Q: What is the right model to use?

A: There is no such thing as the "right" model. The "right model" comes from a combination of goodness of fit and model structure.

The goal is to find the simplest model that explains the experimental data. The Akaike Information Criterion (AIC) and the Schwarz-Bayesian Information Criterion (BIC) are "parsimony criteria" useful to perform model comparison and discrimination, e.g. decide whether a sum of two or three exponentials, or a model with two or three compartments better fit the available data.

To this extent, AIC and BIC are a function both of the goodness of fit(given by the objective function L(p*,u*)), the number of adjustable model parameters P and variance parameters Q, and the total number of data points N. Between two or more rival models, the model with the lowest AIC or BIC (that is, the model that better explains the data with the least number of parameters) is the "best" one.

For example, see Carson, Cobelli and Finkelstein, The Mathematical Modeling of Metabolic and Endocrine Systems, New York: John Wiley, 1983, p.251 for a case study. Note that AIC and BIC are calculated in a slightly different fashion.