PHY 325 and PHY 326

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.  

Syllabus, Biological Physics

Fall 2015

Class: Physics 325, MWF, 8:00–9:07, 378 MSC

Instructor: Brad Roth, Dept. Physics, 166 Hannah Hall, 370-4871, roth@oakland.edu, fax: 370-3408, office hours MWF, 9:15–10:00, https://files.oakland.edu/users/roth/web

Text: Intermediate Physics for Medicine and Biology, 5th Edition, by Hobbie and Roth (An electronic version of this book is available for free through the OU library)
Book Website: https://files.oakland.edu/users/roth/web/hobbie.htm (get the errata!).
Book Blog: http://hobbieroth.blogspot.com

Goal: To understand how physics influences and constrains biology

Grades

Point/Counterpoint		5 %
Exam 1	Feb 5	20 %	Chapters 1–3
Exam 2	March 18	20 %	Chapters 4–6
Exam 3	April 20	20 %	Chapter 7, 8, 10
Final Exam	April 20	10 %	Comprehensive
Homework		25 %

Schedule

Sept 4		Introduction
Sept 9, 11	Chapter 1	Mechanics, Fluid Dynamics
Sept 14–18	Chapter 2	Exponential, Scaling
Sept 21–25	Chapter 3	Thermodynamics
Sept 28–Oct 2		Exam 1
Oct 5–9	Chapter 4	Diffusion
Oct 12–16	Chapter 5	Osmotic Pressure
Oct 19–23	Chapter 6	Electricity and Nerves
Oct 26–30		Exam 2
Nov 2–6	Chapter 7	Extracellular Potentials
Nov 9–13	Chapter 8	Biomagnetism
Nov 16–20	Chapter 10	Heart Arrhythmias, Chaos
Nov 23, 25	Chapter 10	Feedback
Nov 30–Dec 4	Chapter 10	Feedback
Dec 7		Review
Dec 9		Final Exam

Homework

Chapter 1:	6, 7, 8, 16, 17, 33, 40, 42	due Wed, Sept 16
Chapter 2:	3, 5, 10, 29, 42, 46, 47, 48	due Wed, Sept 23
Chapter 3:	29, 30, 32, 33, 34, 40, 47, 48	due Wed, Sept 30
Chapter 4:	7, 8, 12, 20, 22, 23, 24, 41	due Wed, Oct 14
Chapter 5:	1, 3, 5, 6, 7, 8, 10, 16	due Wed, Oct 21
Chapter 6:	1, 2, 22, 28, 37, 41, 43, 61	due Wed, Oct 28
Chapter 7:	1, 10, 15, 24, 25, 36, 42, 47	due Wed, Nov 11
Chapter 8:	3, 10, 24, 25, 27, 28, 29, 32	due Wed, Nov 18
Chapter 10:	12, 16, 17, 18, 40, 41, 42, 43	due Wed, Dec 2

Syllabus, Medical Physics

Winter 2016 

Class: Physics 326, MWF, 10:40–11:47, 204 DH

Instructor: Brad Roth, Department of Physics, 166 HHS, (248) 370-4871, roth@oakland.edu, fax: (248) 370-3408, office hours MWF 9:30–10:30, https://files.oakland.edu/users/roth/web.

Text: Intermediate Physics for Medicine and Biology, 5th Edition, by Hobbie and Roth. An electronic version of the textbook is available through the OU library.
Book Website: https://files.oakland.edu/users/roth/web/hobbie.htm (get the errata!).
Book Blog: http://hobbieroth.blogspot.com

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.

Direct Neural Current Imaging in an Intact Cerebellum with Magnetic Resonance Imaging

I amIn the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a paragraph to Chapter 18 (Magnetic Resonance Imaging) about using MRI to image neural activity.

Much recent research has focused on using MRI to image neural activity directly, rather than through changes in blood flow (Bandettini et al. 2005). Two methods have been proposed to do this. In one, the biomagnetic field produced by neural activity (Chap. 8) acts as the contrast agent, perturbing the magnetic resonance signal. Images with and without the biomagnetic field present provide information about the distribution of neural action currents. In an alternative method, the Lorentz force (Eq. 8.2) acting on the action currents in the presence of a magnetic field causes the nerve to move slightly. If a magnetic field gradient is also present, the nerve may move into a region having a different Larmor frequency. Again, images taken with and without the action currents present provide information about neural activity. Unfortunately, both the biomagnetic field and the displacement caused by the Lorentz force are tiny, and neither of these methods has yet proved useful for neural imaging. However, if these methods could be developed, they would provide information about brain activity similar to that from the magnetoencephalogram, but without requiring the solution of an ill-posed inverse problem that makes the MEG so difficult to interpret.

“Direct Neural Current Imaging in an
Intact Cerebellum with
Magnetic Resonance Imaging,”
by Sundaram et al.

I’m skeptical about most claims of measuring neural currents using MRI. However, a recent paper (Sundaram et al., NeuroImage, Volume 132, Pages 477–490, 2016) from the laboratory of Yoshio Okada has forced me to reconsider. Below I reproduce the introduction to this article (with references removed), which introduces the topic nicely.

Functional study of the human brain has become possible with advances in non-invasive neuroimaging methods. The most widely used technique is blood oxygenation level-dependent functional MRI (BOLD-fMRI). Although BOLD-fMRI is a powerful tool for human brain activity mapping, it does not measure neuronal signals directly. Rather, it images slow local hemodynamic changes correlated with neuronal activity through a complex neurovascular coupling. At present, only electroencephalography (EEG) and magnetoencephalography (MEG) detect signals directly related to neuronal currents with a millisecond resolution. However, they estimate neuronal current sources from electrical potentials on the scalp or from magnetic fields outside the head, respectively. Measurement of these signals outside the brain leads to relatively poor spatial resolution due to ambiguity in inverse source estimation.

Our understanding of human brain function would significantly accelerate if it were possible to noninvasively detect neuronal currents inside the brain with superior spatiotemporal resolution. This possibility has led researchers to look for a method to detect neuronal currents with MRI. Many MRI approaches have been explored in the literature. Of these, the mechanism most commonly used is based on local changes in MR phase caused by neuronal magnetic fields. Electrical currents in active neurons produce magnetic fields (ΔB) locally within the tissue. The component of this field (ΔB_z) along the main field (B_o) of the MR scanner alters the precession frequency of local water protons. This leads to a phase shift ΔΦ of the MR signal. For a gradient-echo (GE) sequence,

ΔΦ = γΔB_zTE

where γ is the gyromagnetic ratio for hydrogen (2π × 42.58 MHz/T for protons) and TE is the echo time. According to Biot-Savart's law, ΔB_z(t) is proportional to the current density J(t) produced by a population of neurons in the local region of the tissue. Thus, measurements of the phase shift ΔΦ can be used to directly estimate neuronal currents in the brain.

Many attempts have been made to detect neuronal currents in human subjects in vivo, but the results so far are inconclusive. The literature contains several reports of positive results which conflict with reports of negative results. This difficulty is presumably due to confounding factors such as blood flow, respiration and motion. Theoretical models, phantoms and cell culture studies indicate that it should be possible to detect neuronal currents with MRI in the absence of physiological noise sources.

Although these studies indicate that MRI technology should have enough sensitivity to detect neural currents, two types of key evidence are still lacking for demonstrating how MRI can be useful for neural current imaging: (1) there are no data showing that the phase shift is timelocked to some measure of population activity and that the phase shift time course matches that of a concurrently recorded local field potential (LFP), and (2) there is still no report showing how the phase shift data can be used to estimate the neuronal current distribution in the brain tissue, even though this should be the goal for neural current imaging.

Our work demonstrates that it is possible to measure an MR phase shift time course matching that of the simultaneously recorded evoked LFP in an isolated, intact whole cerebellum of turtle, free of physiological noise sources. We show how these MR phase maps can be used to estimate the neuronal current distribution in the active region in the tissue. We show that this estimated current distribution matches the distribution predicted based on spatial LFP maps. We discuss how these results can provide an empirical anchor for future development of techniques for in vivo neural current imaging.

After presenting their methods and results, Sundaram et al. write:

We demonstrated that the ΔΦ can be detected reliably in individual cerebelli and that this phase shift is time-locked to the concurrently recorded LFP. The temporal waveform of the ΔΦ matched that of the LFP. Both the MR signal and LFP were produced by neuronal currents mediated by mGluRs. The measured values of ΔΦ in the individual time traces corresponded to local magnetic fields of 0.67–0.93 nT for TE = 26 ms. According to our forward solutions, these values correspond to a current dipole moment density q of 1–2 nA m/mm²,which agrees with the reported current density of 1–2 nA m/mm² determined on the basis of MEG signals measured 2 cm above the cerebellum.

We also show that the MR phase data can be used to estimate the active neuronal tissue. This second step is important if MRI were to be used for imaging neuronal current distributions in the brain. We were able to use the minimum norm estimation technique developed in the field of MEG to estimate the current distribution in the cerebellum responsible for the measured phase shift. The peak values of ΔΦ in the phase map averaged across 7 animals were 0.15° and −0.10°, corresponding to peak ΔB values of +0.37 nT and −0.25 nT, respectively. The empirically obtained group-average ΔΦ of 0.12° and ΔB of 0.30 nT are close to the predicted values of 0.2° and 0.49 nT assuming q = 1 nA m/mm². The slightly smaller group-average ΔΦ and ΔB may be due to variability in the spatial phase map and responses across animals.

They conclude

Our results for metabotropic receptor mediated evoked neuronal activity in an isolated whole turtle cerebellum demonstrate that MRI can be used to detect neuronal currents with a time resolution of 100 ms, approximately ten times greater than for BOLD-fMRI, and with a sensitivity sufficiently high for near single-voxel detection. We have shown that it is possible to detect the MR phase shift with a time course matching that of the concurrently measured local field potential in the tissue. Furthermore, we showed how these MR phase data can be used to accurately estimate the spatial distribution of the current dipole moment density in the tissue.

I’ve been interested in this topic for a while, publishing on the subject with Ranjith Wijesinghe of Ball State University (2009, 2012) and Peter Basser of the National Institutes of Health (2009, 2014). My graduate student Dan Xu (2012) examined the use of MRI to measure electrical activity in the heart, where the biomagnetic fields are largest. I remain skeptical that magnetic resonance imaging can record neural activity of the human brain in a way as accurate as functional MRI using BOLD. Yet, this is the first claim to have measured the magnetic field of neurons using MRI that I believe. It’s a beautiful result and a landmark study. I hope that I’m wrong and the method does have the potential for clinical functional imaging.

An Analytical Example of Filtered Back Projection

One of my hobbies is to find tomography problems that can be solved analytically. I know this is artificial—all tomography for medical imaging uses numerical computation—but as a learning tool analytical analysis helps build insight. I have some nice analytical examples using the Fourier method to solve the tomography problem (see homework problems 26 and 27 in chapter 12 of Intermediate Physics for Medicine and Biology), but I don't have a complete analytical example to illustrate the filtered back projection method (see a previous post for a partial example). Russ Hobbie and I do include a numerical example in section 12.6 of IPMB. I have always wondered if I can do that example analytically. Guess what. I can! Well, almost.

Start with a top-hat function for your object

If we set x = 0, we can plot it as function of y.

The projection of this function is given in IPMB; Homework Problem 36 asks the reader to derive it.

Because the object looks the same from all directions, the projection is independent of the angle. Below is a plot of the projection as a function of x'. It is identical to the top panel of IPMB's Figure 12.22.

The next step is to filter the projection, which means we have to take its Fourier transform, multiply the transform by a high-pass filter, and then do the inverse Fourier transform. The Fourier transform of the projection is

This integral is not trivial, but Abramowitz and Stegun’s Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables contains (Page 360, Equation 9.1.20)

where J₁ is a first-order Bessel function (see Homework Problem 10). Because the projection is an even function, the sine part of the Fourier transform vanishes.

Filtering is easy; multiply by |k|/2π. The result is

To find the inverse Fourier transform, we need

This integral appears in Abramowitz and Stegun (Page 487, Equation 11.4.37)

After some simplification (which I leave to you), the filtered projection becomes

Below is a plot of the filtered projection, which you should compare with the middle panel of Fig. 12.22. It looks the same as the plot in IPMB, except in the numerical calculation there is some ringing near the discontinuity that is not present in the analytical solution

The final step is back projection. Because the projection is independent of the angle, we can calculate the back projection along any radial line, such as along the y axis

If |y| is less than a, the back projection is easy: you just get 1. Thus, the filtered back projection is the same as the object, as it should be. If |y| is greater than a, the result should be zero. This is where I get stuck; I cannot do the integral. If any reader can solve this integral (and presumably show that it gives zero), I would greatly appreciate hearing about it. Below is a plot of the result; the part in red is what I have not proven yet. Compare this plot with the bottom panel of Fig. 12.22.

What happens if you do the back projection without filtering? You end up with a blurry image of the object. I can solve this case analytically too. For |y| less than a, the back projection without filtering is

which is 4a times the complete elliptic integral of the second kind

For |y| greater than a, you get the more complicated expression

which is the incomplete elliptic integral of the second kind

The trickiest part of the calculation is determining the upper limit on the integral, which arises because for some angles the projection is zero (you run into the same situation in homework problem 35, which I highly recommend). Readers who are on the ball may worry that the elliptic integral is tabulated only for kappa less than one, but there are ways around this (see Abramowitz and Stegun, Page 593, Equation 17.4.16). When I plot the result, I get

which looks like Fig. 12.23 in IPMB.

So, now you have an analytical example that illustrates the entire process of filtered back projection. It even shows what happens if you forget to filter before back projecting. For people like me, the Bessel functions and elliptic integrals in this calculation are a source of joy and beauty. I know that for others they may be less appealing. To each his own.

I’ll rely on you readers to fill in the one missing step: show that the filtered back projection is zero outside the top hat.

Five Generations

A five generation picture.

When my first daughter Stephanie was born, we included her in this photo of five generations. From left to right are my maternal grandmother, my great-grandmother (born 1889), my daughter Stephanie (born 1988), me, and my mom. My great grandmother lived to be over 100 years old. I remember playing poker with her when I was young; she generally won and kept the money!

All five editions of
Intermediate Physics for Medicine and Biology.

Recently I took another five-generation photo. There now exist five generations (editions) of Intermediate Physics for Medicine and Biology. My office is one of the few places you can find all five on one bookshelf. I was coauthor on the fourth and fifth editions; the first three editions were authored by Russ Hobbie alone.

Suki with all five editions of
Intermediate Physics for Medicine and Biology.

The yellow book is the first edition of IPMB, published by John Wiley and Sons in 1978. The blue version with the yellow sine wave on the cover is the second edition, again published by Wiley in 1988. The green cover is the third edition, published by Springer with AIP Press in 1997. The blue fourth edition was published by Springer alone in 2007. Finally, the blue/purple fifth edition, again published by Springer, appeared in 2015. My dog Suki seems to like them all.

All five editions of
Intermediate Physics for Medicine and Biology.

I have a special fondness for the first edition, which I bought for a class taught by my PhD advisor John Wikswo at Vanderbilt University in the early 1980s (price: $31.95). That is where I learned much of my biological and medical physics. When Russ was preparing the second edition, he asked John and I to create some three-dimensional figures of the electrical potential and magnetic field of a nerve axon. There figures have appeared in each subsequent edition of IPMB, and are Figs. 7.13 and 8.14 in the fifth. My third edition is pretty beat up. It is the textbook I taught out of for several years after I arrived at Oakland University. The fourth and fifth editions I know best, as I helped write them (although Russ remains the primary force behind every edition).

All five editions of
Intermediate Physics for Medicine and Biology.

IPMB has changed over the years. The first seven chapters are the same in all versions, but Russ added chapters on charged membranes and biomagnetism in the second edition. The first edition’s chapter on signal analysis split into two in the second: one on one-dimensional signal analysis and another on two-dimensional images. The 4th edition picked up a chapter on ultrasound. The first edition’s chapter on x-rays fissioned into a chapter on how x-rays interact with tissue and a chapter on the medical uses of x-rays. Finally, the second edition introduced a chapter on magnetic resonance imaging. Early editions featured a figure on the cover. I particularly like the first edition’s electrocardiogram picture (Fig. 7.16 in the 5th edition). Russ and I planned on using a computed tomography illustration, Fig. 12.12, on the 4th edition cover, but Springer opted to use a generic cover with no figure.

Me holding all five editions of
Intermediate Physics for Medicine and Biology.

Working on revisions of IPMB has been a pleasure and an honor. But really, the five generations of IPMB is a tribute to Russ Hobbie and his vision of advancing the teaching of physics in medicine and biology, which he has pursued over nearly four decades. I hope you find the book as useful as I have.

Trivial Pursuit IPMB

Trivial Pursuit.

Trivial Pursuit is a popular and fun board game invented in the 1980s. While playing it, you learn many obscure facts (trivial, really).

When my daughter Kathy was in high school, she would sometimes test out of a subject by studying over the summer and then taking an exam. Occasionally I would help her study by skimming through her textbook and creating Trivial Pursuit-like questions. We would then play Trivial Pursuit using my questions instead of those from the game. I don’t know if it helped her learn, but she always passed those exams.

Readers of Intermediate Physics for Medicine and Biology may want a similar study aid to help them learn about biological and medical physics. Now they have it! At the book website you can download 100 game cards for Trivial Pursuit: IPMB. To play, you will need the game board, game pieces, and instructions of the original Trivial Pursuit, but you replace the game cards by the ones I wrote.

The game pieces for Trivial Pursuit.

In case you have never played, here are the rules in a nutshell. The board has a circle with spots of six colors. You roll a die and move your game piece around the circle, landing on the spots. Your opponent asks you a question about a topic determined by the color. If you answer correctly you roll again; if you are wrong your opponent rolls. There are special larger spots where a correct answer gets you get a little colored wedge. The first person to get all six colored wedges wins.

The original version of Trivial Pursuit had topics such as sports or literature. The Trivial Pursuit: IPMB topics are

• Numbers and Units (blue)
• People (pink)
• Anatomy and Physiology (yellow)
• Biological Physics (brown)
• Medical Physics (green)
• Mathematics (orange).

One challenge of an interdisciplinary subject like medical and biological physics is that you need a broad range of knowledge. I suspect mathematicians will find the math questions to be simple, but the biologists may find them difficult. Physicists may be unfamiliar with anatomy and physiology, and chemists may find all the topics hard. The beauty of the game is that it rewards a broad knowledge across disciplines.

A game card for Trivial Pursuit.

Many may find the People section most challenging. I suggest you only require the player to know the person’s last name, although the first name is also given on my game card. In Units and Numbers I generally only require numbers to be known approximately. The goal is to have an order-of-magnitude knowledge of biological parameters and physical constants. Many questions ask you to estimate the size of an object, like in Section 1.1 of IPMB. For the math and physics questions you may need a pencil and paper handy, because some of the questions contain equations. You can’t simply show your opponent the equation on the game card, because both the questions and answers are together. This is unlike the real Trivial Pursuit game cards, which had the answers on the back. Unfortunately, such two-sided cards are difficult to make.

I know the game is not perfect. Some questions are truly trivial and others ask for some esoteric fact that no one would be expected to remember. Some questions may have multiple answers of which only one is on the card. You can either print out the game cards (requiring 100 pieces of paper) or use a laptop or mobile device to view the pdf. I try to avoid repetitions, but with 100 game cards some may have slipped in inadvertently.

Trivial Pursuit.

I may try using Trivial Pursuit: IPMB next time I teach Biological Physics (PHY 325) or Medical Physics (PHY 326) here at Oakland University. It would be excellent for, say, the last day of class, or perhaps a day when I know many students will be absent (such as the Wednesday before Thanksgiving). It doesn’t teach important high-level skills, such as learning to use mathematical models to describe biology, or understanding how physics constrains the way organisms evolve. You can’t teach a complex and beautiful subject like tomography using Trivial Pursuit. But for learning a bunch of facts, the game is useful.

Enjoy!

Science Blogging

Science Blogging:
The Essential Guide,
by Wilcox, Brookshire, and Goldman.

After writing this blog for nine years, I decided it is time to figure out what I’m doing. So I read the book Science Blogging: The Essential Guide (Yale University Press, 2016), edited by Christie Wilcox, Bethany Brookshire, and Jason Goldman. Their preface concludes

By bringing together some of the most experienced voices from around the science blogosphere, we hope this book will have something to teach everyone. Whether you’re just getting started, have some blog posts under your belt, or are looking for fresh inspiration, you are not alone. The science communication community may seem overwhelming, but it’s friendly. Dive in and show us what you can do. Seriously. Tweet us and show us your stuff. And use our hashtag, #SciBlogGuide, and find us online at http://www.theopennotebook.com/science-blogging-essential-guide.

I enjoyed Science Blogging, but oddly I didn’t feel connected to what many of the authors discussed. What you are reading now is less a science blog and more an auxiliary resource for the textbook that Russ Hobbie and I wrote: Intermediate Physics for Medicine and Biology. My goal is to provide materials that help instructors use the book in their classes, and extend and update topics that readers of the book are interested in. I view this blog as being similar to the solutions manual and the errata: it augments the book. The closest Science Blogging came to my blog is in the last chapter, “From Science Blog to Book,” by Brian Switek. But his chapter was primarily about using a blog as a springboard to writing a book, and only at the end of his chapter did he add that “there’s no reason to stop blogging when your book comes out.” I did the opposite. My blog began after Russ and I published the 4th edition of IPMB, and my goal was to improve sales. Has it worked? It’s hard to say, because our sales have never been spectacular. I hope it has had some impact.

Matt Shipman’s chapter on “Metrics” inspired me to look into the statistics for my blog. The post with by far the most page views is Frank Netter, Medical Illustrator. While I liked that post, I don’t know why it has more than three times as many page views as the next most viewed entry. In fact, I see no correlation between the number of page views and what I consider quality or relevance.

Bethany Brookshire wrote a chapter on “Science Blogging and Money.” I like money as much as the next guy, but I don’t subject my dear readers to ads. Hobbieroth.blogspot.com is add-free. There is one exception: each blog post contains a reference to IPMB. I guess that is a sort of advertisement.

Several authors talked about building a following using Twitter. I don’t tweet, but should I? Do you want to hear about IPMB several times a day? I don’t think so. I’ll continue posting once a week; every Friday morning, like clockwork. By the way, what’s a hashtag? I always thought I was a hep cat, but I guess not.

My favorite chapter was Ed Yong’s essay about “Building an Audience for Your Blog.” Accumulating a large following is not my goal; I am more a citation man than a page view man. Yong’s advice is that “you have to have something worth writing about, and you have to write it well.” That sums it up nicely. I think that physics applied to medicine and biology is something worth writing about; I hope I write it well. Yong also writes “picture your ideal readers in your head: who are they?” While I hope anyone interested in biological physics or medical physics will find my blog useful, I don’t write it for such a broad audience. I write it for the students and teachers using Intermediate Physics for Medicine and Biology. And, I write it for myself. I hope you enjoy it. I do.

The Four Equations of Old Quantum Theory

Subtle is the Lord,
by Abraham Pais.

In ‘Subtle is the Lord…”: The Science and the Life of Albert Einstein, Abraham Pais illustrates the old quantum theory using four equations:

• Planck’s law for blackbody radiation, the oldest equation in quantum theory,
• Einstein’s 1905 equation for the energy of photoelectrons ejected from a metallic surface irradiated by light, the oldest equation in the quantum theory describing how radiation and matter interact,
• The equation for the Rydberg constant derived by Niels Bohr in 1913, the oldest equation in the quantum theory of atomic structure, and
• Einstein’s 1906 equation for the specific heat of a solid, the oldest equation in the quantum theory of solid state physics.

Does Intermediate Physics for Medicine and Biology introduce students to these four landmark equations? Let us look one by one.

Planck’s law

Planck’s law for blackbody radiation is presented in Sec. 14.8 of IPMB as our Eq. 14.38 (see last week’s post in this blog). Although we don’t delve into the history of this equation, we do analyze it in detail, deriving the Stefan-Boltzmann law and the Wien displacement law (the peak frequency of radiation increases with temperature). Pais writes “It is remarkable that the old quantum theory would originate from the analysis of a problem as complex as blackbody radiation. From 1859 to 1926, this problem remained at the frontier of theoretical physics, first in thermodynamics, then in electromagnetism, then in the old quantum theory, and finally in quantum statistics.”

The Photoelectric Effect

IPMB presents the photoelectric effect equation as Eq. 15.3 in the chapter about the Interaction of Photons and Charged Particles with Matter. However, it is not discussed in the context of light shining on a metal surface. Rather, it describes a photon interacting with tissue. “In the photoelectric effect…the photon is absorbed by the atom and a single electron, called a photoelectron, is ejected. The initial photon energy is equal to…the kinetic energy of the electron…plus the excitation energy of the atom.” The photoelectric effect is the primary mechanism by which low energy photons (soft x-rays, up to photon energies of roughly 100 keV) interact with tissue. It is the main contributor to the tissue cross section at low energies.

The Rydberg Constant

The atomic energy levels of hydrogen, as derived by Niels Bohr, are presented in Eq. 14.8 of IPMB. However, the Rydberg constant is not mentioned in our book except in homework problem 14.4, where the student is asked to “Find an expression for [the Rydberg constant] in terms of fundamental constants.”

The Specific Heat of a Solid

Sorry, but you won’t find Einstein’s equation for the specific heat of a solid in IPMB. In Section 3.1 we do discuss heat capacity. But biology occurs at fairly high temperatures, and human biology is essentially isothermal. The power of Einstein’s equation becomes evident when you examine how the specific heat decreases as the temperature approaches absolute zero. This behavior is critical for understanding low temperature physics, but is irrelevant for physics applied to medicine and biology.

Chernobyl

The Chernobyl nuclear reactor.

The worst nuclear accident ever happened thirty years ago this week: Chernobyl. Below are excerpts from a UNSCEAR (United Nations Scientific Committee on the Effects of Atomic Radiation) website about the disaster.

Summary

The accident at the Chernobyl nuclear reactor that occurred on 26 April 1986 was the most serious accident ever to occur in the nuclear power industry. The reactor was destroyed in the accident and considerable amounts of radioactive material were released to the environment. The accident caused the deaths, within a few weeks, of 30 workers and radiation injuries to over a hundred others. In response, the authorities evacuated, in 1986, about 115,000 people from areas surrounding the reactor and subsequently relocated, after 1986, about 220,000 people from Belarus, the Russian Federation and Ukraine .…

Among the residents of Belarus, the Russian Federation and Ukraine, there had been up to the year 2005 more than 6,000 cases of thyroid cancer reported in children and adolescents who were exposed at the time of the accident, and more cases can be expected during the next decades. Notwithstanding the influence of enhanced screening regimes, many of those cancers were most likely caused by radiation exposures shortly after the accident. Apart from this increase, there is no evidence of a major public health impact attributable to radiation exposure two decades after the accident. There is no scientific evidence of increases in overall cancer incidence or mortality rates or in rates of non-malignant disorders that could be related to radiation exposure. The incidence of leukaemia in the general population, one of the main concerns owing to the shorter time expected between exposure and its occurrence compared with solid cancers, does not appear to be elevated. Although those most highly exposed individuals are at an increased risk of radiation-associated effects, the great majority of the population is not likely to experience serious health consequences as a result of radiation from the Chernobyl accident. Many other health problems have been noted in the populations that are not related to radiation exposure.

Release of Radionuclides

The accident at the Chernobyl reactor happened during an experimental test of the electrical control system as the reactor was being shut down for routine maintenance. The operators, in violation of safety regulations, had switched off important control systems and allowed the reactor, which had design flaws, to reach unstable, low-power conditions. A sudden power surge caused a steam explosion that ruptured the reactor vessel, allowing further violent fuel-steam interactions that destroyed the reactor core and severely damaged the reactor building. Subsequently, an intense graphite fire burned for 10 days. Under those conditions, large releases of radioactive materials took place.

The radioactive gases and particles released in the accident were initially carried by the wind in westerly and northerly directions. On subsequent days, the winds came from all directions. The deposition of radionuclides was governed primarily by precipitation occurring during the passage of the radioactive cloud, leading to a complex and variable exposure pattern throughout the affected region, and to a lesser extent, the rest of Europe.

Exposure of Individuals

The radionuclides released from the reactor that caused exposure of individuals were mainly iodine-131, caesium-134 and caesium-137. Iodine-131 has a short radioactive half-life (eight days), but it can be transferred to humans relatively rapidly from the air and through consumption of contaminated milk and leafy vegetables. Iodine becomes localized in the thyroid gland.….

The isotopes of caesium have relatively longer half-lives (caesium-134 has a half-life of 2 years while that of caesium-137 is 30 years). These radionuclides cause longer-term exposures through the ingestion pathway and through external exposure from their deposition on the ground. Many other radionuclides were associated with the accident, which were also considered in the exposure assessments.

Average effective doses to those persons most affected by the accident were assessed to be about 120 mSv for 530,000 recovery operation workers, 30 mSv for 115,000 evacuated persons and 9 mSv during the first two decades after the accident to those who continued to reside in contaminated areas.… Maximum individual values of the dose may be an order of magnitude and even more …. [As discussed in Chapter 16 of Intermediate Physics for Medicine and Biology, the average annual background dose is about 3 mSv.]

Conclusions

The accident at the Chernobyl nuclear power plant in 1986 was a tragic event for its victims, and those most affected suffered major hardship. Some of the people who dealt with the emergency lost their lives. Although those exposed as children and the emergency and recovery workers are at increased risk of radiation-induced effects, the vast majority of the population need not live in fear of serious health consequences due to the radiation from the Chernobyl accident. For the most part, they were exposed to radiation levels comparable to or a few times higher than annual levels of natural background, and future exposures continue to slowly diminish as the radionuclides decay. Lives have been seriously disrupted by the Chernobyl accident, but from the radiological point of view, generally positive prospects for the future health of most individuals should prevail.

More about the physics of the disaster can be found at this hyperphysics website.

Today the remains of the reactor lie entombed in a concrete sarcophagus, a silent reminder of the Chernobyl nuclear accident.

The Eigenvalue Problem

An image of fiber tracts in the brain
using Diffusion Tensor Imaging.
From: Wikipedia.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I consider many mathematical topics. We analyze partial differential equations, Fourier transforms, vector calculus, probability, and special functions such as Bessel functions and the error function. One mathematical technique we never analyze is the central problem of linear algebra: the eigenvalue problem.

Calculating the eigenvalues and eigenvectors of a matrix has medical and biological applications. For example, in Chapter 18 of IPMB, Russ and I discuss diffusion tensor imaging. In this technique, magnetic resonance imaging is used to measure, in each voxel, the diffusion tensor, or matrix.

This matrix is symmetric, so D_xy = D_yx, etc. It contains information about how easily spins (primarily protons in water) diffuse throughout the tissue, and about the anisotropy of the diffusion: how the rate of diffusion changes with direction. White matter in the brain is made up of bundles of nerve axons, and spins can diffuse down the long axis of an axon much easier than in the direction perpendicular to it.

Suppose you measure the diffusion matrix to be

How do you get the fiber direction from this matrix? That is the eigenvalue and eigenvector problem. Stated mathematically, the fibers are in the direction of the eigenvector corresponding to the largest eigenvalue. In other words, you can determine a coordinate system in which the diffusion matrix becomes diagonal, and the direction corresponding to the largest of the diagonal elements of the matrix is the fiber direction.

The eigenvalue problem starts with the assumption that there are some vectors r = (x, y, z) that obey the equation Dr = Dr, where D in bold is the matrix (a tensor) and D in italics is one of the eigenvalues (a scalar). We can multiply the right side by the identity matrix (1’s along the diagonal, 0’s off the diagonal) and then move this term to the left side, and get the system of equations

One obvious solution is (x, y, z) = (0, 0, 0), the trivial solution. There is a beautiful theorem from linear algebra, which I will not prove, stating that there is a nontrivial solution for (x, y, z) if and only if the determinant of the matrix is zero

I am going to assume you know how to evaluate a determinant. From this determinant, you can obtain the equation

This is a cubic equation for D, which is in general difficult to solve. However, you can show that this equation is equivalent to

Therefore, the eigenvalues of this diffusion matrix are 4, 1, and 1 (1 is a repeated eigenvalue). The largest eigenvalue is D = 4.

To find the eigenvector associated with the eigenvalue D = 4, we solve

The solution is (1, 1, 1), which points in the direction of the fibers. If you do this calculation at every voxel, you generate a fiber map of the brain, leading to beautiful pictures such as you can see at the top of this post, and here or here.

Sometimes anisotropy can be a nuisance. Suppose you just want to determine the amount of diffusion in a tissue independent of direction. You can show (see Problem 49 of Chapter 18 in IPMB) that the trace of the diffusion matrix is independent of the coordinate system. The trace is the sum of the diagonal elements of the matrix. In our example, it is 2+2+2 = 6. In the coordinate system aligned with the fiber axis, the trace is just the sum of the eigenvalues, 4+1+1 = 6 (you have to count the repeated eigenvalue twice). The trace is the same.

Now you try. Here is a new homework problem for Section 13 in Chapter 18 of IPMB.

Problem 49 1/2. Suppose the diffusion tensor in one voxel is

a) Determine the fiber direction.

b) Show explicitly in this case that the trace is the same in the original matrix as in the matrix rotated so it is diagonal.

One word of warning. The examples in this blog post all happen to have simple integer eigenvalues. In general, that is not true and you need to use numerical methods to solve for the eigenvalues.

Have fun!

Darcy’s Law

Table 4.3 of Intermediate Physics for Medicine and Biology contains five transport equations. Each has the form “flux density equals a coefficient times the negative of a gradient of some quantity.” The table includes the flux of particles with the coefficient being the diffusion constant, the flux of heat with the coefficient being the thermal conductivity, the flux of momentum with the coefficient being the viscosity, and the flux of charge with the coefficient being the electrical conductivity. Are there other examples of transport equations important in biology and medicine? Yes. For instance, consider Darcy’s law.

Darcy’s law governs the flow of fluid through a porous medium. It is used to model the movement of groundwater through sedimentary rock, but it also describes the flow of water in tissue's extracellular space. Using a notation consistent with Table 4.3, we can write Darcy’s law as

j_v = - K dp/dx

where j_v is the flux density of fluid volume, p is the pressure, and K is the hydraulic conductivity. The units for j_v are m³ m^-2 s⁻¹, or m s⁻¹; therefore j_v corresponds to the speed of flow. Pressure has units of pascals, so dp/dx is expressed in Pa m⁻¹. Therefore, the units of hydraulic conductivity are m² Pa⁻¹ s⁻¹. Hydraulic conductivity is analogous to electrical conductivity or thermal conductivity; it specifies how well a material permits the transport of a quantity (flow of water) caused by some driving force (pressure gradient).

Russ Hobbie and I don’t discuss Darcy’s law in IPMB, but we come close. In Chapter 5 we analyze the flow of water across a membrane, and define the relationship

j_v = L_p Δp ,    (5.9)

where j_v again is the speed of flow, Δp is the pressure difference across the membrane, and L_p is the hydraulic permeability. If the membrane has a thickness Δx, then we can multiply and divide by Δx and obtain j_v = (L_p Δx) (Δp/Δx). The equation looks just like Darcy’s law (except for a minus sign), where the hydraulic conductivity is the hydraulic permeability times the membrane thickness:

K = L_p Δx.

I first encountered Darcy’s law when reading my friend Peter Basser’s paper about “Interstitial Volume, Pressure, and Flow During Infusion into the Brain” (Microvascular Research, 44:143–165, 1992). He derived a model of swelling in the brain that occurs during infusion of a drug. When Basser combined Darcy’s law with the equations of elasticity, he derived a diffusion equation for volume change of the tissue caused by accumulation of interstitial fluid (swelling), in which the diffusion constant is approximately the hydraulic conductivity times the bulk modulus.

Darcy’s law plays a key role in governing fluid flow in many tissues. A nice summary can be found in “Interstitial Flow and Its Effects in Solft Tissues” by Melody Swartz and Mark Fleury (Annual Review of Biomedical Engineering, 9:229–256, 2007). Below is the abstract to their review.

Interstitial flow plays important roles in the morphogenesis, function, and pathogenesis of tissues. To investigate these roles and exploit them for tissue engineering or to overcome barriers to drug delivery, a comprehensive consideration of the interstitial space and how it controls and affects such processes is critical. Here we attempt to review the many physical and mathematical correlations that describe fluid and mass transport in the tissue interstitium; the factors that control and affect them; and the importance of interstitial transport on cell biology, tissue morphogenesis, and tissue engineering. Finally, we end with some discussion of interstitial transport issues in drug delivery, cell mechanobiology, and cell homing toward draining lymphatics.