Happy Birthday Laser!

Quantum Physics of Atoms,
Molecules, Solids, Nuclei, and Particles,
by Eisberg and Resnick.

This month marks the 50^th anniversary of the invention of the laser. In May 1960, Theodore Maiman built the first device to produce coherent light by the mechanism of “Light Amplification by Stimulated Emission of Radiation” at Hughes Research Laboratories in Malibu, making the laser just slightly older than I am. A special website, called laserfest, is commemorating this landmark event. Eisberg and Resnick discuss lasers in Section 11.7 of their textbook Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (quoted from the first edition, 1974).

In the solid state laser that operates with a ruby crystal, some Al atoms in the Al₂O₃ molecules are replaced by Cr atoms. These “impurity” chromium atoms account for the laser action… The level of energy E1 is the ground state and the level of energy E3 is the unstable upper state with a short lifetime (≈10⁻⁸ sec), the energy difference E3-E1 corresponding to a wavelength of about 5500 Å. Level E2 is an intermediate excited state which is metastable, its lifetime against spontaneous decay being about 3 x 10⁻³ sec. If the chromium atoms are in thermal equilibrium, the population number of the states are such that [n3 is less than n2 is less than n1]. By pumping in radiation of wavelength 5500 Å, however, we stimulate absorption of incoming photons by Cr atoms in the ground state, thereby raising the population number of energy state E3 and depleting energy state E1 of occupants. Spontaneous emission, bringing atoms from state 3 to state 2, then enhances the occupancy of state 2, which is relatively long-lived. The result of this optical pumping is to decrease n1 and increase n2, such that n2 is greater than n1 and population inversion exists. Now, when an atom does make a transition from state 2 to state 1, the emitted photon of wavelength 6943 Å will stimulate further transitions. Stimulated emission will dominate stimulated absorption (because n2 is greater than n1) and the output of photons of wavelength 6943 Å is much enhanced. We obtain an intensified coherent monochromatic beam.

Lasers are an important tool in biology and medicine. Russ Hobbie and I discuss their applications in Chapter 14 (Atoms and Light) the 4th edition of Intermediate Physics for Medicine and Biology. In Section 14.5 (The Diffusion Approximation to Photon Transport) we write

A technique made possible by ultrashort light pulses from a laser is time-dependent diffusion. It allows determination of both μ_s and μ_a [the scattering and absorption attenuation coefficients]. A very short (150-ps) pulse of light strikes a small region on the surface of the tissue. A detector placed on the surface about 4 cm away records the multiply-scattered photons… A related technique is to apply a continuous laser beam, the amplitude for which is modulated at various frequencies between 50 and 800 MHz. The Fourier transform of Eq. 14.29 gives the change in amplitude and phase of the detected signal. Their variation with frequency can also be used to determine μ_a and μ_s.

We also mention lasers in Section 14.10 (Heating Tissue with Light).

Sometimes tissue is irradiated in order to heat it; in other cases tissue heating is an undesired side effect of irradiation. In either case, we need to understand how the temperature changes result from the irradiation. Examples of intentional heating are hyperthermia (heating of tissue as a part of cancer therapy) or laser surgery (tissue ablation). Tissue is ablated when sufficient energy is deposited to vaporize the tissue.

Russ and I give many references about lasers in medicine in our Resource Letter (“Resource Letter MP-2: Medical Physics,” American Journal of Physics, Volume 77, Pages 967–978, 2009):

F. Lasers and optics

Lasers have introduced many medical applications of light, from infrared to the visible spectrum to ultraviolet.

150. Lasers in Medicine, edited by R. W. Waynant (CRC, Boca Raton, 2002). (I)

151. Laser-Tissue Interactions: Fundamentals and Applications, M. H. Niemz (Springer, Berlin, 2007). (I)

152. “Lasers in medicine,” Q. Peng, A. Juzeniene, J. Chen, L. O. Svaasand, T. Warloe, K.-E. Giercksky, and J. Moan, Rep. Prog. Phys. 71, Article 056701, 28 pages
(2008). (A)

A fascinating and fast-growing new technique to image biological tissue is optical coherence tomography “OCT.” It uses reflections like ultrasound but detects the reflected rays using interferometry.

153. Optical Coherence Tomography, M. E. Brezinski (Elsevier, Amsterdam, 2006). Overview of the physics of OCT and applications to cardiovascular medicine, musculoskeletal disease, and oncology. (I)

154. “Optical coherence tomography: Principles and applications,” A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, Rep. Prog. Phys. 66, 239–303 (2003). (I)

With infrared light, scattering dominates over absorption. In this case, light diffuses through the tissue. Optical imaging in turbid media is difficult but not impossible.

155. “Recent advances in diffuse optical imaging,” A. P. Gibson, J. C. Hebden, and S. R. Arridge, Phys. Med. Biol. 50, R1–R43 (2005). (I)

156. “Pulse oximetry,” R. C. N. McMorrow and M. G. Mythen, Current Opinion in Critical Care 12, 269–271 (2006). The pulse oximeter measures the oxygenation of blood and is based on the diffusion of infrared light. (I)

One impetus for medical applications of light has been the development of new light sources, such as free-electron lasers and synchrotrons. In both cases, the light frequency is tunable over a wide range.

157. “Free-electron-laser-based biophysical and biomedical instrumentation,” G. S. Edwards, R. H. Austin, F. E. Carroll, M. L. Copeland, M. E. Couprie, W. E. Gabella, R. F. Haglund, B. A. Hooper, M. S. Hutson, E. D. Jansen, K. M. Joos, D. P. Kiehart, I. Lindau, J. Miao, H. S. Pratisto, J. H. Shen, Y. Tokutake, A. F. G. van der Meer, and A. Xie, Rev. Sci. Instrum. 74, 3207–3245 (2003). (I)

158. “Medical applications of synchrotron radiation,” P. Suortti and W. Thomlinson, Phys. Med. Biol. 48, R1– R35 (2003). (I)

Finally, photodynamic therapy uses light-activated drugs to treat diseases.

159. “The physics, biophysics and technology of photodynamic therapy,” B. C. Wilson and M. S. Patterson, Phys. Med. Biol. 53, R61–R109 (2008). (A)

Happy birthday, laser!

Kalin Lucas and his ruptured Achilles tendon

Basketball fans may recall that in this year's NCAA tournament, Michigan State University (located about a 90 minute drive from where I work here at Oakland University in Rochester Michigan) made it into the final four before losing to Butler. They might have won the entire tournament if they were not without their star, Kalin Lucus, who ruptured his left Achilles tendon in a second round game against the University of Maryland. Coach Tom Izzo, who is much beloved here in southeast Michigan, managed two more wins without Lucus, before losing in the semifinals.

Why did Lucus injure his Achilles tendon? There was no collision or accident, he just landed awkwardly. Readers of the 4th edition of Intermediate Physics for Medicine and Biology won’t be too surprised when they hear about sports injuries to the Achilles tendon. In Section 1.5 of our book, Russ Hobbie and I analyze the forces on the Achilles tendon and show that the tension in this tendon can be nearly twice the body weight.

The Achilles tendon connects the calf muscles (the gastrocnemius and soleus) to the calcaneus at the back of the heal (Fig. 1.9). To calculate the force exerted by this tendon on the calcaneus when a person is standing on the ball of one foot, assume that the entire foot can be regarded as a rigid body. This is our first example of creating a model of the actual situation. We try to simplify the real situation to make the calculation possible while keeping the features that are important to what is happening. In this model the internal forces within the foot are being ignored.

Figure 1.10 shows the force exerted by the tendon on the foot (F_T), the force of the leg bones (tibia and fibula) on the foot (F_B), and the force of the floor upward, which is equal to the weight of the body (W)...

We then go one to solve the equations of translational and rotational equilibrium to find that F_T = 1.8 W and F_B = 2.8 W, and conclude

The tension in the Achilles tendon is nearly twice the person’s weight, while the force exerted on the leg by the talus is nearly three times the body weight. Once can understand why the tendon might rupture.

Many sports injuries result from the laws of biomechanics. The problem is often that a tendon must exert a large force in order to create enough torque to maintain rotational equilibrium. For the Achilles tendon, the moment arm between the joint and the tendon is roughly half the moment arm between the joint and the ball of the foot, so the force on the tendon must be about twice the weight. Our bodies are often built this way: large forces are required to make up for small moment arms. I sometimes give a problem on one of my Biological Physics (PHY 325) exams that illustrates this by calculating the forces on the shoulder of a gymnast performing an “iron cross” on the rings. Here again, the torque exerted by the rings on the arm is huge because of the large moment arm (essentially the entire length of the arm itself), while the moment arm of the pectoral muscle is small because it connects to the humerus (the arm bone) only about 5 cm from the shoulder, at a small angle. The problem suggests that the pectoral muscle must supply a force of over twenty times the body weight! No wonder I was so poor at the rings in my high school physical education class. Readers interested in learning more about this topic might want to read Williams and Lissner's classic textbook Biomechanics of Human Motion. Russ and I cite the 1962 first edition, but I believe that the book has evolved into Biomechanics of Human Motion: Basics and Beyond for the Health Professions, by Barney LeVeau, due out later this year.

Hopefully these insights into biomechanics can help you appreciate how Kalin Lucus could suffer a season-ending injury so easily. I’m glad MSU was able to make it to the final four even without Lucus. However, to be honest, the Spartans were only my 4th favorite team in the tournament this year. Oakland University participated in March Madness for only the second time in the school’s history. Vanderbilt University (where I went for graduate school) also reached the Big Dance, and many predicted that the University of Kansas (where I attended college) would win the entire event. Unfortunately, all three schools lost in the first weekend of play. Russ didn’t fare much better, as the University of Minnesota lost in the first round. Congratulations to Duke University (home to an excellent Biomedical Engineering Department) for their ultimate victory.

Single-Pool Exponential Decomposition Models: Potential Pitfalls in their Use in Ecology Studies

In Section 11.2 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss fitting data using nonlinear least squares. Our first example in this section is a fit using a single exponential decay, y(x) = a e^–bx, where a and b are to be determined. We suggest that the reader

“take logarithms of each side of the equation

log y = log a – b x log e

v = a' – b' x.

This can be fit by the linear [least squares] equation, determining constants a' and b' using Eqs. 11.5.”

This method works fine for ideal data, but in almost any real application the data will be corrupted by noise. In that case, fitting a linear equation to the logarithm of the data may not be wise. I discussed this issue last year in the May 22 entry to this blog, but wish to explore it in more detail this week.

Recently, Russ coauthored a paper about his recent results on this topic, published in the journal Ecology (Volume 91, Pages 1225–1236). In collaboration with his daughter Sarah Hobbie (Associate Professor in the Department of Ecology, Evolution and Behavior at the University of Minnesota), and her former postdoc E. Carol Adair (currently with the National Center for Ecological Analysis and Synthesis at the University of California Santa Barbara), Russ examined “Single-Pool Exponential Decomposition Models: Potential Pitfalls in their Use in Ecology Studies.” The abstract to the paper is given below.

The importance of litter decomposition to carbon and nutrient cycling has motivated substantial research. Commonly, researchers fit a single-pool negative exponential model to data to estimate a decomposition rate (k). We review recent decomposition research, use data simulations, and analyze real data to show that this practice has several potential pitfalls. Specifically, two common decisions regarding model form (how to model initial mass) and data transformation (log-transformed vs. untransformed data) can lead to erroneous estimates of k. Allowing initial mass to differ from its true, measured value resulted in substantial over- or underestimation of k. Log-transforming data to estimate k using linear regression led to inaccurate estimates unless errors were lognormally distributed, while nonlinear regression of untransformed data accurately estimated k regardless of error structure. Therefore, we recommend fixing initial mass at the measured value and estimating k with nonlinear regression (untransformed data) unless errors are demonstrably lognormal. If data are log-transformed for linear regression, zero values should be treated as missing data; replacing zero values with an arbitrarily small value yielded poor k estimates. These recommendations will lead to more accurate k estimates and allow cross-study comparison of k values, increasing understanding of this important ecosystem process.

The authors performed a massive review of the literature, reading and analyzing nearly 500 papers about litter decomposition, most of which fit data to an exponential decay, e^–kt. The bottom line is that doing a linear least squares fit to the logarithm of the data can cause significant errors. Much better is to use a nonlinear least squares fit. The manuscript concludes “We suggest that careful selection of fitting methods, as we have described above, will lead to more accurate and comparable k estimates, thereby increasing our understanding of this important ecosystem process.” Of course, my favorite thing about their paper is that it cites the 4th edition of Intermediate Physics for Medicine and Biology!

One pitfall can be illustrated by considering measurements of the voltage across a resistor in an RC circuit. The voltage decays with an RC time constant. However thermal, or Johnson, noise is also present (see Section 9.8). Once the voltage decays to less than the Johnson noise, the measured voltage fluctuates between positive and negative values. If you take the logarithm of the voltage, the negative values are undefined. In other words, you can’t do a linear least squares fit to the logarithm of the data if the data can be negative. The problem remains even when the data is nonnegative (as in Hobbie’s paper) if the data can be zero. However, if you make a nonlinear least squares fit of the data itself (rather than the logarithm of the data) the problem vanishes.

In order to explain these observations in Intermediate Physics for Medicine and Biology, Russ and I (mainly Russ) wrote an Addendum available at the book’s website. It lists what changes are needed to properly explain least squares fitting of exponential data. Enjoy!

Hysteresis and Bistability in the Direct Transition from 1:1 to 2:1 Rhythm in Periodically Driven Single Ventricular Cells

When preparing the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I added two homework problems (Problems 37 and 38) in Chapter 10 (Feedback and Control) about “cardiac restitution.” These problems contain a fascinating and elegantly simple example of restitution that provides insight into nonlinear dynamics and chaos. Problem 37 begins

Problem 37 The onset of ventricular fibrillation in the heart can be understood in part as a property of cardiac “restitution.” The action potential duration (APD) depends on the previous diastolic interval (DI): the time from the end of the last action potential until the start of the next one. The relationship between APD and DI is called the “restitution curve.” In cardiac muscle, a typical restitution curve has the form

APD_i+1 = 300 (1 – exp(–DI_i/100))

where all times are given in ms. Suppose we apply to the heart a series of stimuli, with period (or “cycle length”) CL. Since APD + DI = CL, we have DI_i+1 = CL – APD_i+1.

The problem then goes on to have the reader do some numerical calculations using various cycle lengths and initial diastolic intervals. Depending on the parameters, you can get (a) a simple 1:1 response between stimulation and action potential, (b) a 2:2 response in which every stimulus triggers an action potential but the APD alternates between long and short, a behavior called “alternans,” (c) a 2:1 response where an action potential is triggered by every second stimulus, with the tissue being refractory and not responding to the other stimuli, and (d) chaos. I have found this model is an excellent way to introduce students to chaotic behavior; even students with a weak mathematics background can understand it. When discussing this mathematical model with students, I often hand out a particularly clear paper to serve as background reading: J. N. Weiss, A. Garfinkel, H. S. Karagueuzian, Z. Qu, and P.-S. Chen (1999) “Chaos and the Transition to Ventricular Fibrillation: A New Approach to Antiarrhythmic Drug Evaluation,” Circulation, Volume 99, Pages 2819–2826.

Problem 38 explores how to understand this behavior by analyzing the slope of the restitution curve. If the slope is too steep, the behavior becomes more complex. Part (d) of Problem 38 says

Suppose you apply a drug to the heart that can change the restitution curve to

APD_i+1 = 300 (1 – b exp(–DI_i/100)) .

Plot APD as a function of DI for b = 0, 0.5, and 1. What value of b ensures that the slope of the restitution curve is always less than 1? Garfinkel et al. (2000) have suggested that one way to prevent ventricular fibrillation is to use drugs to “flatten” the restitution curve.

There is yet another type of behavior that is not discussed in Problems 37 or 38: a bistable response. Below is a new homework problem that discusses bistable behavior.

Problem 38 ½ Use the restitution curve from Problem 38, with b = 1/3 and CL = 250, to analyze the response of the system with initial diastolic intervals of 50, 60, 70, 80, and 90. You should find that the qualitative behavior depends on the initial condition. Which values of the initial diastolic interval give a 1:1 response, and which give 2:1? Determine the initial value of the DI, to three significant figures, for which the system makes a transition from one behavior to the other. When two qualitatively different behaviors can both occur, depending on the initial conditions, the system is “bistable.” To learn more about such behavior, see Yehia et al. (1999).

The full citation to the paper mentioned at the end of the problem is

Yehia, A. R., D. Jeandupeux, F. Alonso, and M. R. Guevara (1999) “Hysteresis and Bistability in the Direct Transition From 1:1 to 2:1 Rhythm in Periodically Driven Single Ventricular Cells,” Chaos, Volume 9, Pages 916–931.

The senior author on this article is Michael Guevara, of the Centre for Applied Mathematics in Bioscience and Medicine at McGill University. The introductory paragraph of their paper is reproduced below.

The majority of cells in the heart are not spontaneously active. Instead, these cells are excitable, being driven into activity by periodic stimulation originating in a specialized pacemaker region of the heart containing spontaneously active cells. This pacemaker region normally imposes a 1:1 rhythm on the intrinsically quiescent cells. However, the 1:1 response can be lost when the excitability of the paced cells is decreased, when there are problems in the conduction of electrical activity from cell to cell, or when the heart rate is raised. When 1:1 synchronization is lost in the intact heart, one of a variety of abnormal cardiac arrhythmias can arise. In single quiescent cells isolated from ventricular muscle, 1:1 rhythm can be replaced by a N+1:N rhythm (N≥2), a period-doubled 2:2 rhythm, or a 2:1 rhythm. We investigate below the direct transition from 1:1 to 2:1 rhythm in experiments on single cells and in numerical simulations of an ionic model of a single cell formulated as a nonlinear system of differential equations. We show that there is hysteresis associated with this transition in both model and experiment, and develop a theory for the bistability underlying this hysteresis that involves the coexistence of two stable fixed-points on a two-branched one-dimensional map.

For those interested in exploring the application of nonlinear dynamics to biology and medicine in more detail, two books Russ and I cite in Intermediate Physics for Medicine and Biology—and which I recommend highly—are From Clocks to Chaos by Leon Glass and Michael Mackey (both also at McGill) and Nonlinear Dynamics and Chaos by Steven Strogatz.