Paul Callaghan (1947-2012)

Principles of Nuclear
Magnetic Resonance Microscopy,
by Pual Callaghan.

Russ Hobbie and I are hard at work on the 5th edition of Intermediate Physics for Medicine and Biology, which has me browsing through many books—some new and some old classics—looking for appropriate texts to cite. The one I’m looking at now is Paul Callaghan’s Principles of Nuclear Magnetic Resonance Microscopy (Oxford University Press, 1991). Callaghan was the PhD mentor of my good friend and Oakland University colleague Yang Xia. You probably won’t be surprised to know that, like Callaghan, Xia is a MRI microscopy expert. He uses the technique to study the ultrastructure of cartilage at a resolution of tens of microns. Xia assigns Callaghan’s book when he teaches Oakland’s graduate MRI class.

Callaghan gives a brief history of MRI on the first page of his book.

Until the discovery of X-rays by Roentgen in 1895 our ability to view the spatial organization of matter depended on the use of visible light with our eyes being used as primary detectors. Unaided, the human eye is a remarkable instrument, capable of resolving separations of 0.1 mm on an object placed at the near point of vision and, with bifocal vision, obtaining a depth resolution of around 0.3 mm. However, because of the strong absorption and reflection of light by most solid materials, our vision is restricted to inspecting the appearance of surfaces. “X-ray vision” gave us the capacity, for the first time, to see inside intact biological, mineral, and synthetic materials and observe structural features.

The early X-ray photographs gave a planar representation of absorption arising from elements right across the object. In 1972 the first X-ray CT scanner was developed with reconstructive tomography being used to produce a two-dimensional absorption image from a thin axial layer.¹ The mathematical methods used in such image reconstruction were originally employed in radio astronomy by Bracewell² in 1956 and later developed for optical and X-ray applications by Cormack³ in 1963. A key element in the growth of tomographic techniques has been the availability of high speed digital computers. These machines have permitted not only the rapid computation of the image from primary data but have also made possible a wide variety of subsequent display and processing operations. The principles of reconstructive tomography have been applied widely in the use of other radiations. In 1973, Lauterbur⁴ reported the first reconstruction of a proton spin density map using nuclear magnetic resonance (NMR), and in the same year Mansfield and Grannell⁵ independently demonstrated the Fourier relationship between the spin density and the NMR signal acquired in the presence of a magnetic field gradient. Since that time the field has advanced rapidly to the point where magnetic resonance imaging (MRI) is now a routine, if expensive, complement to X-ray tomography in many major hospitals. Like X-ray tomography, conventional MRI has a spatial resolution coarser than that of the unaided human eye with volume elements of order (1 mm)³ or larger. Unlike X-ray CT however, where resolution is limited by the beam collimation, MRI can in principle achieve a resolution considerably finer than 0.1 mm and, where the resolved volume elements are smaller than (0.1 mm)³, this method of imaging may be termed microscopic.

1. Hounsfield, G. N. (1973). British Patent No. 1283915 (1972) and Br. J. Radiol. 46, 1016.

2. Bracewell, R. N. (1956). Austr. J. Phys. 9, 109–217.

3. Cormack, A. M. (1963). J. Appl. Phys. 34, 2722–7.

4. Lauterbur, P. C. (1973). Nature 242, 190.

5. Mansfield, P. and Grannell, P. K. (1973). J. Phys. C 6, L422.

Callaghan was an excellent teacher, and he prepared a series of videos about MRI. You can watch them for free here. They really are “must see” videos for people wanting to understand nuclear magnetic resonance. He was a professor at Massey University in Wellington, New Zealand. In 2011 he was named New Zealander of the Year, and you can hear him talk about scientific innovation in New Zealand here.

Callaghan died about two years ago. You can see his obituary here, here and here. Finally, here you can listen to an audio recording of Yang Xia speaking about his mentor at the Professor Sir Paul Callaghan Symposium in February 2013.

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Celebrating the 60th Anniversary of the IEEE TBME

IEEE Transactions on
Biomedical Engineering.

One journal that I have published in several times is the IEEE Transactions on Biomedical Engineering. The May issue of IEEE TBME celebrates the journal’s 60th anniversary. Bin He, editor-in-chief, writes in his introductory editorial

THE IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING (TBME) is celebrating 60 years of publishing biomedical engineering advances. TBME was one of the first journals devoted to biomedical engineering. Thanks to IEEE, all of the TBME papers since January 1964 have been archived and are available to the public. In this special issue, celebrating TBME’s 60th anniversary, we have invited 20 leading groups in biomedical engineering research to contribute review articles. Each article reviews state of the art and trends in an area of biomedical engineering research in which the authors have made important original contributions. Due to limited space, it is not our intention to cover all areas of biomedical engineering research in this special issue, but instead to provide coverage of major subfields within the discipline of biomedical engineering, including biomedical imaging, neuroengineering, cardiovascular engineering, cellular and tissue engineering, biomedical sensors and instrumentation, biomedical signal processing, medical robotics, bioinformatics, and computational biology. These review articles are witness to the development of the field of biomedical engineering, and also reflect the role that TBME has played in advancing the field of biomedical engineering over the past 60 years…

These comprehensive and timely reviews reflect the breadth and depth of biomedical engineering and its impact to engineering, biology, medicine, and the larger society. These reviews aim to serve the readers in gaining insights and an understanding of particular areas in biomedical engineering. Many articles also share perspectives from the authors on future trends in the field. While the intention of this special issue was not to cover all research programs in biomedical engineering, these 20 articles represent a collection of state-of-the-art reviews that highlight exciting and significant research in the field of biomedical engineering and will serve TBME readers and the biomedical engineering community in years to come.

Biomedical Engineering can be thought of as an applied version of medical and biological physics, and many of the topics Russ Hobbie and I discuss in the 4th edition of Intermediate Physics for Medicine and Biology are important to biomedical engineers. We cite nineteen IEEE TBME papers in IPMB:

Tucker, R. D., and O. H. Schmitt (1978) “Tests for Human Perception of 60 Hz Moderate Strength Magnetic Fields,” IEEE Trans. Biomed. Eng. Volume 25, Pages 509–518.

Wiley, J. D., and J. G. Webster (1982) “Analysis and Control of the Current Distribution under Circular Dispersive Electrodes,” IEEE Trans. Biomed. Eng. Volume 29, Pages 381–385. 

Cohen, D., I. Nemoto, L. Kaufman, and S. Arai (1984) “Ferrimagnetic Particles in the Lung Part II: The Relaxation Process,” IEEE Trans. Biomed. Eng. Volume 31, Pages 274–285.

Stark, L. W. (1984) “The Pupil as a Paradigm for Neurological Control Systems,” IEEE Trans. Biomed. Eng. Volume 31, Pages 919–924. 

Barach, J. P., B. J. Roth, and J. P. Wikswo (1985) “Magnetic Measurements of Action Currents in a Single Nerve Axon: A Core Conductor Model,” IEEE Trans. Biomed. Eng. Volume 32, Pages 136–140.

Geddes, L. A., and J. D. Bourland (1985) “The Strength-Duration Curve,” IEEE Trans. Biomed. Eng. Volume 32, Pages 458–459. 

Stanley, P. C., T. C. Pilkington, and M. N. Morrow (1986) “The Effects of Thoracic Inhomogeneities on the Relationship Between Epicardial and Torso Potentials,” IEEE Trans. Biomed. Eng. Volume 33, Pages 273–284. 

Gielen, F. L. H., B. J. Roth and J. P. Wikswo, Jr. (1986) “Capabilities of a Toroid-Amplifier System for Magnetic Measurements of Current in Biological Tissue,” IEEE Trans. Biomed. Eng. Volume 33, Pages 910–921. 

Pickard, W. F. (1988) “A Model for the Acute Electrosensitivity of Cartilaginous Fishes,” IEEE Trans. Biomed. Eng. Volume 35, Pages 243–249. 

Purcell, C. J., G. Stroink, and B. M. Horacek (1988) “Effect of Torso Boundaries on Electrical Potential and Magnetic Field of a Eipole,” IEEE Trans. Biomed. Eng. Volume 35, Pages 671–678.

Trayanova, N., C. S. Henriquez, and R. Plonsey (1990) “Limitations of Approximate Solutions for Computing Extracellular Potential of Single Fibers and Bundle Equivalents,” IEEE Trans. Biomed. Eng. Volume 37, Pages 22–35.

Voorhees, C. R., W. D. Voorhees III, L. A. Geddes, J. D. Bourland, and M. Hinds (1992) “The Chronaxie for Myocardium and Motor Nerve in the Dog with Surface Chest Electrodes,” IEEE Trans. Biomed. Eng. Volume 39, Pages 624–628.

Tan, G. A., F. Brauer, G. Stroink, and C. J. Purcell (1992) “The Effect of Measurement Conditions on MCG Inverse Solutions,” IEEE Trans. Biomed. Eng. Volume 39, Pages 921–927.

Roth, B. J. and J. P. Wikswo, Jr. (1994) “Electrical Stimulation of Cardiac Tissue: A Bidomain Model with Active Membrane Properties,” IEEE Trans. Biomed. Eng. Volume 41, Pages 232–240.

Tai, C., and D. Jiang (1994) “Selective Stimulation of Smaller Fibers in a Compound Nerve Trunk with Single Cathode by Rectangular Current Pulses,” IEEE Trans. Biomed. Eng. Volume 41, Pages 286–291.

Kane, B. J., C. W. Storment, S. W. Crowder, D. L. Tanelian, and G. T. A. Kovacs (1995) “Force-Sensing Microprobe for Precise Stimulation of Mechanoreceptive Tissues,” IEEE Trans. Biomed. Eng. Volume 42, Pages 745–750. 

Esselle, K. P., and M. A. Stuchly (1995) “Cylindrical Tissue Model for Magnetic Field Stimulation of Neurons: Effects of Coil Geometry,” IEEE Trans. Biomed. Eng. Volume 42, Pages 934–941. 

Roth, B. J. (1997) “Electrical Conductivity Values Used with the Bidomain Model of Cardiac Tissue,” IEEE Trans. Biomed. Eng. Volume 44, Pages 326–328.

Roth, B. J., and M. C. Woods (1999) “The Magnetic Field Associated with a Plane Wave Front Propagating through Cardiac Tissue,” IEEE Trans. Biomed. Eng. Volume 46, Pages 1288–1292.

One endearing feature of the IEEE TBME is that at the end of an article they publish a picture and short bio of each author. Over the years, my goal has been to publish my entire CV, piece by little piece, in these short bios. Below is the picture and bio from my very first published paper, which appeared in IEEE TBME [Barach, Roth, and Wikswo (1985), cited above].

Research and Education at the Crossroads of Biology and Physics

The May issue of the American Journal of Physics (my favorite journal) is a “theme issue” devoted to Research and Education at the Crossroads of Biology and Physics. In their introductory editorial, guest editors Mel Sabella and Matthew Lang outline their goals, which are similar to the objectives Russ Hobbie and I have for the 4th edition of Intermediate Physics for Medicine and Biology.

…there is often a disconnect between biology and physics. This disconnect often manifests itself in high school and college physics instruction as our students rarely come to understand how physics influences biology and how biology influences physics. In recent years, both biologists and physicists have begun to recognize the importance of cultivating stronger connections in these fields, leading to instructional innovations. One call to action comes from the National Research Council’s report, BIO2010, which stresses the importance of quantitative and computational training for future biologists and cites that sufficient expertise in physics is crucial to addressing complex issues in the life sciences. In addition, physicists who are now exploring biological contexts in instruction need the expertise of biologists. It is clear that biologists and physicists both have a great deal to offer each other and need to develop interdisciplinary workspaces…

This theme issue on the intersection of biology and physics includes papers on new advances in the fields of biological physics, new advances in the teaching of biological physics, and new advances in education research that inform and guide instruction. By presenting these strands in parallel, in a single issue, we hope to support the reader in making connections, not only at the intersection of biology and physics but also at the intersection of research, education, and education research. Understanding these connections puts us, as researchers and physics educators, in a better position to understand the central questions we face…

The infusion of Biology into Physics and Physics into Biology provides exciting new avenues of study that can inspire and motivate students, educators, and researchers at all levels. The papers in this issue are, in many ways, a call to biologists and physicists to explore this intersection, learn about the challenges and obstacles, and become excited about new areas of physics and physics education. We invite you to read through these articles, reflect, and discuss this complex intersection, and then continue the conversation at the June 2014 Gordon Research Conference titled, “Physics Research and Education: The Complex Intersection of Biology and Physics.”

And guess who has an article in this special issue? Yup, Russ and I have a paper titled “A Collection of Homework Problems About the Application of Electricity and Magnetism to Medicine and Biology.”

This article contains a collection of homework problems to help students learn how concepts from electricity and magnetism can be applied to topics in medicine and biology. The problems are at a level typical of an undergraduate electricity and magnetism class, covering topics such as nerve electrophysiology, transcranial magnetic stimulation, and magnetic resonance imaging. The goal of these problems is to train biology and medical students to use quantitative methods, and also to introduce physics and engineering students to biological phenomena.

Regular readers of this blog know that a “hobby” of mine (pun intended, Russ) is to write new homework problems to go along with our book. Some of the problems in our American Journal of Physics paper debuted in this blog. I believe that a well-crafted collection of homework problems is essential for learning biological and medical physics (remember, for them to be useful you have to do your homework). I hope you will find the problems we present in our paper to be “well-crafted”. We certainly had fun writing them. My biggest concern with our AJP paper is that the problems may be too difficult for an introductory class. The “I” in IPMB stands for “intermediate”, not “introductory”. However, most of the AJP theme issue is about the introductory physics class. Oh well; one needs to learn biological and medical physics at many levels, and the intermediate level is our specialty. If only our premed students would reach the intermediate level (sigh)….

Russ and I are hard at work on the 5th edition of our book, where many of the problems from our paper, along with additional new ones, will appear (as they say, You Ain’t Seen Nothing Yet!).

Anyone interested in teaching biological and medical physics should have a look at this AJP theme issue. And regarding that Gordon Research Conference that Sabella and Lang mention, I’m registered and have purchased my airline tickets! It should be fun. If you are interested in attending, the registration deadline is May 11 (register here). You better act fast.

Bernard Cohen and the Risk of Low Level Radiation

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the work of physicist Bernard Cohen (1924-2012). In Chapter 16 (Medical Use of X Rays), we describe the dangers of radon gas, and show a picture from a 1995 study by Cohen studying lung cancer mortality as a function of radon gas level (our Fig. 16.57). Interestingly, the mortality rate goes down as radon exposure increases: exactly the opposite of what you would expect if you believed radiation exposure from radon caused lung cancer. In this blog entry, I consider two questions: who is this Bernard Cohen, and how is his work perceived today?

Cohen was a professor of physics at the University of Pittsburgh. An obituary published by the university states

Bernard Cohen was born (1924) and raised in Pittsburgh, PA and did his undergraduate work at Case (now Case Western Reserve Univ.). After service as an engineering officer with the U.S. Navy in the Pacific and the China coast during World War II, he did graduate work in Physics at Carnegie Tech (now Carnegie-Mellon Univ.), receiving a Ph.D. in 1950 with a thesis on “Experimental Studies of High Energy Nuclear Reactions” – at that time “high energy” was up to 15 MeV. His next eight years were at Oak Ridge National Laboratory, and in 1958 he moved to the University of Pittsburgh where he spent the remainder of his career except for occasional leaves of absence. Until 1974, his research was on nuclear structure and nuclear reactions .… His nuclear physics research was recognized by receipt of the American Physical Society Tom Bonner Prize (1981), and his election as Chairman of the A.P.S. Division of Nuclear Physics (1974-75).

In the early 1970s, he began shifting his research away from basic physics into applied problems. Starting with trace element analysis utilizing nuclear scattering and proton and X-ray induced X-ray emission (PIXE and XRF) to solve various practical problems, and production of fluorine-18 for medical applications, he soon turned his principal attention to societal issues on energy and the environment. For this work he eventually received the Health Physics Society Distinguished Scientific Achievement Award, the American Nuclear Society Walter Zinn Award (contributions to nuclear power), Public Information Award, and Special Award (health impacts of low level radiation), and was elected to membership in National Academy of Engineering; he was also elected Chairman of the Am. Nuclear Society Division of Environmental Sciences (1980-81). His principal work was on hazards from plutonium toxicity, high level waste from nuclear power (his first papers on each of these drew over 1000 requests for reprints), low level radioactive waste, perspective on risks in our society, society’s willingness to spend money to avert various type risks, nuclear and non-nuclear targets for terrorism, health impacts of radon from uranium mining, radiation health impacts from coal burning, impacts of radioactivity dispersed in air (including protection from being indoors), in the ground, in rivers, and in oceans, cancer and genetic risks from low level radiation, discounting in assessment of future risks from buried radioactive waste, physics of the reactor meltdown accident, disposal of radioactivity in oceans, the iodine-129 problem, irradiation of foods, hazards from depleted uranium, assessment of Cold War radiation experiments on humans, etc.

In the mid-1980s, he became deeply involved in radon research, developing improved detection techniques and organizing surveys of radon levels in U.S. homes accompanied by questionnaires from which he determined correlation of radon levels with house characteristics, environmental factors, socioeconomic variables, geography, etc. These programs eventually included measurements in 350,000 U.S. homes. From these data and data collected by EPA and various State agencies, he compiled a data base of average radon levels in homes for 1600 U.S. counties and used it to test the linear-no threshold theory of radiation-induced cancer; he concluded that that theory fails badly, grossly over-estimating the risk from low level radiation. This finding was very controversial, and for 10 years after his retirement in 1994, he did research extending and refining his analysis and responding to criticisms.

Although he died two years ago, his University of Pittsburgh website is still maintained, and there you can find a list of many of his articles. The first in the list is the article from which our Fig. 16.57 comes from. I particularly like the 4th item in the list, his catalog of risks we face every day. You can find the key figure here. Anyone interested in risk assessment should have a look.

No doubt Cohen’s work is controversial. In IPMB, we cite one debate with Jay Lubin, including articles in the journal Health Physics with titles

Cohen, B. L. (1995) “Test of the Linear-No Threshold Theory of Radiation Carcinogenesis for Inhaled Radon Decay Products.”

Lubin, J. H. (1998) “On the Discrepancy Between Epidemiologic Studies in Individuals of Lung Cancer and Residential Radon and Cohen’s Ecologic Regression.”

Cohen, B. L. (1998) “Response to Lubin’s Proposed Explanations of the Discrepancy.”

Lubin, J. H. (1998) “Rejoinder: Cohen’s Response to ‘On the Discrepancy Between Epidemiologic Studies in Individuals of Lung Cancer and Residential Radon and Cohen’s Ecologic Regression.’”

Cohen, B. L. (1999) “Response to Lubin’s Rejoinder.”

Lubin, J. H. (1999) “Response to Cohen’s Comments on the Lubin Rejoinder.”

Who says science is boring!

What is the current opinion of Cohen’s work. As I see it, there are two issues to consider: 1) the validity of the specific radon study performed by Cohen, and 2) the general correctness of the linear-no threshold model for radiation risk. About Cohen’s study, here is what the World Health Organization had to say in a 2001 publication.

This disparity is striking, and it is not surprising that some researchers have accepted these data at face value, taking them either as evidence of a threshold dose for high-LET radiation, below which no effect is produced, or as evidence that exposure of the lung to relatively high levels of natural background radiation reduces the risk for lung cancer due to other causes. To those with experience in interpreting epidemiological observations, however, neither conclusion can be accepted (Doll, 1998). Cohen’s geographical correlation study has intrinsic methodological difficulties (Stidley and Samet, 1993, 1994) which hamper any interpretation as to causality or lack of causality (Cohen, 1998; Lubin, 1998a,b; Smith et al., 1998; BEIR VI). The probable explanation for the correlation is uncontrolled confounding by cigarette smoking and inadequate assessment of the exposure of a mobile population such as that of the USA.

Needless to say, Cohen did not accept these conclusions. Honestly, I have not looked closely enough into the details of this particular study to provide any of my own insights.

On the larger question of the validity of the linear no-threshold model, I am a bit of a skeptic, but I realize the jury is still out. I have discussed the linear no-threshold model before in this blog here, here, here, and here. The bottom line is shown in our Fig. 16.58, which plots relative risk versus radon concentration for low doses of radiation; the error bars are so large that the data could be said to be consistent with almost any model. It is devilishly hard to get data about very low dose radiation effects.

Right or wrong, you have to admire Bernard Cohen. He made many contributions throughout his long and successful career, and he defended his opinions about low-level radiation risk with courage and spunk. (And, as the 70th anniversary of D-Day approaches, we should all honor his service in World War II). If you want to learn more about Cohen, see his Health Physics Society obituary here, another obituary here, and an interview about nuclear energy here. For those of you who want to hear it straight from the horse’s mouth, you can watch and listen to Cohen's own words in these videos.

The Periodic Table in IPMB

The periodic table of the elements summarizes so much of science, and chemistry in particular. Of course, the periodic table is crucial in biology and medicine. How many of the over one hundred elements do Russ Hobbie and I mention in the 4th edition of Intermediate Physics for Medicine and Biology? Surveying all the elements is too big of a job for one blog entry, so let me consider just the first twenty elements: hydrogen through calcium. How many of these appear in IPMB?

1. Hydrogen. Hydrogen appears many places in IPMB, including Chapter 14 (Atoms and Light) that describes the hydrogen energy levels and emission spectrum.

2. Helium. Liquid helium is mentioned when describing SQUID magnetometers in Chapter 8 (Biomagnetism), and the alpha particle (a helium nucleus) plays a major role in Chapter 17 (Nuclear Physics and Nuclear Medicine).

3. Lithium. Chapter 7 (The Exterior Potential and the Electrocardiogram) mentions lithium-iodide battery that powers most pacemakers, and Chapter 16 (Medical Use of X-rays) mentions lithium-drifted germanium x-ray detectors.

4. Beryllium. I can’t find beryllium anywhere in IPMB.

5. Boron. Boron neutron capture therapy is reviewed in Chapter 16 (Medical Use of X Rays).

6. Carbon. A feedback loop relating the carbon dioxide concentration in the alveoli to the breathing rate is analyzed in Chapter 10 (Feedback and Control).

7. Nitrogen. When working problems about the atmosphere, readers are instructed to consider the atmosphere to be pure nitrogen (rather than only 80% nitrogen) in Chapter 3 (Systems of Many Particles).

8. Oxygen. Oxygen is often mentioned when discussing hemoglobin, such as in Chapter 18 (Magnetic Resonance Imaging) when describing functional MRI.

9. Fluorine. The isotope Fluorine-18, a positron emitter, is used in positron emission tomography (Chapter 17, Nuclear Physics and Nuclear Medicine).

10. Neon. Not present.

11. Sodium. Sodium and sodium channels are essential for firing action potentials in nerves (Chapter 6, Impulses in Nerve and Muscle Cells).

12. Magnesium. Russ and I don’t mention magnesium by name. However, Problem 16 in Chapter 9 (Electricity and Magnetism at the Cellular Level) provides a citation for the mechanism of anomalous rectification in a potassium channel. The mechanism is block by magnesium ions.

13. Aluminum. Chapter 16 (Medical Use of X Rays) tells how sheets of aluminum are used to filter x-ray beams; removing the low-energy photons while passing the high-energy ones.

14. Silicon. Silicon X ray detectors are considered in Chapter 16 (Medical Use of X Rays).

15. Phosphorus. The section on Distances and Sizes that starts Chapter 1 (Mechanics) considers the molecule adenosine triphosphate (ATP), which is crucial for metabolism.

16. Sulfur. The isotope technitium-99m is often combined with colloidal sulfur for use in nuclear medicine imaging (Chapter 17, Nuclear Physics and Nuclear Medicine).

17. Chlorine. Ion channels are described in Chapter 9 (Electricity and Magnetism at the Cellular Level), including chloride ion channels.

18. Argon. In Problem 32 of Chapter 16 (Medical Use of X rays), we ask the reader to calculate the stopping power of electrons in argon.

19. Potassium. The selectivity and voltage dependence of ion channels have been studied using the Shaker potassium ion channel (Chapter 9, Electricity and Magnetism at the Cellular Level).

20. Calcium. After discussing diffusion in Chapter 4 (Transport in an Infinite Medium), in Problem 23 we ask the reader to analyze calcium diffusion when a calcium buffer is present.

Bilinear Interpolation

If you know the value of a variable at a regular array of points (x_i,y_j), you can estimate its value at intermediate positions (x,y) using an interpolation function. For bilinear interpolation, the function f(x,y) is

f(x,y) = a + b x + c y + d x y

where a, b, c, and d are constants. You can determine these constants by requiring that f(x,y) is equal to the known data at points (x_i,y_j), (x_i+1,y_j), (x_i,y_j+1), and (x_i+1,y_j+1):

f(x_i,y_j) = a + b x_i + c y_j + d x_i y_j

f(x_i+1,y_j) = a + b x_i+1 + c y_j + d x_i+1 y_j

f(x_i,y_j+1) = a + b x_i + c y_j+1 + d x_i y_j+1

f(x_i+1,y_j+1) = a + b x_i+1 + c y_j+1 + d x_i+1 y_j+1 .

Solving these four equations for the four unknowns a, b, c, and d, plugging those values into the equation for f(x,y), and then doing a bit of algebra gives you

f(x,y) = [ f(x_i,y_j)  (x_i+1 – x) (y_j+1 – y) + f(x_i+1,y_j) (x – x_i) (y_j+1 – y) 

                            + f(x_i,y_j+1) (x_i+1 – x) (y – y_j) + f(x_i+1,y_j+1) (x – x_i) (y – y_j) ] /(ΔxΔy)

where x_i+1 = x_i + Δx and y_j+1 = y_j + Δy. To see why this makes sense, let x = x_i and y = y_j. In that case, the last three terms in this expression go to zero, and the first term reduces to f(x_i,y_j), just as you would want an interpolation function to behave. As you can check for yourself, this is true of all four data points. If you hold y fixed then the function is a linear function of x, and if you hold x fixed then the function is a linear function of y. If you assume y = e x, then the function is quadratic.

If you want to try it yourself, see http://www.ajdesigner.com/phpinterpolation/bilinear_interpolation_equation.php

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce bilinear interpolation in Problem 20 of Chapter 12, in the context of computed tomography. In CT, you obtain the Fourier transform of the image at points in a polar coordinate grid k_i, θ_j. In other words, the points lie on concentric circles in the spatial frequency plane, each of radius k_i. In order to compute a numerical two-dimensional Fourier reconstruction to recover the image, one needs the Fourier transform on a Cartesian grid k_x,n, k_y,m. Thus, one needs to interpolate from data at k_i, θ_j to k_x,n, k_y,m. In Problem 20, we suggest doing this using bilinear interpolation, and ask the reader to perform a numerical example.

I like bilinear interpolation, because it is simple, intuitive, and often “good enough.” But it is not necessarily the best way to proceed. Tomogrphic methods arise not only in CT but also in synthetic aperture radar (SAR) (see: Munson, D. C., J. D. O’Brien, and W. K. Jenkins (1983) “A Tomographic Formulation of Spotlight-Mode Synthetic Aperture Radar,” Proceedings of the IEEE, Volume 71, Pages 917–925). In their conference proceeding paper “A Comparison of Algorithms for Polar-to-Cartesian Interpolation in Spotlight Mode SAR” (IEEE International Conference on Acoustics, Speech and Signal Processing '85, Volume 10, Pages 1364–1367, 1985), Munson et al. write

Given the polar Fourier samples, one method of image reconstruction is to interpolate these samples to a cartesian grid, apply a 2-D inverse FFT, and to then display the magnitude of the result. The polar-to-cartesian interpolation operation must be of extremely high quality to prevent aliasing . . . In an actual system implementation the interpolation operation may be much more computationally expensive than the FFT. Thus, a problem of considerable importance is the design of algorithms for polar-to-cartesian interpolation that provide a desirable quality/computational complexity tradeoff.

Along the same lines, O’Sullivan (“A Fast Sinc Function Gridding Algorithm for Fourier Inversion in Computer Tomography,” IEEE Trans. Medical Imaging, Volume 4, Pages 200–207, 1985) writes

Application of Fourier transform reconstruction methods is limited by the perceived difficulty of interpolation from the measured polar or other grid to the Cartesian grid required for efficient computation of the Fourier transform. Various interpolation schemes have been considered, such as nearest-neighbor, bilinear interpolation, and truncated sinc function FIR interpolators [3]-[5]. In all cases there is a tradeoff between the computational effort required for the interpolation and the level of artifacts in the final image produced by faulty interpolation.

There has been considerable study of this problem. For instance, see

Stark et al. (1981) “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoustics, Speech, and Signal Processing, Volume 29, Pages 237–245.

Moraski, K. J. and D. C. Munson (1991) “Fast tomographic reconstruction using chirp-z interpolation,” 1991 Conference Record of the Twenty-Fifth Asilomar Conference on Signals, Systems and Computers, Volume 2, Pages 1052–1056.

Going into the details of this topic would take me into more deeply into signal processing than I am comfortable with. Hopefully, Problem 20 in IPMB will give you a flavor for what sort of interpolation needs to be done, and the references given in this blog entry can provide an entry to more detailed analysis.

17 Equations that Changed the World

In Pursuit of the Unknown:
17 Equations that Changed the World,
by Ian Stewart.

Ian Stewart’s book In Pursuit of the Unknown: 17 Equations that Changed the World “is the story of the ascent of humanity, told through 17 equations.” Of course, my first thought was “I wonder how many of those equations are in the 4th edition of Intermediate Physics for Medicine and Biology?” Let’s see.

1. Pythagorean theorem: a²+b²=c². In Appendix B of IPMB, Russ Hobbie and I discuss vectors, and quote Pythagoras’ theorem when relating a vector’s x and y components to its magnitude.

2. Logarithms: log(xy)=log(x)+log(y). In Appendix C, we present many of the properties of logarithms, including this sum/product rule as Eq. C6. Log-log plots are discussed extensively in Chapter 2 (Exponential Growth and Decay).

3. Definition of the derivative: df/dt = limit h → 0 (f(t+h)-f(t))/h. We assume the reader has taken introductory calculus (the preface states “Calculus is used without apology”), so we don’t define the derivative or consider what it means to take a limit. However, in Appendix D we present the Taylor series through its first two terms, which is essentially the same equation as the definition of the derivative, just rearranged.

4. Newton’s law of gravity: F = Gm₁m₂/d². Russ and I are ruthless about focusing exclusively on physics that has implications for biology and medicine. Almost all organisms live at the surface of the earth. Therefore, we discuss the acceleration of gravity, g, starting in Chapter 1 (Mechanics), but not Newton’s law of Gravity.

5. The square root of minus one: i² = -1. Russ and I generally avoid complex numbers, but they are mentioned in Chapter 11 (The Method of Least Squares and Signal Analysis) as an alternative way to formulate the Fourier series. We write the equation as i = √-1, which is the same thing as i² = -1.

6. Euler’s formula for polyhedra: F – E + V = 2. We never come close to mentioning it.

7. Normal distribution: P(x) = 1/√(2πσ) exp[-(x-μ)²/2σ²]. Appendix I is about the Gaussian (or normal) probability distribution, which is introduced in Eq. I.4.

8. Wave equation: ∂²u/∂t² = c² ∂²u/∂x². Russ and I introduce the wave equation (Eq. 13.5) in Chapter 13 (Sound and Ultrasound).

9. Fourier transform: f(k) = ∫ f(x) e^-2πixk dx. In Chapter 11 (The Method of Least Squares and Signal Analysis) we develop the Fourier transform in detail (Eq. 11.57), and then use it in Chapter 12 (Images) to do tomography.

10. Navier-Stokes equation: ρ (∂v/∂t + v ⋅∇ v) = -∇ p + ∇ ⋅ T + f. Russ and I analyze biological fluid mechanics in Chapter 1 (Mechanics), and write down a simplified version of the Navier-Stokes equation in Problem 28.

11. Maxwell’s equations: ∇ ⋅ E = 0, ∇ × E = -1/c ∂H/∂t, ∇ ⋅ H = 0, and ∇ × H = 1/c ∂E/∂t. Chapter 6 (Impulses in Nerve and Muscle Cells), Chapter 7 (The Exterior Potential and the Electrocardiogram), and Chapter 8 (Biomagnetism) discuss each of Maxwell’s equations. In Problem 22 of Chapter 8, Russ and I ask the reader to collect all these equations together. Yes, I own a tee shirt with Maxwell’s equations on it.

12. Second law of thermodynamics: dS ≥ 0. In Chapter 3 (Systems of Many Particles), Russ and I discuss the second law of thermodynamics. We derive entropy from statistical considerations (I would have chosen S = k_B lnΩ rather than dS ≥ 0 to sum up the second law). We state in words “the total entropy remains the same or increases,” although we don’t actually write dS ≥ 0.

13. Relativity: E = mc². We don’t discuss special relativity in much detail, but we do need E = mc² occasionally, most notably when discussing pair production in Chapter 15 (Interaction of Photons and Charged Particles with Matter).

14. Schrödinger’s equation: i ħ ∂Ψ/∂t = Ĥ Ψ. Russ and I don’t write down or analyze Schrödinger’s equation, but we do mention it by name, particularly at the start of Chapter 3 (Systems of Many Particles).

15. Information theory: H = - Σ p(x) log p(x). Not mentioned whatsoever.

16. Chaos theory: x_i+1 = k x_i (1-x_i). Russ and I analyze chaotic behavior in Chapter 10 (Feedback and Control), including the logistic map x_i+1=kx_i(1-x_i) (Eq. 10.36).

17. Black-Scholes equation: ½ σ²S² ∂²V/∂S² + rS ∂V/∂S + ∂V/∂t – rV = 0. Never heard of it. Something about economics and the 2008 financial crash. Nothing about it in IPMB.

Seventeen is a strange number of equations to select (a medium sized prime number). If I were to round it out to twenty, then I would have three to select on my own. My first thought is Newton’s second law, F=ma, but Stewart mentions that this relationship underlies both the Navier-Stokes equation and the wave equation, so I guess it is already present implicitly. Here are my three:

18. Exponential equation with constant input: dy/dt = a – by. Chapter 2 of IPMB (Exponential Growth and Decay) is dedicated to the exponential function. This equation appears over and over throughout the book. Stewart discusses the exponential function briefly in his chapter on logarithms, but I am inclined to add the differential equation leading to the exponential function to the list. Among its many uses, this function is crucial for understanding the decay of radioactive isotopes in Chapter 17 (Nuclear Physics and Nuclear Medicine).

19. Diffusion equation: ∂C/∂t = D ∂²C/∂x². To his credit, Stewart introduces the diffusion equation in his chapter on the Fourier transform, and indeed it was Fourier’s study of the heat equation (the same as the diffusion equation, with T for temperature replacing C for concentration) that motivated the development of the Fourier series. Nevertheless, the diffusion equation is so central to biology, and discussed in such detail in Chapter 4 (Transport in an Infinite Medium) of IPMB, that I had to include it. Some may argue that if we include both the wave equation and the diffusion equation, we also should add Laplace’s equation, but I consider that a special case of Maxwell’s equations, so it is already in the list.

20. Light quanta: E = hν: Although Stewart included Schrodinger’s equation of quantum mechanics, I would include this second equation containing Planck’s constant h. It summarizes the wave-particle duality of light, and is crucially important in Chapters 14 (Atoms and Light), 15 (Interaction of Photons and Charged Particles with Matter), and 16 (Medical Uses of X Rays).

Runners up include the Bloch equations since I need something from Chapter 18 (Magnetic Resonance Imaging), the Boltzmann factor (except that it is a factor, not an equation), Stokes law, the ideal gas law and its analog the van’t Hoff’s law from Chapter 5 (Transport through Neutral Membranes), the Hodgkin and Huxley equations, the Poisson-Boltzmann equation in Chapter 9 (Electricity and Magnetism at the Cellular Level), the Poisson probability distribution, and Planck’s blackbody radiation law (perhaps in place of E=hν).

Overall, I think studying the 4th edition of Intermediate Physics for Medicine and Biology introduces the reader to most of the critical equations that have indeed changed the world.

The Correspondence Between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs

I recently obtained through interlibrary loan a copy of The Correspondence Between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, edited by David Wilson. This wonderful collection of over 650 letters spans the years 1846 to 1901. Robert Purrington, in his book Physics in the Nineteenth Century, claims that “the Thomson-Stokes correspondence is one of the treasures of nineteenth-century scientific communication.” I thought I would share a few excerpts from these letters with the readers of this blog.

Stokes (1819-1903) was the older of the two men. For years he served as the Lucasian Professor of Mathematics at the University of Cambridge. I have discussed him in this blog before. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I mention Stokes in connection to Stokes’ law for the drag force of a sphere moving in a viscous fluid, and the Navier-Stokes equations of fluid dynamics.

Kelvin (1824-1907) was five years younger than Stokes. He was born with the name William Thomson, but was made a Lord in 1892 and was thereafter referred to as Lord Kelvin. Russ and I don’t mention Kelvin in IPMB, but we do mention the unit of absolute temperature named after him. He spent his career at the University of Glasgow in Scotland, and is remembered for many accomplishments, but primarily for his contributions to thermodynamics.

Stokes’ and Kelvin’s letters were full of mathematics (Stokes was primarily a mathematical physicist) and critiques of the many famous physicists of their era. They spent a lot of time trying to obtain copies of papers. In the days before the internet, or even the Xerox machine, making a copy of a scientific paper was not easy, and they were constantly lending out the few copies they possessed. In some years, their letters were primarily about reviewing manuscripts, as both men served as editors of journals at one time and as reviewers for journals at another. Most commonly Stokes, as editor, was trying to coax Kelvin to complete his reviews on time.

A letter of April 7, 1847—between two relatively young and up-and-coming physicists—highlights the different areas of interest of the two men.

My Dear Stokes,

Many thanks for your letters…..I have been for a long time thinking on subjects such as those you write about, and helping myself to understand them by illustrations from the theories of heat, electricity, magnetism, and especially galvanism; sometimes also water. I can strongly recommend heat for clearing the head on all such considerations, but I suppose you prefer cold water….

Yours very truly, William Thomson

In this letter from Oct 25, 1849, Kelvin congratulates Stokes on becoming the Lucasian Professor.

My Dear Professor

I have been daily expecting to hear of the election of a Lucasian Professor and whenever the Times has been in my hands I have looked for such a proceeding in the University Intelligence, and now I am glad to be able to congratulate you on the result…

Yours sincerely, William Thomson

Sometimes the two engaged in a bit of trash-talking about other scientists. In a Jan 6, 1851 letter, Stokes adds a pugnacious postscript.

My Dear Thomson,

…..

Yours very truly, G. G. Stokes.

P.S. Have you seen Prof Challis’s awful heterodoxy in the present no. of the Phil. Mag. I am half inclined to take up arms, but I fear the controversy would be endless.

Readers of IPMB will recall the cable equation in Chapter 6 that describes the electrical properties of a nerve axon. This equation was not originally derived to model an axon, but instead was proposed by Kelvin to describe a submarine telegraph cable. Kelvin was deeply involved with the trans-Atlantic cable, and in this Oct 30, 1854 letter he described this work to Stokes.

My Dear Stokes,

An application of the theory of the transmission of electricity along a submarine telegraph wire, which I omitted to mention in the haste of finishing my letter on Saturday, shows how the question raised by Faraday as to the practicability of sending distinct signals along such a length as the 2000 or 3000 miles of wire that would be required for America, may be answered. The general investigations will show exactly how much the sharpness of the signals will be worn down, and will show what the maximum strength of current through the apparatus in America, would be produced by a specified battery action on the end in England, with wire of given dimensions etc.

The following form of solution of the general equation

σ²kc dv/dt = d²v/dx² - hv

which is the first given by Fourier, enables us to compare the times until a given strength of current shall be obtained, with different dimensions etc of wire….

Yours always truly, William Thomson

Neither scientist could properly be called a biological physicist. Medicine and biology almost never appear in their letters (unless one of them is sick). Kelvin once brought up a biological topic in his Jan 28, 1856 letter.

My Dear Stokes,

…Have you seen Clerk Maxwell’s paper in the Trans R S E [Transactions of the Royal Society of Edinburgh] on colour as seen by the eye?...do you believe that the whites produced by various combinations, such as two homogenous colours, three homogeneous colours, etc. are absolutely indistinguishable from one another and from solar white by the best eye?...Are you at all satisfied with Young’s idea of triplicity in the perceptive organ?

Yours very truly, William Thomson

Stokes’ curt reply on Feb 4 indicated he could not have been less interested in the topic.

My Dear Thomson,

….I have not made any experiments on the mixture of colours, nor attended particularly to the subject….

Yours very truly, G. G. Stokes

Once Kelvin was late getting some page proofs sent to Stokes, and in a Jan 20, 1857 letter he received a stern tongue lashing (one suspects, tongue-in-cheek).

My Dear Thomson,

You are a terrible fellow and I must write you a scolding…Hoping you will be more punctual for the future I remain

Yours most sincerely, G. G. Stokes

This was not the last time Kelvin was slow in responding to Stokes, and he often apologized for being late. He was tardy once when reviewing one of Maxwell’s papers for a journal. To me, Maxwell is a giant of physics, to be spoken of in the same category as Newton and Einstein. But for Kelvin, reviewing one of Maxwell’s papers was just another chore he needed to find time to do.

Although most of the letters were about science, personal matters were sometimes mentioned. Kelvin wrote a Dec 27, 1863 letter of consolation after scarlet fever took the life of Stokes’ infant child. Stokes himself was also ill with the disease.

My Dear Stokes,

I am very sorry to hear of the loss you have had and I feel much concerned about the danger you have yourself suffered. I hope you are still improving steadily, and that you will soon be quite strong again….

Yours always truly, W. Thomson

I hope the others of your family have perfectly recovered, if not escaped the scarlet fever.

When Kelvin became a Lord, Stokes’ Jan 2, 1892 letter had a bit of fun with the event. But afterwards, his letters were always addressed to Kelvin rather than Thomson.

My Dear Lord (What?)

I write to congratulate you on the great honour Her Majesty has bestowed on you, and through you on science, by creating you a Peer. At the same time I may add my congratulations to those of my wife to The Lady Thomson, or whatever she is to be called. I was speculating whether you would be Lord Thomson, or Lord Netherhall, or Lord Largs, or what. Time will tell….

Yours sincerely, G. G. Stokes

They didn’t always agree on scientific issues. In an Oct 27, 1894 letter, the 75-year-old Stokes’ humerously addresseed a disagreement about the behavior of a fluid in some container.

My Dear Lord Kelvin,

…..perhaps you think to demolish me by saying, Let the vessel be rigid but massless. Well. There is life in the old dog yet….

Yours sincerely, G. G. Stokes

I found it fascinating to listen as they corresponded about the important physics of their era. For instance, they discussed Rontgen’s discovery of X-rays extensively in 1896, and considered writing a joint note about their electromagnetic nature. They debated Becquerel’s discovery of radioactivity from uranium in 1897. Their last letter, in 1901, analyzed a problem from fluid dynamics and contained mathematical equations.

I didn’t have time to read all the letters, but I did spend most of a Saturday sampling many of them. The letters provide a valuable glimpse into the relationship between two intelligent yet human scientists. Wilson lists a few quotes at the start of his book, with the last one by Arthur Schuster (1932) stating

I shall always remember Lord Kelvin, as he stood at the open grave, almost overcome by his emotion, saying in a low voice: “Stokes is gone and I shall never return to Cambridge again.”

A Dozen New Homework Problems

Russ Hobbie and I are hard at work on the 5th edition of Intermediate Physics for Medicine and Biology. Sometimes we consider adding new material, try things out, debate its merits, but in the end it doesn’t make the cut. For instance, we thought about adding a section on elasticity theory to Chapter 1, but that was going to be too long (we are constantly battling between adding important topics and keeping the book from getting too fat), so we tried writing some new homework problems to teach the material that way. But it was also too much, and eventually we gave up on the idea. For those wanting to learn more about the biological applications of elasticity theory, I recommend Y. C. Fung’s book Biomechanics: Mechanical Properties of Living Tissue (cited in IPMB), or his more general textbook First Course in Continuum Mechanics.

I don't want to see good homework problems go to waste, so I offer them here in this blog. Twelve new homework problems. Free. They offer a way to learn a bit of elasticity theory. Enjoy.

Problem 1 Consider the rod in Fig. 1.20; the x-axis is along the rod’s length and x=0 is where the rod meets the wall. Let the displacement u(x,y) be the change in position of each point in the material in response to the force F. Express the displacement as u_x=Ax, u_y=0, and u_z=0, where A is a constant.
(a) Calculate the normal strain ε_n using Eq. 1.24 and Fig. 1.20.
(b) Calculate ε_n using the definition ε_n=∂u_x/∂x . Is it the same as in (a)?

Problem 2 Consider the rod in Fig. 1.23; x is horizontal, y is vertical, and y=0 is where the rod meets the floor. Express the displacement as u_x=By, u_y=0, and u_z=0, where B is a constant.
(a) Calculate the shear strain ε_s using Eq. 1.27 and Fig. 1.23 (assume B « 1)
(b) Calculate ε_s using the definition ε_s=∂u_x/∂y+∂u_y/∂x. Is it the same as in (a)?

Problem 3 The normal and shear strains can be combined into a strain tensor, which a 3 x 3 matrix. Define the diagonal components of this tensor (ε_xx, ε_yy, ε_zz) as the normal strain in each direction, and the off-diagonal components (ε_xy, ε_yz, ε_zx) as one half of the shear strain in each direction. For example, ε_xy=(∂u_x/∂y+∂u_y/∂x)/2.
(a) Derive expressions relating each component of the strain tensor to the displacement.
(b) Show that the strain tensor is symmetric (e.g., ε_xy=ε_yx).
Note: This expression for the strain tensor is correct for small strains. For large strains it is a more complicated nonlinear function of the displacement (Fung, 1993).

Problem 4 The dilatation is defined as the change in volume over the original volume, ΔV/V. For small strains, the dilatation is ε_xx+ε_yy+ε_zz .
(a) Calculate the dilatation for the displacement in Problem 1. Does the volume change?
(b) Calculate the dilatation for the displacement in Problem 2. Does the volume change?
(c) Calculate the dilatation for the displacement u_x=Cx, u_y=Cy, u_z=Cz, where C is a constant. Does the volume change?
(d) Show that the dilatation is equal to the trace of the strain tensor (the trace of a matrix is the sum of the diagonal components) and also is equal to the divergence of the displacement (the divergence is defined in Chapter 4).

Problem 5 Consider the displacement u_x=Dx, u_y=-Dy, u_z=0, where D is a constant.
(a) Sketch a plot of the displacement distribution by drawing the displacement vectors over a 5 x 5 grid centered at the origin.
(b) Sketch how a small square in the x-y plane centered at the origin is deformed.
(c) Calculate the strain tensor (defined in Problem 3) for this displacement.
(d) Calculate the dilatation (defined in Problem 4) for this displacement.

Problem 6 Repeat the analysis of Problem 5 for the displacement u_x=Fy, u_y=Fx, u_z=0, where F is a constant.

Problem 7 Repeat the analysis of Problem 5 for the displacement u_x=Hy, u_y=-Hx, u_z=0, where H is a constant. This is a special case of rigid body motion. Interpret this displacement physically.

Problem 8 Like the strain, the stress can be written as a 3 x 3 symmetric tensor. For an isotropic material, the relationship between the components of the stress tensor, s_ij, and the strain tensor, ε_ij, is s_ij=λδ_ij(ε_xx+ε_yy+ε_zz)+2με_ij, where λ and μ are the Lame parameters, and δ_ij is the Kronecker delta (1 if i=j, and 0 otherwise).
(a) Show that for the case in Problem 1, this relationship reduces to Eq. 1.25 where s_n=s_xx and ε_n=ε_xx. Express the Young’s modulus E in terms of the Lame parameters.
(b) Show that for the case in Problem 2, this relationship reduces to Eq. 1.28 where s_s=s_xy and ε_s=2ε_xy. Express the shear modulus G in terms of the Lame parameters.
(c) Show that for the case in Problem 4c, this relationship reduces to Eq. 1.32 where the diagonal components of the stress tensor are given in terms of the pressure p as s_xx=s_yy=s_zz=-p. Express the compressibility κ in terms of the Lame parameters.

Problem 9 Figure 1.25 shows that pressure p will exert a net force on an element of fluid only if p is not uniform. Similarly, a stress will exert a net force on an element of tissue only if the stress is not uniform. The equations of mechanical equilibrium (zero net force) are ∂s_ix/∂x+∂s_iy/∂y+∂s_iz/∂z=0, where i is either x, y, or z.
(a) Substitute the relationship from Problem 8 into these equations, and derive three equations of mechanical equilibrium written in terms of the strain tensor.
(b) Substitute the relationships between the components of the strain tensor and the displacement found in Problem 3 and derive the equations of mechanical equilibrium in terms of the displacement.

Problem 10 Figure 1.20, showing a rod subject to a force along its length, is a simplification. Actually, the cross-sectional area of the rod shrinks as the rod lengthens. A better representation of the displacement than that given in Problem 1 would be u_x=Ax, u_y=-Aνy, and u_z=-Aνz, where A is a constant and ν is the Poisson’s ratio.
(a) Use the results of Problem 4 to calculate the dilatation.
(b) What value of Poisson’s ratio corresponds to an incompressible material (zero dilatation)?
(c) For an isotropic material, -1 « ν « 0.5. How would a material with negative ν behave?
Elliott et al. (2002) measured Poisson’s ratio for articular (joint) cartilage under tension and found 1 « ν « 2. This large value is possible because cartilage is anisotropic: its properties depend on direction.

Problem 11 Many biological tissues are composed mainly of water and are therefore nearly incompressible. To analyze such a tissue, start with the stress-strain relationship in Problem 8. (Assume u_z=0 and all derivatives in the z direction are zero; the case of “plane strain”.)
(a) For an incompressible tissue, ε_xx+ε_yy goes to zero and λ goes to infinity such that λ(ε_xx+ε_yy) is finite. Set it equal to –p, where p is the pressure.
(b) The displacement can be found from a stream function φ(x,y), where u_x=∂φ/∂y and u_y=-∂φ/∂x. Show that these definitions ensure that the dilatation is zero. Express the strain tensor in terms of φ.
(c) Write the components of the stress tensor (s_xx, s_yy, s_xy) in terms of p and φ.
(d) Use the analysis of Problem 9 to derive the equations of mechanical equilibrium in terms of p and φ.
(e) Manipulate these equations to find two new equations, one for p only and one for φ only. (Hint: try taking derivatives of the equations).

Problem 12 Start with the stress-strain relationship in Problem 8 and modify it to describe a two-dimensional sheet of cardiac muscle (Ohayon and Chadwick 1988). (Assume u_z=0 and all derivatives in the z direction are zero; the case of “plane strain”.)
(a) Cardiac tissue is nearly incompressible. For an incompressible tissue, ε_xx+ε_yy goes to zero and λ goes to infinity such that λ(ε_xx+ε_yy) is finite. Set it equal to –p, where p is the pressure.
(b) Cardiac muscle can develop an active tension T along the myocardial fibers caused by the interaction of actin and myosin molecules. Assume the fibers lie along the x direction, and add the term T to the expression for s_xx.
(c) The extracellular space consists of collagen fibers that can exert a shear force. Assume the collagen is isotropic, and interpret μ in Problem 8 as the collagen’s shear modulus.
(d) Derive expressions for s_xx, s_yy, and s_xy in terms of p, μ, T, and the strain tensor.
(e) Assume a solution u_x=-Ax, u_y=Ay, and p=P, where A and P are constants. If the tissue is free at its edges, then it must have zero stress throughout. Use this condition to derive expressions for A and P in terms of T and μ.
(f) Let T=3 x 10⁴ Pa and μ=10⁴ Pa (typical for cardiac tissue). Calculate values for A and P. Are the strains small? Sketch qualitatively the displacement distribution.

Ohayon, J. and R. S. Chadwick (1988) “Effects of Collagen Microstructure on the Mechanics of the Left Ventricle,” Biophys. J., Volume 54, Pages 1077–1088.

Light Scattering

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I often discuss the scattering of light. We mention four types of scattering, each differentiated by the name of the brilliant scientist who first studied it: Compton scattering, Thomson scattering, Rayleigh scattering, and Raman scattering. Let’s see if we can get these all straight.

Compton Scattering

In Chapter 15 (Interaction of Photons and Charged Particles with Matter) of IPMB, Russ and I analyze Compton scattering. This is a particularly simple case: a photon interacts with a free electron, resulting in a scattered photon of lower energy and a recoiling electron. This type of scattering is particularly important for x-rays. You might be wondering how often do we encounter a free electron? Aren’t most electrons bound to atoms? If the incident photon has an energy much greater than the binding energy, then the electron is to a first approximation free and Compton scattering occurs. In the interaction of x-rays with biological tissue, Compton scattering is the dominant mechanism contributing to the interaction cross-section at intermediate energies; say, one tenth to a few MeV. Since the electrons act almost as if they were free, the atomic number of the target atom is unimportant and scattering depends only on how many electrons are present (meaning the mass attenuation coefficient is nearly independent of atomic number). You don’t really want to do imaging of tissue when Compton scattering is the dominate interaction because you don’t get much discrimination between different tissues (the weak dependence on atomic number) and, well, you get a lot of scattering that blurs the image.

Compton scattering is named after Arthur Holly Compton (1892–1962), an American physicist who played a key role in the Manhattan Project. Compton scattering was important in the development of quantum mechanics. The light quanta hypothesis had been developed by Planck and Einstein, but was not widely embraced until 1923, when Compton analyzed his x-ray scattering data by treating the x-ray photon as a particle with energy and momentum, interacting with another particle, the electron. Compton won the 1927 Nobel Prize in Physics for his discovery.

Thomson Scattering

When Compton scattering occurs at such a low energy that we can ignore the difference in energy between the incident and scattered photons, the process is called Thomson scattering. We can analyze Thomson scattering by treating the incident light as an electromagnetic wave rather than a photon. The electric field accelerates the electron, causing it to radiate an electromagnetic wave at the same frequency. The direction of the electric field is important for determining the distribution of the outgoing dipole radiation, so Thomson scattering depends on the polarization of the incident light. This type of scattering is particularly important in plasma physics, where many free charged particles are present. It is not too important in biology and medicine, because usually either the photon energy is so high that Compton scattering occurs, or else the photon energy is so low that one cannot treat the electron as being free. Because the frequency of the light (and therefore the energy of the photons) does not change, Thomson scattering is a type of elastic scattering.

Thomson scattering was first analyzed by, and was named after, J. J. Thomson (1856–1940), the British physicist who discovered the electron, for which he received the Nobel Prize in Physics in 1906. I have my own connection to Thomson: academically speaking, he is my great-great-great-great-great-grandfather.

Rayleigh Scattering

Rather than scattering from a single electron, light can also scatter from an entire atom or molecule, and even larger particles. When the wavelength of the light is much larger than the size of the particle, we get Rayleigh scattering. Like for Thomson scattering, in Rayleigh scattering the light is treated as an electromagnetic wave. However, unlike Thomson scattering, in Rayleigh scattering the scatterer is not a single particle, but instead can be represented by a continuous, polarizable medium. The electric field of the light causes the induced charge distribution to oscillate at the same frequency as the incident light, resulting in the scattered light having the same frequency as the incident light. In IPMB, Russ and I refer to Rayleigh scattering as coherent scattering, because the atom responds coherently as a whole, rather than as individual charged particles. In tissue, coherent scattering dominates Compton scattering at low energies (say, below 1 keV), but such low energy photons also interact by the more important photoelectric effect, so Rayleigh scattering is often not very important. It is crucial for understanding how sunlight scatters off the molecules of the air, causing the blue color of the sky.

When I was an undergraduate at the University of Kansas, I had my first research experience in Professor Wes Unruh’s laboratory studying light scattering off of colloidal impurities in crystals. We were able to determine the size of the impurities by measuring the scattered light as a function of angle. However, these colloids tended to be large, so that you could not ignore interference between light scattered from different parts of the particle. In that case, you must use a more advanced theory, called Mie theory, to calculate the distribution of scattered light. I recall struggling to learn Mie theory from Milton Kerker’s book The Scattering of Light and Other Electromagnetic Radiation. I didn’t work much with Unruh himself, but rather was mentored by then-graduate student Robert Bunch. The first item in my CV is an abstract resulting from that research (Bunch, Roth, and Unruh, 1983, “Size Distributions of Ni and Co Colloids Within MgO,” March Meeting of the American Physical Society).

Rayleigh scattering is named after English physicist John William Strutt (1842-1919), also known as Lord Rayleigh. He was awarded the Nobel Prize for Physics in 1904 for the discovery of argon. Because one of Rayleigh’s students was J. J. Thomson, Rayleigh is my academic great-great-great-great-great-great-grandfather. Rayleigh was the second Cavendish Professor of Physics at the University of Cambridge, following Maxwell and succeeded by J. J. Thomson, Ernest Rutherford, and William Bragg; quite an impressive bunch.

Raman Scattering

In IPMB, Russ and I discuss Raman scattering in Chapter 14 (Atoms and Light). The mechanism of Raman scattering is similar to Rayleigh scattering, in that the scattering occurs off an entire molecule. However, it is unlike Rayleigh scattering in that the scattered light does not have the same frequency as the incident light (inelastic scattering). Instead, some of the energy induces transitions between different vibrational energy levels. These transitions result in the scattered light having a lower energy (Stokes) or a higher energy (Anti-Stokes). Also, because the vibrational energy levels are quantized, the spectrum of Raman scattered light consists of a series of discrete lines. This spectrum contains information about the vibrations within the molecule, and therefore about the chemical bonds.

The description of Raman scattering given above (and in IPMB) is a quantum view that depends on the presence of discrete energy levels. However, one can also develop a classical model of Raman scattering. For instance, treat a simple diatomic molecule as two atoms attached by a spring, so that the molecule has its own natural frequency of oscillation, f_o. If an electric field of frequency f is incident on the atom, it will respond by not only oscillating both at frequency f (Rayleigh scattering) but also at frequencies f+f_o and f-f_o (Raman scattering). The frequency difference between adjacent lines is f_o, which is the same frequency as one would expect in the infrared absorption spectrum. (For those who have read Appendix F of IPMB and are wondering why the the scattered light oscillates with a component at the natural frequency, realize that the charge induced by polarization depends on the electric field, so the force on the charge--charge times electric field--depends on the square of the electric field and the problem is nonlinear.)

Raman scattering was named after Indian physicist C. V. Raman (1888–1970), whose discovery led to the 1930 Nobel Prize for Physics.


Four types of scattering, named after four Nobel Prize winners. Here are some ways to keep them straight: Compton and Thomson scattering is off a single charged particle (usually an electron), whereas Rayleigh and Raman scattering is off an entire atom or molecule or particle. Thomson and Rayleigh scattering are elastic, whereas Compton and Raman scattering are inelastic. Thomson and Rayleigh scattering are most commonly described using the classical wave theory of light, whereas Compton and Raman scattering are typically analyzed using quantum mechanics (although Raman scattering is sometimes analyzed with classical theory).

I admire all four scientists: Compton, Thomson, Rayleigh, and Raman. Who is my favorite? I like Rayleigh best. Love those Victorians.