This document provides an overview of pharmacokinetic models and parameters. It discusses one-compartment models for intravenous bolus and intravenous infusion administration. For intravenous bolus, the elimination rate constant, half-life, apparent volume of distribution, and clearance are defined. For intravenous infusion, the equations for drug concentration over time are presented. Compartmental models are used to mathematically describe drug behavior in the body and calculate pharmacokinetic parameters.

1. B.Pharmacy
Subject-Biopharmaceutics and Pharmacokinetics
Sub Code-BP604T
MODULE-3,4
PHARMACOKINETIC
MODELS
SUBMITTED BY
ALKA MARAL
Assistant Professor (Pharmaceutics)
NARAINA VIDHYAPEETH GROUP OF INSTITUTIONS
FACULTY OF PHARMACY, PANKI, KANPUR (U.P.)

2. Objective of course;
Understand various pharmacokinetic parameters, their significance
& applications.
Learning Outcomes;
Students will learn about different type of pharmacokinetic models used
to determine various pharmacokinetic parameters like Vd,t1/2,AUC,Ka
etc.

3. OVERVIEW
• Basic considerations in pharmacokinetics
• Compartment models
• One compartment model
• Assumptions
• Intravenous bolus administration
• Intravenous infusion
• Extravascular administration (zero order and first order absorption
model)
• Multi-compartment model

4. BASIC CONSIDERATIONS IN
PHARMACOKINETICS
• Pharmacokinetic parameters
• Pharmacodynamic parameters
• Zero, first order & mixed order kinetic
• Rates and orders of kinetics
• Plasma drug conc. Time profiles
• Compartmental models – physiological model
• Applications of pharmacokinetics
• Non compartment model

5. Common units in Pharmacokinetics
S.no Pharmacokinetic parameter Abbreviation Fundamental units Units example
1. Area under the curve AUC Concentration x time µg x hr/mL
2. Total body clearance ClT Volume x time Litres/time
3. Renal clearance ClR Volume x time Litres/time
4. Hepatic clearance ClH Volume x time Litres/time
5. Apparent volume of distribution VD Volume Litres
6. Vol. of distribution at steady state VSS Volume Litres
7. Peak plasma drug concentration CMAX Concentration mg/L
8. Plasma drug concentration CP Concentration mg/L
9. Steady-state drug concentration Css Concentration mg/L
10. Time for peak drug concentration TMAX Time Hr
11. Dose DO Mass mg
12. Loading dose DL Mass mg
13. Maintenance dose DM Mass mg
14. Amount of drug in the body DB Mass Mg
15. Rate of drug infusion R Mass/time mg/hr
16. First order rate constant for drug absorption Ka 1/time 1/hr
17. Zero order rate constant for drug absorption KO Mass/time mg/hr
18. First order rate constant for drug elimination K 1/time 1/hr
19. Elimination half-life t Time hr

6. A TYPICAL PLASMA DRUG CONC. AND TIME CURVE
OBTAINED AFTER A SINGLE ORAL DOSE OF A
DRUG, SHOWING VARIOUS P'KINETIC AND
P’DYNAMIC PARAMETERS DEPICTED IN BELOW
FIG

7. PHARMACOKINETIC PARAMETERS
Three important parameters useful in assessing the bioavailability of a drug
from its formulation are:
1. Peak plasma concentration ( cmax )
the point at which, maximum concentration of drug in plasma.
Units : µg/ml
• Peak conc. Related to the intensity of pharmacological response, it
should be above MEC but less than MSC.
• The peak level depends on administered dose and rate of absorption
and elimination.

8. 2. Time of peak concentration (tmax )
the time for the drug to reach peak concentration in plasma
(after extra vascular administration).
Units : hrs
• Useful in estimating onset of action and rate of absorption.
• Important in assessing the efficacy of single dose drugs used to treat acute
conditions (pain, insomnia ).

9. 3. Area under curve (AUC)
It represents the total integrated area under the plasma level-time profile and
expresses the total amount of the drug that comes into systemic circulation after
its administration.
Units : µg/ml x hrs
• Represents extent of absorption – evaluating the bioavailability of drug from its
dosage form.
• Important for drugs administered repetitively for treatment of chronic conditions
(asthma or epilepsy).

10. PHARMACODYNAMIC PARAMETERS
1. Minimum effective concentration (MEC)
Minimum concentration of drug in plasma/receptor site required to produce
therapeutic effect.
• Concentration below MEC – sub therapeutic level
• Antibiotics - MEC
2. Maximum safe concentration (MSC)
Concentration in plasma above which adverse or unwanted effects are
precipitated.
• Concentration above MSC – toxic level

11. 3. Onset time
Time required to start producing pharmacological response.
Time for plasma concentration to reach mec after administrating drug
4. Onset of action
The beginning of pharmacologic response.
It occurs when plasma drug concentration just exceeds the required mec.
5. Duration of action
The time period for which the plasma concentration of drug remains above MEC
level.
6. Intensity of action
It is the minimum pharmacologic response produced by the peak plasma conc. Of
drug.
7. Therapeutic range the drug conc. Between MEC and MSC

12. CONCEPT OF “HALF LIFE”
 ½ Life = how much time it takes for blood levels of drug to decrease to half
of what it was at equilibrium
 There are really two kinds of ½ life…
“Distribution” ½ life = when plasma levels fall to half what they were
at equilibrium due to distribution to/storage in body’s tissue reservoirs.
 “Elimination” ½ life = when plasma levels fall to half what they were
at equilibrium due to drug being metabolized and eliminated.
 It is usually the elimination ½ life that is used to determine dosing
schedules, to decide when it is safe to put patients on a new drug.

13. PHARMACOKINETIC MODELS AND
COMPARTMENTS

14. Pharmacokinetic Modelling
Compartment
Models
Non-Compartment
Models
Physiologic
Models
Caternary
Model
Mamillary
Model
AUC, MRT, MAT, Cl
One compt
i v
bolus
i v
infusion
Multiple
doses
Multi compt Two compt
Single oral Dose
Intermittent i v infusion
i v bolus
Oral drug

15. PHARMACOKINETIC MODELS
 Means of expressing mathematically or quantitatively, time course of drug
through out the body and compute meaningful pharmacokinetic parameters.
Useful in :
• Characterize the behavior of drug in patient.
• Predicting conc. Of drug in various body fluids with dosage regimen.
• Calculating optimum dosage regimen for individual patient.
• Evaluating bioequivalence between different formulation.
• Explaining drug interaction.
Pharmacokinetic models are hypothetical structures that are used to describe the
fate of a drug in a biological system following its administration.
Model
• Mathematical representation of the data.
• It is just hypothetical

16. WHY MODEL THE DATA?
There are three main reasons due to which the data is subjected to modelling.
1. Descriptive: to describe the drug kinetics in a simple way.
2. Predictive: to predict the time course of the drug after multiple dosing based
on single dose data, to predict the absorption profile of the drug from the iv
data.
3. Explanatory: to explain unclear observations.

17. PHARMACOKINETIC MODELING IS USEFUL
IN :-
• Prediction of drug concentration in plasma/ tissue/ urine at any point of time.
• Determination of optimum dosage regimen for each patient.
• Estimation of the possible accumulation of drugs/ metabolites.
• Quantitative assessment of the effect of disease on drug’s adme.
• Correlation of drug concentration with pharmacological activity.
• Evaluation of bioequivalence.
• Understanding of d/i.

18. COMPARTMENTAL MODELS
• A compartment is not a real physiological or anatomic region
but an imaginary or hypothetical one consisting of tissue/ group
of tissues with similar blood flow & affinity.
• Our body is considered as composed of several compartments
connected reversibly with each other.

19. ADVANTAGES
• Gives visual representation of various rate processes involved in drug
disposition.
• Possible to derive equations describing drug concentration changes in each
compartment.
• One can estimate the amount of drug in any compartment of the system after
drug is introduced into a given compartment.
DISADVANTAGES
• Drug given by IV route may behave according to single compartment model
but the same drug given by oral route may show 2 compartment behaviour.
• The type of compartment behaviour i.E. Type of compartment model may
change with the route of administration.

20. TYPES OF COMPARTMENT
1. Central compartment
Blood & highly perfused tissues such as heart, kidney, lungs, liver, etc.
2. Peripheral compartment
Poorly per fused tissues such as fat, bone, etc.
MODELS:
“OPEN” and “CLOSED” models:
• The term “open” itself mean that, the administered drug dose is removed from
body by an excretory mechanism ( for most drugs, organs of excretion of drug is
kidney)
• If the drug is not removed from the body then model refers as “closed” model.

22. LOADING DOSE
• A drug dose does not show therapeutic activity unless it reaches the desiredsteady
state.
• It takes about 4-5 half lives to attain it and therefore time taken will be too long if
the drug has a long half-life.
• Plateau can be reached immediately by administering a dose that gives the desired
steady state instantaneously before the commencement of maintenance dose x0.
• Such an initial or first dose intended to be therapeutic is called as priming dose or
loading dose x0,l.

23. CALCULATION OF LOADING
DOSE
• After e.V. Administration, cmax is always smaller than that achieved after i.V.
And hence loading dose is proportionallysmaller.
• For the drugs having a low therapeutic indices, the loading dose may be
divided into smaller doses to be given at a various intervals before the first
maintenance dose.
• A simple equation for calculating loading dose is :
xo,l = css,av vd
F

24. CALCULATION….,
• When vd is not known, loading dose may be calculated by the following
equation :
xo,l = 1_
Xo (1 – e-ket) (1 – e-kat)
• Given equation applies when ka >> ke and drug is distributed rapidly.
• When drug is given i.V. Or when absorption is extremely rapid, the
absorption phase is neglected and the above equation reduces to
accumulation index:

25. ASSUMPTIONS
1. One compartment
 The drug in the blood is in rapid equilibrium with drug in the extra-vascular
tissues. This is not an exact representation however it is useful for a number
of drugs to a reasonable approximation.
2. Rapid mixing
 We also need to assume that the drug is mixed instantaneously in blood or
plasma.
3. Linear model
 We will assume that drug elimination follows first order kinetics.

26. LINEAR MODEL - FIRST ORDER
KINETICS
• FIRST-ORDER
KINETICS

27. MATHEMATICALLY
• This behavior can be expressed mathematically as :

28. ONE COMPARTMENT MODEL
One compartment model can be defined :
• One com. Open model – i.V. Bolus.
• One com. Open model - cont. Intravenous infusion.
• One com. Open model - extra vas. Administration (zero-orderabsorption)
• One com. Open model - extra vas. Administration (First-order absorption )
• INTRAVENOUS (IV) BOLUS ADMINISTRATION

29. RATE OF DRUG PRESENTATION TO BODY
IS:
• Dx =rate in (availability)–rate out( Eli)
Dt
• Since rate in or absorption is absent, equation becomes
dx = - rate out
dt
• If rate out or elimination follows first order kinetic
Dx/dt = -kex (eq.1)
ELIMINATION PHASE:
Elimination phase has three parameters:
• Elimination rate constant
• Elimination half life
• Clearance

30. ELIMINATION RATE CONSTANT
• Integration of equation (1)
• In x = ln xo – ke t (eq.2)
Xo = amt of drug injected at time t = zero i.E. Initial amount of drug injected
X=xo e-ket ( eq.3)
• Log x= log xo – ke t
2.303 (eq.4)
• Since it is difficult to directly determine amount of drug in body x, we use relationship
that exists between drug conc. In plasma C and X; thus
• X = vd C
• So equation-8 becomes
log c = log co – ke t
(eq. 5)
2.303 (eq.6)

31. KE = KE + KM +KB +KL+….. (Eq.7)
(KE is overall elimination rateconstant)

32. ELIMINATION HALF LIFE
T1/2 = 0.693
KE (eq.8)
• Elimination half life can be readily obtained from the graph of log c
versus t
• Half life is a secondary parameter that depends upon the primary
parameters such as clearance and volume of distribution.
• T1/2 = 0.693 Vd
Cl T (eq.9)

33. APPARENT VOLUME OF
DISTRIBUTION
• Defined as volume of fluid in which drug appears to be distributed.
• Vd = amount of drug in the body =
Plasma drug concentration
x
C (eq.10)
Vd = xo/co
=I.V.Bolus dose/co (eq.11)
• Example: 30 mg i.V. Bolus, plasma conc.= 0.732 mcg/ml.
• Vol. Of dist. = 30mg/0.732mcg/ml
= 41 liter.
• For drugs given as i.V.Bolus,
Vd (area)=xo/KE.Auc
• For drugs admins. Extra. Vas.
=30000mcg/0.732mcg/ml
…….12.A
Vd (area)=fxo/ke.Auc .................................... 12.B

34. CLEARANCE
Clearance = rate of elimination
Plasma drug conc.. (Or) cl= dx /dt
C ……., (eq.13)
Thus, renal clearance
Hepatic clearance =
= rate of elimination by kidney
C
rate of elimination by liver
C
Other organ clearance = rate of elimination by organ
C
Total body clearance:
Clt = clr + clh + clother ……, (eq.14)

35. • According to earlier definition
cl = dx /dt
C
• Submitting eq.1 dx/dt = KE X , above eq. Becomes ,clt = KE X/ C .., (Eq 15)
• By incorporating equation 1 and equation for vol. Of dist. ( Vd= X/C ) we can
get
clt =KE vd (eq.16)
• Parallel equations can be written for renal and hepatic clearance.
Clh =km vd
Clr =ke vd
• But, KE= 0.693/t1/2
(eq.17)
(eq.18)
• So, clt = 0.693 vd (eq.19)
t1/2

36. • For non compartmental method which follows one compartmental
kinetic is :
• For drug given by i.V. Bolus
clt = xo....................... 20.A
Auc
• For drug administered by e.V.
Clt = f xo....................20.B
Auc
• For drug given by i.V. Bolus
renal clearance = xu∞
auc
…….(eq. 21)

37. ORGAN CLEARANCE
• Rate of elimination by organ= rate of presentation to the organ – rate of exit
from the organ.
• Rate of elimination =q. Cin- Q.Cout
(Rate of extraction) =Q (cin- cout)
Clorgan=rate of extraction/cin
=q(cin-cout)/cin
=Q.Er
• Extraction ratio:
ER= (cin- cout)/ cin
…………….(eq 22)
• ER is an index of how efficiently the eliminating organ clear the blood
flowing through it of drug.

38. According to ER, drugs can be classified as
• Drugs with high ER (above 0.7)
• Drugs with intermediate ER (between 0.7-0.3)
• Drugs with low ER (below 0.3)
• The fraction of drug that escapes removal by organ is expressed as
F= 1- ER
• Where f=systemic availability when the eliminating organ is liver.

39. HEPATIC CLEARANCE
Clh = clt –clr
 Can also be written down from eq 22
 Clh= QH ERH
 QH= hepatic blood flow. ERH = hepatic extraction ratio.
 Hepatic clearance of drug can be divided into two groups :
1. Drugs with hepatic blood flow rate-limited clearance
2. Drugs with intrinsic capacity- limited clearance

40. HEPATIC BLOOD FLOW
• F=1-erh
= AUC oral
AUC i.V

41. INTRINSIC CAPACITY CLEARANCE
• Denoted as clint, it is defined as the inherent ability of an organ to
irreversibly remove a drug in the absence of any flow limitation.

42. Blood & other
Body tissues
ONE COMPARTMENT OPEN MODEL:
INTRAVENOUS INFUSION
• Model can be represent as : ( i.v infusion)
Drug R0 KE
Zero order
Infusion
rate
Dx/dt =ro-kex
X=ro/ke(1-e-ket)
Since X =vdc
…eq 23
…eq 24
C= ro/kevd(1-e-ket) …eq 25
= Ro/clt(1-e-ket) …eq 26

43. • At steady state. The rate of change of amount of drug in the body is zero,eq
23 becomes
Zero=ro-kexss
Kexss=ro
Css=ro/kevd
=Ro/clt i.E
…27
…28
…29
infusion rate..... 30
Clearance
Substituting eq. 30 in eq. 26
•C=css(1-e-ket)
Rearrangement yields:
…31
• [Css-c]=e-ket .................................................32
Css
Log CSS-C
Css
= -ket
2.303
…33

44. • If n is the no. Of half lives passed since the start of infusion(t/t1/2)
• Eq. Can be written as
• C=CSS [1-(1/2)n] …34

45. INFUSION PLUS LOADING
DOSE
XO,L=CSSVD …35
• SUBSTITUTION OF CSS=RO/KEVD
• XO,L=RO/KE …36
• C=XO,L/VD E-KET+ RO/KEVD(1-E-KET) …37

46. ONE COMPARTMENT OPEN MODEL
EXTRA VASCULARADMINISTRATION
• When drug administered by extra vascular route (e.G. Oral, i.M, rectal ),
absorption is prerequisite for its therapeutic activity.

47. n
ONE COMPARTMENT MODEL: EXTRA VASCULAR
ADMIN ( ZERO ORDER ABSORPTION)
• This model is similar to that for constant rate infusion.
Drug at site
R0
Absorptio
zero order elimination
o Rate of drug absorption as in case of CDDS , is constant and continues until
the amount of drug at the absorption site (Ex. GIT) is depleted.
o All equations for plasma drug conc. Profile for constant rate i.V. Infusion
are also applicable to this model.
Blood & other
Body tissues

48. ONE COMPARTMENT MODEL: EXTRA
VASCULAR ADMIN ( FIRST ORDER
ABSORPTION)
• Drug that enters the body by first order absorption process gets distributedin
the body according to one compartment kinetic and is eliminated by first
order process.
• The model can be depicted as follows and final equation is as follows
C=Ka F Xo/Vd (Ka-KE) [e -Ket-e-Kat] …41
Drug at
site
Ka
First order
absorption
K
elimination
Blood & other
Body tissues
E