Friday, April 27, 2012

Physics and Medicine

Readers of Intermediate Physics for Medicine and Biology already know how important physics is to medicine. Now, subscribers to the famed British medical journal The Lancet are learning this too. The April 21-27 issue (Volume 379, Issue 9825) of The Lancet contains a series of articles under the heading "Physics and Medicine". In his editorial introducing this series, Peter Knight (president of the Institute of Physics) calls for UK medical schools to reinstate an undergraduate physics requirement for aspiring premed students. The English don't require their premed students to take physics? Yikes!

Richard Horton, editor-in-chief of The Lancet, seconds this call for better physics education. He concludes that “Young physicists need to be nurtured to ensure a sustainable supply of talented scientists who can take advantage of the opportunities for health-related physics research in the future. Schools, indeed all of us interested in the future of health care, should declare and implement a passion for physics. Our Series is our commitment to do so.” Bravo! Below I reproduce the abstracts to the five articles in the Physics and Medicine series. In brackets I indicate the chapter or section in the 4th edition of Intermediate Physics for Medicine and Biology where a particular topic is discussed.

Physics and Medicine: a Historical Perspective
Stephen F Keevil

Nowadays, the term medical physics usually refers to the work of physicists employed in hospitals, who are concerned mainly with medical applications of radiation, diagnostic imaging, and clinical measurement. This involvement in clinical work began barely 100 years ago, but the relation between physics and medicine has a much longer history. In this report, I have traced this history from the earliest recorded period, when physical agents such as heat and light began to be used to diagnose and treat disease. Later, great polymaths such as Leonardo da Vinci and Alhazen used physical principles to begin the quest to understand the function of the body. After the scientific revolution in the 17th century, early medical physicists developed a purely mechanistic approach to physiology, whereas others applied ideas derived from physics in an effort to comprehend the nature of life itself. These early investigations led directly to the development of specialties such as electrophysiology [Chpts 6, 7], biomechanics [Secs 1.5-1.7] and ophthalmology [Sec 14.12]. Physics-based medical technology developed rapidly during the 19th century, but it was the revolutionary discoveries about radiation and radioactivity [Secs 17.2-17.4] at the end of the century that ushered in a new era of radiation-based medical diagnosis and treatment, thereby giving rise to the modern medical physics profession. Subsequent developments in imaging [Chpt 12] in particular have revolutionised the practice of medicine. We now stand on the brink of a new revolution in post-genomic personalised medicine, with physics-based techniques again at the forefront. As before, these techniques are often the unpredictable fruits of earlier investment in basic physics research.

Diagnostic Imaging
Peter Morris, Alan Perkins

Physical techniques have always had a key role in medicine, and the second half of the 20th century in particular saw a revolution in medical diagnostic techniques with the development of key imaging instruments: x-ray imaging [Chpt 16] and emission tomography [Secs 12.4-12.6] (nuclear imaging [Secs 17.12-17.13] and PET [Sec 17.14]), MRI [Chpt 18], and ultrasound [Chpt 13] These techniques use the full width of the electromagnetic spectrum [Sec 14.1], from gamma rays to radio waves, and sound [Secs 13.1-13.3]. In most cases, the development of a medical imaging device was opportunistic; many scientists in physics laboratories were experimenting with simple x-ray images within the first year of the discovery of such rays, the development of the cyclotron and later nuclear reactors created the opportunity for nuclear medicine, and one of the co-inventors of MRI was initially attempting to develop an alternative to x-ray diffraction for the analysis of crystal structures. What all these techniques have in common is the brilliant insight of a few pioneering physical scientists and engineers who had the tenacity to develop their inventions, followed by a series of technical innovations that enabled the full diagnostic potential of these instruments to be realised. In this report, we focus on the key part played by these scientists and engineers and the new imaging instruments and diagnostic procedures that they developed. By bringing the key developments and applications together we hope to show the true legacy of physics and engineering in diagnostic medicine.

The Importance of Physics to Progress in Medical Treatment
Andreas Melzer, Sandy Cochran, Paul Prentice, Michael P MacDonald, Zhigang Wang, Alfred Cuschieri

Physics in therapy is as diverse as it is substantial. In this review, we highlight the role of physics—occasionally transitioning into engineering—through discussion of several established and emerging treatments. We specifically address minimal access surgery, ultrasound [Sec 13.7], photonics [Chpt 14], and interventional MRI, identifying areas in which complementarity is being exploited. We also discuss some of the fundamental physical principles involved in the application of each treatment to medical practice.

Future Medicine Shaped by an Interdisciplinary New Biology
Paul O'Shea

The projected effects of the new biology on future medicine are described. The new biology is essentially the result of shifts in the way biological research has progressed over the past few years, mainly through the involvement of physical scientists and engineers in biological thinking and research with the establishment of new teams and task forces to address the new challenges in biology. Their contributions go well beyond the historical contributions of mathematics, physical sciences, and engineering to medical practice that were largely equipment oriented. Over the next generation, the entire fabric of the biosciences will change as research barriers between disciplines diminish and eventually cease to exist. The resulting effects are starting to be noticed in front-line medicine and the prospects for the future are immense and potentially society changing. The most likely disciplines to have early effects are outlined and form the main thrust of this paper, with speculation about other disciplines and emphasis that although physics-based and engineering-based biology will change future medicine, the physical sciences and engineering will also be changed by these developments. Essentially, physics is being redefined by the need to accommodate these new views of what constitutes biological systems and how they function.

The Importance of Quantitative Systemic Thinking in Medicine
Geoffrey B West

The study and practice of medicine could benefit from an enhanced engagement with the new perspectives provided by the emerging areas of complexity science [Secs 10.7-10.8] and systems biology. A more integrated, systemic approach is needed to fully understand the processes of health, disease, and dysfunction, and the many challenges in medical research and education. Integral to this approach is the search for a quantitative, predictive, multilevel, theoretical conceptual framework that both complements the present approaches and stimulates a more integrated research agenda that will lead to novel questions and experimental programmes. As examples, the importance of network structures and scaling laws [Sec 2.10] are discussed for the development of a broad, quantitative, mathematical understanding of issues that are important in health, including ageing and mortality, sleep, growth, circulatory systems [Sec 1.17], and drug doses [Sec 2.5]. A common theme is the importance of understanding the quantifiable determinants of the baseline scale of life, and developing corresponding parameters that define the average, idealised, healthy individual.

Friday, April 20, 2012

Frequency versus Wavelength

I am currently reading The Optics of Life: A Biologist’s Guide to Light in Nature, by Sonke Johnsen. I hope to have more to say about this fascinating book when I finish it, but today I want to consider a point made in Chapter 2 (Units and Geometry), which addresses the tricky issue of measuring light intensity as a function of either frequency or wavelength. Johnsen favors using wavelength whenever possible.
“However, one critical issue must be discussed before we put frequency away for good. It involves the fact that light spectra are histograms. Suppose you measure the spectrum of daylight, and that the value at 500 nm is 15 photons/cm2/s/nm. That doesn’t mean that there are 15 photons/cm2/s with a wavelength of exactly 500 nm. Instead, it means that, over a 1-nm-wide interval centered on a wavelength of 500 nm, you have 15 photons/cm2/s. The bins in a spectrum don’t have to be 1 nm wide, but they all must have the same width.

Let’s suppose all the bins are 1 nm wide and centered on whole numbers (i.e., one at 400 nm, one at 401 nm, etc.). What happens if we convert these wavelength values to their frequency counterparts? Let’s pick the wavelengths of two neighboring bins and call them λ1 and λ2. The corresponding frequencies ν1 and ν2 are equal to c/λ1 and c/λ2, where c is the speed of light. We know that λ12 equals 1 nm, but what does ν12 equal?
ν12 = … = -c/λ12
…So the width of the frequency bins depends on the wavelengths they correspond to, which means they won’t be equal! In fact, they are quite unequal. Bins at the red end of the spectrum (700 nm) are only about one-third as wide as bins at the blue end (400 nm). This means that a spectrum generated using bins with equal frequency intervals would look different from one with equal wavelength intervals. So which one is correct? Neither or both. The take-home message is that the shape of a spectrum depends on whether you have equal frequency bins or equal wavelength bins.”
Johnsen goes on to note that the wavelength at which the spectrum is maximum depends on if you use equal frequency or equal wavelength bins. It does not make sense to say that the spectrum of, say, sunlight peaks at a particular wavelength, unless you specify the type of spectrum you are using. Furthermore, you cannot unambiguously say light is “white” (a uniform spectrum). White light using equal wavelength bins is not white using equal frequency bins. Fortunately, if you integrate the spectrum, you get the same value regardless of if you express it in terms of wavelength or frequency.

Russ Hobbie and I discuss this issue in Chapter 14 (Atoms and Light) of the 4th edition of Intermediate Physics for Medicine and Biology.
“Early measurements of the radiation function were done with equipment that made measurements vs. wavelength. It is also possible to measure vs. frequency. To rewrite the radiation function in terms of frequency, let λ1 and λ2= λ1+dλ be two slightly different wavelengths, with power Wλ(λ, T)dλ emitted per unit surface area at wavelengths between λ1 and λ2. The same power must be emitted between frequencies ν1 = c/λ1 and ν2 = c/λ2:

Wν(ν,T) dν = Wλ(λ,T) dλ.     (14.35)

Since ν = c/λ, dν/dλ = −c/λ2, and

|dν| = + c λ2 |dλ| .                 (14.36)

...This transformation is shown in Fig. 14.24. The amount of power per unit area radiated in the 0.5 μm interval between two of the vertical lines in the graph on the lower right is the area under the curve of Wλ between these lines. The graph on the upper right transforms to the corresponding frequency interval. The radiated power, which is the area under the Wν curve between the corresponding frequency lines on the upper left, is the same. We will see this same transformation again when we deal with x rays. Note that the peaks of the two curves are at different frequencies or wavelengths.”
Students who prefer visual explanations should see Fig. 14.24, which Russ drew. It is one of my favorite pictures in our book, and provides an illuminating comparison of the two spectra.

One detail I should mention: why in Eq. 14.36 do we use absolute values to eliminate the minus sign introduced by the derivative dν/dλ? Typically, when you integrate a spectrum, you start from the lower frequency and go to the higher frequency (say, zero to infinity), and you start from the shorter wavelength and go to the longer wavelength (again, zero to infinity). However, zero frequency corresponds to an infinite wavelength, and an infinite frequency corresponds to zero wavelength. So, really one case should be integrated forward (zero to infinity) and the other backwards (infinity to zero). If we keep the convention of always integrating from zero to infinity in both cases, we introduce an extra minus sign, which cancels the minus sign introduced by dν/dλ.

Sometimes it helps to have an elementary example to illustrate these ideas. Therefore, I have developed a new homework problem that introduces an extremely simple spectrum for which you can do the math fairly easily, thereby allowing you to focus on the physical interpretation. Enjoy.
Section 14.7

Problem 23 ½ Let Wν(ν) = A ν (νο - ν) for ν less than νο, and Wν(ν) = 0 otherwise.
(a) Plot Wν(ν) versus ν.
(b) Calculate the frequency corresponding to the maximum of Wν(ν), called νmax.
(c) Let λο = c/νο and λmax = c/νmax. Write λmax in terms of λο.
(d) Integrate Wν(ν) over all ν to find Wtot.
(e) Use Eqs. 14.35 and 14.36 to calculate Wλ(λ).
(f) Plot Wλ(λ) versus λ.
(g) Calculate the wavelength corresponding to the maximum of Wλ(λ), called λ*max, in terms of λο.
(h) Compare λmax and λ*max. Are they the same or different? If λο is 400 nm, calculate λmax and λ*max? What part of the electromagnetic spectrum is each of these in?

(i) Integrate Wλ(λ) over all λ to find W*tot. Compare Wtot and W*tot. Are they the same or different?

Friday, April 13, 2012

Stirling’s Formula!

Factorials are used in many branches of mathematics and physics, and particularly in statistical mechanics. One often needs the natural logarithm of a factorial, ln(n!). In Chapter 3 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I use Stirling’s approximation to compute ln(n!). We analyze this approximation in Appendix I.
“There is a very useful approximation to the factorial, called Stirling’s approximation:
ln(n!) = n ln nn .
To derive it, write ln(n!) as
ln(n!) = ln 1 + ln 2 + … + ln n = ∑ ln m
The sum is the same as the total area of the rectangles in Fig. I.1, where the height of each rectangle is ln m and the width of the base is one. The area of all the rectangles is approximately the area under the smooth curve, which is a plot of ln m. The area is approximately
1nln m dm = [m ln mm]1n = n ln nn + 1.
This completes the proof.” 

David Mermin—one of my favorite writers among physicists—has much more to say about Stirling’s approximation in his American Journal of Physics paper Stirling's Formula! (leave it to Mermin to work an exclamation point into his title). He writes Stirling’s approximation as n! = √(2 π n) (n/e)n. Taking the natural logarithm of both sides gives ln(n!) = ln(2 π n)/2 + n ln nn . For large n, the first term is small, and the result is the same as Russ and I present. I wonder what affect the first term has on the approximation? For small n, it makes a big difference! In Table I.1 of our textbook, we compute the accuracy of n ln nn for n = 5. In that case, n! = 120, so ln(n!) = ln(120) = 4.7875 and 5 ln 5 – 5 = 3.047, giving a 36% error. But ln(10 π)/2 + 5 ln 5 – 5 = 4.7708, implying an error of 0.35 %, so Mermin’s formula is much better than ours (I shouldn't call it Mermin's formula; I believe Stirling himself derived n! = √(2 π n) (n/e)n.).

Mermin doesn’t stop there. He analyzes the approximation in more detail, and eventually derives an exact formula for n! that looks like Stirling’s approximation given above, except multiplied by a infinite product. In the process, he looks at the approximation for the base of the natural logarithms, e, presented in Chapter 2 of Intermediate Physics for Medicine and Biology, e = (1 + 1/N)N, and shows that a “spectacularly better” approximation for e is (1 + 1/N)N+1/2. He then goes on to derive an improved approximation for n!, which is his expression for Stirling’s formula times e(1/12n). Perhaps getting carried away, he then derives even better approximations.

All of this matters little in applications to statistical mechanics, where n is on the order of Avogadro’s number, in which case the first term in Stirling’s formula is utterly negligible. Nevertheless, I urge you to read Mermin’s paper, if only to enjoy the elegance of his writing. To learn more about Mermin’s views on writing physics, see his essay Writing Physics.

Friday, April 6, 2012

Stokes' Law

Stokes' law appears in Chapter 4 of the 4th edition of Intermediate Physics for Medicine and Biology. Russ Hobbie and I write
“For a Newtonian fluid … with viscosity η, one can show (although it requires some detailed calculation6) that the drag force on a spherical particle of radius a is given by

 Fdrag = - β v = -6 π η a v.

This equation is valid when the sphere is so large that there are many collisions of fluid molecules with it and when the velocity is low enough so that there is no turbulence. The result is called Stokes’ law.” 
Footnote 6 says “This is an approximate equation. See Barr (1931, p. 171).”

We can derive the form of Stokes’s law from dimensional reasoning. For a spherical particle of radius a in a fluid moving with speed v and having viscosity η, try to create a quantity having dimensions of force from some combination of a (meter), v (meter/second), and η (Newton second/meter2; see Sec. 1.14). The only way to do this is the combination η a v. You get the form of Stokes’ law, except for the dimensionless factor of 6π. Calculating the 6π was Stokes’ great accomplishment.

In order to learn how Stokes obtained his complete solution, I turn to one of my favorite books on fluid dynamics: Boundary-Layer Theory, by Hermann Schlichting. Consider a sphere of radius R placed into in an unbounded fluid moving with speed U. Assume that the motion occurs at low Reynolds number (a “creeping motion”), so that inertial effects are negligible compared to viscous forces. The Navier-Stokes equation (see Problem 28 of Chapter 1 in Intermediate Physics for Medicine and Biology) reduces to ∇ p = μ ∇2 v, where p is the pressure, μ the viscosity, and v the fluid speed. Assume further that the fluid is incompressible, so that div v = 0 (see Problem 35 of Chapter 1), and that far from the sphere the fluid speed is v=U. Finally, assume no-slip boundary conditions at the sphere surface, so that v=0 at r=R. At this point, let us hear the results in Schlichting’s own words (translated from the original German, of course):
“The oldest known solution for a creeping motion was given by G. G. Stokes who investigated the case of parallel flow past a sphere [17]. The solution of eqns. (6.3) [Navier-Stokes equation] and (6.4) [div v=0] for the case of a sphere of radius R, the centre of which coincides with the origin, and which is placed in a parallel stream of uniform velocity U, Fig. 6.1, along the x-axis can be represented by the following equations for the pressure and velocity components [Eqs. 6.7, which are slightly too complicated to reproduce in this blog, but which involve no special functions or other higher mathematics]. . . The pressure distribution along a meridian of the sphere as well as along the axis of abscissae, x, is shown in Fig. 6.1 [a plot with a peak positive pressure at the upstream edge and a peak negative pressure at the downstream edge]. The shearing-stress distribution over the sphere can also be calculated from the above formula. It is found that the shearing stress has its largest value [at a point along the equator in the sphere center] . . . Integrating the pressure distribution and the shearing stress over the surface of the sphere we obtain the total drag D = 6 π μ R U This is the very well known Stokes equation for the drag of a sphere. It can be shown that one third of the drag is due to the pressure distribution and that the remaining two thirds are due to the existence of shear. . . the sphere drags with it a very wide layer of fluid which extends over about one diameter on both sides.”
Reference 17 is to Stokes, G. G. (1851) On the effect of internal friction of fluids on the motion of pendulums. Phil. Trans. Cambr. Phil. Soc., 9(II):8-106.

Schlichting goes on to analyze the flow around a sphere for high Reynolds number, which is particularly fascinating because in that case viscosity is negligible everywhere except near the sphere surface where the no-slip boundary condition holds. This results in a thin boundary layer forming at the sphere surface. In his introduction, Schlichting writes
“In a paper on ‘Fluid Motion with Very Small Friction’, read before the Mathematical Congress in Heidelberg in 1904, L. Prandtl showed how it was possible to analyze viscous flows precisely in cases which had great practical importance. With the aid of theoretical considerations and several simple experiments, he proved that the flow about a solid body can be divided into two regions: a very thin layer in the neighbourhood of the body (boundary layer) where friction plays an essential part, and the remaining region outside this layer, where friction may be neglected.” 
The book that Russ and I cite in footnote 6 is A Monograph of Viscometry by Guy Barr (Oxford University Press, 1931). I obtained a yellowing and brittle copy of this book through interlibrary loan. It doesn’t describe the derivation of Stokes law in as much detail as Schlichting, but it does consider many corrections to the law, including Oseen’s correction (a first order correction when expanding the drag force in powers of the Reynold’s number), corrections for the effects of walls, consideration of the ends of tubes, and even the mutual effect of two spheres interacting. I found the following sentence, discussing cylinders as opposed to spheres, to be particularly interesting: “Stoke’s approximation leads to a curious paradox when his system of equations is applied to the movement of an infinite cylinder in an infinite medium, the only stable condition being that in which the whole of the fluid, even at infinity, moves with the same velocity as the cylinder.” You can't derive a Stoke's law in two dimensions.

While boundary layer theory and high Reynolds number flow is important for many engineering applications, much of biology takes place a low Reynolds number, where Stokes law applies. (For more about life at low Reynolds number, see Life at Low Reynolds Number by Edward Purcell.)

Stokes’ life is described in Asimov's Biographical Encyclopedia of Science and Technology
Stokes, Sir George Gabriel
British mathematician and physicist
Born: Skreen, Sligo, Ireland, August 13, 1819
Died: Cambridge, England, February 1, 1903

Stokes was the youngest child of a clergyman. He graduated from Cambridge in 1841 at the head of his class in mathematics and his early promise was not belied. In 1849 he was appointed Lucasian professor of mathematics at Cambridge; in 1854, secretary of the Royal Society; and in 1885, president of the Royal Society. No one had held all three offices since Isaac Newton a century and a half before. Stokes’s vision is indicated by the fact that he was one of the first scientists to see the value of Joule’s work.

Between 1845 and 1850 Stokes worked on the theory of viscous fluids. He deduced an equation (Stokes’s law) that could be applied to the motion of a small sphere falling through a viscous medium to give its velocity under the influence of a given force, such as gravity. This equation could be used to explain the manner in which clouds float in air and waves subside in water. It could also be used in practical problems involving the resistance of water to ships moving through it. In fact such is the interconnectedness of science that six decades after Stokes’s law was announced, it was used for a purpose he could never have foreseen—to help determine the electric charge on a single electron in a famous experiment by Millikan. . .”