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A useful compilation describing common pharmacokinetic equations as used by PK Solutions.

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Introduction to the Pharmacokinetic Equations

Source

This document is taken from the pharmacokinetics equation section of the
PK Solutions 2 User Guide, and describes the equations used by the software.

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PK Solutions offers a fast and easy way to compute, graph, and analyze the most commonly used pharmacokinetic parameters associated with blood (plasma, serum) concentration-time data following extravascular (such as oral) and intravenous dosing.

Approximately 75 pharmacokinetic parameters are computed for each data set, including tables and graphs projecting multiple dose results based on single dose values.

The equations in this list are those used by PK Solutions. These are sometimes referred to as "noncompartmental" because they do not require curve-fitting or make any assumption concerning compartmental models. The results, however, are the same as those produced by curve-fitting programs.

The PK parameters are based on two standard methods of analysis: (1) curve-stripping (or method of residuals) to derive the exponential terms that describe the blood level curve, and (2) area under the curve calculations. PK Solutions applies both methods where applicable and compares the results side-by-side.

 

Hyperlinked Table of Contents   

Single Dose Pharmacokinetics Single Dose Pharmacokinetics

Multiple Intravenous Dose Pharmacokinetics Multiple Intravenous Dose Pharmacokinetics

Multiple Oral Dose Pharmacokinetics Multiple Oral Dose Pharmacokinetics

 

Equation Table

Single Dose Pharmacokinetics
General Disposition Parameters and Constants 

Dose Amount

D

Fraction of dose absorbed

Used to correct dose amount for some oral dose calculations.

F

Exponential Summation

Expression for sum of 1st order kinetic terms.

for n exponential terms

Y-Intercept

Coefficient of each exponential term. Note the sign of the absorption coefficient is negative.

Slope

Rate constant

Elimination rate constant

Half-life

Descriptive Curve Parameters 

Cinitial

Initial concentration extrapolated to time zero for i.v. dose.

Tmax (obs)

Usually applies to oral doses only.

Cmax (calculated)

For biexponential oral data only.

where V is Vd (area).

Tmax (calculated)

For biexponential oral data only.

whereand are the apparent absorption and elimination rate constants, respectively.

Lag time

For biexponential oral data only.

whereand are the apparent absorption and elimination rate constants, respectively.

Curve Area Calculations 

AUC(0-t) (obs area)

Trapezoid calculation of AUC using observed data points only (not extrapolated to infinity). Useful when final concentration values tend to exaggerate total AUC.

where n is the number of data points.

AUC (area)

Total AUC computed by combining AUC(0-t) with an extrapolated value.

where is the last concentration.

AUC (expo)

Total AUC computed using exponential terms.

% of AUC (expo)

Percent each exponential term contributes to the total AUC.

Statistical Moment Calculations 

AUMC (area)

Calculation of total area under the first-moment curve (plot of Ct vs t) by combining trapezoid calculation of AUMC(0-t) and extrapolated area.

+

AUMC (expo)

Total AUMC computed using exponential terms.

% of AUMC (expo)

Percent each exponential term contributes to the total AUMC.

MRT (area)

Mean Residence Time calculated using trapezoid area calculations extrapolated to infinity.

where both area terms use trapezoidal calculations

MRT (expo)

Mean Residence Time calculated using exponential terms.

Volume of Distribution Calculations 

Vc (initial central compartment)

Apparent volume of the central compartment for i.v. doses only.

Vd (obs area)

Apparent volume of distribution based on AUC(0-t) trapezoid calculation and elimination rate. Use when total AUC (area) is exaggerated due to high terminal concentration values.

Vd (area)

Apparent volume of distribution based on trapezoid AUC (area) and elimination rate. Applies mainly to i.v., but also to oral if complete absorption (F=1) is assumed.

Vd (area) / kg

Apparent volume of distribution normalized by animal weight. Uses same equation as Vd (area).

Vd (expo)

Apparent volume of distribution calculated from exponential terms.

where is the elimination rate

Vss (area)

Apparent volume of distribution at steady state estimated graphically from trapezoidal total area measurements. Applies to iv dose.

Vss (expo)

Apparent volume of distribution at steady state estimated from exponential terms. Applies only after iv and assumes elimination from central compartment.

Systemic Clearance Calculations 

CL(sys) (obs area)

Systemic clearance based on AUC(0-t) trapezoid calculation. Use when total AUC (area) is exaggerated due to high last concentration.

CL (area)

Systemic clearance based on trapezoid AUC (area). Applies mainly to i.v. data. Limited to oral data only if complete absorption (F=1) is assumed.

CL (area) / kg

Systemic clearance normalized by animal weight. Uses same equation as CL (area).

CL (expo)

Systemic clearance calculated using exponential terms.

Half-life based on Vd and CL

Alternate calculation of half-life using Vd (area) and CL (area). For i.v. data only.

Two-compartment Open Model Microconstants 

k12

Microconstant calculated using exponentials. Applies to 2 compartment i.v. dose data only.

k21

Microconstant calculated using exponentials. Applies to 2 compartment i.v. dose data only.

k10

Microconstant calculated using exponentials. Applies to 2 compartment i.v. dose data only.

Multiple Intravenous Dose Pharmacokinetics
General 

Dose Interval (tau)

Time span between dosing intervals. Distinguish from time after dose (t).


tau

Assume constant dose interval

First Dose Concentration Calculations 

C1(max)

Maximum concentration after first dose interval (tau). Equal to Cinitial

C1(min)

Minimum concentration at end of first dose interval (tau).

C1(ave)

Average concentration during first dose interval (tau).

Prediction of Steady State Parameters 

Css(min)

Minimum concentration during any dosing interval at steady state.

Css(min)

Minimum concentration during any dosing interval at steady state. Included on graph.

Css(max) - Css(min)

Difference between peak and trough concentration during steady state.

Css(ave)

Average concentration at steady state.

Css(ave) (area)

Average concentration at steady state calculated from trapezoidal AUC data for a single dose.

Accumulation Factors 

R based on Css(max)/C1(max)

Accumulation ratio based on maximum concentrations after first dose and at steady state.

R based on Css(min)/C1(min)

Accumulation ratio based on minimum concentrations after first dose and at steady state.

R based on Css(ave)/C1(ave)

Accumulation ratio based on average concentrations after first dose and at steady state.

Time to Reach Percent of Steady State 

To reach 95% Css(ave)

Time required to reach 95% of average steady state concentration. Assumes one-compartment characteristics apply.

where is the fraction of the steady state concentration.

To reach 99% Css(ave)

Time required to reach 95% of average steady state concentration. Assumes one-compartment characteristics apply.

where is the fraction of the steady state concentration.

Ad Hoc Calculations 

Calculated loading dose

Loading dose required to produce an immediate steady state minimum concentration, Css(min).

Total time through Nth dose

Total time elapsed between first dose (t=0) and specified dose (N).

C(ave) during Nth dose

Average concentration during any dose interval (N). Becomes Css(ave) when steady state reached.

Fraction of Css(ave) after N doses

Fraction of the ultimate average steady state concentration reached after N doses.

where is the fraction of the steady state concentration

Css at t after ss dose

Steady state concentration at any time (t) during a dosing interval at steady state.

Conc. at any time and dose

Computes the concentration at any time during a dosing interval. Enter both time (t) and dose interval (N).

Multiple Oral Dose Pharmacokinetics
General and Graphing Functions 

Dose Interval (tau)

Constant time span between dosing intervals. Distinguish from time after dose (t).

(tau)  Assumes equal dose intervals

Graphing Function

The graphing function is based on a mathematical generalization of the graphical superimposition principle. It involves the addition of a decay function (CN) to the initial concentration (C1)at repeated time points for a progressive series of doses (N). Assumes constant dose intervals during the postdistribution phase.

where

and

First Dose Concentration Values 

C1(max)

Observed maximum concentration taken from data set.

C1(min)

Minimum concentration at end of first dose interval (tau).

C1(ave)

Average concentration during first dose interval (tau).

Prediction of Steady State Parameters 

Css(max)

Computed from a simplification of the graphing function to a steady state form as shown. The Css(max) is evaluated as the maximum concentration during the steady state dosing interval.

where

Css(min)

Computed using same steady state equation as Css(max) and evaluating the minimum concentration during a steady state dose interval.

Same as above.

Css(max) - Css(min)

Difference between peak and trough concentration during steady state.

Css(ave)

Average concentration at steady state.

Css(ave) (area)

Average concentration at steady state calculated from trapezoidal AUC data for a single dose.

Accumulation Factors 

R based on Css(min)/C1(min)

Accumulation factor based on elimination rate constant.

R based on Css(ave)/C1(ave)

Accumulation ratio based on average concentrations after first dose and at steady state.

Additional Oral Dose Calculations 

Tmax (1st dose, observed)

Observed time of largest concentration value from data set.

Tmax (1st dose, calculated)

Calculation of time at which maximum concentration occurs after a single dose. Applies to 1-compartment characteristics, but calculated also to illustrate magnitude for 2-compartments.

whereis the absorption rate

and is the elimination rate.

Tmax(ss)

Calculation of time at which maximum concentration occurs after dosing during steady state. Applies to 1-compartment characteristics, but calculated also to illustrate magnitude for 2-compartments.


whereis the absorption rate

and is the elimination rate.

 

Bibliography

Milo Gibaldi and Donald Perrier, Pharmacokinetics , Second edition (Marcel Dekker, New York, NY), 1982.

Robert E. Notari, Biopharmaceutics and Clinical Pharmacokinetics, Fourth edition (Marcel Decker, New York, NY), 1987.

Malcolm Roland and Thomas N. Tozer, Clinical Pharmacokinetics - Concepts and Applications, Second edition (Lea & Febiger, Malvern, PA), 1989.

Francis L.S. Tse and James M. Jaffe, Preclinical Drug Disposition - A Laboratory Handbook (Marcel Dekker, New York, NY), 1991.

Peter G. Welling, Pharmacokinetics: Processes and Mathematics (American Chemical Society, Washington, DC), 1986.

Peter G. Welling, Pharmacokinetics: Principles and Applications (American Chemical Society, Washington, DC), 1987.

 

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