principles of pharmacokinetics pharmacokinetics of oral administration, 1-compartment karunya...
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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, Ph.D.Assistant Professor, Pharmaceutics
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
→
↑
↑
←
↓
↓