Proton Therapy

Section 16.11.3 in the 4th edition of Intermediate Physics for Medicine and Biology discusses proton therapy.

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.

Proton therapy has become popular recently: see articles in US News and World Report and on MSNBC. There even exists a National Association for Proton Therapy. Their website explains the main advantage of protons over X-rays.

Both standard x-ray therapy and proton beams work on the principle of selective cell destruction. The major advantage of proton treatment over conventional radiation, however, is that the energy distribution of protons can be directed and deposited in tissue volumes designated by the physicians in a three-dimensional pattern from each beam used. This capability provides greater control and precision and, therefore, superior management of treatment. Radiation therapy requires that conventional x-rays be delivered into the body in total doses sufficient to assure that enough ionization events occur to damage all the cancer cells. The conventional x-rays lack of charge and mass, however, results in most of their energy from a single conventional x-ray beam being deposited in normal tissues near the body’s surface, as well as undesirable energy deposition beyond the cancer site. This undesirable pattern of energy placement can result in unnecessary damage to healthy tissues, often preventing physicians from using sufficient radiation to control the cancer.

Protons, on the other hand, are energized to specific velocities. These energies determine how deeply in the body protons will deposit their maximum energy. As the protons move through the body, they slow down, causing increased interaction with orbiting electrons.

Figure 16.51 of the 4th edition of Intermediate Physics for Medicine and Biology shows the dose versus depth from a 150 MeV proton beam, including the all-important Bragg peak located many centimeters below the tissue surface. If you want to understand better why proton energy is deposited in the Bragg peak rather than being spread throughout the tissue, solve Problem 31 in Chapter 16.

To learn more about the pros and cons of proton therapy, I suggest several “point/counterpoint” articles from the journal Medical Physics: “Within the Next Decade Conventional Cyclotrons for Proton Radiotherapy will Become Obsolete and Replaced by Far Less Expensive Machines using Compact Laser Systems for the Acceleration of the Protons,” Chang-Ming Ma and Richard Maughan (Medical Physics, Volume 33, Pages 571–573, 2006), “Proton Therapy is the Best Radiation Treatment Modality for Prostate Cancer,” Michael Moyers and Jean Pouliot (Medical Physics, Volume 34, Pages 375–378, 2007), and “Proton Therapy is Too Expensive for the Minimal Potential Improvements in Outcome Claimed,” Robert Schulz and Alfred Smith (Medical Physics, Volume 34, Pages 1135–1138, 2007).

The Diffusion Approximation to Photon Transport

Chapter 14 in the 4th edition of Intermediate Physics for Medicine and Biology contains a section describing the diffusion approximation to photon transport.

When photons enter a substance, they may scatter many times before being absorbed or emerging from the substance. This leads to turbidity, which we see, for example, in milk or clouds. The most accurate studies of multiple scattering are done with “Monte Carlo” computer simulation, in which probabilistic calculations are used to follow a large number of photons as they repeatedly interact in the tissue being simulated. However, Monte Carlo techniques use lots of computer time. Various approximate analytical solutions also exist... One of the approximations, the diffusion approximation, is described here. It is valid when many scattering events occur for each photon absorption.

Optical Mapping of Cardiac
Excitation and Arrhythmias,
edited by Rosenbaum and Jalife.

Today, I would like to present a new homework problem about the diffusion approximation, based on a brief communication I published in the August 2008 issue of IEEE Transactions on Biomedical Engineering (Volume 55, Pages 2102–2104). I was interested in the problem because of its role in optical mapping of transmembrane potential in the heart, discussed briefly at the end of Sec 7.10 and reviewed exhaustively in the excellent book Optical Mapping of Cardiac Excitation and Arrhythmias, edited by David Rosenbaum and Jose Jalife. Enjoy the problem, which belongs at the bottom of the left column of page 394.

Section 14.5

Problem 16 ½ Consider light with fluence rate φ₀ continuously and uniformly irradiating a half-infinite slab of tissue having an absorption coefficient μ_a and a reduced scattering coefficient μ'_s. Divide the photons into two types: the incident ballistic photons that have not yet interacted with the tissue, and the diffuse photons undergoing multiple scattering. The diffuse photon fluence rate, φ, is governed by the steady state limit of the photon diffusion equation (Eq. 14.26). The source of diffuse photons is the scattering of ballistic photons, so the source term in Eq. 14.26 is s = μ'_s exp(-z/λ_unatten), where z is the depth below the tissue surface. At the surface (z=0), the diffuse photons obey the boundary condition φ = 2 D dφ/dz.

(a) Derive an analytical expression for the diffuse photon fluence rate in the tissue, phi(z). 

(b) Plot φ(z) versus z for μ_a=0.08 mm⁻¹ and μ'_s=4 mm⁻¹.  

(c) Evaluate λ_unatten and λ_diffuse for these parameters.

The most interesting aspect of this calculation is that the diffuse photon fluence rate is not maximum at the tissue surface, but rather it builds up to a peak below the surface, somewhat like the imparted energy from 10 MeV photons shown in Fig. 15.32. This has some interesting implications for optical mapping of the heart: subsurface tissue may contribute more to the optical signal than surface tissue.

If you want the solution, send me an email (roth@oakland.edu) and I will gladly supply it.

We Should All Congratulate Professor Hobbie For This Excellent Text

Peter Kahn reviewed the third edition of Intermediate Physics for Medicine and Biology in the American Journal of Physics (volume 67, Pages 457–458, 1999). He wrote:

As a professor of physics I am upset that our biology students have such brief and superficial exposure to physics and mathematics, and that, at the same time, our physics students go through a curriculum that ignores the important role that biology is playing in modern science. We should all congratulate Professor Hobbie for this excellent text. Now it is up to us to initiate the dialogue that builds on this solid foundation.

Div, Grad, Curl, and All That

Russ Hobbie and I assume that readers of the 4th edition of Intermediate Physics for Medicine and Biology know the basics of calculus (our preface states that “calculus is used without apology”). We even introduce some concepts from vector calculus, such as the divergence, gradient, and curl. Although these vector derivatives are crucial for understanding topics such as diffusion and electricity, many readers may be unfamiliar with them. These functions are even more complicated in curvilinear coordinate systems, and in Appendix L we summarize how to write the divergence, gradient, curl, and Laplacian in rectangular, cylindrical, and spherical coordinates.

Div, Grad, Curl, and All That,
by H. M. Schey.

When I was a young physics student at the University of Kansas, Dr. Jack Culvahouse gave me a book that helped explain vector calculus: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, by H. M. Schey. For me, this book made clear and intuitive what had been confusing and complicated. By defining the divergence and curl in terms of surface and line integrals, I suddenly could understand what these seemingly random collections of partial derivatives meant. One can hardly make sense of Maxwell’s equations of electromagnetism without vector calculus (try reading a textbook from Maxwell’s era before vector calculus was invented if you don't believe me). In fact, Schey introduces vector calculus using electromagnetism as his primary example:

In this text the subject of vector calculus is presented in the context of simple electrostatics. We follow this procedure for two reasons. First, much of vector calculus was invented for use in electromagnetic theory and is ideally suited to it. This presentation will therefore show what vector calculus is and at the same time give you an idea of what it's for. Second, we have a deep-seated conviction that mathematics—in any case some mathematics—is best discussed in a context that is not exclusively mathematical. Thus, we will soft-pedal mathematical rigor, which we think is an obstacle to learning this subject on a first exposure to it, and appeal as much as possible to physical and geometric intuition.

For readers of Intermediate Physics for Medicine and Biology who get stuck when we delve into vector calculus, I suggest setting our book aside for a few days (but only a few!) to read Div, Grad, Curl, and All That. Not only will you be able to understand our book better, but you’ll find this background useful in many other fields of physics, math, and engineering.