Friday, September 26, 2014

The First Steps in Seeing

Russ Hobbie and I discuss the eye and vision in Chapter 14 of the 4th edition of Intermediate Physics for Medicine and Biology. But we just barely begin to describe the complexities of how we perceive light. If you want to learn more, read The First Steps in Seeing, by Robert Rodieck. This excellent book explains how the eye works. The preface states
“This book is about the eyes—how they capture an image and convert it to neural messages that ultimately result in visual experience. An appreciation of how the eyes work is rooted in diverse areas of science—optics, photochemistry, biochemistry, cellular biology, neurobiology, molecular biology, psychophysics, psychology, and evolutionary biology. This gives the study of vision a rich mixture of breadth and depth.

The findings related to vision from any one of these fields are not difficult to understand in themselves, but in order to be clear and precise, each discipline has developed its own set of words and conceptual relations—in effect is own language—and for those wanting a broad introduction to vision, these separate languages can present more of an impediment to understanding than an aid. Yet what lies beneath the words usually has a beautiful simplicity.

My aim in this book is to describe how we see in a manner understandable to all. I’ve attempted to restrict the number of technical terms, to associate the terms that are used with a picture or icon that visually express what they mean, and to develop conceptual relations according to arrangements of these icons, or by other graphical means. Experimental findings have been recast in the natural world whenever possible, and broad themes attempt to bring together different lines of thought that are usually treated separately.

The main chapters provide a thin thread that can be read without reference to other books. They are followed by some additional topics that explore certain areas in greater depth, and by notes that link the chapters and topics to the broader literature.

My intent is to provide you with a framework for understanding what is known about the first steps in seeing by building upon what you already know.”
Rodieck explains things in a quantitative, almost “physicsy” way. For instance, he imagines a person staring at the star Polaris, and estimates the number of photons (5500) arriving at the eye each tenth of a second (approximately the time required for visual perception), then determines their distribution on the retina, finds how many are at each wavelength, and how many per cone cell.

Color vision is analyzed, as are the mechanisms of how rhodopsin responds to a photon, how the photoreceptor produces a polarization of the neurons, how the retina responds with such a large dynamic range (“the range of vision extends from a catch rate of about one photon per photoreceptor per hour to a million per second”), and how eye movements hold an image steady on the retina. There is even a discussion of photometry, with a table similar to the one I presented last week in this blog. I learned that the unit of retinal illuminance is the troland (td), defined as the luminance (candelas per square meter) times the pupil area (square millimeters).

Like IPMB, Rodieck ends his book with several appendices, including a first one on angles. His appendix on blackbody radiation includes in a figure showing the Planck function versus frequency plotted on log-log paper (I've always seen it plotted on linear axes, but the log-log plot helps clairfy the behavior at very large and small frequencies). The photon emission from the surface of a blackbody as a function of temperature is 1.52 x 1015 T3 photons per second per square meter (Rodieck does everything in terms of the number of photons). The factor of temperature cubed is not a typo; Stefan's law contains a T3 rather than T4 when written in terms of photon number. A lovely appendix analyzes the Poisson distribution, and another compares frequency and wavelength distributions.

The best feature of The First Steps in Seeing are the illustrations. This is a beautiful book. I suspect Rodieck read Edward Tufte’s the Visual Display of Quantitative Information, because his figures and plots elegantly make his points with little superfluous clutter. I highly recommend this book.

Friday, September 19, 2014

Lumens, Candelas, Lux, and Nits

In Chapter 14 (Atoms and Light) of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss photometry, the measurement of electromagnetic radiation and its ability to produce a human visual sensation. I find photometry interesting mainly because of all the unusual units.

Let’s start by assuming you have a source of light emitting a certain amount of energy per second, or in other words with a certain power in watts. This is called the radiant power or radiant flux, and is a fundamental concept in radiometry. But how do we perceive such a source of light? That is a question in photometry. Our perception will depend on the wavelength of light. If the light is all in the infrared or ultraviolet, we won’t see anything. If in the visible spectrum, our perception depends on the wavelength. In fact, the situation is even more complicated than this, because our perception depends on if we are using the cones in the retina of our eye to see bright light in color (photopic vision), or we are using rods to see dim light in black and white (scotopic vision). Moreover, our ability to see varies among individuals. The usual convention is to assume we are using photopic vision, and to say that a source radiating a power of one watt of light at a wavelength of 555 nm (green light, the wavelength that the eye is most sensitive to) has a luminous flux of 683 lumens.

The light source may emit different amounts of light in different directions. In radiometry, the radiant intensity is the power emitted per solid angle, in units of watt per steradian. We can define an analogous photometric unit for the luminous intensity to be the luman per steradian, or the candela. The candela is one of seven “SI base units” (the others are the kilogram, meter, second, ampere, mole, and kelvin). Russ and I mention the candela in Table 14.6, which is a large table that compares radiometric, photometric and actinometric quantities. We also define it in the text, using the old-fashioned name “candle” rather than candela.

Often you want to know the intensity of light per unit area, or irradiance. In radiometry, irradiance is measured in watts per square meter. In photometry, the illuminance is measured in lumens per square meter, also called the lux.

Finally, the radiance of a surface is the radiant power per solid angle per unit surface area (W sr-1 m-2). The analogous photometric quantity is the luminance, which is measured in units of lumen sr-1 m-2, or candela m-2, or lux sr-1, or nit. The brightness of a computer display is measured in nits.

In summary, below is an abbreviated version of Table 14.6 in IPMB

Radiometry Photometry
Radiant power (W) Luminous flux (lumen)
Radiant Intensity (W sr-1) Luminous intensity (candela)
Irradiance (W m-2) Illuminance (lux)
Radiance (W sr-1 m-2) Luminance (nit)

Where did the relationship between 1 W and 683 lumens come from? Before electric lights, a candle was a major source of light. A typical candle emits about 1 candela of light. The relationship between the watt and the lumen is somewhat analogous to the relationship between absolute temperature and thermal energy, and the relationship between a mole and the number of molecules. This would put the conversion factor of 683 lumens per watt in the same class as Boltzmann's constant (1.38 x 10-23 J per K) and Avogadro's number (6.02 x 1023 molecules per mole).

Friday, September 12, 2014

More about the Stopping Power and the Bragg Peak

The Bragg peak is a key concept when studying the interaction of protons with tissue. In Chapter 16 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I write
Protons are also used to treat tumors. Their advantage is the increase of stopping power at low energies. It is possible to make them come to rest in the tissue to be destroyed, with an enhanced dose relative to intervening tissue and almost no dose distally (‘downstream’) as shown by the Bragg peak in Fig. 16.51 [see a similar figure here]. Placing an absorber in the proton beam before it strikes the patient moves the Bragg peak closer to the surface. Various techniques, such as rotating a variable-thickness absorber in the beam, are used to shape the field by spreading out the Bragg peak (Fig. 16.52) [see a similar figure here].”
Figure 16.52 is very interesting, because it shows a nearly uniform dose throughout a region of tissue produced by a collection of Bragg peaks, each reaching a maximum at a different depth because the protons have different initial energies. The obvious question is: how many protons should one use for each energy to produce a uniform dose in some region of tissue? I have discussed the Bragg peak before in this blog, when I presented a new homework problem to derive an analytical expression for the stopping power as a function of depth. An extension of this problem can be used to answer this question. Russ and I considered including this extended problem in the 5th edition of IPMB (which is nearing completion), but it didn’t make the cut. Discarded scraps from the cutting room floor make good blog material, so I present you, dear reader, with a new homework problem.

Problem 31 3/4 A proton of kinetic energy T is incident on the tissue surface (x = 0). Assume its stopping power s(x) at depth x is given by
where C is a constant characteristic of the tissue.
(a) Plot s(x) versus x. Where does the Bragg peak occur?

(b) Now, suppose you have a distribution of N protons. Let the number with incident energy between T and T+dT be A(T)dT, where
Determine the constant B by requiring
Plot A(T) vs T.
(c) If x is greater than T22/2C what is the total stopping power? Hint: think before you calculate; how many particles can reach a depth greater than T22/2C?

(d) If x is between T12/2C and T22/2C, only particles with energy from (2Cx)1/2 to T2 contribute to the stopping power at x, so
Evaluate this integral. Hint: let u = T2 - (2Cx + T22)/2.
(e) If x is less than T12/2C, all the particles contribute to the stopping power at x, so

Evaluate this integral.

(f) Plot S(x) versus x. Compare your plot to that found in part a, and to Fig. 16.52.

One reason this problem didn’t make the cut is that it is rather difficult. Let me know if you need the solution. The bottom line: this homework problem does a pretty good job of explaining the results in Fig. 16.52, and provides insight into how to apply proton therapy to an large tumor.

Friday, September 5, 2014

Raymond Damadian and MRI

The 2003 Nobel Prize in Physiology or Medicine was awarded to Paul Lauterbur and Sir Peter Mansfield "for their discoveries concerning magnetic resonance imaging". In Chapter 18 of the 4th Edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss MRI and the work behind this award. Choosing Nobel Prize winners can be controversial, and in this case some suggest that Raymond Damadian should have shared in the prize. Damadian himself famously took out an ad in the New York Times claiming his share of the credit. Priority disputes are not pretty events, but one can gain some insight into the history of magnetic resonance imaging by studying this one. The online news source The Why Files tells the story in detail. The controversy continues even today (see, for instance, the website of Damadian's company FONAR). Unfortunately, Damadian’s religious beliefs have gotten mixed up in the debate.

I think the issue comes down to a technical matter about MRI. If you believe the variation of T1 and T2 time constants among different tissues is the central insight in developing MRI, then Damadian has a valid claim. If you believe the use of magnetic field gradients for encoding spatial location is the key insight in MRI, his claim is weaker than Lauterbur and Mansfield's. Personally, I think the key idea of magnetic resonance imaging is using magnetic field gradients. IPMB states “Creation of the images requires the application of gradients in the static magnetic field Bz which cause the Larmor frequency to vary with position.” My understanding of MRI history is that this idea originated with Lauterbur and Mansfield (and was also earlier discovered by Hermann Carr).

To learn more, I suggest you read Naked to the Bone, which I discussed previously in this blog. This book discusses both the Damadian controversy, and a similar controversy centered around William Oldendorf and the development of Computed Tomography (which is mentioned in IPMB).