Developing models for pharmacokinetic or pharmacodynamic data requires creativity, patience, and hard work. Sometimes that creativity ends up violating mathematical principles which can lead to poorly fitting models and frustration for a modeler. One common area where mistakes are made is related to the analysis of parent and metabolite data simultaneously. Because data are available for both analytes (parent drug and metabolite), it seems like it should be a piece of cake … and sometimes it is. But other times we forget about a key mathematical principle called identifiability.
You can look up a complex definition of identifiability on wikipedia (link), but I much prefer a simple explanation that my father gave when helping me with Algebra many years ago …
You can only solve the set of equations if you have the same number of equations as unknowns. Two unknowns require two equations, three unknowns require three equations, and so on.
Some people are visual learners, so here is what this looks like in equation form:
Equation Set 1
There are 2 equations and 2 unknowns (x and y) in that set of equations. This set of equations is identifiable. It is also solvable using basic algebra. Now here is a second set of equations to consider:
Equation Set 2
Now we have 2 equations and 3 unknowns (x, y, and z) in this set of equations. This set of equations has an identifiability problem. Everything will have to be solved in terms of z because we don’t have enough information to solve each unknown. We could insert values for z and then determine what x and y are for each specific value of z; however, we cannot “solve” the set of equations like we did in Equation Set 1.
How does this relate to parent/metabolite modeling? Assume we have concentration-time data for both parent and metabolite following intravenous (IV) administration of the parent drug. We have no data on the metabolite being administered alone. Also, let’s assume that both the parent drug and metabolite follow first order elimination processes. In this situation, we would want to estimate the following parameters:
Not all of these parameters are independent of one another. In particular, the volume of distribution for the metabolite (Vm) represents the relationship between the amount of metabolite in the body to the concentration of metabolite being measured in the blood/plasma/serum. And the amount of metabolite in the body is estimated by the fraction of parent drug converted to metabolite (fm). Thus we cannot estimate both Vm and fm. We have more variables than equations!
We can estimate the Vp and CLp for the parent drug because we know the dose. We don’t know the dose of the metabolite (i.e., fm) which creates the identifiability problem. There are three possible solutions to this dilemma for parent/metabolite modeling work:
- Estimate the volume of distribution for the metabolite following administration of the metabolite directly. Use the estimated value for Vm in the model and only estimate fm.
- If the fraction of parent drug converted to metabolite is known from a previous study (e.g. mass balance study), use that estimated fraction fm in the model and only estimate Vm.
- Assume the the volume of distribution for the parent and metabolite are identical and estimate both fm and Vm, where Vm=Vp.
Solutions 1 and 2 reduce the number of unknowns, and solution 3 adds one more equation. In essence, all three solutions make the number of equations equivalent to the number of unknowns, allowing for the set of equations to be identifiable.
While I have only presented one example, there are many more situations where creative model-builders construct models that are not identifiable. So if you have an unstable model, or seem to have parameters that are highly sensitive to initial estimates, step back and look to see if you have an identifiability problem.
Identifiability is a common problem with complex QSP models. Watch this webinar to learn how why reduction methods are a potent and necessary tool in the modeler’s arsenal; how reduction methods can be applied to QSP models; and, how model reduction can be used to extract scientific and business insights from complex models.