- The document discusses nonlinear pharmacokinetics where parameters like clearance and volume of distribution change with dose. This occurs when transporters or enzymes involved in absorption, distribution, metabolism and excretion get saturated at high drug concentrations. - The Michaelis-Menten equation is used to describe saturation kinetics and estimate parameters Km and Vmax. Various methods like Lineweaver-Burk, direct linear and graphical plots are presented to determine these values using in vivo or in vitro concentration and rate data. - Estimation of Km and Vmax is also described at steady-state concentrations achieved after constant rate infusion or multiple dosing, through plots of steady-state concentration versus dosing rate.

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Mohi-ud-din islaMic instituteMohi-ud-din islaMic institute
ofof
PharMaceutical sciences.PharMaceutical sciences.

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-:PRESENTED BY:-
Muslim Khan
Class: 4th
Year Pharm-D
Roll No# 19

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CONTENTS
Introduction
 Nonlinearity Pharmacokinetic
Causes of nonlinearity
Michaelis – Menten equation
Estimation of Km and Vmax
Estimation of Km and Vmax at steady-state concentration

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» Dose independent paharmacokinetic
Pharmacokinetic paraméters dont change with increse or
decrease in dose
» In such situation the rate processes are follw first order or
linear kinetics and all semilog plots of C Vs t for different
doses are superimposable.
LINEAR PHARMACOKINETICS

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NONLINEAR PHARMACOKINETICS
» Dose dependent paharmacokinetic
Pharmacokinetic paraméters change with increse or decrease
in dose
Pharmacokinetic paraméters(ADME ) are depend upon carrier
or enzymes that are substrate specific, have definite
capacities and are susceptible to saturation at a high drug
concentration.
» In such cases, first-order kinetics transform into a mixture of
first-order and zero-order rate processes
»

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Linear Pharmacokinetics Non Linear
Pharmacokinetics
Pharmacokinetic parameters
for a drug would not change
with change in dose
Pharmacokinetic
parameters for a drug can
change with change in dose.
Dose Independent Dose dependent
First Order kinetics Also called as Mixed order,
Saturated kinetics, capacity
limited
All semilog plots of C vs t for
diff. doses are
superimposable.
Not superimposable

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CAUSES OF NON-LINEARITY
• Three causes:-
• I) Solubility / dissolution of drug is rate-limited;
Griseofulvin - at high concentration in intestine.
II) Carrier - mediated transport system; Ascorbic
acid - saturation of transport system.
III) Presystemic gut wall / hepatic metabolism
attains saturation; Propranolol.
Drug absorption

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At high doses non-linearity due to
• Two causes:-
• I) Binding sites on plasma proteins get
saturated; Phenylbutazone.
• II) Tissue binding sites get saturated.
• In both cases there is increase in plasma drug
concentration.
.
Drug distribution

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• Non-linearity occurs due to capacity limited metabolism, small
changes in dose administration - large variations in plasma
concentration at steady state.
• Two imp causes:-
• I) Capacity - limited metabolism - enzyme &/
cofactor saturation; Phenytoin, Alcohol.
• II) Enzyme induction - decrease in plasma
concentration; Carbamazepine.
Drug metabolism

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Drug excretion
• Two active processes which are saturable,
I) Active tubular secretion - Penicillin G
II) Active tubular reabsorption - Water soluble
vitamins & Glucose.
• Saturation of carrier systems - decrease in renal clearance
& increase in Half life.
• Other reasons like forced diuresis, change in urine pH,
nephrotoxicity & saturation of binding sites.
• In case of biliary excretion non - linearity due to saturation -
Tetracycline & Indomethacin.

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Drugs that demonstrate saturation kinetics
usually show the following characteristics:
1. Elimination of drug does not follow simple first-order kinetics—
that is, elimination kinetics are nonlinear.
2. The elimination half-life changes as dose is increased. Usually,
the elimination half-life increases with increased dose due to
saturation of an enzyme system.

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3. The area under the curve (AUC) is not proportional to
the amount of bioavailable drug.
4. The saturation of capacity-limited processes may be
affected by other drugs that require the same
enzyme or carrier-mediated system
(ie,competition effects).

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Saturable Enzymatic Elimination
Processes
• Michaelis–Menten kinetics:
dCp/dt = rate of decline of drug concentration with time.
Vmax = theoretical maximum rate of the process.
Km = Michaelis constant.

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C p
>> K m
C p
= K m
C p
<< K m
• saturation of the
enzymes occurs.
•The value for K M
is negligible.
•The rate of
elimination
proceeds at a
constant rate
•Rate of process is
= one half of its
max. rate
• -dc/dt = Vmax/2
•rate of drug
elimination becomes
a first-order process

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Drug Elimination by CapacityLimitedPharmacokinetics:
One-Compartment Model, IV Bolus Injection:
• If a single IV bolus injection of drug (D 0
) is
given at t = 0, the drug concentration (C p
) in
the plasma at any time t may be calculated by
an integrating equation,

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• Amount of drug in the body after IV Bolus ,
• Where, D 0
is the amount of drug in the body at
t = 0.
• To calculate time for dose in the boy
• Rearranging equation:

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Determination of K M
and V max

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ESTIMATION OF Vmax & Km
In enzymatic kinetic work, the classic Michaelis-Menten
equation:
Vmax . C ………..(1)
KM + C
where, V= reaction rate,
C= substrate conc. both are
used to determine Vmax
& Km.
The velocity of the reaction(V) at various concentration levels of
drug(C) are determined either by in-vitro experiments or in-vivo
experiments at constant enzyme levels.
V =

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Method 1
By inverting equation (1), we get :
1 Km . 1 1 ……..(2)
V Vmax . C Vmax
When 1/V is plotted against 1/C, a straight line is
obtained with a slope of Km/Vmax and an Intercept of
1/Vmax.
= +

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Figure 2
Plot of 1/V vs 1/C for determining Km & Vmax
-1/ Km
Km/Vmax
1/ Vmax

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Multiplying eq. 2 by C, we get :
C Km C
= + ………..... (3)
V Vmax Vmax
A plot of C /V vs C gives a straight line with 1 / Vmax as the slope and
Km / Vmax as the intercept (shown in the fig. 3).
Method 2

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Figure 3.
Plot of C/V vs C for determining Km & Vmax
ax
ax

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The equation can also be written
as :
V = - Km V + Vmax …………(4)
C
A plot of V vs V / C gives a straight line with a slope of –Km & an Intercept
of Vmax. (shown in the fig. 4)
Method 3

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Figure 4
Plot of V vs V / C for determining Km & Vmax

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CALCULATION OF KM & VMAX
STEADY- STATE CONCENTRATION
• If drug is administered for constant rate IV infusion/ in a multiple
dosage regimen, the steady-state conc. is given in terms of
dosing rate (DR):
DR = Css
• If the steady-state is reached, then the dosing rate = the rate of
decline in plasma drug conc. & if the decline occurs due to a
single capacity-limited process then eq. I become as:
• From a plot of Css vs. DR, a typical curve having a shape of
hocky-stick is obtained which is shown in fig. 5.
……………….. (1)
=
Vmax Css
KM+ Css
DR ……………….. (2)

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Css
Km
Vmax / 2 Vmax
DR (in mg/hr or mg/day )
Curve for a drug following nonlinear kinetics
By plotting the steady-state concentration against dosing rates
Figure 5

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METHODS USED TO
DETERMINE THE
KM & VMAX AT
STEADY-STATE

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• There are three methods which are used to define the KM
& Vmax at steady-state with appreciable accuracy:
1) Lineweaver-Burk Plot:- the reciprocal of eq. (2) we get
• If 1/DR is plotted against 1/Css a straight line is obtained
having slope KM/Vmax & y-intercept 1/Vmax.
2) Direct linear plot:-
• Plotting a pair of Css, i.e.Css1,&Css2 against corresponding
dosing rates DR1 & DR2 we get following fig. 6 which
gives values KM &Vmax
1
DR
=
KM
Vmax Css
1
Vmax
+ ……………….. (3)

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Css
Css 1 Css 2
0 KM
KM
DR
DR1
DR2
Vmax
Direct linear plot for estimation of KM & Vmax
at steady-state conc. Of a drug, when it is
administered at different dosing rates
Figure 6

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3) Graphical method:-
• In this method by rearranging eq. (2) we get
• In graph DR is plotted against DR/Css, a straight line is
obtained with slope –KM & y - intercept Vmax.
• KM & Vmax can be estimated by simultaneous eq. as
DR
DR
Css
KM
= Vmax
……………….. (4)
=DR1
=DR2
……………….. (6)
…………….…...(5)
-
Vmax
Vmax
-
-
KM
KM
DR1
DR2
Css1
Css2

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• On solving above eq. 5 & 6 we get,
• By substituting values of DR1, DR2, Css1 & Css2 we get value of
KM & from KM we can found value of Vmax at steady-state
concentration.
• From experimental observations, it shows that KM is much
less variable than Vmax.
KM =
DR2- DR1
Css 1 Css 2
DR1 DR2
-
……………….. (7)
DR
3) Graphical method:-

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