The Wien Exponential Law

In Section 14.8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss blackbody radiation. Our analysis is similar to that in many modern physics textbooks. We introduce Planck’s law for W_λ(λ,T) dλ, the spectrum of power per unit area emitted by a completely black surface at temperature T and wavelength λ

where c is the speed of light, h is Planck’s constant, and k_B is Boltzmann’s constant. We then 1) express this function in terms of frequency ν instead of wavelength λ, 2) integrate over all wavelengths to derive the Stefan-Boltzmann law, and 3) show that the wavelength of peak emission decreases with temperature, often known as the Wien displacement law.

Russ and I like to provide homework problems that reinforce the concepts in the text. Ideally, the problem requires the reader to repeat many of the same steps carried out in the book, but for a slightly different case or in a somewhat different context. Below I present such a homework problem for blackbody radiation. It is based on an approximation to Planck’s law at short wavelengths derived by Wilhelm Wien.

Problem 25 ½. Consider the limit of Planck’s law, Eq. 14.33, when hc/λ is much greater than k_BT, an approximation known as the Wien exponential law.

(a) Derive the mathematical form of W_λ(λ,T) in this limit.

(b) Convert Wien’s law from a function of wavelength to a function of frequency, and determine the mathematical form of W_ν(ν,T).

(c) Integrate W_ν(ν,T) over all frequencies to obtain the total power emitted per unit area. Compare this result with the Stefan-Boltzmann law (Eq. 14.34). Derive an expression for the Stefan-Boltzmann constant in terms of other fundamental constants.

(d) Determine the frequency ν_max corresponding to the peak in W_ν(ν,T). Compare ν_max/T with the value obtained from Planck’s law.

Subtle is the Lord,
by Abraham Pais.

The Wien exponential law predated Planck’s law by several years. In his landmark biography ‘Subtle is the Lord…’: The Science and the Life of Albert Einstein, Abraham Pais discusses 19th century attempts to describe blackbody radiation theoretically.

Meanwhile,proposals for the correct form of [W_λ(λ,T)] had begun to appear as early as the 1860s. All these guesses may be forgotten except one, Wien’s exponential law, proposed in 1896…

Experimental techniques had sufficiently advanced by then to put this formula to the test. This was done by Friedrich Paschen from Hannover, whose measurements (very good ones) were made in the near infrared, λ = 1-8 μm (and T = 400 -1600 K). He published his data in January 1897. His conclusion: “It would seem very difficult to find another function…that represents the data with as few constants.” For a brief period, it appeared that Wien’s law was the final answer. But then, in the year 1900, this conclusion turned out to be premature…

And the rest, as they say, is history.

Chemostat Homework Problems

In the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added a section on the chemostat.

2.6  The Chemostat

The chemostat is used by bacteriologists to study the growth of bacteria (Hagen 2010). It allows the rapid growth of bacteria to be observed over a longer time scale. Consider a container of bacterial nutrient of volume V. It is well stirred and contains y bacteria with concentration C = y/V. Some of the nutrient solution is removed at rate Q and replaced by fresh nutrient. The bacteria in the solution are reproducing at rate b. The rate of change of y is

Therefore the growth rate is slowed to

and can be adjusted by varying Q.

However, Russ and I didn’t write any new homework problems for this section. If a topic is worth discussing in the text, then it’s worth creating homework problems to reinforce and extend that discussion. So, here are some new problems about the chemostat.

Problem 21.1.  Often a chemostat is operated in steady state.

(a) Determine the solution removal rate Q required for steady state, as a function of the bacteria reproduction rate b and the container volume V, using Eq. 2.22. Determine the units of b, Q, and V, and verify that your expression for Q has the correct dimensions.

(b) If the rate Q is larger than the steady-state value, what is happening physically?

(c) Sometimes b varies with some external parameter (for example, temperature or glucose concentration), and you want to determine b as a function of that parameter. Suppose you can control Q and you can measure the number of bacteria y. Qualitatively design a way to determine b as your external parameter changes, assuming that for each value of the parameter your chemostat reaches steady state. (If unsure how to begin, take a look at Sec. 6.13.1 about the voltage clamp used in electrophysiology.)

Problem 21.2.  Consider an experiment using a chemostat in which the bacteria's reproduction rate b slows as the number of bacteria y increases.

(a) Modify Eq. 2.22 so that “b” becomes “b (1 − y/y_∞),” analogous to the logistic model (Sec. 2.10).

(b) Determine the value of y once the chemostat reaches steady state, as a function of Q, V, b, and y_∞.

(c) Suppose your chemostat has a volume of 1.7 liters. You measure the steady state value of y (arbitrary units) for different values of Q (liters per hour), as shown in the table below. Plot y versus Q, and determine b and y_∞.

Q	y
0.2	11.64
0.4	9.47
0.6	7.31
0.8	5.14
1.0	2.98

Problem 21.3.  Let the growth rate of the bacteria in your chemostat be limited by a small, constant amount of some essential metabolite, so the term “by” in Eq. 2.22 is replaced by a constant “a.”

(a) Find an expression for the solution removal rate Q in terms of a, the number of bacteria y, and the chemostat volume V, when the chemostat is in steady state.

(b) Determine the time constant governing how quickly the chemostat reaches steady state (Hint: see Sec. 2.8).

“Exponential Growth of Bacteria:
Constant Multiplication Through Division,”
by Stephen Hagen.

Russ and I cite an American Journal of Physics article about the exponential growth of bacteria, written by Stephen Hagen (Volume 78, Pages 1290-1296, 2010). Here’s what Hagen says about the chemostat.

Because the growth rate of the cell determines its size and chemical composition, a device that allows us to fine tune the growth rate will select the physiological properties of the cells. The bacterial chemostat is such a machine. In the chemostat a bacterial culture grows in a well-stirred vessel while a supply of fresh growth medium is fed into the vessel at a fixed flow rate Q (volume/time). At the same time, the medium (containing bacteria) is continuously withdrawn from the vessel at the same rate so as to maintain constant volume V. Thus, the bacterial population is continuously diluted at a rate D = Q/V. If this dilution rate exceeds the growth rate k [our b], the population is diluted, which allows its growth to accelerate until it matches the dilution rate, k = Q/V. (If D is too large, the culture will be diluted away entirely.) Therefore, the chemostat allows the experimenter to select the growth rate by selecting Q. Because it harnesses an exponential growth process to produce a tunable, steady output, we might think of the chemostat as the microbiological analog of a nuclear fission reactor. Interestingly, the chemostat reactor was first described by the physicist Leo Szilard (with Aaron Novick), who also (with Enrico Fermi) patented the nuclear reactor."

I like the analogy to the nuclear reactor. Adjusting the flow rate in a chemostat is like pulling the cadmium control rods in and out of an atomic pile (except it’s less dangerous).

Neural Lacing

One feature of blogging that I like are the comments. I don’t get many, but I appreciate those I do get. Each week I share my new blog entry on the Intermediate Physics for Medicine and Biology Facebook page, which provides another venue for comments, likes, and other interactions with readers. A couple weeks ago I received the following on Facebook:

Neeraj Kapoor

June 3 at 1:36pm

Yesterday, during a conference with Elon Musk at a coding conference, he mentioned something about Neural Lacing (this group at harvard seems to be one of the few major groups working on it...http://cml.harvard.edu/) . I'm wondering if you have any knowledge of this Brad Roth and if so, if you could do a blog post on it.

After a bit of googling, I found a Newsweek article about neural lacing, Elon Musk, and the coding conference.

Billionaire polymath Elon Musk has warned that humans risk being treated like house pets by artificial intelligence (AI) unless they implant technology into their brains.

Musk believes that a technology concept known as “neural lace” could act as a wireless brain-computer interface capable of augmenting natural intelligence.

Speaking at the Code Conference in California on Wednesday, Musk said a neural lace could work “well and symbiotically” with the rest of a human’s body.

“I don’t love the idea of being a house cat, but what’s the solution? I think one of the solutions that seems maybe the best is to add an AI layer,” Musk said.

So what does all this talk about neural lacing mean, and how does it relate to Intermediate Physics for Medicine and Biology? As best I can tell, neural lacing would be used to monitor and excite nerves. The technology to stimulate nerves already exists, and is described in Chapter 7 of IPMB.

The information that has been developed in this chapter can also be used to understand some of the features of stimulating electrodes. These may be used for electromyographic studies; for stimulating muscles to contract called functional electrical stimulation (Peckham and Knutson 2005); for a cochlear implant to partially restore hearing (Zeng et al. 2008); deep brain stimulation for Parkinson’s disease (Perlmutter and Mink 2006); for cardiac pacing (Moses and Mullin 2007); and even for defibrillation (Dosdall et al. 2009). The electrodes may be inserted in cells, placed in or on a muscle, or placed on the skin.

The best example of what I think Mr. Musk is talking about is the cochlear implant. A microphone records sound and analyzes it with a computer, which decides what location along the auditory nerve it should stimulate in order to fool the brain into thinking the ear heard that sound. For this technique to work, electrode arrays must be implanted in the cochlea so different spots can be stimulated, mimicking the sensitivity of different locations along the cochlea to different frequencies of sound.

What is different between a cochlear implant and a neural lace? Musk talks about the stimulating electrodes being wireless. Wireless neural stimulation is fairly common, and most cochlear implants are wireless (no wire passing through the skin). Most wireless systems work by transferring energy and information using electromagnetic induction. Chapter 8 of IPMB discusses induction, mainly in the context of magnetic stimulation. In fact, transcranial magnetic stimulation could be thought of as a low-spatial-resolution precursor to neural lacing. It allows neurons to be excited with no wires penetrating the body so the method is completely noninvasive. The problem is, transcranial magnetic stimulation provides a resolution of about 1 cm—some claim as low as 1 mm—which is a factor of a hundred to a thousand too coarse to stimulate individual neurons. If you could somehow build very small magnetic stimulators (there are enormous technical challenges in doing this), you still would not be able to excite deep neurons without simultaneously activating shallow neurons even more strongly. To make something like neural lacing work, you would need to use electromagnetic induction to transfer energy to a stimulator implanted in the body, and then distribute the excitation using small wires or some other technology that provides the necessary spatial resolution and the ability to excite deep neurons. Wireless deep brain stimulation is one example.

Spatial scale is a key factor in developing the technology of neural lacing. Cochlear implants only work because the electrodes are small enough that individual sites along the auditory nerve can be excited locally. I believe that neural lacing would require miniaturization to be increased dramatically. If you are going to stimulate the brain in a truly selective way, you need to be able to excite individual neurons. This means you need electrodes spaced by about ten microns or closer, and you need a lot of them. Neural lacing would therefore require advances in electrode array miniaturization. This is where the Lieber group at Harvard—which Kapoor mentioned in his Facebook comment—enters the picture. They are developing the arrays of microelectrodes that would be necessary to provide a fine-grained interaction between a computer and the human brain. For example, their paper “syringe-injectable electronics” (Nature Nanotechnology, Volume 10, Pages 629–636, 2015) discusses small scale arrays of electrodes that can be injected through a syringe.

Seamless and minimally invasive three-dimensional interpenetration of electronics within artificial or natural structures could allow for continuous monitoring and manipulation of their properties. Flexible electronics provide a means for conforming electronics to non-planar surfaces, yet targeted delivery of flexible electronics to internal regions remains difficult. Here, we overcome this challenge by demonstrating the syringe injection (and subsequent unfolding) of sub-micrometre-thick, centimetre-scale macroporous mesh electronics through needles with a diameter as small as 100 μm. Our results show that electronic components can be injected into man-made and biological cavities, as well as dense gels and tissue, with [greater than] 90% device yield. We demonstrate several applications of syringe-injectable electronics as a general approach for interpenetrating flexible electronics with three-dimensional structures, including (1) monitoring internal mechanical strains in polymer cavities, (2) tight integration and low chronic immunoreactivity with several distinct regions of the brain, and (3) in vivo multiplexed neural recording. Moreover, syringe injection enables the delivery of flexible electronics through a rigid shell, the delivery of large-volume flexible electronics that can fill internal cavities, and co-injection of electronics with other materials into host structures, opening up unique applications for flexible electronics.

Is neural lacing science or science fiction? Hard to say. I am skeptical that in the future we will all have electrode arrays hardwired into our brains. But 50 years ago I would have been skeptical that cochlear implants could restore hearing to the deaf. I will reserve judgment, except to say that if neural lacing is developed, I am certain it will be based on the basic concepts Russ Hobbie and I discuss in Intermediate Physics for Medicine and Biology. That is the beauty of the book: it teaches the fundamental principles upon which you can build the amazing biomedical technologies of the future.

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.