Chapter Five, toxicology, is challenging for educators and students alike. Here I present some guidance based on my experience teaching from it. The Integrated Toxicology and Epidemiology Lecture covers a lot of the material in Chapters Five and Six.

Overview. It is easy to come away from this chapter feeling that we really have little idea about what we are doing when modeling low-dose carcinogenesis. This is true...and it isn't. One big problem is the misconception that human health conservative or regulatory assumptions are "antiscientific" and invariably end up overestimating risks. They may underestimate risk. They may overestimate risk. They may be irrelevant. They certainly do not calculate "actual risk." Nonetheless, having some set of tools with which to scope out the potential range of risks is enormously helpful, and far better than nothing at all. Challenge your students to avoid spurious precision, and to integrate quantitative and qualitative discussions of uncertainty into their problem sets. Also challenge them NOT to just recreate "official" methods...these can always be picked up by someone who understands the underlying assumptions and methods.

Why Cancer? In our graduate and undergraduate programs, we have an excellent course on non-cancer toxicology, so in my risk classes I only touch on those and focus on carcinogen risk analysis. If your students are not getting it elsewhere, be sure to discuss non-cancer endpoints extensively.

Fancy math tricks. For mathematical types, the middle of chapter five is lots of fun...but it is also a lesson in how uncertain we are about these extrapolation methods, and students should be DISCOURAGED from thinking that they've found the right answer just because they get the models right! For the big picture: compare the three q*'s using different mechanistic assumptions, then look at where Tony Cox takes these data using a non-mechanistic curve fitting, and the q* modeled on the EPA Proposed Guidelines for Carcinogen Risk Analysis, which you can find in the Integrated Toxicology and Epidemiology Lecture.

Animal to human dose. (Pages 187 to 189). I think the explanation in the Integrated Toxicology and Epidemiology Lecture is a bit clearer than the version in the book. But students will probably do best by seeing it BOTH ways.

Bootstrapping. Anyone can, with patience, set up a spreadsheet and run a bootstrap program. But there is an important element of artistry to ensuring that one finds true, and not local minima and maxima.

• Many bootstrap programs don't like numbers that are either far from or close to zero: taking logs, multiplying or dividing by multiples of 10 are handy ways to keep parameters to reasonable sizes.
• Take students through bootstrapping one step at a time:
• First try to get a linear function to get as close as possible to two data points by minimizing squares. No problem, since a line is defined by two points!
• Then move on to fitting a line to multiple points, then maximizing likelihoods instead of minimizing squares, then fitting nonlinear equations.
• DON'T let students fit curves using Excel or other curve fitting functions. They don't learn a thing that way! (Alternatively, let them do it...but as part of a problem set make them explain in their own words how the function works!)

Errata in Chapter Five (Paperback Edition)

• Page 175, table 5-3 title. Crump and Howe used mice, not rats.
• Page 178, table 5-5. Here, we used the conversion for rat data rather than those for mouse data...so the values should be 175, 350 and 700 mg/kg/day in the equivalent human dose column. Mercifully, this factor of two is less significant than the uncertainty associated with the different extrapolation methods. One handy shortcut (be sure to be explicit) is to just go ahead and use the data as shown, and acknowledge the error.

• Page 181-182. At the bottom of 181 we say the q* is 0.0025, while on the next page we use 0.0015. I believe the latter is correct, and I will rebuild the model and check when I get a chance. In the meantime, note that this difference is noise compared to the differences between the one-hit, two-hit and two-stage models.

• Page 188. About two-thirds of the way down the page: should be ...ADDlifetime/0.090). (not /0.90). This is a typo: the calculations are correct.