Detection of non-linearity in pharmacokinetics Causes of nonlinearity Michaelis – Menten equation Estimation of Km and Vmax

1. Non-linear Pharmacokinetics
Submitted to:-
Dr. D.C.Bhatt
Dean Faculty of Medical
Sciences
Submitted by:-
Shashi Yadav
Reg.No.- 190121220012
M.Pharma 2nd Sem
Dept. of Pharmaceutical Sciences
Guru Jambheshwar University of Science and
Technology

2. CONTENTS
• Introduction
• Linear & Nonlinearity Pharmacokinetics
• Detection of non-linearity in
pharmacokinetics
• Causes of nonlinearity
• Michaelis – Menten equation
• Estimation of Km and Vmax

3. Linear Pharmacokinetics
• At therapeutic doses, the change in the amount of drug in the body
or the change in its plasma concentration due to absorption,
distribution, binding, metabolism or excretion, is proportional to its
dose, whether administered as a single dose or as multiple doses.
• In such situation the rate processes are said to follow first order or
linear kinetics and all semilog plots of Conc. vs Time for different
doses when collected for dose administered, are superimposable.
• The important pharmacokinetic parameters viz. F, Ka, KE, Vd, Clr, Clh
which describes the time course of a drug in the body remain
unaffected by the dose.
• Pharmacokinetics is dose independent.

4. Non-linear Pharmacokinetics
• The rate process of drug’s 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, an essentially first-order kinetics transform into a
mixture of first-order and zero-order rate processes and the
pharmacokinetic parameters are changed with the size of the
administered dose.
• Pharmacokinetics of such drugs are said to be dose-dependent.
Terms synonymous with it are mixed-order, nonlinear and capacity-
limited kinetics.

5. • A number of drugs demonstrate saturation or capacity-limited
metabolism in humans.
• Examples of these saturable metabolic processes include
- glycine conjugation of salicylate
- sulphate conjugation of salicylamide
- acetylation of p-aminobenzoic acid
- elimination of phenytoin

6. Drugs that demonstrate saturation kinetics usually
show the following characteristics:
• Elimination of drug does not follow simple first-order kinetics- that is,
elimination kinetics are nonlinear.
• 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. However, the elimination half-life might decrease due to
“self”-induction of liver biotransformation enzymes, as is observed for
carbamazepine.
• The area under the curve (AUC) is not proportional to the amount of
bioavailable drug.
• The saturation of capacity-limited processes may be affected by other drugs
that require the same enzyme or carrier-mediated system (i.e., competition
effects).
• The composition and/or ratio of the metabolites of a drug may be affected
by a change in the dose.

7. Detection of Non-Linearity in Pharmacokinetics
•There are several tests to detect non–linearity in
pharmacokinetics but the simplest ones are:
1) First test:- Determination of steady state plasma concentration
at different doses.
2) 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 is indicative to non-linearity which are usually
constant.

8. Causes of Non-Linearity
Drug absorption
 Three causes:-
• Solubility/dissolution of drug is rate-limited; Griseofulvin - at high
concentration in intestine.
• Carrier - mediated transport system; Ascorbic acid - saturation of transport
system.
• Presystemic gut wall/hepatic metabolism attains saturation; Propranolol.
 These parameters affected F, Ka, Cmax and AUC.
 A decrease in these parameters is observed in former two causes and an
increase in latter cause.

9. 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 concentration.
• Increase in Vd only in (I)
• Clearance with high ER get increased due to saturation of binding
sites.

10. Drug metabolism
• Non-linearity occurs due to capacity limited metabolism, small
changes in dose administration - large variations in plasma
concentration at steady state - large intersubject variability.
• Two imp causes:-
I) Capacity-limited metabolism – enzyme/cofactor saturation; Phenytoin,
Alcohol.
II) Enzyme induction - decrease in plasma concentration;
Carbamazepine.
• Autoinduction in dose dependent concentration.
• Saturation of enzymes - decrease in Clh - increase in Css.
• In case of enzyme induction reverse condition.
• Other reasons includes saturation of binding sites, inhibitory effects of
the metabolites on the action of enzymes.

11. 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.
• In case of biliary excretion non - linearity due to saturation -
Tetracycline & Indomethacin.

12. Examples of drugs showing nonlinear pharmacokinetics

13. Michaelis-Menten Enzyme Kinetics
It is also called as Capacity-limited metabolism or
Mixed order kinetics.
E + D <—> ED —> E + M
Enzymes usually react with the substrate to form
enzyme substrate complexes; then the product is
formed. The enzyme can go back to react with
another substrate to form another molecule of the
product.

14. Michaelis-Menten equation
• The kinetics of capacity limited or saturable processes is best described by Michaelis-
Menten equation.
−
𝑑𝐶
𝑑𝑡
= 𝑉max.C/ Km + C………….I
• 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 in the plot of dc/dt vs. C. shown in fig. 1

15. 2) 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 occurs at constant rate Vmax and is independent of
drug conc.
E.g. metabolism of ethanol

16. A plot of Michaelis- Menten eq. (Fig. -1)

17. Estimation of Km and Vmax from Steady State Conc.
• When a drug is administered as a constant rate i.v. infusion or in a multiple
dose regimen, the steady-state concentration Css is given in terms of
dosing rate DR as:
• where DR = Ro when the drug is administered as zero-order i.v. infusion
and it is equal to FXo/τ when administered as multiple oral dosage
regimen (F is fraction bioavailable, Xo is oral dose and is dosing interval).
• At steady-state, the dosing rate equals rate of decline in plasma drug
concentration and if the decline (elimination) is due to a single capacity-
limited process (for e.g. metabolism), then;

18. • A plot of Css versus DR yields a typical hockey-stick shaped curve as
shown in Fig.-2
Curve for a drug with nonlinear kinetics obtained by plotting the steady-state
concentration versus dosing rates. (Fig.-2)

19. • To define the characteristics of the curve with a reasonable degree of
accuracy, several measurements must be made at steady-state during
dosage with different doses.
• Practically, one can graphically compute Km and Vmax in 3 ways:
1. Lineweaver-Burke Plot/Klotz Plot
2. Direct Linear Plot
3. Graphical method

20. Lineweaver-Burke Plot/Klotz Plot
• Taking reciprocal of DR equation, we
get:
• A plot of 1/DR versus 1/Css yields a
straight line with slope Km/Vmax and y-
intercept 1/Vmax (Fig.-3). Lineweaver-Burke/Klotz plot for estimation of
Km andVmax at steady-state concentration of
drug.

21. Direct Linear Plot
• A pair of Css viz. Css,1 and Css,2
obtained with two different dosing rates
DR1 and DR2 is plotted. The points
Css,1 and DR1 are joined to form a line
and a second line is obtained similarly by
joining Css,2 and DR2. The point where
these two lines intersect each other is
extrapolated on DR axis to obtain Vmax
and on x-axis to get Km. Direct linear plot for estimation of Km andVmax
at steady-state concentrations of a drug given at
different dosing rates. (Fig.-4)

22. Graphical method
• The graphical method of estimating Km and Vmax involves
rearranging DR equation to yield:
• A plot of DR versus DR/Css yields a straight line with slope -Km and Y-
intercept Vmax.
• Km and Vmax can also be calculated numerically by setting up
simultaneous equations as shown below:

23. • Combination of the above two equations
yields:
• After having computed Km, its subsequent
substitution in any one of the two
simultaneous equations will yield Vmax.

24. Nonlinear Pharmacokinetics (Phenytoin)
• Phenytoin is an example of a drug which commonly has a Km value
within or below the therapeutic range.
– The average Km value is about 4 mg/L.
– The normally effective plasma concentrations for phenytoin are
between 10 and 20 mg/L.
• Therefore it is quite possible for patients to be overdosed due to drug
accumulation.
• At low concentration the apparent half-life is about 12 hours, whereas
at higher concentration it may well be much greater than 24 hours.

25. • Dosing every 12 hours, the normal half-
life, can rapidly lead to dangerous
accumulation.
• At concentrations above 20 mg/L
elimination maybe very slow in some
patients. Dropping for example from 25 to
23 mg/L in 24 hours, whereas normally
you would expect it to drop from -25> -
12.5> 6 mg/L in 24 hours.
• Typical Vm values are 300 to 700 mg/day.
These are the maximum amounts of drug
which can be eliminated by these patients
per day. Giving doses approaching these
values or higher would cause very
dangerous accumulation of drug.
Cpss profile following different doses
of Phenytoin.

26. REFERENCE
• Applied biopharmaceutics and pharmacokinetics by Shargel L.,
Andrew B.C.
• Biopharmaceutics and Pharmacokinetics a treatise by Brahmankar
DM, Jaiswal SB.