The Bragg Peak (Continued)

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 it’s 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.

Section 16.10

Problem 31 ½. Assume the stopping power of a particle, S = −dT/dx, as a function of kinetic energy, T, is S = S_o e^−T/T_o. 

(a) What are the units of S_o and T_o? 

(b) If the initial kinetic energy at x = 0 is T_i, calculate T(x). 

(c) Determine the range R of the particle as a function of T_o, S_o, and T_i. 

(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 T_i and T_o. You might consider adding a part (f)

(f) Plot T(x), S(x), and R(T_i) for T_i >> T_o and for T_i << T_o.

The case T_i >> T_o has a conspicuous Bragg peak; the case T_i << T_o 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.

 Enjoy!