Pharmacokinetic models

PHARMACOKINETIC
SUBMITTED BY:
MODELS
VYJAYANTHI RAO VALLABHANENI
REG NO: 256213886030
DEPT. OF PHARMACEUTICS
Submitted To:
Dr. Satyabrata Bhanja

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

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

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 t½ Time hr

Mixed Order Kinetics
 Kinetics of a pharmacokinetic process changes from First order to Zero order
with increasing dose or chronic medication.
 Deviations from original Linear kinetic profile – Non Linear kinetics.
 Dose dependent kinetics
 Seen when P’kinetic process Carriers / Substrates
Capacity Limited –
get saturated at
Higher drug Conc.
Michaelis – Menten
Kinetics
 Describes velocity of Capacity limited, enzyme reactions and non
linear pharmacokinetics

Some examples;
Absorption (Vitamin C), Distribution (Naproxen), and Elimination
MICHAELIS MENTON EQUATION
-DC/DT = VMAX . C / KM + C
KM = Michaelis constant
VMAX = Theoretical maximum
Rate of process
(Riboflavin)

PLASMA DRUG CONCENTRATION – TIME
PROFILE
Effectiveness of Dosage
Regimen
Concentration of Drug in the Body
Conc. at Site of
action
Conc. in whole Blood (Plasma,
Serum), Saliva, Urine, CSF
PK Parameters determine drug
Conc.

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
8

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.

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 ).

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).

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

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

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.

PHARMACOKINETIC MODELS AND
COMPARTMENTS

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

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

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.

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.

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.

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.

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.

LOADING DOSE
• A drug dose does not show therapeutic activity unless it reaches the desired steady
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.

CALCULATION OF LOADING
DOSE
• After e.V. Administration, cmax is always smaller than that achieved after i.V.
And hence loading dose is proportionally smaller.
• 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

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:

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.

LINEAR MODEL - FIRST ORDER
KINETICS
• FIRST-ORDER
KINETICS

MATHEMATICALLY
• This behavior can be expressed mathematically as :

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-order absorption)
• One com. Open model - extra vas. Administration (First-order absorption )
• INTRAVENOUS (IV) BOLUS ADMINISTRATION

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

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 (eq. 5)
• So equation-8 becomes
log c = log co – ke t
2.303 (eq.6)

KE = KE + KM +KB +KL +….. (Eq.7)
(KE is overall elimination rate constant)

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 V d
Cl T (eq.9)

APPARENT VOLUME OF
DISTRIBUTION
• Defined as volume of fluid in which drug appears to be distributed.
• Vd = amount of drug in the body = x
Plasma drug concentration 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 =30000mcg/0.732mcg/ml
= 41 liter.
• For drugs given as i.V.Bolus,
Vd (area)=xo/KE.Auc …….12.A
• For drugs admins. Extra. Vas.
Vd (area)=f xo/ke.Auc ……..12.B

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

• 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 (eq.17)
Clr =ke vd (eq.18)
• But, KE= 0.693/t1/2
• So, clt = 0.693 vd (eq.19)
t1/2

• 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∞ …….(eq. 21)
auc

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 …………….(eq 22)
• Extraction ratio:
ER= (cin- cout)/ cin
• ER is an index of how efficiently the eliminating organ clear the blood
flowing through it of drug.

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.

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

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

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.

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

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

• 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

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

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

ONE COMPARTMENT MODEL: EXTRA VASCULAR
ADMIN ( ZERO ORDER ABSORPTION)
• This model is similar to that for constant rate infusion.
Drug at site
zero order elimination
Blood & other
Body tissues
R0
Absorption
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.

ONE COMPARTMENT MODEL: EXTRA
VASCULAR ADMIN ( FIRST ORDER
ABSORPTION)
• Drug that enters the body by first order absorption process gets distributed in
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
Blood & other
Body tissues
Drug at
site
Ka
KE
First order
absorption
elimination
C=Ka F Xo/Vd (Ka-KE) [e -Ket-e-Kat] …41

• Ideally a true pharmacokinetic model should be the one with a rate constant for
each tissue undergoing equilibrium.
• Therefore best approach is to pool together tissues on the basis of similarity in
their distribution characteristics.
• The drug disposition occurs by first order.
• Multi-compartment characteristics are best described by administration as i.v
bolus and observing the manner in which the plasma concentration declines with
time.
The no. Of exponentials required to describe such a plasma level-time profile
determines the no. Of kinetically homogeneous compartments into which a
drug will distribute.
The simplest and commonest is the two compartment model which classifies the
body tissues in two categories :
1. Central compartment or compartment 1
2. Peripheral or tissue compartment or compartment 2.

TWO COMPARTMENT OPEN MODEL-IV BOLUS
ADMINISTRATION:
Elimination from central compartment
Fig:
1
Central
2
peripheral
• After the iv bolus of a drug the decline in the plasma conc. Is bi-exponential.
• Two disposition processes- distribution and elimination.
• These two processes are only evident when a semi log plot of C vs. T is
made.
• Initially, the conc. Of drug in the central compartment declines rapidly, due
to the distribution of drug from the central compartment to the peripheral
compartment. This is called distributive phase.

Extending the relationship X= vd C
Dcc = K21 xp – K12 xc – KE xc
Dt vp vc vc
X= Amt. Of drug in the body at any time t remaining to be eliminated
C=drug conc in plasma
Vd =proportionality const app. Volume of distribution
Xc and xp=amt of drug in C1 and C2
Vc and vp=apparent volumes of C1 and C2
= K12 xc – K21 xp
Vc vp On integration equation gives conc of drug in central and
peripheral compartments at any given time t
Cp = xo [( K21 – a)e-at + (K12 – b)e-bt]
Vc b – a a – b
Xo = iv bolus dose

• The relation between hybrid and microconstants is given as :
a + b = K12 + K21 + KE
A b = K21 KE
Cc = a e-at + be-bt
Cc=distribution exponent + elimination
exponent
A and B are hybrid constants for two exponents and can be resolved by graph
by method of residuals.
A = X0 [K21 - A] = CO [K21 – A]
VC B – A B – A
B = X0 [K21 - B] = CO [K21 – B]
VC A – B A – B
CO = Plasma drug concentration immediately after i.v. Injection

• Method of residuals : the biexponential disposition curve obtained after i. V.
Bolus of a drug that fits two compartment model can be resolved into its
individual exponents by the method of residuals.
C = a e-at + b e-bt
From graph the initial decline due to distribution is more rapid than the terminal
decline due to elimination i.E. The rate constant a >> b and hence the term e-at
approaches zero much faster than e –bt
C = B e-bt
Log C = log B – bt/2.303 C = back extrapolated pl. Conc.
• A semilog plot of C vs t yields the terminal linear phase of the curve having
slope –b/2.303 and when back extrapolated to time zero, yields y-intercept log
B. The t1/2 for the elimination phase can be obtained from equation
• t1/2 = 0.693/b.
• Residual conc values can be found as-
Cr = C – C = ae-at
Log cr = log A – at
2.303
A semilog plot cr vs t gives a straight line.

Ke = a b c
A b + B a
K12 = a b (b - a)2
C0 (A b + B a)
K21 = A b + B a
C0
• For two compartment model, KE is the rate constant for elimination of drug
from the central compartment and b is the rate constant for elimination from
the entire body. Overall elimination t1/2 can be calculated from b.
Area under (auc) = a + b
The curve a b
App. Volume of central = X0 = X0
compartment C0 KE (AUC)

App. Volume of = VP = VC K12
Peripheral compartment K21
Apparent volume of distribution at steady state or equilibrium
Vd,ss = VC +VP
Vd,area = X0
B AUC
Total systemic clearence= clt = b vd
Renal clearence= clr = dxu = KE VC
Dt
The rate of excretion of unchanged drug in urine can be represented by :
dxu = KEA e-at + KE B e-bt
Dt
The above equation can be resolved into individual exponents by the method of residuals.

TWO – COMPARTMENT OPEN MODEL- I.V.
INFUSION
The plasma or central compartment conc of a drug when administered as constant rate (0 order) i.V. Infusion is
given as:
C = R0 [1+(KE - b)e-at +(KE - a)e-bt]
VCKE b – a a - b
At steady state (i.E.At time infinity) the second and the third term in the bracket becomes zero and the equation
reduces to:
Css = R0
Vc ke
Now VC KE = vd b
Css = r0 = r0
Vdb clt
The loading dose X0,L = css vc = R0
Ke
1
Central
2
Peripheral

TWO-COMPARTMENT OPEN MODEL-EXTRAVASCULAR
ADMINISTRATION
• First - order absorption :
• For a drug that enters the body by a first-order absorption process and
distributed according to two compartment model, the rate of change in drug
conc in the central compartment is described by three exponents :
• An absorption exponent, and the two usual exponents that describe drug
disposition.
The plasma conc at any time t is
C = n e-kat + l e-at + m e-bt
C = absorption + distribution + elimination
Exponent exponent exponent
• Besides the method of residuals, ka can also be found by loo-riegelman method
for drug that follows two-compartment characteristics.
• Despite its complexity, the method can be applied to drugs that distribute in any
number of compartments.

CALCULATING Ka using Wagner-nelson
method(Bioavailability
parameters)

WAGNER-NELSONS METHOD
THEORY: The working equations can be derived from the mass balance
equation: Gives the following eqaution with time and mass balance
• Above equation Integrating gives
• To the equation amount
absorbed VERSUS TIME

• Taking this to infinity where cp equals 0
• Finally (Amax - A), the amount remaining to be absorbed can also be
expressed as the amount remaining in the GI, xg
• We can use this equation to look at the absorption process. If, and only if,
absorption is a single first order process

• Example data for the method of wagner-nelson kel (from IV data) = 0.2 hr-
Time
(hr)
Plasma
Concentratio
n
(mg/L)
Column
3
ΔAUC
Column
4
AUC
Column 5
kel * AUC
A/V
[Col2 +
Col5]
(Amax - A)/V
0.0 0.0 0.0 0.0 0.0 0.0 4.9
1.0 1.2 0.6 0.6 0.12 1.32 3.58
2.0 1.8 1.5 2.1 0.42 2.22 2.68
3.0 2.1 1.95 4.05 0.81 2.91 1.99
4.0 2.2 2.15 6.2 1.24 3.44 1.46
5.0 2.2 2.2 8.4 1.68 3.88 1.02
6.0 2.0 2.1 10.5 2.1 4.1 0.8
8.0 1.7 3.7 14.2 2.84 4.54 0.36
10.0 1.3 3.0 17.2 3.44 4.74 0.16
12.0 1.0 2.3 19.5 3.9 4.9 -
∞ 0.0 5.0 24.5 4.9 4.9 -

• The data (Amax-A)/V versus time can be plotted on semi-log and linear
graph paper

• Plotting (Amax-A)/V versus time produces a straight line on semi-log graph paper and a
curved line on linear graph paper. This would support the assumption that absorption can be
described as a single first process. The first-order absorption rate constant, ka, can be
calculated to be 0.306 hr-1 from the slope of the line on the semi-log graph paper.
ADVANTAGES:
• The absorption and elimination processes can be quite similar and accurate determinations of
ka can still be made.
• The absorption process doesn't have to be first order. This method can be used to investigate
the absorption process.
DISADVANTAGES:
• The major disadvantage of this method is that you need to know the elimination rate constant,
from data collected following intravenous administration.
• The required calculations are more complex.

RESIDUAL METHOD OR
FEATHERING TECHNIQUE
• Absowhen a drug is administered by extravascular route, absorption is a
prerequisite for its therapeutic activity.
• The absorption rate constant can be calculated by the method of
residuals.
• The technique is also known as feathering, peeling and stripping.

φ It is commonly used in pharmacokinetics to resolve a
multiexponential curve into its individual components.
φ For a drug that follows one-compartment kinetics and
administered extravascularly, the concentration of drug
in plasma is expressed by a biexponential equation.
C=
퐾푎퐹푋0
[e-K
푉푑(퐾푎−퐾퐸)
t – e-K
E
a
t] (1)
If KaFX0/Vd(Ka-KE) = A, a hybrid constant, then:
C = A e-KEt – A e-Kat (2)

φ During the elimination phase, when absorption is
almost over, Ka<<KE and the value of second
exponential e-Kat approaches zero whereas the first
exponential e-KEt retains some finite value.
φ At this time, the equation (2) reduces to:
퐶
−
= 퐴 푒
− 퐾퐸푡(3)
φ In log form, the above equation is:
Log C− = log A -
퐾퐸푡
2.303
(4)

Where ,
C− = back extrapolated plasma concentration values
φ A plot of log C versus t yield a biexponential curve with a
terminal linear phase having slope –KE/2.303
φ Back extrapolation of this straight line to time zero yields y-intercept
equal to log A.
70

Plasma conc.-Time profile after oral administration of a single dose of a drug

φ Subtraction of true plasma concentration values i.e.
equation (2) from the extrapolated plasma
concentration values i.e. equation (3) yields a series
of residual concentration value Cτ.
(C− - C) = Cτ = A e-Kat(5)
φ In log form , the equation is:
log Cτ= log A -
퐾푎푡
2.303
(6)

φ A plot of log Cτ versus t yields a straight line with slope -
Ka /2.303 and y-intercept log A.
φ Thus, the method of residual enables resolution of the
biexponential plasma level-time curve into its two
exponential components.
φ The technique works best when the difference between
Ka and KE is large (Ka/KE ≥ 3).

THREE COMPARTMENT MODEL
AND APPLICATIONS OF
PHARMACOKINETIC
PARAMETERS IN DOSAGE
DEVELOPMENT

THREE COMPARTMENT MODEL
• Gibaldi & feldman described a three compartment open model to
explain the influence of route of administration .I.E. Intravenous
vs. Oral, on the area under the plasma concentration vs. Time
curve.
• Portman utilized a three compartment model which included
metabolism & excretion of hydroxy nalidixic acid.

CENTRAL
COMPARTMENT
TISSUE
COMPARTMENT
DEEP
TISSUE
COMPARTMENT
DRUG INPUT
K10
THREE COMPARTMENT CATENARY MODEL
TISSUE
COMPARTMENT
RAPID IV
DRUG INPUT
K21 K13
CENTRAL
COMPARTMENT
DEEP
TISSUE
COMPARTMENT
K10
K12
DRUG OUTPUT
K31
THREE COMPARTMENT MAMMILLARY MODEL

Three compartment model consist of the following compartments .
Central compartment.
Tissue compartment.
Deep tissue compartment.
 In this compartment model drug distributes most rapidly in to first or central
compartment.
 Less rapidly in to second or tissue compartment .
Very slowly to the third or deep tissue compartment. The third compartment is poor
in tissue such as bone & fat.
• Each compartment independently connected to the central compartment.
• Notari reported the tri exponential equation
c=a e-t+ b e-βt+ c e-γt
• A,B,C are the y-intercept of extrapolated lines.
• Α,β,γ are the rate constants

RAPID I.V BOLUS
ADMINISTRATIONS
• When the drug is administered by i.V the drug will rapidly distributed in c.C
,less rapidly in to t.C. Very slowly in to deep tissue compartment.
Plasma profile
• When the drug is administered by i.V the plasma conc. Will increased in c.C
this is first order release.
• The conc. Of drug in c.C. Exhibits an initial distribution this is very rapid.
• Drug in central compartment exhibits an initial distribution this is very rapid .

Pharmacokinetic parameters
Bioloigical half-life ::
• It is defined as the time taken for the amount of drug in the body as well as
plasma to decline by one half or 50% its initial value.
• Concentration of drug in plasma as a function of time is
c=a e - t+ b e -β t+ c e -γ t
• In this equation α>β>γ some time after the distributive phase (i.e. When time
become large) the two right hand side terms values are equal to zero.
• The eq.. Is converted in to
c=a e-αt
Taking the natural logarithm on both sides
the rate constant of this straight line is ‘α’ and biological half life is
t1/2 =0.693/α

VOLUME OF CENTRAL COMPARTMENT
• At time=0
C=A e –α t+ B e –β t+ C e –γ t
This equation becomes
CO = A+B+C -----1
CO =conc. Of plasma immediately after the i.V administration
• When administered the dose is not distributed in tissue compartment.
• Therefore the drug is present in c.C only .
• If D is dose administered then CO = D /V C---------2
Vc=volume of drug in c.C
Combining the 1&2 eq.. We get Vc = d/co (c o----- conc. Of drug in plasma)
ELIMINATION RATE CONSTANT:
 Drug that follows three compartment kinetics and administered by i.V injection the decline
in the plasma drug conc. Is due to elimination of drug from the three compartments.
Ke=(a+b+c) α β γ/a β γ +b α γ+ cα β

PHYSIOLOGICALLY BASED
PHARMACOKINETIC MODELS
• Blood flow rate limited or perfusion rate
limited model.
• Drawn on the basis of anatomic and
physiologic data.(More realistic)
• Organs or tissues having no perfusion are
excluded.
• Drug movement to a particular region is
much more rapid than its rate of delivery to
that region by blood - perfusion rate limited
model.
• Thus, applicable to highly membrane
permeable drugs, i.e. Low molecular weight,
poorly ionized and highly lipophilic drugs.
• For highly polar, ionized and charged drugs,
the model is referred to as membrane
permeation rate limited. 81

TISSUE
DOSIMETRY
• Measure of the level of some reactive metabolites reaching the target
tissue provide a better dose parameter for risk assessment purpose
than administered doses.
• The effects of growth and ageing(since the fat increasing proportional
to the body weight, as animal grows), topical adsorption( in inhalation
studies), pregnancy and lactation(for example changes in body
weight, total body water, plasma proteins, body fat and cardiac output
will alter the distribution of many drugs and their metabolites.)And
competitive multiple metabolites are illustrated in PBPK modelling.
83

REFERENCES :
BIOPHARMACEUTICS AND PHARMACOKINETICS.
P L MEDAN, 1ST EDN
BIOPHARMACEUTICS AND PHARMACOKINETICS.
D.M BRAHMANKAR AND SUNIL. B .JAISWAL, 1ST EDN
 APPLIED BIOPHARMACEUTICS AND PHARMACOKINETICS
LEON SHARGEL AND ANDREW YU,
4TH EDN.
BIOPHARMACEUTICS AND CLINICAL PHARMACOKINETICS BY MILO
GIBALDI, 4TH EDN.
WWW.GOOGLE.COM
WWW.BOOKS.GOOGLE.COM

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