no new results, but many old results are clarified. My hope is that the reader will develop the intuition necessary to understand qualitatively how cardiac tissue behaves, without having to resort to lengthy mathematical derivations or numerical calculations.

Parts of this article have worked their way into Intermediate Physics for Medicine and Biology. For instance, the article explains how a wave front propagating through cardiac tissue creates a magnetic field. This analysis is reproduced as Problem 19 in Chapter 8 on biomagnetism.

Problem 50 in Chapter 7 examines the transmembrane potential induced in cardiac tissue when an electric shock is applied in the presence of an insulating obstacle. I love how this example highlights the importance of unequal anisotropy ratios.

Consider an insulating cylinder in an otherwise uniform tissue with straight fibers (Fig. 7). An electric field is applied from left to right. Far from the insulator, the current is in the x-direction and is distributed equally between the intracellular and extracellular spaces. As current approaches the insulator, it turns left to circle around the obstacle. The current then is flowing approximately perpendicular to the fibers, so most of the current will be extracellular. As the current turns right to flow once again in the x-direction, it will be parallel to the fibers and will again be distributed more or less equally between the two spaces. As current leaves and then reenters the intracellular space, it causes depolarization and then hyperpolarization. The transmembrane potential distribution surrounding the insulator is even in y and odd in x. The result is the complex pattern of polarization surrounding an insulator in cardiac tissue during electrical stimulation.

Fig. 7. Distribution: Polarization caused by an insulating obstacle.
This figure explains the results observed in [18].

The role of theoretical analysis in biology and medicine is to make predictions that can be tested experimentally. My former PhD advisor John Wikswo and his team used optical mapping to measure the transmembrane potential around an obstacle during a shock. Their results are shown in the picture below. The bottom line: the prediction and the experiment are consistent. Physics works!

Optical mapping to measure the transmembrane potential around an obstacle during a shock,
from: Woods et al. (Heart Rhythm, 3:751-752, 2006).

One graduate student, Marcella Woods, was involved in both of the projects I mentioned. She performed the theoretical analysis of the magnetic field produced by wave fronts in cardiac muscle under my direction when I was on the faculty of Vanderbilt University. After I left, she worked with Wikswo and carried out the experiments shown above.

Have We Reached the Athletic Limits of the Human Body?

The Olympics are in full swing this week, giving us in the United States a brief respite from our nasty presidential campaign. As you might guess, I view the Olympics through the lens of biological physics. One question that physics can help answer is: Have we reached the athletic limits of the human body? Can sprinters run faster and faster, or have we reached the physical and physiological limit? Can pole vaulters vault higher? Can long jumpers jump longer? Can swimmers swim quicker? An article in last week’s Scientific American by Bret Stetka tries to answer these questions.

At this month’s summer Olympic Games in Rio, the world's fastest man, Usain Bolt—a six-foot-five Jamaican with six gold medals and the sinewy stride of a gazelle—will try to beat his own world record of 9.58 seconds in the 100-meter dash. If he does, some scientists believe he may close the record books for good. Whereas myriad training techniques and technologies continue to push the boundaries of athletics, and although strength, speed and other physical traits have steadily improved since humans began cataloguing such things, the slowing pace at which sporting records are now broken has researchers speculating that perhaps we’re approaching our collective physiological limit—that athletic achievement is hitting a biological brick wall.

The article cites a 2008 paper by Mark Denny, the author of Air and Water, a book often cited in Intermediate Physics for Medicine and Biology. Denny suggests that there are limits, and we are closing in on them.

Are there absolute limits to the speed at which animals can run? If so, how close are present-day individuals to these limits? I approach these questions by using three statistical models and data from competitive races to estimate maximum running speeds for greyhounds, thoroughbred horses and elite human athletes. In each case, an absolute speed limit is definable, and the current record approaches that predicted maximum. While all such extrapolations must be used cautiously, these data suggest that there are limits to the ability of either natural or artificial selection to produce ever faster dogs, horses and humans. Quantification of the limits to running speed may aid in formulating and testing models of locomotion.

Yet Denny was not overly cautious in his paper. He predicted minimum times for many races, including the 100 m dash. Stetka writes

Bolt hopes to beat the researcher’s [that is, Denny’s] fastest predicted 100-meter dash time of 9.48 seconds. Unfortunately, according to Denny, the now notably older sprinter may have missed his chance. The sprinter was a chasm ahead of the pack in a semifinals race at the 2008 Beijing Olympics when he slowed up before crossing the finish line. “I think had he kept going at full speed he would’ve set an all-time, unbeatable world record,” Denny speculates.

Then Stetka quotes Denny as saying

“When I published my paper, the feedback I got was that this was going to destroy the Olympics,” he recollects. “That’s like saying the 1962 Brazilian soccer team was the best ever so no one’s ever going to watch the World Cup again. But if Bolt can run the 100 in 9.47 seconds and beat my prediction, then hats off to him. I think there’s always going to be the lure of ‘maybe someone’s going to do better.’”

I plan to watch the Olympics and see if humans can run faster than ever before. I’m a big fan of Mark Denny, but I’ll be routing for Bolt (or Gatlin) to beat Denny's prediction.

Enjoy!


P.S. A long frustrated sigh goes to Michael Phelps and the other USA swimmers engaged in “cupping” therapy pseudoscience. Oh, where is Bob Park when we need him! Ignore the quackery and gibberish and focus on the swimming.

Zapping Their Brains at Home

“Zapping Their Brains at Home,”
by Anna Wexler.

A couple weeks ago, Anna Wexler published an article in the New York Times titled “Zapping Their Brains at Home.”

Earlier this month, in the journal Annals of Neurology, four neuroscientists published an open letter to practitioners of do-it-yourself brain stimulation. These are people who stimulate their own brains with low levels of electricity, largely for purposes like improved memory or learning ability. The letter, which was signed by 39 other researchers, outlined what is known and unknown about the safety of such noninvasive brain stimulation, and asked users to give careful consideration to the risks.

I worked on brain stimulation when at the National Institutes of Health, and Russ Hobbie and I analyze neural stimulation in Intermediate Physics for Medicine and Biology. So what is my reaction to these do-it-yourselfers? My first thought was “Yikes…this sounds like trouble!” But the more I think about it, the less worried I am.

We are talking about transcranial direct current stimulation, which uses weak currents applied to the scalp. I have always been surprised that such tiny currents have any effect at all; see my editorial “What Does the Ratio of Injected Current to Electrode Area Not Tell Us About tDCS?” (Clinical Neurophysiology, Volume 120, Pages 1037–1038, 2009). My advice to the do-it-yourselfers is not so much “be careful” but rather “don’t get your hopes up.”

Of the four coauthors on the letter in Annals of Neurology, the only one I know is Alvaro Pascual-Leone, who I worked with while at NIH and who we cite several times in IPMB. Below I list the main points raised in the letter:

• Stimulation affects more of the brain than a user may think 
• Stimulation interacts with ongoing brain activity, so what a user does during tDCS changes its effects 
• Enhancement of some cognitive abilities may come at the cost of others 
• Changes in brain activity (intended or not) may last longer than a user may think 
• Small differences in tDCS parameters can have a big effect 
• tDCS effects are highly variable across different people 
• The risk/benefit ratio is different for treating diseases versus enhancing function

What do I think of do-it-yourselfers in general? I have mixed feelings. Heaven help us if they start fooling around with heart defibrillators, which could be suicidal. For transcranial magnetic stimulation, I think the biggest risk would be the construction of a device that sends kiloamps of current through a coil. I have always thought that TMS is more dangerous for the physician (who often holds the coil) than for the patient. Moreover, the induced current in the brain is larger for TMS than for tDCS. I would be wary of do-it-yourself magnetic stimulation. But for D.I.Y.ers using relatively low-level electrical current applied to the scalp, if someone educates themself on the technique and follows reasonable safety recommendations, then I don’t see it as a problem.

Wexler ends her letter

The open letter this month is about safety. But it also a recognition that these D.I.Y. practitioners are here to stay, at least for the time being. While the letter does not condone, neither does it condemn. It sticks to the facts and eschews paternalistic tones in favor of measured ones. The letter is the first instance I’m aware of in which scientists have directly addressed these D.I.Y. users. Though not quite an olive branch, it is a commendable step forward, one that demonstrates an awareness of a community of scientifically involved citizens.

If you want to read more by Wexler, look here and here.

My final, and admittedly self-serving, advice to the D.I.Y.ers: go buy a copy of Intermediate Physics for Medicine and Biology, so you can learn the scientific principles behind this and other techniques.