1. SEMINAR ON
Nonlinear pharmacokinetics
1ST Year M. Pharma
Dept. Of Pharmaceutics,
KLES’s College Of Pharmacy
Belgaum .
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2. contents
• Introduction
• Pharmacokinetics & Pharmacodynamic
• Nonlinearity
• Causes of nonlinearity
• Michaelis – menten equation
• Estimation of Km and Vmax
• Estimation of Km and Vmax steady-state conc.
• Bioavailability dependent nonlinear
pharmacokinetics
• Non-linear pharmacokinetics due to drug-
protein binding
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3. Phama
r cokinet &. Phama
ics r codynamics
Phama
r cokinet
ics Phama
r codynamics
A ion oft bodyon t chemica
ct he he l A ion oft chemica on t body
ct he l he
Syst A pt distibut
em: bsor ion, r ion, Syst Biol ll nds orot t r s in
em: ogica iga her aget
met bol el t (A E
a ism, iminaion DM ) t biopha
he se.
Out : Concentaion-ime
put rt t Out : Biol lr
put ogica esponse
r aionships
el t
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4. Objectives:
• To understand the schemes & differential equations associated
with non linear pharmacokinetic models.
• To understand the effect of parallel pathways.
• To estimate the parameters KM & VM.
• To design appropriate dosage regimen for drugs with non linear
elimination.
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5. Linear and Nonlinear Pharmacokinetics
• In case of therapeutic dose of drug, at single or multiple doses, the
process follows first order or linear kinetics if drug in the body/ the
change in plasma conc. due to ADME is proportional to dose.
• In first order/ linear kinetics processes, the plot of log C vs. t for
different doses on a semilog paper, are superimposable. This is
known as principle of superposition.
• But there are some drugs that do not follow first order/ linear
kinetics. In such situations, a first order kinetic transformed in to a
mixture of first order and zero order rate processes and the
pharmacokinetic parameters change with the size of administered
dose.
• The pharmacokinetic s of such drugs are said to be dose
dependent.
• The other term synonymous with it are mixed-order, non linear,
capacity limited kinetics.
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6. Detection of non-linearity in pharmacokinetics
• There are several tests to detect non –linearity in
pharmacokinetics but the simplest ones are:
 First test:- Determination of steady state plasma
concentration at different doses.
 Second test:- Determination of some important
pharmacokinetic parameters such as fraction
bioavailability, elimination half life or total systemic
clearance at different doses of drug .any change in these
parameters which are usually constant, is indicative to
non-linearity.
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7. Reasons for non
linearity:
• Saturation of the enzymes and of the active transport systems for the
drug.
Drugs which show capacity limited elimination by the kidneys &
capacity limited biotransformation exhibit non linearity at high doses.
EX: salicylates & naproxen-capacity limited elimination.
vitamin C- capacity limited absorption.
• Changes in protein binding characteristics.
• Variations in blood flow around the site of absorption.
• Low solubility of drug, erratic dissolution behavior from dosage
form.
• Changes in Ph at the site of absorption.
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8. Causes of non-linearity
• Drug absorption:-
• Three causes:- I) solubility/ dissolution of drug is rate-
limited
Griseofulvin- at high conc. in intestine.
II) carrier-mediated transport system
ascorbic acid- saturation of transport
system.
III) Presystemic gut wall /hepatic
metabolism
attains saturation ; Propranolol
• Above three factors affects F, Ka,KE,ClR,ClH
• First two factors causes decrease in above parameters,
while last factor causes er T ain parameters
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9. Drug distribution
• Drug distribution:- 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 conc.
• Increase in Vd only in I)
• Clearance with high ER get increased due to saturation of
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binding sites. Pa

10. Drug metabolism
• Drug metabolism:- non-linearity occurs due to capacity limited
metabolism, small changes in dose administrations- large
variations in plasma conc. at steady state- large intersubject
variability.
• Two imp causes:- I) capacity-limited metabolism- enzyme &/
cofactor saturation; phenytoin, alcohol.
II) enzyme induction-decrease in plasma
conc.
Carbamazepine.
• Autoinduction in dose dependent conc.
• Saturation of enzymes- decrease in ClH- increase in Css.
• In case of enzyme induction reverse condition.
• Other reasons includes saturation oftbinding sites, inhibitory10 ge 10
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11. Drug excretion
• 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
in case of I & increase in II. Half life also increases.
• Other reasons like forced diuresis, change in urine pH,
nephrotoxicity, & saturation of binding sites.
• The AUC is not proportional to amount of bioavailable
drug.
• In case of biliary excretion non-linearity due to saturation-
tetracycline & indomethacin. T aes
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12. Examples of drugs showing nonlinear pharmacokinetics
causes drugs
GI absorption:-
Saturable transport in gut wall Riboflavin, gabapentin
Saturable GI decomposition Penicillin G, omeprazole
Intestinal metabolism Propranolol, salicylamide
Distribution:-
Saturable plasma protein binding Phenylbutazone, lidocaine
Tissue binding Imipramine
Metabolism:-
Saturable metabolism Phenytion, salicylic acid
Enzyme induction Carbamazepine
Metabolite inhibition Diazepam
Renal elimination:-
Active secretion Para- aminohippuric acid
Tubular reabsorption Ascorbic acid, riboflavin
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13. • In order to determine whether a drug is following dose
dependent kinetics.
• The drug is given at various dose levels and a plasma
concentration level time curve is obtained for each dose.
• The curve should exhibit parallel slopes if the drug fallows a
dose dependent kinetics.
• Alternatively, a plot of the areas under the plasma level time
curves at various doses should be linear.
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14. MICHAELIS MENTEN EQUATION
• The kinetics of capacity limited or saturable processes is best
described by Michaelis-Menten equation.
dC V xC
ma
= … … … .. I
………
dt KM+ C
• Where ,
• -dC/dt= rate of decline of drug conc. with time
• Vmax= theoretical maximum rate of the process
• KM=Michaelis constant
• Three situation can now be considered depending upon the value of
Km and C.
1) when KM=C:
• under this situation , eq I reduces to,
• -dC/dt=Vmax/2 ...................II
• The rate of process is equal to half of its maximum rate.
• This process is represented er the empl t dc/dt vs. C. shown in fig. 1
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15. 1) If a drug at low conc. Undergoes a saturable biotransformation
then KM>>C:
• here , KM +C =KM and eq. I reduces to,
-dC/dt =Vmax C /KM………………III
• above eq. is identical to the one that describe first order elimination
of
drug, where Vmax/KM= KE.
3) when KM<<C:
• Under this condition ,KM +C= C and eq. I will become,
-dC/dt =Vmax …………….IV
above eq. is identical to the one that describe a zero order process
i.e.
the rate process occursPowconstant aes Vmax and is independent of
at er T rate
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drug conc.

16. Zer or r t a high doses
o der ae t
M or r t a
ixed der ae t
int mediaed doses
er t
dcdt F stor r t a l doses
ir der ae t ow
C
fig.1 Apl ofM E
ot M
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17. Estimation of Vmax & KM
• The parameters Km and Vmax can be assessed from the
plasma conc.-time data collected after i.v. bolus
administration of a drug with non linear characteristics
• Rewriting eq. I
dC Vmax C
- = … … … .. X
………
dt KM +C
• integration of above eq. we get
Co-
C V xt
ma
InC = InCo + - … … … .. XI
………
KM KM
• If above eq. is converted to log base 10, we get
L C= l 0 +
og ogC (C0-C) – Vmax t … … … .. XII
………
2.303 KM 2.303 KM
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18. • A semi log plot of C versus t yields a curve with a terminal
linear portion having slope –Vmax/ 2.303 Km and when back
extrapolated to time zero gives y- intercept log Co . The
equation that describe this line is:
• l = l Co –
ogC og Vmax t
… … … .. XIII
………
2.303 Km
• At low plasma conc. , eq. XII and XIII are identical .
equating the two and simplifying further ,
we got: (Co-
C) Co
= log … … … .. XIV
………
2.3 3Km
0 Co
• from above eq. KM can be obtained. The value KM on
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substituting in the slope value gives Vmax Pa

19. • a plot of drug given a IV bolus with nonlinear elimination
L Co
og
Y
L Co
og
T mina l rporion of
er l inea t
L C
og T cur e ha ing sl
he v v ope=
-V x/ 0 KM
ma 2.3 3
t X
Fig.4
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20. Lineweaver-Burk Plot
• An alternation approach of estimating Vmax and KM is
determining the rate of change of plasma drug conc. at
different times and using the reciprocal of equation I thus:
- dC Vmax C
=
dt KM +C
• Ignoring the – ve sign & writing reciprocal of the above eq. we
get
• If CM is the plasma conc. at the midpoint of the sampling
interval, then on replacing C=Cm in above eq.
1 KM 1
= + … … … .. XV
………
dC/dt V x Cm
ma Vxma
• The Lineweaver-Burk Plot is obtained by ploting1/(dC/dt) vs.1/
Cm of the above eq. a straight line is obtained having slope =
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KM/Vmax & y-intercept = 1/Vmax. Pa

21. Hanes-Woolf Plot
Cm KM Cm
= + … … … .. XV
……… I
dC/dt V x Cm
ma Vx
ma
W f-A insson-Hofst Pl
ool ugust ee ot
dC dC/
dt KM … … … .. XV
… … … II
= Vx
ma
dt Cm
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22. 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 ClT … … … .. (1)
………
• 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:
V x Css
ma
DR = … … … .. (2)
………
KM+ Css
• From a plot of Css vs. DR, a typical curve having a shape of
hocky-stic is obtained which is shown in fig. 5
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23. Cur e foradr folow nonl rkinet
v ug l ing inea ics
Bypl t t st dy- ae conc. a instdosing r t
oting he ea st t ga aes
Css
Km
V x/
ma 2
Vx
ma
DR(in mg/ ormg/ y)
hr da
Fig.5
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24. • There are three methods which are used to define the KM
& Vmax at steady-state with appreciable accuracy:
• Lineweaver-Burk Plot: the reciprocal of eq. (2) we get
1 KM 1
= + … … … .. (3
……… )
DR V x Css
ma Vxma
• 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. 5 which
gives values KM &Vmax
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25. Dir l rpl forest t ofKM &V x
ect inea ot imaion ma
a st dy- ae conc. Ofadr
t ea st t ug,
w itis a
hen dminist ed a differ dosing r t
er t ent aes
DR
Vx
ma
DR1
DR2
Css1 Css 2 KM
Css 0 KM
F 6
ig.
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26. 1) Graphical method:-
• In this method by rearranging eq.(2) we get
KM DR … … … .. (4
……… )
DR = Vx
ma
Css
• If 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
V x Css1
ma
DR1
=
KM+ Css1
V x Css2
ma
DR =
2
… … … .. (6)
………
KM+ Css2
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27. • On solving above eq. 5& 6 we get
DR-DR1 2
KM = … … … .. (7)
………
DR 1 DR 2
-
Css 1 Css 2
• By substituting values of DR1, DR2,Css1 & Css2 we get value
of KM & from KM we can found value of Vmax at steady-state
conc.
• From experimental observations, it shows that KM is much
less variable than Vmax.
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28. Bioavailability dependent nonlinear
pharmacokinetics
• The bioavailability of drugs that follows nonlinear kinetics is
difficult to estimate accurately.
• In such cases in the presence of saturable pathways in drug
ADME, drug bioavailability changes within single dose or
subsequent doses.
• The extent of bioavailability can also estimate from [AUC]0Φ
Non-linear pharmacokinetics due to drug-protein
binding
… … … .. (1)
………
• In case of protein binding drug the conc. of free drug can be
obtained by
• Cf = Cp (1- fraction bound)
• in case of one compartmentpointT aes protein bounded drugge 28
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29. 1
Cb Kd Cr … … … .. (2)
………
= 1
p 1+ ( K ) Cr
d
• Where Cb= plasma conc. Of bound drug.
p= protein conc. in plasma
Kd= K2/K1= dissociation constant of the protein drug
complex.
K1& K2= rate constants
• The above eq. can be rearranged as
p Cr
Cb = = Cp - Cr … … … .. (3
……… )
Kd Cr
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30. References
• Biopharmaceutics and clinical pharmacokinetics by Milo
Gibaldi 4th edition 1991
• Biopharmaceutics and pharmacokinetics by D.M .
Brahmankar and sunil.B. Jaiswal
• Text book of Biopharmaceutics and pharmacokinetics by
Dr.shobha rani r. hiremath ,2000
• Biopharmaceutics and pharmacokinetics by
G.R.chatwal, 1st edition.
• Applied Biopharmaceutics and pharmacokinetics by
Leon Shargel/ Andrew B.C 4th edition.
• Drug disposition and pharmacokinetics by Curry
Blackwell 3rd edition.
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31. Pow pointT aes
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1
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