Nonlinear regression for the difference of two exponentials

I wanted to provide an overview of how you analyze a classic nonlinear regression model. It is a difference of two exponential functions. This nonlinear function is used commonly in pharmocokinetic models and is a simply way to model the oral administration of a drug. I want to show how the model works in a mathematical sense and then how you fit it using R.

Here are some simple examples of nonlinear regression. We will use the built-in data set Theoph for all our examples.

head(Theoph)

  Subject   Wt Dose Time  conc
1       1 79.6 4.02 0.00  0.74
2       1 79.6 4.02 0.25  2.84
3       1 79.6 4.02 0.57  6.57
4       1 79.6 4.02 1.12 10.50
5       1 79.6 4.02 2.02  9.66
6       1 79.6 4.02 3.82  8.58

tail(Theoph)

    Subject   Wt Dose  Time conc
127      12 60.5  5.3  3.52 9.75
128      12 60.5  5.3  5.07 8.57
129      12 60.5  5.3  7.07 6.59
130      12 60.5  5.3  9.03 6.11
131      12 60.5  5.3 12.05 4.57
132      12 60.5  5.3 24.15 1.17

look at just the first patient.

Earlier versions are here and here.