How to obtain exponential equation parameters from Simbiology fit of IV bolus pharmacokinetic data?

I have used Simbiology to fit a pharmacokinetic data set (IV bolus administration of a drug to a sample population) to both two- and three-compartment models with linear, first-order rate expressions. Rather than the fit parameters produced by Simbiology (e.g. clearance, inter-compartmental clearance, compartment volumes, etc.), I am most interested in the set of parameters that enter into the poly-exponential equation defining the concentration-time profile. That is the A_i's and alpha_i's in

c(t) = \sum_{i=1}^{i=n} A_i exp(-\alpha_i t)

for an n-compartment model. Other than deriving or looking up these relationships in a book, is there a way to get Simbiology to produce them with (importantly) their associated error estimates?

Answer 1

[Arthur Goldsipe]

If your goal is to estimate the coefficients of a poly-exponential and their corresponding error estimates, then I think it's easier to do that in directly rather than try to relate these coefficients to a SimBiology model. For example, you could just use non-linear regression with the poly-exponential as the model function. Statistics toolbox functions like nlinfit and fitnlm support this, and the Curve Fitting Toolbox also provides convenient ways to do this sort of problem.

I'm a developer on the SimBiology team, and this is the first time I personally have heard of someone being interested in the coefficients of a poly-exponential. I always thought folks were more interested in the actual PK parameters than the poly-exponential coefficients. I would like to hear more about why you're doing this. If you want, send me a note telling me a little more about what you're doing and why this functionality is important. We can't add the functionality until we know what our customers need and why.

[Alex Bukoski]

Thanks for your response Arthur. As a consumer of PK data (I'm a veterinary anesthesiologist), I am at times interested in classical PK parameters (e.g. clearance) and at other times the exponential c(t) function. My current interest is the latter as my ultimate goal is to design an infusion scheme to maintain plasma concentrations within a narrow window. The method boils down to a convolution of the poly-exponential c(t) function with a function describing a piecewise continuous infusion rate. Relationships between the c(t) poly-exponential parameters and the PK parameters do exist or can be derived by solving the first-order system analytically but I was hoping to avoid doing the error propagation.

Compartmental models have somewhat limited utility in clinical pharmacokinetics given the identifiability issue with linear first-order systems and their lack of correspondence to true physiological spaces. For this reason I am actually rather surprised that Simbiology does not have this functionality built in. I will admit that I erroneously assumed this functionality would exist when I purchased it. Although there is a trend in the literature toward reporting NCA results, one often still sees the poly-exponential equation reported for basic PK work.

The issue I have with using nlinfit directly (I do have the statistics toolbox) is that it appears I would then need to purchase yet another toolbox to obtain the commonly used AIC and BIC information criteria for model discrimination. These already exist in Simbiology. I understand that Simbiology is a tool with broader application areas in mind than just pharmacokinetics and can see why it approaches the PK-modelling area the way it does. Any further advice you can offer would be greatly appreciated.

--Alex Bukoski

[Arthur Goldsipe]

Thanks for the additional information. As you guessed, SimBiology is primarily designed to model systems that are more general than linear first-order systems. We have done some additional work to support linear systems, but obviously there is more we could do to support the kind of work you're doing.

In the meantime, I did think of one other idea that could allow you to stay in SimBiology. Essentially, you can do the same nonlinear regression I suggested you do in nlinfit using SimBiology. I think it would be more work than doing the same thing directly with nlinfit, but you could create a SimBiology model with a repeated assignment that's a poly-exponential. For example, you could create a rule something like this:

c = A1*exp(-alpha1*time) + A2+exp(-alpha2*time)

where c is a species, and A1, A2, alpha1, and alpha2 are SimBiology parameters.