Principles of Pharmacokinetics

Pharmacokinetics of Oral Administration, 1-Compartment

Principles of Pharmacokinetics

Pharmacokinetics of Oral Administration, 1-Compartment

Karunya Kandimalla, Ph.D.Assistant Professor, Pharmaceutics

Karunya.kandimalla@famu.edu

Karunya Kandimalla, Ph.D.Assistant Professor, Pharmaceutics

Karunya.kandimalla@famu.edu

2

ObjectivesObjectives

• 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 ReadingsRecommended Readings

• Chapter 7, p. 161-71, p. 176• Pharmacokinetics of drug absorption

• Zero order absorption model

• First order absorption model

• Absorption rate constants

• Skip:

• Wagner-Nelson method (p. 171-73)

• Estimation of ka from urinary data (p. 174-75)

• Two-compartment determination of ka

• Chapter 7, p. 161-71, p. 176• Pharmacokinetics of drug absorption

• Zero order absorption model

• First order absorption model

• Absorption rate constants

• Skip:

• Wagner-Nelson method (p. 171-73)

• Estimation of ka from urinary data (p. 174-75)

• Two-compartment determination of ka

4

Intravascular vs. Extravascular (Oral) AdministrationIntravascular 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) AdministrationIntravascular 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 1st 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 1st order kinetics

6

Extravascular (Oral) AdministrationExtravascular (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 ReviewOrders of Reaction: Quick Review

Zero Order First Order

Differential rate expression

-dc/dt = k -dc/dt = kC

Plasma [C] at time t

Cp = ka (1 - e -kel • t)

Vd • kel

Cp = F•D•ka (e-kel•t - e-ka•t)

Vd•(ka - kel)

Half-life Co/2kel 0.693/kel

Elimination Constant amount per unit time

Constant fraction per unit time

Units of kel/ka Amount per unit time Reciprocal of time (h-1)

Absorption Independent of [C] at absorption site

Proportional to [C] at absorption site

[C] vs. t Graph Linear decrease Exponential decrease

8

Orders of Reaction: Quick ReviewOrders of Reaction: Quick Review

Cp

Time Time

Cp

Ln Cp vs. t: Slope = -k

First Order Drug EliminationZero Order Drug Elimination

Slope = -k

Keeping the math straight: 1. When Cp plotted on semilog

paper, slope = -k/2.3032. 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

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

0.0 20.0 40.0 60.0Time (hr)

Cp

(mg/

dL)

The One-Compartment Extravascular Administration Model, Single DoseThe 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

• 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

t1 t2 t3

Absorption phase

Elimination phase

10

First Order Absorption/EliminationFirst 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:

tkatkel

d

eekelkaV

kaDoseFCp

)( tkatkel

d

eekelkaV

kaDoseFCp

)(

Hybrid Constant

Difference between 2 exponential terms

Here ka must be greater than kel

11

0

2

4

6

8

10

12

14

0 5 10 15 20

Time (Hours)

Pla

sma

con

cen

trat

ion

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 kelFirst 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)

• General 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)

• General integrated [C] equation becomes:

tkeltka

d

eekakelV

kaDoseFCp

)(

Hybrid Constant

Difference between 2 exponential terms

13

Drugs Products with Flip-Flop CharacteristicsDrugs 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 AbsorptionFlip-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

15Journal of Veterinary Pharmacology & Therapeutics 2004;27(6):427-39

Deciphering Atypical AbsorptionDeciphering 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

• 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 FunctionConcentration at Any Time t is A Bi-Exponential Function

tkatkel

dt ee

kelkaV

kaDoseFCp

)( tkatkel

dt ee

kelkaV

kaDoseFCp

)(

17

The Bi-Exponential First Order PlotThe 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 CurvePlasma-Concentration Time Curve

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0Time (hr)

Cp

(m

g/d

L)

Cp mg/dL

Cpt = 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 CurvePlasma-Concentration Time Curve

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Time (hr)

Ln

Cp

• Using log or natural log of [C] data “linearizes” the first order plot

• Using log or natural log of [C] data “linearizes” the first order plot

Slope = ln Cp1 – ln Cp2 = -kel t1 - t2

lnCpt = A – kel • t

(Postabsorption)

T1/2 = 22 hr

Absorption Time

A

20

“Archaic” Determination of kel“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 = C1 – C3 = -kel/2.303

t1 – t3

21

Determination of ka—Method of “Residuals”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 (x1, x2, x3)

• You should be in the absorptive phase

• Calculate difference between extrapolated concentrations (e.g., x’1, x’2) and measured concentrations (e.g., x1, x2)

• 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 (x1, x2, x3)

• You should be in the absorptive phase

• Calculate difference between extrapolated concentrations (e.g., x’1, x’2) and measured concentrations (e.g., x1, x2)

• Plot differences at corresponding time points

22

Determination of ka—Application of Method of ResidualsDetermination of ka—Application of Method of Residuals

Time (hr) Observed [C] Extrapolated [C] Residual

0.5

1.0

2.0

4.0

8.0

12.0

18.0

24.0

36.0

48.0

5.36

9.95

17.18

25.78

29.78

26.63

19.40

13.26

5.88

2.56

57.14

55.36

51.95

45.78

--

--

--

--

--

--

51.74

45.36

34.75

19.98

--

--

--

--

--

--

23

1.0

10.0

100.0

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Time (hr)

[C]

Cp mg/dL Extrapolated Line Residual Line

Determination of Ka: Application of Method of ResidualsDetermination of Ka: Application of Method of Residuals

ka • F• Dose = A Vd(ka – kel)

Slope = -kel/2.3 = -0.064

• • •

•

•

•

Slope = -ka/2.3 = -0.254(Residual Line)

24

Relevance of Absorption Rate ConstantsRelevance of Absorption Rate Constants• Useful in designing multiple dose regimen

• Prediction of tmax (time to Cmax)

• Prediction of peak plasma [C] (Cmax)

• Prediction of trough plasma [C] (Cmin)

• 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 tmax (time to Cmax)

• Prediction of peak plasma [C] (Cmax)

• Prediction of trough plasma [C] (Cmin)

• 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

25http://www.jantoven.com/hcp/bioequiv.html

Jantoven-Coumadin Bioequivalency(5-mg Dose)

Jantoven-Coumadin Bioequivalency(5-mg Dose)

Parameter Ratio of Means (Jantoven/Coumadin)

90% Confidence Intervals

AUC 0-t

AUC 0-∞

Cmax

98.9%

98.3%

96.9%

95.5-102.4%

94.3-102.4%

92-102.2%

Question:

•Do similar AUC and Cmax imply a similar tmax?

•Check for yourself at http://www.cop.ufl.edu/cgi-bin/hh2.exe

26

Notes on Volume of DistributionNotes 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 <, =, or > 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 <, =, or > than body volume

27

Volume of Distribution—The ConceptVolume 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 DistributionQuiz 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 DataCalculation of Vd From Oral Absorption Data

• 1 compartment, y intercept method (requires IV study to determine F):

• 1 compartment, y intercept method (requires IV study to determine F):

• Model-independent method (works regardless of model fitting drug’s kinetics)

• Model-independent method (works regardless of model fitting drug’s kinetics)

0AUCkel

DoseVd

)( kelkaA

DoseFkaVd

30

Calculation of AUC (Model-IndependentCalculation of AUC (Model-Independent

tntn AUCAUCTotalAUC 00

• Calculated by linear trapezoidal rule and extrapolation to infinity

• Units = [C] • time

• Calculated by linear trapezoidal rule and extrapolation to infinity

• Units = [C] • time

1

01

10 2

n

iii

iitn CCtt

AUC

kelCnAUCtn

tntn AUCAUCTotalAUC 00

31

Oral Bioavailability (F)Oral Bioavailability (F)

• Defined as fraction of orally-administered drug that reaches systemic circulation

• 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

• Also expressed in relative terms (e.g., bioavailability of a generic relative to a brand

IVoral

oralIV

AUCD

AUCDF

ba

ab

b

a

AUCD

AUCD

F

F

May be affected by hepatic enzyme induction or inhibition (increased or decreased 1st pass metabolism or change in formulation excipients

32

Calculation of Cp at AnytimeCalculation of Cp at Anytime

• We can calculate plasma concentration at anytime if we know values of all parameters:

• We can calculate plasma concentration at anytime if we know values of all parameters:

• Cp can then be calculated from the following equations:

• Cp can then be calculated from the following equations:

)( tkatkelt eeACp

atktkel

dt ee

kelkaV

DoseFkaCp

)(

33

Calculation of Cmax and tmaxCalculation of Cmax and tmax

• We can also calculate the time of peak concentration if we know ka and kel:

• 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

• We can calculate maximal plasma concentration if we know kel:

• Note: Direct measurement of Cmax is difficult, so calculation is necessary

kel

ka

kelkat ln

1max

maxmax

tkeleVd

DoseFC

34

Knowledge in Action—Understanding the Effects of Dose, F, ka, kel and Vd on CpKnowledge 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?

35http://www.cop.ufl.edu/cgi-bin/hh2.exe

Influence of Dose on Plasma LevelsInfluence of Dose on Plasma Levels

IN OUT

Dose 60 120 Tmax

(h)

1.53 1.53

F 1 1 Cmax

(mg/L)

0.33 0.65

ka (1/h)

1 1 kel

(1/h)

0.4 0.4

Vd

(L)

100 100 t½

(h)

1.73 1.73

CL (L/h)

40 40 AUC (mg/L•h)

1.5 3

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?

37http://www.cop.ufl.edu/cgi-bin/hh2.exe

Influence of Bioavailability on Plasma LevelsInfluence of Bioavailability on Plasma Levels

IN OUT

Dose 60 60 Tmax

(h)

1.53 1.53

F 1 0.5 Cmax

(mg/L)

0.33 0.16

ka (1/h)

1 1 kel

(1/h)

0.4 0.4

Vd

(L)

100 100 t½

(h)

1.73 1.73

CL (L/h)

40 40 AUC (mg/L•h)

1.5 0.75

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 LevelsInfluence of Absorption Rate on Plasma Levels

IN OUT

Dose 60 60 Tmax

(h)

1.53 4.62

F 1 1 Cmax

(mg/L)

0.33 0.09

ka (1/h)

1 0.1 kel

(1/h)

0.4 0.4

Vd

(L)

100 100 t½

(h)

1.73 1.73

CL (L/h)

40 40 AUC (mg/L•h)

1.5 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 LevelsInfluence of Vd on Plasma Levels

IN OUT

Dose 60 60 Tmax

(h)

1.53 1.8

F 1 1 Cmax

(mg/L)

0.33 0.25

ka (1/h)

1 1 kel

(1/h)

0.4 0.27

Vd

(L)

100 150 t½

(h)

1.73 2.6

CL (L/h)

40 40 AUC (mg/L•h)

1.5 1.5

Everything else held constant, increasing Vd will increase Tmax, diminish Cmax and ke and increase half-life

42

Bonus QuestionBonus Question

Which of all the parameters reviewed affect the area under the curve?

43

Putting it All TogetherPutting it All Together

Change in

kel Unchanged ka Unchanged

ka ↓ ka ↑ kel ↓ kel ↑

Tmax

Cmax

AUC

→

↓

Same

←

↑

Same

→

↑

↑

←

↓

↓