Principles of Pharmacokinetics Pharmacokinetics of Oral Administration, 1-Compartment Karunya Kandimalla, Ph.D. Assistant Professor, Pharmaceutics Karunya Kandimalla, Ph.D. Assistant Professor, Pharmaceutics

2 Objectives Be able to: Describe 1-compartment pharmacokinetic models with first order absorption/elimination Define and calculate absorption & elimination rate constants, volume of distribution, area under the curve and bioavailability from concentration-time data Understand influence of all these parameters on plasma concentration versus time curves Recognize and use working equations for 1-compartment models Be able to: Describe 1-compartment pharmacokinetic models with first order absorption/elimination Define and calculate absorption & elimination rate constants, volume of distribution, area under the curve and bioavailability from concentration-time data Understand influence of all these parameters on plasma concentration versus time curves Recognize and use working equations for 1-compartment models

3 Recommended Readings Chapter 7, p , p. 176 Pharmacokinetics of drug absorption Zero order absorption model First order absorption model Absorption rate constants Skip: Wagner-Nelson method (p ) Estimation of ka from urinary data (p ) Two-compartment determination of ka Chapter 7, p , p. 176 Pharmacokinetics of drug absorption Zero order absorption model First order absorption model Absorption rate constants Skip: Wagner-Nelson method (p ) Estimation of ka from urinary data (p ) Two-compartment determination of ka

4 Intravascular vs. Extravascular (Oral) Administration IV administration (bolus or infusion): Drugs are injected directly into central compartment (plasma, highly perfused organs, extracellular water) No passage across membranes Population or individual elimination rate constants (kel) and volumes of distribution (Vd) enable us to calculate doses or infusion rates that produce target (desired) concentrations IV administration (bolus or infusion): Drugs are injected directly into central compartment (plasma, highly perfused organs, extracellular water) No passage across membranes Population or individual elimination rate constants (kel) and volumes of distribution (Vd) enable us to calculate doses or infusion rates that produce target (desired) concentrations

5 Intravascular vs. Extravascular (Oral) Administration Oral administration: Drug not placed in central compartment but absorbed through at least 1 membrane Significant inter- and intra-patient variability in rate + extent of absorption Stomach emptying rate Surface area of GI tract/blood flow Peristaltic rate (intestinal motility) First pass extraction (metabolism by liver) Food, disease (e.g., diarrhea), other factors Typically follows 1 st order kinetics Oral administration: Drug not placed in central compartment but absorbed through at least 1 membrane Significant inter- and intra-patient variability in rate + extent of absorption Stomach emptying rate Surface area of GI tract/blood flow Peristaltic rate (intestinal motility) First pass extraction (metabolism by liver) Food, disease (e.g., diarrhea), other factors Typically follows 1 st order kinetics

6 Extravascular (Oral) Administration Schematically, the simplest model can be represented as: Where Xa is the amount of drug to be absorbed, Xp is the amount of drug in the body, Vd is the volume in which the drug distributes, ka is the first order absorption rate constant, and kel is the first order elimination rate constant Schematically, the simplest model can be represented as: Where Xa is the amount of drug to be absorbed, Xp is the amount of drug in the body, Vd is the volume in which the drug distributes, ka is the first order absorption rate constant, and kel is the first order elimination rate constant Drug in GI Tract Drug in Body Drug Eliminated Xa Xp = Vd Cp ka kel

7 Orders of Reaction: Quick Review Zero OrderFirst Order Differential rate expression -dc/dt = k-dc/dt = kC Plasma [C] at time t Cp = ka (1 - e -kel t ) Vd kel Cp = FDka (e -kelt - e -kat ) Vd(ka - kel) Half-lifeCo/2kel0.693/kel EliminationConstant amount per unit time Constant fraction per unit time Units of kel/kaAmount per unit timeReciprocal of time (h -1 ) AbsorptionIndependent of [C] at absorption site Proportional to [C] at absorption site [C] vs. t GraphLinear decreaseExponential decrease

8 Orders of Reaction: Quick Review Cp Time Cp Ln Cp vs. t: Slope = -k First Order Drug Elimination Zero Order Drug Elimination Slope = -k Keeping the math straight: 1.When Cp plotted on semilog paper, slope = -k/ Log Cp vs time: slope = -k/2.303 Think of zero order processes as “saturated” (e.g., ethanol metabolism) or “limited” (e.g., controlled release) processes

9 The One-Compartment Extravascular Administration Model, Single Dose Absorption phase: dXa/dt > dXel/dt Peak concentration: dXa/dt = dXel/dt Elimination phase: dXa/dt < dXel/dt Plasma level-time curve for a single oral dose, first order (concentration-dependent) kinetics t 1 t 2 t 3 Absorption phase Elimination phase

10 First Order Absorption/Elimination At any time t, plasma concentration is a function of “rate in” minus “rate out” dXp/dt = dXa/dt – dXel/dt General integrated equation for calculation of drug concentration in plasma at time t is: At any time t, plasma concentration is a function of “rate in” minus “rate out” dXp/dt = dXa/dt – dXel/dt General integrated equation for calculation of drug concentration in plasma at time t is: Hybrid Constant Difference between 2 exponential terms Here ka must be greater than kel

11 Plasma concentration Influence of Variations in Relative Rates of Absorption and Elimination on Plasma Concentration, Single Oral Dose ka/kel=10 ka/kel=0.1* ka/kel=0.01* ka/kel=1 ka decreases, kel increases *Note: Flip-flop modelling applies

12 First Order Input & Elimination: Flip- Flop of ka and kel When kel > ka, slope of terminal elimination phase is governed by ka Slope = -ka/2.303 (semilog paper, log [C] vs t) G eneral integrated [C] equation becomes: When kel > ka, slope of terminal elimination phase is governed by ka Slope = -ka/2.303 (semilog paper, log [C] vs t) G eneral integrated [C] equation becomes: Hybrid Constant Difference between 2 exponential terms

13 Drugs Products with Flip-Flop Characteristics Fast elimination (kel > ka) kel typically >> 0.69 hr -1 ka typically << 1.38 hr -1 Not often suitable for oral drug products Extended release drug products Most marketed drugs have elimination half-lives that are longer than their absorption half-lives, i.e., their kel < ka Fast elimination (kel > ka) kel typically >> 0.69 hr -1 ka typically << 1.38 hr -1 Not often suitable for oral drug products Extended release drug products Most marketed drugs have elimination half-lives that are longer than their absorption half-lives, i.e., their kel < ka

14 Flip-Flop of ka and kel: Deciphering Atypical Drug Absorption Requires an IV bolus study After injection, decline in plasma level represents true elimination rate Calculate IV kel and compare with kel from oral profile (terminal phase of ln Cp vs time) If mismatch, assume a case of flip flop kinetics Requires an IV bolus study After injection, decline in plasma level represents true elimination rate Calculate IV kel and compare with kel from oral profile (terminal phase of ln Cp vs time) If mismatch, assume a case of flip flop kinetics

15 Journal of Veterinary Pharmacology & Therapeutics 2004;27(6): Deciphering Atypical Absorption 2, high ka: Slope of terminal phase is parallel to i.v.’s and represents a true rate of drug elimination (controlled by Vd and clearance) 3, low ka: Slope of terminal phase not parallel to i.v.’s, reflecting rate limiting absorption

16 Concentration at Any Time t is A Bi-Exponential Function

17 The Bi-Exponential First Order Plot Cp can be plotted as a function of the difference between the two exponential curves If we plot each exponential separately… Cp can be plotted as a function of the difference between the two exponential curves If we plot each exponential separately…

18 Plasma-Concentration Time Curve Cp t = ka F Dose (e –kel t – e –ka t ) Vd (ka – kel) A function of difference between ka and kel Cmax

19 Plasma-Concentration Time Curve Using log or natural log of [C] data “linearizes” the first order plot Slope = ln Cp 1 – ln Cp 2 = -kel t 1 - t 2 lnCp t = A – kel t (Postabsorption) T 1/2 = 22 hr Absorption Time A

20 “Archaic” Determination of kel Sample plasma drug concentration at multiple times Plot concentrations vs. time on semilog paper, with concentrations on y axis Draw straight line through 3 points along terminal elimination phase Avoid points close to Cmax Calculate slope (“rise over run”) and solve for kel: Sample plasma drug concentration at multiple times Plot concentrations vs. time on semilog paper, with concentrations on y axis Draw straight line through 3 points along terminal elimination phase Avoid points close to Cmax Calculate slope (“rise over run”) and solve for kel: Slope = C 1 – C 3 = -kel/2.303 t 1 – t 3

21 Determination of ka—Method of “Residuals” Read any 3 points (x’ 1, x’ 2, x’ 3 ) on upper part of back-extrapolated elimination line Drop essentially vertically and read 3 corresponding points on concentration-time curve (x 1, x 2, x 3 ) You should be in the absorptive phase Calculate difference between extrapolated concentrations (e.g., x’ 1, x’ 2) and measured concentrations (e.g., x 1, x 2 ) Plot differences at corresponding time points Read any 3 points (x’ 1, x’ 2, x’ 3 ) on upper part of back-extrapolated elimination line Drop essentially vertically and read 3 corresponding points on concentration-time curve (x 1, x 2, x 3 ) You should be in the absorptive phase Calculate difference between extrapolated concentrations (e.g., x’ 1, x’ 2) and measured concentrations (e.g., x 1, x 2 ) Plot differences at corresponding time points

22 Determination of ka—Application of Method of Residuals Time (hr)Observed [C]Extrapolated [C]Residual

23 Determination of Ka: Application of Method of Residuals ka F Dose = A Vd(ka – kel) Slope = -kel/2.3 = Slope = -ka/2.3 = (Residual Line)

24 Relevance of Absorption Rate Constants Useful in designing multiple dose regimen Prediction of t max (time to C max ) Prediction of peak plasma [C] (C max ) Prediction of trough plasma [C] (C min ) Useful in bioequivalence studies Pharmaceutical equivalents must demonstrate nearly identical rates of absorption AUC (area under the curve), Cmax, and tmax must be the same within statistical limits Useful in designing multiple dose regimen Prediction of t max (time to C max ) Prediction of peak plasma [C] (C max ) Prediction of trough plasma [C] (C min ) Useful in bioequivalence studies Pharmaceutical equivalents must demonstrate nearly identical rates of absorption AUC (area under the curve), Cmax, and tmax must be the same within statistical limits

25 Jantoven-Coumadin Bioequivalency (5-mg Dose) ParameterRatio of Means (Jantoven/Coumadin) 90% Confidence Intervals AUC 0-t AUC 0-∞ C max 98.9% 98.3% 96.9% % % % Question: Do similar AUC and C max imply a similar t max ? Check for yourself at

26 Notes on Volume of Distribution Definition: Size of a compartment necessary to account for total amount of drug at the concentration found in plasma Different tissues may contain different drug concentration (differing binding affinities) Anatomically speaking, does not have true physiological meaning Represents result of dynamic drug distribution May be than body volume Definition: Size of a compartment necessary to account for total amount of drug at the concentration found in plasma Different tissues may contain different drug concentration (differing binding affinities) Anatomically speaking, does not have true physiological meaning Represents result of dynamic drug distribution May be than body volume

27 Volume of Distribution—The Concept Plasma [C] Tissue [C] “Apparent” Vd NB: For lipid-soluble drugs, Vd changes with body size and age (decreased lean body mass, increased fat)

28 Quiz Yourself: Volume of Distribution In general, if a drug is confined in vascular region (i.e., it is highly bound to plasma protein), volume of distribution is _________. On the other hand, if it distributes into tissues extensively, Vd becomes ____________. Why would certain drugs have different Vds? In general, if a drug is confined in vascular region (i.e., it is highly bound to plasma protein), volume of distribution is _________. On the other hand, if it distributes into tissues extensively, Vd becomes ____________. Why would certain drugs have different Vds?

29 Calculation of Vd From Oral Absorption Data 1 compartment, y intercept method (requires IV study to determine F): Model-independent method (works regardless of model fitting drug’s kinetics)

30 Calculation of AUC (Model- Independent Calculated by linear trapezoidal rule and extrapolation to infinity Units = [C] time

31 Oral Bioavailability (F) Defined as fraction of orally-administered drug that reaches systemic circulation Also expressed in relative terms (e.g., bioavailability of a generic relative to a brand May be affected by hepatic enzyme induction or inhibition (increased or decreased 1 st pass metabolism or change in formulation excipients

32 Calculation of Cp at Anytime We can calculate plasma concentration at anytime if we know values of all parameters: Cp can then be calculated from the following equations:

33 Calculation of Cmax and tmax We can also calculate the time of peak concentration if we know ka and kel: We can calculate maximal plasma concentration if we know kel: Note: Direct measurement of Cmax is difficult, so calculation is necessary

34 Knowledge in Action—Understanding the Effects of Dose, F, ka, kel and Vd on Cp Investigate the effect of changing The Dose Bioavailability (F) Absorption rate constant (ka) Elimination rate constant (kel) Apparent volume of distribution (V) …. on Cmax and AUC… Investigate the effect of changing The Dose Bioavailability (F) Absorption rate constant (ka) Elimination rate constant (kel) Apparent volume of distribution (V) …. on Cmax and AUC… How would doubling the dose affect the Cp curve?

35 Influence of Dose on Plasma Levels INOUT Dose60120Tmax (h) 1.53 F11Cmax (mg/L) ka (1/h) 11kel (1/h) 0.4 Vd (L) 100 t½ (h) 1.73 CL (L/h) 40 AUC (mg/Lh) 1.53 Everything else held constant, doubling the dose doubles Cmax and the AUC

36 How would a reduction in F from 1 to 0.5 affect the Cp curve?

37 Influence of Bioavailability on Plasma Levels INOUT Dose60 Tmax (h) 1.53 F10.5Cmax (mg/L) ka (1/h) 11kel (1/h) 0.4 Vd (L) 100 t½ (h) 1.73 CL (L/h) 40 AUC (mg/Lh) Everything else held constant, diminishing F will diminish Cmax and the AUC

38 How would a reduction in ka from 1 to 0.1 affect the Cp curve? What happens if ka becomes smaller than kel?

39 Influence of Absorption Rate on Plasma Levels INOUT Dose60 Tmax (h) F11Cmax (mg/L) ka (1/h) 10.1kel (1/h) 0.4 Vd (L) 100 t½ (h) 1.73 CL (L/h) 40 AUC (mg/Lh) 1.5 Everything else held constant, diminishing ka will increase Tmax and diminish Cmax (as in slow-release preparations)

40 How would an increase in Vd from 100 to 150 liters affect the Cp curve?

41 Influence of Vd on Plasma Levels INOUT Dose60 Tmax (h) F11Cmax (mg/L) ka (1/h) 11kel (1/h) Vd (L) t½ (h) CL (L/h) 40 AUC (mg/Lh) 1.5 Everything else held constant, increasing Vd will increase Tmax, diminish Cmax and ke and increase half-life

42 Bonus Question Which of all the parameters reviewed affect the area under the curve?

43 Putting it All Together Change in kel Unchangedka Unchanged ka ↓ka ↑kel ↓kel ↑ Tmax Cmax AUC → ↓ Same ← ↑ Same →↑↑→↑↑ ←↓↓←↓↓