In a pharmacometrics model, certain compartments can have an associated lag time. For example, if I give a patient at time t = 0 a drug in compartment 1, which has tlag = 1, the drug amount in that compartment increases not a t = 0, but t = 1.
A bolus dose causes an instantaneous increase. So there is a discontinuity with respect to time, but also to the parameter tlag. The derivative is ill-defined and the function is only half-continuous.
The way lag times are handled is by augmenting the event schedule. A dose at t = 0 with lag time tlag is really a dose at t = tlag with no lag time. So I create another event
if (tlag != 0). While the derivative with respect to tlag is ill-defined at the dosing time, it should be defined at other times, i.e. when the body clears the drug. No problem with finite-diff, but autodiff does not handle the boolean
tlag != 0 when differentiating with respect to tlag. The IF statement doesn’t get “auto-diffed”.
Is the derivative mathematically ill-defined when evaluated at tlag = 0? I don’t think this is a major issue. When tlag = 0, it is usually fixed, and rarely a parameter. Then again, who knows what a user may come up with…
Reasons for using the IF statement
I could remove the if statement and create a new dosing event at t + tlag even when tlag = 0. The auto-diff then agrees with the finite diff. However, the new dosing event occurs after the original event, so the predicted amount at the original event is wrong. If I change the order of the events, I create a conditional statement that doesn’t get auto-diffed.