Population Simulation Outputs Equations

When running VBE trials using the Population Simulator, the following outputs are produced:

Arithmetic and geometric means for the following pharmacokinetic endpoints:
- F_a - The fraction of drug absorbed.
- FD_p - The fraction of the dose that reaches the portal vein.
- F - The bioavailability of the drug.
- C_max- The maximum concentration of the drug in plasma.
- T_(max) - The time to C_max of the drug.
- AUC_(t) - The measure of the total exposure of the drug to the body at time t.
- AUC_inf - The measure of the total exposure of the drug to the body from. administration until all the drug is eliminated.
Coefficients of variation.
90% confidence intervals, calculated from both ln-transformed and untransformed data.
Minimum and maximum values for all pharmacokinetic endpoints.

Equation 6-1 through Equation 6-6 show the relevant calculations for these outputs.

Equation 6-1: Geometric mean of x values

Equation 6-2: Lower (L) and upper (U) bounds for 90% confidence intervals calculated from untransformed data

Equation 6-3: Lower (L) and upper (U) bounds for 90% confidence intervals calculated from ln-transformed data

where for all the above equations:

Variable	Definition
	The analyzed pharmacokinetic endpoint (F_a, FD_p, F, C_max, C(t)_plasma(max), AUC_(0-t), AUC_(0-inf).
	The arithmetic mean and geometric mean, respectively, for the given pharmacokinetic endpoint.
	The standard deviation calculated from an untransformed x value.
	The standard deviation calculated from an ln-transformed x value.
	A critical t value.

The comparison of the experimental and simulated profiles is done based on 90% confidence intervals on the difference between experimental and simulated means of C_max, AUC_(t), and AUC_inf, which are calculated based on the assumption of independent samples (unpaired t-test).

Equation 6-4: Lower (L) bound and upper (U) bound for 90% confidence interval on the difference between means of test (T, simulated profiles) and reference (R, experimental profiles) as calculated from untransformed data

Equation 6-5: Lower bound expressed as percentage (L%) and upper bound expressed as percentage (U%) for 90% confidence interval on the difference between means of test (T, simulated profiles) and reference (R, experimental profiles) calculated from ln-transformed data

where for Equation 6-4 and Equation 6-5, and the given pharmaceutical endpoint:

Variable	Definition
	The arithmetic mean.
	The pooled variance calculated from untransformed values.
	The pooled variance calculated from ln-transformed data.
	The number of values in the test population.
	The number of values in the reference population.
	A critical t value.

Equation 6-6: Pooled variances

where:

Variable	Definition
	The standard deviation calculated from either untransformed or ln-transformed data the for the test population and the reference population, respectively.
	The number of values in the test population.
	The number of values in the reference population.

If the experimental data for individual subjects are available, then the relevant means and standard deviations of the experimental C_max and AUC values are calculated from the data, and n_R represents the number of individual subject profiles.

If only an average C_p-time profile is available, then the exact means, geometric means, and relevant standard deviations for C_max and AUC cannot be obtained from individual subject data. In such cases, the following assumptions are made in the calculation of the confidence intervals:

The standard deviation for the reference formulation (σ_R) and the number of subjects for the reference formulation (n_R) are the same as the values that were obtained from simulated profiles (σ_T and n_T).

In the calculation of the confidence intervals on untransformed data, the assumption is made that the C_max and AUC of the mean C_p-time profile represent the arithmetic mean of values from individual subjects.

In the calculation of the confidence intervals of ln-transformed data, the assumption is that the C_max and AUC of the mean C_p-time profile represent the geometric mean of values from individual subjects.

It is up to you, the user, to know what type of mean C_p-time profile (arithmetic or geometric) was entered and decide whether the analysis on untransformed or ln-transformed data is more relevant.