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.
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)
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.
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.
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.
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.
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.
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.
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.
We first consider the interaction of the projectile with a target
electron, which leads to the electronic stopping power,
Se. Many authors call it the collision stopping power, Scol.
There can be interactions in which a single electron is ejected
from a target atom or interactions with the electron cloud as a
whole (a plasmon excitation). The stopping power at higher
energies, where it is nearly proportional to β−2 [β = v/c, where v is the speed of the projectile and c is the speed of light], has been
modeled by Bohr, by Bethe, and by Bloch (see the review
by Ahlen 1980).
Niels Bohr's Times: In Physics, Philosophy, and Polity,
by Abraham Pais.
Bohr finished his paper on this subject [the energy loss of alpha particles when traversing matter] only after he had left Manchester; it appeared in 1913. The problem of the stopping of electrically charged particles remained one of his lifelong interests. In 1915 he completed another paper on that subject, which includes the influence of effects due to relativity and to straggling (that is, the fluctuations in energy and in range of individual particles)…
Bohr’s 1913 paper on α-particles, which he had begun in Manchester, and which had led him to the question of atomic structure, marks the transition to his great work, also of 1913, on that same problem. While still in Manchester, he had already begun an early sketch of these entirely new ideas. The first intimation of this comes from a letter, from Manchester, to Harald [Niels’ brother]: “Perhaps I have found out a little about the structure of atoms. Don’t talk about it to anybody…It has grown out of a little information I got from the absorption of α-rays.”
I leave the discussion of these beginnings to the next chapter.
On 24 July 1912 Bohr left Manchester for his beloved Denmark. His postdoctoral period had come to an end.
So the alpha particle stopping power calculation Russ and I discuss in Chapter 15 led directly to Bohr’s model of the hydrogen atom, for which he got the Nobel Prize in 1922.
Chapter 3 of Intermediate Physics for Medicine and Biology discusses the Boltzmann factor. In the homework exercises at the end of the chapter, we include a problem in which you apply the Boltzmann factor to estimate the error rate during the copying of DNA.
Problem 30. The DNA molecule consists of two intertwined linear chains. Sticking out from each monomer (link in the chain) is one of four bases: adenine (A), guanine (G), thymine (T), or cytosine (C). In the double helix, each base from one strand bonds to a base in the other strand. The correct matches, A-T and G-C, are more tightly bound than are the improper matches. The chain looks something like this, where the last bond shown is an “error.”
A DNA molecule containing an error.
The probability of an error at 300 K is about 10−9 per base pair. Assume that this probability is determined by a Boltzmann factor e−U/kBT, where U is the additional energy required for a mismatch.
(a) Estimate this excess energy.
(b) If such mismatches are the sole cause of mutations in an organism, what would the mutation rate be if the temperature were raised 20° C?
This is a nice simple homework problem that provides practice with the Boltzmann factor and insight into the thermodynamics of base pair copying. Unfortunately, reality is more complicated.
Biophysics:
Searching for Principles,
by William Bialek.
William Bialek addresses the problem of DNA copying in his book Biophysics: Searching for Principles (Princeton University Press, 2012). He notes that the A typically binds to T. If A were to bind with G, the resulting base pair would be the wrong size and grossly disrupt the DNA double helix (A and G are both large double-ring molecules). However, if A were to bind incorrectly with C, the result would fit okay (C and T are about the same size) at the cost of eliminating one or two hydrogen bonds, which have a total energy of about 10 kBT. Bialek writes
An energy difference of ΔF ~ 10 kBT means that the probability of an incorrect base pairing should be, according to the Boltzmann distribution, e-ΔF/kBT ~ 10−4. A typical protein is 300 amino acids long, which means that it is encoded by about 1000 bases; if the error probability is 10-4, then replication of DNA would introduce roughly one mutation in every tenth protein. For humans, with a billion base pairs in the genome, every child would be born with hundreds of thousands of bases different from his or her parents. If these predicted error rates seem large, they are—real error rates in DNA replication vary across organisms [see the vignette “what is the error rate in transcription and translation” in Cell Biology by the Numbers], but are in the range of 10−8–10−12, so the entire genome can be copied without almost any mistakes.
So, how is the does the error rate become so small? There are enzymes called DNA polymerases that proofread the copied DNA and correct most errors. Because of these enzymes, the overall error rate is far smaller than the 10−4 rate you would estimate from the Boltzmann factor alone.
Our homework problem is therefore a little misleading, but it has redeeming virtues. First, the error we show in the figure is G-A, which would more severely disrupt the DNA's double helix structure. That specific error may well have a higher energy and therefore a lower error rate from the Boltzmann factor alone. Second, the problem illustrates how sensitive the Boltzmann factor is to small changes in energy. If ΔE = 10 kBT, the Boltzmann factor is e−10 = 0.5 × 10−4. If ΔE = 20 kBT, the Boltzmann factor is e−20 = 2 × 10−9. A factor of two increase in energy translates into more than a factor of 10,000 reduction in error rate. Wow!
I have always wondered about those funny-looking collections of different-sized, different-colored words: the word cloud. This week I learned how to create a word cloud from any text I choose using the free online software at www.wordclouds.com. Of course, I chose Intermediate Physics for Medicine and Biology. Here is what I got.
A word cloud based on Intermediate Physics for Medicine and Biology.
The word cloud speaks for itself, but let me add a few comments. First, I deleted the preface, the table of contents, and the index from a pdf copy of IPMB before submitting it. The software was having trouble with such a large input file, and reducing the size seemed to help. After the list of words and their frequencies was created, I edited it. The software is smart enough to not include common words like “the”
and “is,” but I deleted others that seemed generic to me, like “consider” and “therefore.” I kept words that appeared at least 250 times, which was about 65 words. The most common word was “Fig,” as in “...spherical air sacs called alveoli (Fig. 1.1b).” The third most common was “Problem” as in “Problem 1. Estimate the number of....” I considered removing these, but illustrations and end-of-chapter exercises are an important part of the book, so they stayed. I was surprised by the second most common word: “energy.” Russ Hobbie and I did not set out to make this a unifying theme in the book, but apparently it is.
I’ll let you decide if this word cloud is profound or silly. It was fun, and I like to share fun things with the readers of IPMB. Enjoy!
One of the central missions of our book is to serve as an entry point that invites the reader to explore some of the key numbers of cell biology. We hope to attract readers of all kinds—from seasoned researchers, who simply want to find the best values for some number of interest, to beginning biology students, who want to supplement their introductory course materials. In the pages that follow, we provide a broad collection of vignettes, each of which focuses on quantities that help us think about sizes, concentrations, energies, rates, information content, and other key quantities that describe the living world.
• 1 dalton (Da) = 1 g/mol ~ 1.6 × 10−24 g.
• 1 nM is about 1 molecule per bacterial volume [E. coli has a volume of about 1 μm3].
• 1 M is about one per 1 nm3.
• Under standard conditions, particles at a concentration of 1 M are ~ 1 nm apart.
• Water molecule volume ~ 0.03 nm3, (~0.3 nm)3.
• A small metabolite diffuses 1 nm in ~1 ns.
The book consists of a series of vignettes, each phrased as a question. Here is an excerpt form one.
Which is bigger, mRNA or the protein it codes for?
The role of messenger RNA molecules (mRNAs), as epitomized in the central dogma, is one of fleeting messages for the creation of the main movers and shakers of the cell—namely, the proteins that drive cellular life. Words like these can conjure a mental picture in which an mRNA is thought of as a small blueprint for the creation of a much larger protein machine. In reality, the scales are exactly the opposite of what most people would guess. Nucleotides, the monomers making up an RNA molecule, have a mass of about 330 Da. This is about three times heavier that the average amino acid mass, which weighs in at ~110 Da. Moreover, since it takes three nucleotides to code for a single amino acid, this implies an extra factor of three in favor of mRNA such that the mRNA coding a given protein will be almost an order of magnitude heavier.
It’s obvious once someone explains it to you. Here is another that I liked.
What is the pH of a cell?
…Even though hydrogen ions appear to be ubiquitous in the exercise sections of textbooks, their actual abundance inside cells is extremely small. To see this, consider how many ions are in a bacterium or mitochondrion of volume 1 μm3 at pH 7. Using the rule of thumb that 1 nM corresponds to ~ 1 molecule per bacterial cell volume, and recognizing that pH 7 corresponds to a concentration of 10−7 M (or 100 nM), this means that there are about 100 hydrogen ions per bacterial cell…This should be contrasted with the fact that there are in excess of a million proteins in that same cellular volume.
This one surprised me.
What are the concentrations of free matabolites in cells?
…The molecular census of metabolites in E. coli reveals some overwhelmingly dominant molecular players. The amino acid glutamate wins out…at about 100 mM, which is higher than all other amino acids combined…Glutamate is negatively charged, as are most of the other abundant metabolites in the cell. This stockpile of negative charges is balanced mostly by a corresponding positively changed stockpile of free potassium ions, which have a typical concentration of roughly 200 mM.
Somehow, I never realized how much glutamate is in cells. I also learned all sorts of interesting facts. For instance, a 5% by weight mixture of alcohol in water (roughly equivalent to beer) corresponds to a 1 M concentration. I guess the reason this does not wreak havoc on your osmotic balance is that alcohol easily crosses the cell membrane. Apparently yeast use the alcohol they produce to inhibit the growth of bacteria. This must be why John Snow found that during the 1854 London cholera epidemic, the guys working (and, apparently, drinking) in the brewery were immune.
I’ll give you one more example. Milo and Phillips analyze how long it will take a substrate to collide with a protein.
…Say we drop a test substrate molecule into a cytoplasm with a volume equal to that of a bacterial cell. If everything is well mixed and there is no binding, how long will it take for the substrate molecule to collide with one specific protein in the cell? The rate of enzyme substrate collisions is dictated by the diffusion limit, which as shown above, is equal to ~ 109 s−1M−1 times the concentration. We make use of one of our tricks of the trade, which states that in E. coli, a single molecule (say, our substrate) has an effective concentration of about 1 nM (that is, 10−9 M). The rate of collisions is thus 109 s−1M−1 × 10−9 M. That is, they will meet within a second on average. This allows us to estimate that every substrate molecule collides with each and every protein in the cell on average about once per second.
Each and every one, once per second! The beauty of this book, and the value of making these order-of-magnitude estimates, is to provide such insight. I cannot think of any book that has provided me with more insight than Cell Biology by the Numbers.
Readers of IPMB will enjoy CBbtN. It is well written and the illustrations by Nigel Orme are lovely. It may have more cell biology than readers of IPMB are used to (Russ Hobbie and I are macroscopic guys), but that is fine. For those who prefer video over text, listen to Rob Phillips and Ron Milo give their views of life in the videos below.
I’ll give Milo and Phillips the last word, which could also sum up our goals for IPMB.
We leave our readers with the hope that they will find these and other questions inspiring and will set off on their own path to biological numeracy.
Russ and I like to provide homework problems that reinforce the concepts in the text. Ideally, the problem requires the reader to repeat many of the same steps carried out in the book, but for a slightly different case or in a somewhat different context. Below I present such a homework problem for blackbody radiation. It is based on an approximation to Planck’s law at short wavelengths derived by Wilhelm Wien.
Problem 25 ½. Consider the limit of Planck’s law, Eq. 14.33, when hc/λ is much greater than kBT, an approximation known as the Wien exponential law.
(a) Derive the mathematical form of Wλ(λ,T) in this limit.
(b) Convert Wien’s law from a function of wavelength to a function of frequency, and determine the mathematical form of Wν(ν,T).
(c) Integrate Wν(ν,T) over all frequencies to obtain the total power emitted per unit area. Compare this result with the Stefan-Boltzmann law (Eq. 14.34). Derive an expression for the Stefan-Boltzmann constant in terms of other fundamental constants.
(d) Determine the frequency νmax corresponding to the peak in Wν(ν,T). Compare νmax/T with the value obtained from Planck’s law.
Meanwhile,proposals for the correct form of [Wλ(λ,T)] had begun to appear as early as the 1860s. All these guesses may be forgotten except one, Wien’s exponential law, proposed in 1896…
Experimental techniques had sufficiently advanced by then to put this formula to the test. This was done by Friedrich Paschen from Hannover, whose measurements (very good ones) were made in the near infrared, λ = 1-8 μm (and T = 400 -1600 K). He published his data in January 1897. His conclusion: “It would seem very difficult to find another function…that represents the data with as few constants.” For a brief period, it appeared that Wien’s law was the final answer. But then, in the year 1900, this conclusion turned out to be premature…
The chemostat is used by bacteriologists to study the growth of bacteria (Hagen 2010). It allows the rapid growth of bacteria to be observed over a longer time scale. Consider a container of bacterial nutrient of volume V. It is well stirred and contains y bacteria with concentration C = y/V. Some of the nutrient solution is removed at rate Q and replaced by fresh nutrient. The bacteria in the solution are reproducing at rate b. The rate of change of y is
Therefore the growth rate is slowed to
and can be adjusted by varying Q.
However, Russ and I didn’t write any new homework problems for this section. If a topic is worth discussing in the text, then it’s worth creating homework problems to reinforce and extend that discussion. So, here are some new problems about the chemostat.
Problem 21.1. Often a chemostat is operated in steady state.
(a) Determine the solution removal rate Q required for steady state, as a function of the bacteria reproduction rate b and the container volume V, using Eq. 2.22. Determine the units of b, Q, and V, and verify that your expression for Q has the correct dimensions.
(b) If the rate Q is larger than the steady-state value, what is happening physically?
(c) Sometimes b varies with some external parameter (for example, temperature or glucose concentration), and you want to determine b as a function of that parameter. Suppose you can control Q and you can measure the number of bacteria y. Qualitatively design a way to determine b as your external parameter changes, assuming that for each value of the parameter your chemostat reaches steady state. (If unsure how to begin, take a look at Sec. 6.13.1 about the voltage clamp used in electrophysiology.)
Problem 21.2. Consider an experiment using a chemostat in which the bacteria's reproduction rate b slows as the number of bacteria y increases.
(a) Modify Eq. 2.22 so that “b” becomes “b (1 − y/y∞),”
analogous to the logistic model (Sec. 2.10).
(b) Determine the value of y once the chemostat reaches steady state, as a function of Q, V, b, and y∞.
(c) Suppose your chemostat has a volume of 1.7 liters. You measure the steady state value of y (arbitrary units) for different values of Q (liters per hour), as shown in the table below. Plot y versus Q, and determine b and y∞.
Q
y
0.2
11.64
0.4
9.47
0.6
7.31
0.8
5.14
1.0
2.98
Problem 21.3. Let the growth rate of the bacteria in your chemostat be limited by a small, constant amount of some essential metabolite, so the term “by” in Eq. 2.22 is replaced by a constant “a.”
(a) Find an expression for the solution removal rate Q in terms of a, the number of bacteria y, and the chemostat volume V, when the chemostat is in steady state.
(b) Determine the time constant governing how quickly the chemostat reaches steady state (Hint: see Sec. 2.8).
“Exponential Growth of Bacteria:
Constant Multiplication Through Division,”
by Stephen Hagen.
Russ and I cite an American Journal of Physics article about the exponential growth of bacteria, written by Stephen Hagen (Volume 78, Pages 1290-1296, 2010). Here’s what Hagen says about the chemostat.
Because the growth rate of the cell determines its size and
chemical composition, a device that allows us to fine tune the
growth rate will select the physiological properties of the
cells. The bacterial chemostat is such a machine. In the
chemostat a bacterial culture grows in a well-stirred vessel
while a supply of fresh growth medium is fed into the vessel
at a fixed flow rate Q (volume/time). At the same time, the
medium (containing bacteria) is continuously withdrawn
from the vessel at the same rate so as to maintain constant
volume V. Thus, the bacterial population is continuously diluted
at a rate D = Q/V. If this dilution rate exceeds the
growth rate k [our b], the population is diluted, which allows its
growth to accelerate until it matches the dilution rate, k
= Q/V. (If D is too large, the culture will be diluted away
entirely.) Therefore, the chemostat allows the experimenter
to select the growth rate by selecting Q. Because it harnesses
an exponential growth process to produce a tunable, steady
output, we might think of the chemostat as the microbiological
analog of a nuclear fission reactor. Interestingly, the
chemostat reactor was first described by the physicist Leo Szilard (with Aaron Novick), who also (with Enrico Fermi)
patented the nuclear reactor."
I like the analogy to the nuclear reactor. Adjusting the flow rate in a chemostat is like pulling the cadmium control rods in and out of an atomic pile (except it’s less dangerous).
One feature of blogging that I like are the comments. I don’t get many, but I appreciate those I do get. Each week I share my new blog entry on the Intermediate Physics for Medicine and Biology Facebook page, which provides another venue for comments, likes, and other interactions with readers. A couple weeks ago I received the following on Facebook:
Neeraj Kapoor
June 3 at 1:36pm
Yesterday, during a conference with Elon Musk at a coding conference, he mentioned something about Neural Lacing (this group at harvard seems to be one of the few major groups working on it...http://cml.harvard.edu/) . I'm wondering if you have any knowledge of this Brad Roth and if so, if you could do a blog post on it.
After a bit of googling, I found a Newsweek article about neural lacing, Elon Musk, and the coding conference.
Billionaire polymath Elon Musk has warned that humans risk being treated like house pets by artificial intelligence (AI) unless they implant technology into their brains.
Musk believes that a technology concept known as “neural lace” could act as a wireless brain-computer interface capable of augmenting natural intelligence.
Speaking at the Code Conference in California on Wednesday, Musk said a neural lace could work “well and symbiotically” with the rest of a human’s body.
“I don’t love the idea of being a house cat, but what’s the solution? I think one of the solutions that seems maybe the best is to add an AI layer,” Musk said.
So what does all this talk about neural lacing mean, and how does it relate to Intermediate Physics for Medicine and Biology? As best I can tell, neural lacing would be used to monitor and excite nerves. The technology to stimulate nerves already exists, and is described in Chapter 7 of IPMB.
The best example of what I think Mr. Musk is talking about is the cochlear implant. A microphone records sound and analyzes it with a computer, which decides what location along the auditory nerve it should stimulate in order to fool the brain into thinking the ear heard that sound. For this technique to work, electrode arrays must be implanted in the cochlea so different spots can be stimulated, mimicking the sensitivity of different locations along the cochlea to different frequencies of sound.
What is different between a cochlear implant and a neural lace? Musk talks about the stimulating electrodes being wireless. Wireless neural stimulation is fairly common, and most cochlear implants are wireless (no wire passing through the skin). Most wireless systems work by transferring energy and information using electromagnetic induction. Chapter 8 of IPMB discusses induction, mainly in the context of magnetic stimulation. In fact, transcranial magnetic stimulation could be thought of as a low-spatial-resolution precursor to neural lacing. It allows neurons to be excited with no wires penetrating the body so the method is completely noninvasive. The problem is, transcranial magnetic stimulation provides a resolution of about 1 cm—some claim as low as 1 mm—which is a factor of a hundred to a thousand too coarse to stimulate individual neurons. If you could somehow build very small magnetic stimulators (there are enormous technical challenges in doing this), you still would not be able to excite deep neurons without simultaneously activating shallow neurons even more strongly. To make something like neural lacing work, you would need to use electromagnetic induction to transfer energy to a stimulator implanted in the body, and then distribute the excitation using small wires or some other technology that provides the necessary spatial resolution and the ability to excite deep neurons. Wireless deep brain stimulation is one example.
Spatial scale is a key factor in developing the technology of neural lacing. Cochlear implants only work because the electrodes are small enough that individual sites along the auditory nerve can be excited locally. I believe that neural lacing would require miniaturization to be increased dramatically. If you are going to stimulate the brain in a truly selective way, you need to be able to excite individual neurons. This means you need electrodes spaced by about ten microns or closer, and you need a lot of them. Neural lacing would therefore require advances in electrode array miniaturization. This is where the Lieber group at Harvard—which Kapoor mentioned in his Facebook comment—enters the picture. They are developing the arrays of microelectrodes that would be necessary to provide a fine-grained interaction between a computer and the human brain. For example, their paper “syringe-injectable electronics” (Nature Nanotechnology, Volume 10, Pages 629–636, 2015) discusses small scale arrays of electrodes that can be injected through a syringe.
Seamless and minimally invasive three-dimensional interpenetration of electronics within artificial or natural structures could allow for continuous monitoring and manipulation of their properties. Flexible electronics provide a means for conforming electronics to non-planar surfaces, yet targeted delivery of flexible electronics to internal regions remains difficult. Here, we overcome this challenge by demonstrating the syringe injection (and subsequent unfolding) of sub-micrometre-thick, centimetre-scale macroporous mesh electronics through needles with a diameter as small as 100 μm. Our results show that electronic components can be injected into man-made and biological cavities, as well as dense gels and tissue, with [greater than] 90% device yield. We demonstrate several applications of syringe-injectable electronics as a general approach for interpenetrating flexible electronics with three-dimensional structures, including (1) monitoring internal mechanical strains in polymer cavities, (2) tight integration and low chronic immunoreactivity with several distinct regions of the brain, and (3) in vivo multiplexed neural recording. Moreover, syringe injection enables the delivery of flexible electronics through a rigid shell, the delivery of large-volume flexible electronics that can fill internal cavities, and co-injection of electronics with other materials into host structures, opening up unique applications for flexible electronics.
Is neural lacing science or science fiction? Hard to say. I am skeptical that in the future we will all have electrode arrays hardwired into our brains. But 50 years ago I would have been skeptical that cochlear implants could restore hearing to the deaf. I will reserve judgment, except to say that if neural lacing is developed, I am certain it will be based on the basic concepts Russ Hobbie and I discuss in Intermediate Physics for Medicine and Biology. That is the beauty of the book: it teaches the fundamental principles upon which you can build the amazing biomedical technologies of the future.
One reason I write this blog is to help instructors who adopt Intermediate Physics for Medicine and Biology as their textbook. I teach classes from IPMB myself; here at Oakland University we have a Biological Physics class (PHY 325) and a Medical Physics class (PHY 326). Instructors might benefit from seeing how I structure these classes, so below are my most recent syllabi.
Goal: To understand how physics contributes to medicine
Grades
Point/Counterpoint
5 %
Exam 1
Feb 5
20 %
Chapters 13–15
Exam 2
March 18
20 %
Chapters 16, 11–12
Exam 3
April 20
20 %
Chapter 17, 18
Final Exam
April 20
10 %
Homework
25 %
Schedule
Jan 6, 8
Introduction
Jan 11, 13, 15
Chpt 13
Sound and Ultrasound
Jan 20, 22
Chpt 14
Atoms and Light
Jan 25, 27, 29
Chpt 15
Interaction of Photons and Matter
Feb 1, 3, 5
Exam 1
Feb 8, 10, 12
Chpt 16
Medical Uses of X rays
Feb 15, 17, 19
Chpt 11
Least Squares and Signal Analysis
Feb 22, 24, 26
Winter Recess
Feb 29, March 2, 4
Chpt 12
Images
March 7, 9, 11
Chpt 12
Images
March 14, 16, 18
Exam 2
March 21, 23, 25
Chpt 17
Nuclear Medicine
March 28, 30, Apr 1
Chpt 17
Nuclear Medicine
April 4, 6, 8
Chpt 18
Magnetic Resonance Imaging
April 11, 13, 15
Chpt 18
Magnetic Resonance Imaging
April 18
Conclusion
April 20
Final Exam
Homework
Chapter 13:
7, 10, 12, 21, 22, 27, 30, 36
due Fri, Jan 22
Chapter 14:
4, 5, 16, 21, 22, 47, 48, 49
due Wed, Jan 27
Chapter 15:
2, 4, 5, 10, 12, 14, 15, 16
due Wed, Feb 3
Chapter 16:
4, 5, 7, 16, 19, 20, 22, 31
due Wed, Feb 17
Chapter 11:
9, 11, 15, 20, 21, 36, 37, 41
due Wed, Mar 2
Chapter 12:
7, 9, 10, 23
due Wed, Mar 9
Chapter 12:
25, 32, 34, 35, and 27 (extra credit)
due Wed, Mar 16
Chapter 17:
1, 2, 7, 9, 14, 17, 20, 22
due Wed, Mar 30
Chapter 17:
29, 30, 40, 54, 57, 58, 59, 60
due Wed, Apr 6
Chapter 18:
9, 10, 13, 14, 15, 18, 35, 49
due Wed, Apr 13
Point/Counterpoint articles
Jan 8: The 2014 initiative is not only unnecessary but it constitutes a threat to the future of medical physics. Med Phys, 38:5267–5269, 2011.
Jan 15: Ultrasonography is soon likely to become a viable alternative to x-ray mammography for breast cancer screening. Med Phys, 37:4526–4529, 2010.
Jan 22: High intensity focused ultrasound may be superior to radiation therapy for the treatment of early stage prostate cancer. Med Phys, 38:3909–3912, 2011.
Jan 29: The more important heavy charged particle radiotherapy of the future is more likely to be with heavy ions rather than protons. Med Phys, 40:090601, 2013.
Feb 12: The disadvantages of a multileaf collimator for proton radiotherapy outweigh its advantages. Med Phys, 41:020601, 2014.
Feb 19: Low-dose radiation is beneficial, not harmful. Med Phys, 41:070601, 2014.
March 4: Recent data show that mammographic screening of asymptomatic women is effective and essential. Med Phys, 39:4047–4050, 2012.
March 11: PDT is better than alternative therapies such as brachytherapy, electron beams, or low-energy x rays for the treatment of skin cancers. Med Phys, 38:1133–1135, 2011.
March 25: Submillimeter accuracy in radiosurgery is not possible. Med Phys, 40:050601, 2013.
April 1: Within the next ten years treatment planning will become fully automated without the need for human intervention. Med Phys, 41:120601, 2014.
April 8: Medical use of all high activity sources should be eliminated for security concerns. Med Phys, 42:6773, 2015.
April 15: MRI/CT is the future of radiotherapy treatment planning. Med Phys, 41:110601, 2014.
Notes:
The OU library has an electronic version of IPMB that students can download. If they are willing to read pdfs, they have no textbook expense in either class.
I skip Chapter 9. I have nothing against it. There just isn’t time for everything.
I cover Chapters 13-16 before the highly mathematical Chapters 11-12. I don’t like to start the semester with a week or two of math.
In Medical Physics, we spend the last 15 minutes of class each Friday discussing a point/counterpoint article from the journal Medical Physics. The students seem to really enjoy this.
I let the students work together on the homework, but they cannot simply copy someone else’s work. They must turn in their own assignment.
Both PHY 325 and PHY 326 are aimed at upper-level undergraduates. The prerequisites are a year of introductory physics and a year of introductory calculus. The students tend to be physics majors, medical physics majors, bioengineering majors, plus a few biology, chemistry, math, and mechanical engineering majors. The typical enrollment is about ten.
I encourage premed students to take these classes. Occasionally one does, but not too often. I wish more would, because I believe it provides an excellent preparation for the MCAT. Unfortunately, they have little room in their busy schedule for two extra physics classes.
OU offers a medical physics major. It consists of many traditional physics classes, these two specialty classes (PHY 325 and PHY 326), plus some introductory and intermediate biology.
I am a morning person, so I often teach at 8 A.M. The students hate it, but I love it. Sometimes, however, I can’t control the time of day for the class and I teach at a later time.
I am an emeritus professor of physics at Oakland University, and coauthor of the textbook Intermediate Physics for Medicine and Biology. The purpose of this blog is specifically to support and promote my textbook, and in general to illustrate applications of physics to medicine and biology.
