# Appendix: Origin of Mathematical Models for Anesthetic Drug–Drug Interactions

Modeling is the technique of defining a mathematical equation that fits a data set as accurately as possible. Linear regression could be considered as a rudimentary form of modeling. A linear regression analysis of a data set determines the best-fitting linear correlation between data couples by minimizing the sum of squares of all perpendicular distances from each observation to the proposed line (least-squares principle). As such, the optimal value for parameters *a* and *b* in an equation of the form *y* = *a* ⋅ *x* + *b* can be determined from an observational data set. The resulting linear equation describes the correlation of the data couples within the population with the least possible scatter (or residual error). Consequently, the resulting equation can be used to predict prospectively which *y* can be expected for every known *x*, with the smallest possible error.

In many pharmacologic processes, the equations that best describe the data are nonlinear in nature. For example, the classical dose- or concentration-response curve is generally described as a sigmoidal E_{max} curve, also called the Hill equation. This equation includes 4 parameters that need to be estimated from data: baseline value of the effect variable (E_{0}), maximal possible effect obtained at high doses (E_{max}), the effective dose or concentration related to 50% of E_{max} (ED_{50} or EC_{50}), and the slope of the dose or concentration versus effect curve (γ). The ED_{50} or EC_{50} represents potency of the drug, and E_{max} reflects efficacy of the drug.

In order to fit nonlinear equations (also called structural models) on a data set, a statistical approach called *nonlinear mixed effects modeling* (as performed by the software package NONMEM (ICON Development Solutions, Hanover, MD) has become the standard methodology. Several structural models have been applied on anesthesia interaction data. We will review the major differences between structural models later.

In NONMEM, the optimal structural model is selected when it results in the lowest objective function value (OFV), that is, the minimum value of the “–2 log likelihood” function. A low OFV reflects the ability of the model to describe the data with the least amount of scatter. Mixed effects refers to the combination of fixed effects and random effects. Fixed effects are sources of variability in the observation that are the result of covariates that we can measure. For example, we can categorize the studied subjects into relevant groups according to differences in fixed effects (eg, age, weight). If the inclusion of a fixed effect in the structural model improves the fit on the data set significantly, this covariate is considered a relevant improvement for the predictive performance of the model. On the other hand, even after having defined a number of significant fixed effects in the structural model, a residual amount of error will always remain. This is caused by random effects—for example, ...