When more than one pharmacokinetic model is fitted to the same patient and the same measured drug concentrations, compare the models using the same observations, dosing history, and time window. A model is useful when it describes the measured concentrations accurately without adding unnecessary complexity.

1. Sum of Squared Errors

The Sum of Squared Errors measures the total difference between each measured concentration and the concentration predicted by a model. For observation i:

Sum of Squared Errors = Σ_i=1ⁿ (Measured concentration_i − Predicted concentration_i)²

n is the number of measured concentrations. Squaring makes all errors positive and gives greater influence to large errors. Therefore, a lower Sum of Squared Errors means that the model's concentration predictions are closer overall to the patient's measured concentrations. It is meaningful only when the models are evaluated with the same patient observations.

2. Error calculation in the Matzke-Pai model

The Matzke-Pai model is a one-compartment model. It represents the body as one kinetically uniform space and estimates patient-specific parameters, such as volume of distribution and clearance, from the measured concentrations. Its fitting objective may give each concentration a weight based on expected assay variability:

Weighted Sum of Squared Errors = Σ [(Measured concentration_i − Predicted concentration_i)² ÷ (Expected standard deviation_i)²]

With a coefficient-of-variation assumption, the expected standard deviation is often expressed as coefficient of variation × measured concentration. Weighting prevents high-concentration observations from dominating the fit and reflects that measurement uncertainty commonly changes with concentration. The weighted value is the quantity optimized by the fitting procedure; the displayed root-mean-square error remains useful for an easily interpretable concentration-scale summary.

When the fit is Bayesian, the concentration error is combined with a population-parameter penalty. This penalty keeps the patient-fitted parameters reasonably close to the population values unless the patient's measured concentrations provide evidence for a meaningful difference:

One-compartment total error = Weighted concentration error + [((Population volume of distribution − Fitted volume of distribution) ÷ (Parameter variability × Population volume of distribution))² + ((Population clearance − Fitted clearance) ÷ (Parameter variability × Population clearance))²]

The bracketed term is the parameter penalty. For the Matzke-Pai model, the parameters shown are volume of distribution and clearance when both are fitted. If only clearance is fitted, the volume term is omitted. In general, each parameter contributes ((population value − fitted value) ÷ expected parameter standard deviation)². A fitted parameter that is two expected standard deviations from its population value contributes four penalty units. This does not mean that the patient fit is wrong; it means that the improved concentration fit must justify the departure from the population expectation.

3. Error calculation in the two-compartment models

The Goti and Carreno models are two-compartment vancomycin models. They represent an initial central compartment and a peripheral compartment, allowing the model to describe both early distribution and later elimination. Their concentration error is evaluated using the same basic equation:

Two-compartment Sum of Squared Errors = Σ (Measured concentration_i − Two-compartment predicted concentration_i)²

When the fitting procedure uses measurement weighting, the weighted form above is applied to the two-compartment predictions as well. The model structures differ, but the observations and error definition must be matched before comparing results. A lower error for a two-compartment model is especially informative when the patient has early post-dose samples that show distribution or when the one-compartment model systematically misses the early or late concentrations.

A Bayesian two-compartment fit adds the same type of population-parameter penalty, but includes the parameters used by the selected model:

Two-compartment total error = Weighted concentration error + [central volume penalty + peripheral volume penalty + inter-compartmental clearance penalty + systemic clearance penalty]

The central volume is the volume in the circulation-facing compartment, the peripheral volume represents the distribution compartment, inter-compartmental clearance describes movement between the compartments, and systemic clearance describes drug removal. The equation includes only parameters estimated for that specific Goti or Carreno fit; a fixed population parameter contributes no fitted-parameter term. Thus, the two-compartment model receives credit for explaining a patient's concentrations better, but it also pays for every additional parameter that moves away from its population expectation.

4. Akaike Information Criterion

The Akaike Information Criterion combines fit quality with a penalty for the number of estimated parameters:

Akaike Information Criterion = n × ln(Sum of Squared Errors ÷ n) + 2 × k

Here, n is the number of measured concentrations, k is the number of independently estimated model parameters, and ln is the natural logarithm. The parameter penalty matters because a more complex model can always improve its fit by adding flexibility. The Akaike Information Criterion asks whether that improvement is large enough to justify the additional parameters.

5. Practical decision rule

1. Compare models fitted to the same patient data, doses, infusion times, and observation times.
2. Prefer a lower Sum of Squared Errors or root-mean-square error when the difference is clinically meaningful.
3. Use the Akaike Information Criterion to balance that improvement against model complexity. Lower is better.
4. An Akaike Information Criterion difference of less than 2 provides little evidence of a meaningful distinction; a difference above 2 supports the lower-value model, and a difference above 10 provides strong evidence.
5. Confirm the choice using the concentration-time plot, early and late prediction errors, convergence status, and clinical plausibility. Do not select a model from one statistic alone.