1. Karunya Kandimalla, Ph.D. Assistant Professor, Pharmaceutics
Principles of Pharmacokinetics Pharmacokinetics of Oral Administration, 1-Compartment
Karunya Kandimalla, Ph.D.
Assistant Professor, Pharmaceutics
CP Clarke,L FD/MB

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
CP Clarke,L FD/MB

3. Recommended 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 )
Estimation of ka from urinary data (p )
Two-compartment determination of ka
CP Clarke,L FD/MB

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
CP Clarke,L FD/MB

5. Intravascular vs. Extravascular (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
With intravenously administered drugs, if we know kel and V for a particular patient, we can calculate appropriate doses or dosing rates (infusion rates) to produce the necessary therapeutic concentrations.
Most of the routes of administration are extravascular; for example IM, SC, and most importantly oral. With this type of drug administration the drug isn't placed in the central compartment but must be absorbed through at least one membrane. This has a considerable effect on drug pharmacokinetics and may cause a reduction in the actual amount of drug which is absorbed. The process of absorption typically follows first order kinetics (i.e., a constant fraction/proportion/% is absorbed). With zero order absorption kinetics, a constant amount of drug would be absorbed per unit time.
Zero order absorption is used only if modeling improves significantly or if it has been verified experimentally or if the drug is packaged in a dosage form that limits absorption to zero order kinetic.
CP Clarke,L FD/MB

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
kel
Drug
Eliminated ka
Drug in GI
Tract
Drug in
Body
Xa Xp = Vd • Cp
The rate of change in systemic drug level is dependent on the balance between the rates of drug absorption and elimination
For both zero and first order processes, the net rate of drug accumulation is equal to the rate of drug absorption minus the rate of drug elimination
The absorption (ka) and elimination (kel) rate constants describe how quickly serum concentrations rise (ka) or decline (kel) after administration.
The elimination rate of a drug can be computed by taking the product of the elimination rate constant (kel) and the amount of drug in the body (Xp)
In general, low molecular weight, high lipid solubility and lack of charge encourage absorption.
CP Clarke,L FD/MB

7. Orders 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
When a drug follows first order pharmacokinetics, serum concentrations decline in a curvilinear fashion. When the data are plotted on a semilogarithmic axis, serum concentrations decrease in a linear fashion after drug absorption and distribution phases are complete. This part of the curve is know as the elimination phase.
CP Clarke,L FD/MB

8. Orders of Reaction: Quick Review
Zero Order Drug Elimination
First Order Drug Elimination
Ln Cp vs. t: Slope = -k Cp Cp
Slope = -k
Keep in mind that a first order process (e.g., absorption) can become a zero order process (e.g., saturation of B12 or calcium absorption at higher doses; here the active transport mechanism is overloaded)
Time
Time
Think of zero order processes as “saturated” (e.g., ethanol metabolism) or “limited” (e.g., controlled release) processes
Keeping the math straight:
When Cp plotted on semilog paper, slope = -k/2.303
Log Cp vs time: slope = -k/2.303
CP Clarke,L FD/MB

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
Absorption phase
Elimination phase
When absorption is completed, ka•e-kat approaches zero. The plasma level-time curve is then governed by kel, and the amount of drug in the systemic circulation can be described as a first order process
The time it takes for serum concentrations to decrease by half in the elimination phase is a constant and is called the half-life and has dimensions of time (h, min, d).
The half-life and elimination rate constant are related to each other: t1/2 = 0.693/kel.
Half-life is important because it determines the time to steady state during continuous dosing and also determines the dosing interval.
t1 t t3
Plasma level-time curve for a single oral dose, first order (concentration-dependent) kinetics
CP Clarke,L FD/MB

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:
Here ka must be greater than kel
Where F is the fraction of drug that is actually absorbed. The value of F varies from 1 (drug fully absorbed) to 0 (drug completely unabsorbed). Here, we can see that the concentration of an orally absorbed drug (single dose) is a function of how much drug the patient takes (dose), how much drug reaches the systemic circulation (F), the volume in which the drug distributes (V), and the difference between how fast it is absorbed (ka) and how fast it is eliminated (kel)
Hybrid
Constant
Difference between 2 exponential terms
CP Clarke,L FD/MB

11. ka decreases, kel increases
Influence of Variations in Relative Rates of Absorption and Elimination on Plasma Concentration, Single Oral Dose
ka/kel=10
ka decreases, kel increases
ka/kel=1
ka/kel=0.1*
Plasma concentration
ka/kel=0.01*
The relative sizes of ka and kel determine the shape of the plasma concentration-time curve
Here the two right-most curves display a phenomenon called “flip-flop.” The rate at which the drug can be eliminated is limited by the rate at which the drug is absorbed, i.e., elimination is governed by ka
The second, orange, curve from the left illustrates a situation where ka = kel. In this instance, the terminal slope of the log concentration-time curve is not linear and it cannot be used to determine a proper rate constant. Further, the method of residuals cannot be used to determine ka.
*Note: Flip-flop modelling applies
CP Clarke,L FD/MB

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)
General integrated [C] equation becomes:
Notice the flip-flop of ka and kel
Hybrid
Constant
Difference between 2 exponential terms
CP Clarke,L FD/MB

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
Notice the emphasis on “often.” Some drugs are eliminated rapidly, and rapid elimination can at times be an advantage. For example, the ACE inhibitor captopril is rapidly eliminated
CP Clarke,L FD/MB

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
CP Clarke,L FD/MB

15. 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
For the rapid rate constant of absorption (2) the slope of the terminal phase is parallel to that of the i.v. disposition curve and represents a true rate of drug elimination (i.e. it is controlled by drug clearance and extent of distribution).
In contrast, for a slow rate constant of absorption (3), the slope of the terminal phase is no longer parallel to that of the i.v. administration, reflecting the actual rate limiting step which is now related to drug absorption
Journal of Veterinary Pharmacology & Therapeutics 2004;27(6):427-39
CP Clarke,L FD/MB

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…
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18. Plasma-Concentration Time Curve
A function of difference between ka and kel
Cmax
Cpt = ka • F• Dose • (e –kel • t – e –ka • t)
Vd (ka – kel)
When only postabsorption, postdistribution serum concentrations are obtained for a drug administered extravascularly, the equation for Cpt simplifies to Cpt = [(FD)/V] • e –kel • t, where t is any postabsorption, postdistribution time.
CP Clarke,L FD/MB

19. Plasma-Concentration Time Curve
Using log or natural log of [C] data “linearizes” the first order plot A
Slope = ln Cp1 – ln Cp2 = -kel
t t2
lnCpt = A – kel • t
(Postabsorption)
Plotting the natural log of the exact same data on the y axis yields a terminal elimination curve that is straight and thus easy to work with.
When absorption phase is over, e –ka • t approaches zero and this term can therefore be ignored from the Cp equation. The remaining concentration is then a function of the initial concentration (y intercept A) and the term kel • t. Note that this describes the equation of a straight line.
The back-extrapolated y intercept is a hybrid constant termed A. It can be used to determined volume of distribution
T1/2 = 22 hr
Absorption Time
CP Clarke,L FD/MB

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:
You should avoid points close to Cmax as some absorption may still be occurring. With drugs that follow 2-compartment models, you must also make sure that the 3 points are past the distribution phase.
Slope = C1 – C3 = -kel/2.303
t1 – t3
CP Clarke,L FD/MB

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 (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
You should plot the drug concentrations versus time data on semilog paper
CP Clarke,L FD/MB

22. Determination 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
CP Clarke,L FD/MB

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)
Extrapolation of terminal elimination phase (ln C vs. time) gives:
A = ka•F•Dose/Vd(ka-kel); A can be used to estimate Vd
Slope = -kel/2.303
Determination of residual line (via method of residuals) of ln C vs. time curve gives:
Slope = -ka/2.303
Intercepts of 2 lines should be equal, but due to experimental error, they rarely are
CP Clarke,L FD/MB

24. Relevance 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
If bioequivalency is established, steady state concentrations and therapeutic efficacy are expected to remain unchanged. Keep in mind that the ka for a given drug may be affected by the drug formulations (solutions, capsules, tablets, and sustained release tablets all have different drug release rates). ka may also be affected by disease: malabsorption, intestinal atrophy, Crohn’s disease, etc.
CP Clarke,L FD/MB

25. 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% % % %
Question:
Do similar AUC and Cmax imply a similar tmax?
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 <, =, or > than body volume
CP Clarke,L FD/MB

27. Volume of Distribution—The Concept
Plasma [C] Tissue [C] “Apparent” Vd
• • ••
• • • • • • •
• • • • •
• • • • • •
• • •
• • •
Volume of distribution is important in that it determines the size of loading doses needed to achieve a particular steady state concentrations.
Elimination rate constants and half-lives are known as dependent parameters, because their value depend on the clearance and volume of distribution of drugs. These parameters can change, either because of a change in clearance or a change in the volume of distribution. Because the values for clearance and volume of distribution depend solely on physiologic parameters and can vary independently of each other, there are known as independent parameters.
• • • • •
• • • •
NB: For lipid-soluble drugs, Vd changes with body size and age (decreased lean body mass, increased fat)
CP Clarke,L FD/MB

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?
Factors which tend to keep the drug in plasma or increase Cp (therefore decreasing Vd) include low lipid solubility (high water solubility, low pKa), high plasma protein binding, or decreased tissue binding.
Factors that tend to increase volume of distribution include low plasma protein biding, increased tissue binding, and increased lipid solubility.
CP Clarke,L FD/MB

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)
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30. Calculation of AUC (Model-Independent
Calculated by linear trapezoidal rule and extrapolation to infinity
Units = [C] • time
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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
Note that oral bioavailability may be reduced by hepatic enzyme induction.
May be affected by hepatic enzyme induction or inhibition (increased or decreased 1st pass metabolism or change in formulation excipients
CP Clarke,L FD/MB

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:
Note that Vd can easily be calculated if we know the y intercept, hybrid constant, A.
CP Clarke,L FD/MB

33. Calculation of Cmax and tmax
We can calculate maximal plasma concentration if we know kel:
Note: Direct measurement of Cmax is difficult, so calculation is necessary
We can also calculate the time of peak concentration if we know ka and kel:
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34. How would doubling the dose affect the Cp curve?
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…
How would doubling the dose affect the Cp curve?
CP Clarke,L FD/MB

35. Influence of Dose on Plasma Levels
OUT
Dose 60
120
Tmax
(h)
1.53 F 1
Cmax
(mg/L)
0.33
0.65
ka (1/h)
kel
(1/h)
0.4 Vd
(L)
100 t½
1.73
CL (L/h) 40
AUC (mg/L•h)
1.5 3
Everything else held constant, doubling the dose doubles Cmax and the AUC
CP Clarke,L FD/MB

36. How would a reduction in F from 1 to 0.5 affect the Cp curve?
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37. Influence of Bioavailability on Plasma Levels
OUT
Dose 60
Tmax
(h)
1.53 F 1
0.5
Cmax
(mg/L)
0.33
0.16
ka (1/h)
kel
(1/h)
0.4 Vd
(L)
100 t½
1.73
CL (L/h) 40
AUC (mg/L•h)
1.5
0.75
Everything else held constant, diminishing F will diminish Cmax and the AUC
CP Clarke,L FD/MB

38. How would a reduction in ka from 1 to 0.1 affect the Cp curve?
What happens if ka becomes smaller than kel?
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39. Influence of Absorption Rate on Plasma Levels
OUT
Dose 60
Tmax
(h)
1.53
4.62 F 1
Cmax
(mg/L)
0.33
0.09
ka (1/h)
0.1
kel
(1/h)
0.4 Vd
(L)
100 t½
1.73
CL (L/h) 40
AUC (mg/L•h)
1.5
Everything else held constant, diminishing ka will increase Tmax and diminish Cmax (as in slow-release preparations)
CP Clarke,L FD/MB

40. How would an increase in Vd from 100 to 150 liters affect the Cp curve?
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41. Influence of Vd on Plasma Levels
OUT
Dose 60
Tmax
(h)
1.53
1.8 F 1
Cmax
(mg/L)
0.33
0.25
ka (1/h)
kel
(1/h)
0.4
0.27 Vd
(L)
100
150 t½
1.73
2.6
CL (L/h) 40
AUC (mg/L•h)
1.5
Everything else held constant, increasing Vd will increase Tmax, diminish Cmax and ke and increase half-life
CP Clarke,L FD/MB

42. Which of all the parameters reviewed affect the area under the curve?
Bonus Question
Which of all the parameters reviewed affect the area under the curve?
F and dose
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43. Putting it All Together
Change in
kel Unchanged
ka Unchanged
ka ↓
ka ↑
kel ↓
kel ↑
Tmax
Cmax
AUC → ↓
Same ← ↑