In last week’s post, I discussed the Bragg peak: protons passing through tissue lose most of their energy near the end of their path. In Chapter 16 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I present a homework problem in which the student calculates the stopping power (energy lost per distance traveled), S, as a function of depth, x, given a relationship between stopping power and energy, T. This problem is a toy model illustrating the physical origin of the Bragg peak. Often its helpful to have two such exercises; one to assign as homework and one to work in class (or put on an exam). Here’s a new homework problem similar to the one in IPMB, but with a different assumption about how stopping power depends on energy.
Problem 31 ½. Assume the stopping power of a particle, S = −dT/dx, as a function of kinetic energy, T, is S = So exp(−T/To).
(a) What are the units of So and To?
(b) If the initial kinetic energy at x = 0 is Ti, calculate T(x).
(c) Determine the range R of the particle as a function of To, So, and Ti.
(d) Plot S(x) vs. x. Does this plot contain a Bragg peak?
(e) Discuss the implications of the shape of S(x) for radiation treatment using this particle.
The answer to part (d) is difficult, because your conclusion is different depending on the relative magnitude of Ti and To. You might consider adding a part (f)
(f) Plot T(x), S(x), and R(Ti) for Ti >> To and for Ti << To.
The case Ti >> To has a conspicuous Bragg peak; the case Ti << To doesn’t.
The homework problem in IPMB is more realistic than this new one, because Fig. 15.17 indicates that the stopping power decreases as 1/T (assumed in the original problem) rather than exponentially (assumed in the new problem). This changes the particle s behavior, particularly at low energies (near the end of its range, in the Bragg peak). Nevertheless, having multiple versions of the problem is useful.
The answer to part (e) is given in IPMB.
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.