PK Solutions 2.0 - Noncompartmental Pharmacokinetic Equations

PK Solutions 2.0 Noncompartmental Pharmacokinetics Data Analysis
Copyright © 2000 by David S. Farrier, Summit Research Services
All Rights Reserved Worldwide.

Copyrighted and licensed materials manufactured in the United States of America. This document and any accompanying software are protected by United States and International Copyright Laws. This document is an excerpt taken from the software manual, PK Solutions 2.0 User Guide, and has been modified for independent distribution and publication. The document may be reproduced and distributed provided that the original contents are not altered in any way and that it is not sold. Please contact Summit Research Services if you wish to use the document for other purposes.

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About PK Solutions 2.0

Pharmacokinetics is the study of the time course of absorption, distribution, metabolism, and excretion of a drug or other substance in the body. PK Solutions is designed to provide a fast and easy means of computing and graphing the basic and most commonly reported pharmacokinetic parameters associated with blood (plasma, serum) concentration-time data following extravascular (oral) and intravenous dosing. Approximately 60 pharmacokinetic parameters are computed for each data set, including tables and graphs projecting multiple dose regimes based on single dose results. No programming or construction of equations are required. The program is automated, operating entirely by click-and-point methods. In addition to concentration-time data, PK Solutions can independently or simultaneously compute and graph results using disposition constants imported, for example, from a curve-fitting program or from the literature. Once the calculation mode is selected, a few mouse selections are all that is needed to analyze either type of data and produce graphs and parameter tables ready for printing and saving. Dynamic updating of tables and graphs provides a means of trying out "what if" cases and makes PK Solutions an ideal tool for learning the principles of pharmacokinetic analysis.

Methodology

PK Solutions relies on the use of noncompartmental methods of analysis for the estimation of pharmacokinetic parameters. Two noncompartmental techniques are employed and their results compared where appropriate in the parameter tables. One technique is based on the estimation of the area associated with the curve described by the concentration-time profile. In this case, the classical trapezoidal rule is used to compute the area under the curve (AUC).

The second noncompartmental technique is based on the method of residuals (also called curve stripping or feathering) which resolves a curve into a series of up to three exponential terms corresponding to the absorption, distribution, and elimination phases occurring during the time course of the drug in the blood. These exponential terms are used to calculate the various single and multiple dose pharmacokinetic parameters following well established textbook calculations. The curve stripping approach assumes that the disposition phases of the drug follow apparent first-order rate processes, which is evidenced by linearity in the terminal portion of a semi-log plot. This is, in fact, the case for the overwhelming majority of drugs, making PK Solutions a widely useful tool.

Calculation of PK parameters based on curve area and curve stripping data are called "model-independent" because they are free of any assumptions about the underlying compartmental model that the drug obeys. Nonetheless, noncompartmental methods can yield results that confer specific model characteristics on a drug's behavior. The model-independent approach, which serves as the basis for PK Solutions, can be contrasted with mathematical curve-fitting programs that are specifically designed to elaborate compartmental models and their descriptive equations. PK Solutions focuses on producing graphs and tables representing model-independent pharmacokinetic solutions rather than on compartmental equations. While compartmental programs serve special needs, the types of calculations produced by PK Solutions are those most commonly reported in the tables of drug metabolism and pharmacokinetics literature and will suffice as an accurate model-independent description of drug pharmacokinetics.

Typical Users

PK Solutions is intended to be used by researchers, by those who need to determine and publish basic pharmacokinetic parameters, by students who can make use of the program's flexibility to study the principles of PK analysis, and by pharmacists and physicians to view the effects of dosing regimens on blood levels. It is designed to augment rather than replace or compete with compartmental analysis, experimental curve-fitting, or clinical dose prediction software. Any one who works with blood level data in pharmaceutical or agrochemical product development, veterinary and medical sciences, or other research and teaching areas will benefit from the quick solutions and range of results afforded by PK Solutions 2.0.

Excel-Based Program

PK Solutions is intentionally developed as a Microsoft Excel workbook in order to make use of Excel's extensive feature set, its cross-platform compatibility between Windows-PCs and Macintosh computers, and its ready integration with other Microsoft Office software, intranets, external databases, and other software. Unlike proprietary programs that require long learning curves and are limited in scope to the effort programmers expend on the design and features, PK Solutions gives you easy access to pharmacokinetic analysis as well as the power of Excel to enhance, customize, and integrate your results with the rest of your working world.

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 1^st 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
C_initial Initial concentration extrapolated to time zero for i.v. dose.
Tmax (obs) 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 (for comparison)
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 C_initial
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 (C_N) 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 (1^st dose, observed) Observed time of largest concentration value from data set.
Tmax (1^st 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.