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

2. Pharmacokinetics & Pharmacodynamics
ADME R R
Target
organ R R R R

3. Kinetics From the Blood or Plasma Data
Pharmacokinetics of a drug in plasma or blood
Absorption (Input)
Disposition
Distribution
Elimination
Excretion
Metabolism

4. Objectives Be able to: To understand the properties of linear models
To understand assumptions associated with first order kinetics and one compartment models
To define and calculate various one compartment model parameters (kel, t½, Vd, AUC and clearance)
To estimate the values of kel, t½, Vd, AUC and clearance from plasma or blood concentrations of a drug following intravenous administration.
CP Clarke,L FD/MB

5. Recommended Readings Chapter 3, p. 47-62 IV route of administration
Elimination rate constant
Apparent volume of distribution
Clearance
CP Clarke,L FD/MB

6. Intravascular 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

7. Disposition Analysis (Dose Linearity)

8. Disposition Analysis (Time Variance)

9. Linear Disposition
The disposition of a drug molecule is not affected by the presence of the other drug molecules
Demonstrated by:
Dose linearity
Saturable hepatic metabolism may result in deviations from the dose linearity
Time invariance
Influence of the drug on its own metabolism and excretion may cause time variance

10. Disposition Modeling A fit adequately describes the experimental data
A model not only describes the experimental data but also makes extrapolations possible from the experimental data
A fit that passes the tests of linearity will be qualified as a model

11. One Compartment Model (IV Bolus)
Schematically, one compartment model can be represented as:
Where Xp is the amount of drug in the body, Vd is the volume in which the drug distributes and kel is the first order elimination rate constant
kel
Drug
Eliminated
Drug in
Body
Xp = Vd • C
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

12. One Compartment Data (Linear Plot)

13. One Compartment Data (Semi-log Plot)

14. Two Compartment Model (IV Bolus)
K 12
Blood, kidneys, liver
Drug in
Central
Compartment
Drug in
Peripheral
Compartment
K 21
kel
Fat, muscle
Drug
Eliminated
For both 1- and 2-compartment models, elimination takes place from central compartment

15. Two Compartment Data (Linear Plot)

16. Two Compartment Data (Semi-log Plot)

17. One Compartment Model-Assumptions
1-Compartment—Intravascular drug is in rapid equilibrium with extravascular drug
Intravascular drug [C] proportional to extravascular [C]
Rapid Mixing—Drug mixes rapidly in blood and plasma
First Order Elimination Kinetics:
Rate of change of [C]  Remaining [C]
Notice here that the log-transformed equation represents the equation of a straight line
CP Clarke,L FD/MB

18. Derivation-One Compartment Model
Bolus IV
Central Compartment (C)
Kel
This is the simplest of the intravascular administration models
The entire dose enters the systemic circulation immediately and the body is depicted as a single homogenous unit
Plasma concentrations are not necessarily equal to tissue concentrations, but they are proportional
The advantage of this model is that it permits calculation of drug concentrations at any time (no absorption phase, no distribution phase)
The disadvantage is that the kinetic parameters do not have physiologic meaning
CP Clarke,L FD/MB

19. Concentration versus time, semilog paper
IV Bolus Injection: Graphical Representation Assuming 1st Order Kinetics
C0 = Initial [C]
C0 is calculated by back-extrapolating the terminal elimination phase to time = 0
C0 = Dose/Vd
C0 = Dose/Vd
Slope = -K/2.303
Slope = -Kel/2.303
Note that half-life can also be read from the graph
Concentration versus time, semilog paper
CP Clarke,L FD/MB

20. Elimination Rate Constant (Kel)
Kel is the first order rate constant describing drug elimination (metabolism + excretion) from the body
Kel is the proportionality constant relating the rate of change of drug concentration and the concentration
The units of Kel are time-1, for example hr-1, min-1 or day-1

21. Half-Life (t1/2)
Time taken for the plasma concentration to reduce to half its original concentration
Drug with low half-life is quickly eliminated from the body
t/t1/2
% drug remaining 1 50 2 25 3
12.5 4
6.25 5
3.125

22. Change in Drug Concentration as a Function of Half-Life

23. Apparent Volume of Distribution (Vd)
Vd is not a physiological volume
Vd is not lower than blood or plasma volume but for some drugs it can be much larger than body volume
Drug with large Vd is extensively distributed to tissues
Vd is expressed in liters and is calculated as:
Distribution equilibrium between drug in tissues to that in plasma should be achieved to calculate Vd

24. 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

25. Area Under the Curve (AUC)
AUC is not a parameter; changes with Dose
Toxicology: AUC is used as a measure of drug exposure
Pharmacokinetics: AUC is used as a measure of bioavailability and bioequivalence
Bioavailability: criterion of clinical effectiveness
Bioequivalence: relative efficacy of different drug products (e.g. generic vs. brand name products)
AUC has units of concentration  time (mg.hr/L)

26. Calculation of AUC using trapezoidal rule

27. Clearance (Cl)
The most important disposition parameter that describes how quickly drugs are eliminated, metabolized and distributed in the body
Clearance is not the elimination rate
Has the units of flow rate (volume / time)
Clearance can be related to renal or hepatic function
Large clearance will result in low AUC

28. Clearance -The Concept
Cinitial
Cfinal
ORGAN
elimination
If Cfinal < Cinitial, then it is a clearing organ

29. Practical Example Time (hr) Plasma Conc. (mg/L) ln (PlasmaConc.) 1
9.46
2.25 2
7.15
1.97 3
5.56
1.71 4
4.74
1.56 6
3.01
1.10 10
1.26
0.23 12
0.83
-0.19
IV bolus administration
Dose = 500 mg
Drug has a linear disposition

30. Linear Plot

31. Natural logarithm Plot
Kel
ln (C0)

32. Half-Life and Volume of Distribution
t1/2 = / Kel = hrs
Vd = Dose / C0 = 500 / = 44.66
ln (C0) =
C0 = Inv ln (2.4155) = mg/L

33. Clearance
Cl = D/AUC
Cl = VdKel
Cl =  = 9.73 L/hr

34. Home Work
Determine AUC and
Calculate clearance from AUC

35. Karunya Kandimalla, Ph.D
Principles of Pharmacokinetics Pharmacokinetics of IV Administration, 2-Compartment
Karunya Kandimalla, Ph.D
CP Clarke,L FD/MB

36. Objectives
Be able to:
Describe assumptions associated with multi-compartment models
Describe processes that take place during distribution and terminal elimination
Define and calculate , β, t½, Vi, VdSS, Cl and AUC
Understand influence of Volume of distribution on loading doses and toxicity
Design appropriate experiments to determine proper modeling of drug disposition
CP Clarke,L FD/MB

37. Recommended Readings Chapter 4, p. 73-92, 95-97
Multicompartment model assumptions (73-4)
Two-compartment open model (75-9)
Method of residuals (79-81)
Digoxin simulation (81-84)
Apparent volume of distribution (84-90)
Drug in tissue compartment (90-91)
Clearance and elimination constant (92)
Determination of compartment models (95-7)
CP Clarke,L FD/MB

38. Physiological Perspective
One
compartment
Quick
One compartment models are used when the drug appears to distribute into tissues instantaneously and uniformly.
Multicompartment drugs are delivered concurrently to one or more peripheral compartments composed of groups of tissues with lower blood perfusion and different affinity for the drug.
The one compartment model is the simplest of all multicompartment models.
k12
Two
compartments
Quick
Slow
CP Clarke,L FD/MB

39. Notes on Two-Compartment Modeling
Ideal model should mimic distribution and disposition
Full set of rate processes seldom taken into account
Tissue [C] often unknown
Because tissue [C] correlates with plasma [C], response often (but not always) correlates with plasma [C]
Invasive nature of tissue sampling limits sophistication
Blood or Plasma Pharmacokinetics
(2 compartment model)
Absorption (Input)
Disposition
Distribution
Elimination
Excretion
Metabolism

40. Multicompartment Modeling
k12 Vi Vt
Two compartment
model
k21
k10
Three compartment
model
k31
Vt3 Vi
k12
Vt2
k13
k21
k10
Vi = Volume of central compartment
Vt 2 or 3 = Volume of peripheral compartments

41. Assumptions (Two-Compartment Model)
k12 Vi Vt
k21
k10
Drug in peripheral compartment (bone, fat, muscle etc.) equilibrates with drug in central compartment
Plasma, highly perfused organs, extracellular water
[C] in a given compartment is uniform
Two-compartment drugs distribute into various tissues at different, first order rates
Elimination follows a single 1st order rate process only after distribution equilibrium is reached
Different peripheral compartments will accumulate multicompartment drugs at different rates
Distribution in different compartments may be uneven, i.e., drugs generally concentrate unevenly in different groups of tissues
CP Clarke,L FD/MB

42. Two-Compartment Model (Mathematical Perspective)
Ct is a bi-exponential decaying function that depends on 2 hybrid constants (A and B), which can be determined graphically, and the distribution () and elimination (β) rate constants
Ct = A • e -t + B • e –βt
Because  >> than β, this term
goes to zero at greater t values
A function of k10, k12 and k21

43. 2-Compartment Data (Linear Plot)
Concentration-Time Course of Caffeine IV Bolus
Clinical Pharmacology and Therapeutics. 1993;53:6-14

44. 2-Compartment Data (Semi-log Plot)
Concentration-Time Course of Caffeine IV Bolus
Clinical Pharmacology and Therapeutics. 1993;53:6-14

45. 2-Compartment Data (Semi-Log Plot)
Concentration-Time Course of Caffeine IV Bolus
Distribution or Alpha Phase A
Elimination or Beta Phase
Slope = β/2.303 B
Drug elimination and distribution occur concurrently and at varying degrees during the distribution phase, yet there is a net transfer of drug from the central compartment to the tissue compartment
Notice that the plasma concentration time curve has 2 distinct slopes—one that is a hybrid constant made up of all compartment distribution constants, and another that describes both the dynamic process of drug transfer between tissue and central compartments and elimination from the central compartment
Slope = /2.303
Note the bi-exponential decline in drug concentration
CP Clarke,L FD/MB

46. Calculation of Micro-constants
k21 =  • B + β • A
A + B
k10 =  • β
k21
k12 =  + β – k10 – k21
k12 Vi Vt
k21
k10
Note: Micro-constants cannot be calculated by direct means

47. Two-Compartment Elimination Rate Constants
k10 represents elimination from central compartment only
Larger than β
Not dependent on drug transfer into tissue compartment
β represents elimination when distribution equilibrium attained
Influenced by drug transfer into deep tissues
Clinically more useful than k10

48. Initial Concentration (Time = 0)
Question 1:
Based on the information gathered
thus far, what is the drug
concentration at time Zero?
Answer: The initial concentration at time = 0 is equal to the sum of the intercepts A and B

49. Half-Life
Compounds demonstrating two compartment kinetics will have t1/2 estimates for each exponential phases
Distribution half-life
t ½ Dist = ln2/
Elimination half-life
t ½ Elim = ln2/β
Terminal Half life is the elimination half life for most of drugs

50. What is the Elimination Half-Life (Aspirin Vs. Gentamicin)?
Distribution phase accounts for 31% of the dose
Elimination phase accounts for 69% of the dose
Terminal half life is the elimination half-life for aspirin
Gentamicin
Distribution phase accounts for 98% of the dose
Elimination phase accounts for 2% of the dose
Distribution half life is the appropriate half-life for gentamicin
Clinical Pharmacokinetics Concepts and Applications, Third edition, Lippincott Williams & Wilkins, Media, PA 19063

51. Volume of Distribution (Vd)
One compartment model Vd is constant:
Two compartment model Vd changes with time and reaches a plateau at the distribution equilibrium

52. Two Compartment Model (Vd vs. Time)Vt
Vdss Vi

53. Determination of Vi, Vdss and Vd From Hybrid Constants
Vi = Dose
A + B
VdSS = Vi [1 + k12]
k21
VdSS = Dose
β • AUC 0  ∞
Vt = Vi k12
k21
Note that VdSS is a function of transfer rate constants
The more extensively a drug distributes (i.e., the higher k12) the larger the volume of distribution

54. Vdss- The Concept
Vt is mostly influenced by the elimination rate and doesn’t reflect distribution
Vdss is mostly influenced by distribution
Volume term of the steady state when a drug is infused at a constant rate
Lies between Vi and Vt
Generally, difference between Vdss and Vt is small
Aspirin
Vdss = 10.4 L, Vt = 10.5 L
Gentamicin
Vdss = 345 L, Vt = 56 L
Substantially eliminated before distribution equilibrium achieved

55. Loading Doses
Loading doses are designed to achieve therapeutic concentrations faster
A: 45 mg/h constant IV infusion
B: Plasma [C]
C: Drug remaining from 530 mg IV loading dose
DL = Cp target • Vd F

56. Two-Compartment Distribution, Loading Doses & Site of Action
Some 2-compartment drugs exert their therapeutic and toxic effects on target organs located in the central compartment
Lidocaine, quinidine, procainamide
Question 2:
How should loading doses
for these drugs be handled?
In this instance, the concentration of drug delivered to target organs could be much higher than expected and produce toxicity if the loading dose is not adjusted appropriately.
The problem can be circumvented by calculating the total loading dose based on the total volume of distribution (Vd). The loading dose should be administered at a rate slow enough to allow for drug distribution into Vt; or the total loading dose should be given in sufficiently small bolus doses such that Cp in Vi does not exceed some predetermined critical concentration.
CP Clarke,L FD/MB

57. Loading Doses for Two-Compartment Drugs Acting in Vi
Answer:
Slow administration to allow for drug
distribution into Vt
OR…
Small bolus doses such that
Cp does not exceed
predetermined concentrations
DL = VC • CSS

58. Two-Compartment Distribution, Loading Doses & Site of Action
Some 2-compartment drugs exert their therapeutic and toxic effects on target organs located in Vt
Digoxin has a myocardium distribution half-life of 35 min and requires 8 to 12 h to completely distribute
In this instance, the concentration of drug delivered to target organs could be much higher than expected and produce toxicity if the loading dose is not adjusted appropriately.
The problem can be circumvented by calculating the total loading dose based on the total volume of distribution (Vd). The loading dose should be administered at a rate slow enough to allow for drug distribution into Vt; or the total loading dose should be given in sufficiently small bolus doses such that Cp in Vi does not exceed some predetermined critical concentration.
Question 3:
How should loading doses
for these drugs be handled?
CP Clarke,L FD/MB

59. Loading Doses for Two-Compartment Drugs Acting in Vt
Answer:
Quick administration is fine since
the initially observed high Cps are
not dangerous.
These concentrations, however,
cannot be used to predict
therapeutic effects.
DL = VSS • CSS

60. Loading Doses: The Case of Lidocaine
A: Loading dose + Infusion using Vi (volume of central compartment)
B: Loading dose + Infusion using VSS
Doted line: Constant infusion with no loading dose
Dashed line: Loading dose using Vi, no infusion
Seizures
Hypotension

61. Tip
If Vdss is unknown, use a value that falls between Vt and Vi

62. AUC by Trapezoidal Method
Estimation of AUC
From hybrid constants:
AUC0∞ = A + B
 β
Area t2 t3 = ½ (t3 – t2)(C2 + C3)
AUC by Trapezoidal Method

63. Clearance –Two Compartment Model
Q.CA
Q.Cv
ORGAN
elimination
CA = arterial blood concentration; Cv= Venous blood concentration; Q = blood flow
Clearance = Q(Ca-Cv) Ca

64. Clearance (Two Compartment Model)
Question 4:
Clearance (1- compartment Model):
Vd • Kel
Clearance (2- compartment model): ?
Answer: Clearance is model independent. However we need to use different rate constants depending on the choice of volume term
Example: Cltotal = k10 • Vi

65. Model-Independent Calculation of Clearance
Cl = Dose
AUC 0  ∞
No modeling consideration needed, but requires accurate measurement of AUC
Early & frequent sampling essential
Units = Volume/time
Theoretical volume of blood or plasma completely cleared of drug per unit time

66. One vs. Two Compartment Dilemma
Distribution phase may be missed entirely if blood is sampled too late or at wide intervals after drug administration

67. Use of One Compartment Modeling for Two-Compartment Drugs
If no concentration-time data points lie above back-extrapolated terminal line (semilog paper), assume one-compartment kinetics
One-compartment modeling can be used in place of two-compartment modeling provided:
Duration of distribution is small compared with elimination half-life
Elimination is minimal during distribution
Referred to as “non-significant” 2-compartment kinetics
Pharmacokinetic parameters must be computed after distribution is over

68. Tips For Solving the Problem Set

69. Plot Cp against time on semilog paper
Extrapolate terminal phase to t = 0
Intercept = B
Slope = b/2.303
Read at least 3 extrapolated [C]s during distribution
Calculate residual [C]s
Measured – extrapolated
Plot residuals against time (semilog paper)
Intercept of “feathered” line = A
Slope = /2.303