A Toy Model for Straggling

One of the homework problems in Intermediate Physics for Medicine and Biology (Problem 31 in Chapter 16) introduces a toy model for the Bragg peak. I won’t review that entire problem, but students derive an equation for the stopping power, S, (the energy per unit distance deposited in tissue by a high energy ion) as a function of the depth below the tissue surface, x

where S₀ is the ion’s stopping power at the surface (x = 0) and R is the ion’s range. At a glance you can see how the Bragg peak arises—the denominator goes to zero at x = R so the stopping power goes to infinity. That, in fact, is why proton therapy for cancer is becoming so popular: Energy is deposited primarily at one spot well below the tissue surface where a tumor is located, with only a small dose to upstream healthy tissue. 

One topic that comes up when discussing the Bragg peak is straggling. The idea is that the range is not a single parameter. Instead, protons have a distribution of ranges. When preparing the 6th edition of Intermediate Physics for Medicine and Biology, I thought I would try to develop a toy model in a new homework problem to illustrate straggling. 

Section 16.10 

Problem 31 ½. Consider a beam of protons incident on a tissue. Assume the stopping power S for a single proton as a function of depth x below the tissue surface is

Furthermore assume that instead of all the protons having the same range R, the protons have a uniform distribution of ranges between R – δ/2 and R + δ/2, and no protons have a range outside this interval. Calculate the average stopping power by integrating S(x) over this distribution of ranges. 

This calculation is a little more challenging than I had expected. We have to consider three possibilities for x. 

x < R — δ/2

In this case, all of the protons contribute so the average stopping power is

We need to solve the integral 

First, let

With a little analysis, you can show that

So the integral becomes

This new integral I can look up in my integral table

Finally, after a bit of algebra, I get

Well, that was a lot of work and the result is not very pretty. And we are not even done yet! We still have the other two cases. 

 R — δ/2 <  x <  R + δ/2

In this case, if the range is less than x there is no contribution to the stopping power, but if the range is greater than x there is. So, we must solve the integral

I’m not going to go through all those calculations again (I’ll leave it to you, dear reader, to check). The result is 

x   >  R + δ/2

This is the easy case. None of the protons make it to x, so the stopping power is zero. 

Well, I can’t look at these functions and tell what the plot will look like. All I can do is ask Mr. Mathematica to make the plot (he’s much smarter than I am). Here’s what he said: 

The peak of the “pure” (single value for the range) curve (the red one) goes to infinity at x = R, and is zero for any x greater than R. As you begin averaging, you start getting some stopping power past the original range, out to R + δ/2. To me the most interesting thing is that for x = R – δ/2, the stopping power is larger than for the pure case. The curves all overlap for x  > R + δ/2 (of course, they are all zero), and for fairly small values x (in these cases, about x <  0.5) the curves are all nearly equal (indistinguishable in the plot). Even a small value of δ (in this case, for a spread of ranges equal to one tenth the pure range), the peak of the stopping power curve is suppressed. 

The curves for straggling that you see in most textbooks are much smoother, but that’s because I suspect they assume a smoother distribution of range values, such as a normal distribution. In this example, I wanted something simple enough to get an analytical solution, so I took a uniform distribution over a width δ . 

Will this new homework problem make it into the 6th edition? I’m not sure. It’s definitely a candidate. However, the value of toy models is that they illustrate the physical phenomenon and describe it in simple equations. I found the equations in this example to be complicated and not illuminating. There is still some value, but if you are not gaining a lot of insight from your toy model, it may not be worth doing. I’ll leave the decision of including it in the 6th edition to my new coauthor, Gene Surdutovich. After all, he’s the expert in the interaction of ions with tissue.