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