Friday, September 30, 2016

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.
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 na3/2 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 na3/2 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 ii along an axon is related to the potential v and the resistance per unit length ri 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 gm is the membrane conductance per unit area and cm 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
An equation for the length constant of a nerve axon.
and the time constant as
An equation for the time constant of a nerve axon.
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
The cable equation.
Now, lets 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 ri now varies with position (ri = 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 ri is independent of x, the new term disappears and you get the plain old cable equation.” Its 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

An equation representing Rall’s equivalent cylinder model of a dendritic tree.
If λri 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 ri and λ in terms of a and n, you find that this condition is equivalent to saying that na3/2 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.

Friday, September 23, 2016

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 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
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.

Friday, September 16, 2016

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
Mathematical expressions for the differential cross section
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
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
The leading constant factor in the Rutherford scattering formula.
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, superimposed on Intermediate Physics for Medicine and Biology.
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.”

Friday, September 9, 2016

The Biomechanics of Solids and Fluids: The Physics of Life

The first page of The Biomechaics of Solids and Fluids: The Physics of Life, by David Alexander, superimposed on Intermediate Physics for Medicine and Biolog.
“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.

Friday, September 2, 2016


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 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:

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!

A photograph of our car, after the accident. Its left rear is smashed in. The car was totaled. My wife and I were OK, but could have suffered from whiplash.
Our car, after the accident.