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Seminar on NON-LINEAR PHARMACOKINETICS By KARTHEEK THUMULURI M.Pharm (1st Sem) Pharmaceutics St.Peter’s Institute of Pharmaceutical Sciences Vidyanagar Hanamkonda,Warangal 506 001


CONTENTS Introduction Saturable Enzymatic Elimination Processes Drug Elimination by Capacity-Limited Pharmacokinetics : One-Compartment Model, IV Bolus Injection. Determination of km and vmax. Dependence of Clearance on Dose. Equations for Drugs Distributed as One-Compartment Model and Eliminated by Non-Linear Pharmacokinetics. Chronopharmacokinetics and Time-Dependent Pharmacokinetics.

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Non-Linear Pharmacokinetics due to Drug-Protein Binding. Conclusion. References.


INTRODUCTION Pharmacokinetics is the study of rate processes associated with the absorption, distribution and excretion of drug or metabolites level in the body fluids, tissues and excrete at any point in time from the moment of administration until elimination from the body is complete. In most cases, drugs administered at therapeutic doses, the rates of change in amount/conc of drug are proportional to its dose, whether administered as a single dose or as multiple doses which is said to follow FIRST ORDER or LINEAR KINETICS and also follows the PRINCIPLE OF SUPERPOSITION.

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For some drugs given at therapeutic level, one of the specialized processes like enzymes or carrier mediated systems become saturated because these systems are relatively drug specific,, have definite capacities and susceptible to saturation at high drug concentration. In these cases, an essentially first-order kinetics transform into a mixture of first-order and zero-order processes and the pharmacokinetic parameters change with the size of administered dose. The pharmacokinetics of such drugs are said to be DOSE DEPENDENT or MIXED-ORDER or NON-LINEAR or CAPACITY-LIMITED KINETICS.

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Plasma level–time curves for a drug that exhibits a saturable elimination process. Curves A and B represent high and low doses of durg, respectively, given in a single IV bolus. The terminal slopes of curves A and B are the same. Curve C represents the normal first-order elimination of a different drug. Area under the plasma level–time curve versus dose for a drug that exhibits a saturable elimination process. Curve A represents dose-dependent or saturable elimination kinetics. Curve C represents dose-independent kinetics.

Saturable Enzymatic Elimination Processes : 

Saturable Enzymatic Elimination Processes This process can be described by Michaelis-Menten kinetics. Where cp is the conc of drug in plasma vmax is the maximum elimination rate km is the Michaelis constant

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aK M = 0.1μg/mL, V max = 0.5μg/mL per hour. bThe ratio of the elimination rate to the concentration is equal to the rate constant

Drug Elimination by Capacity-Limited Pharmacokinetics: One-Compartment Model, IV Bolus Injection : 

Drug Elimination by Capacity-Limited Pharmacokinetics: 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 integrated form of Eq no 1 described by

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In order to calculate the time for the dose of the drug to decline to a certain amount of drug in the body, Eq no 4 must be rearranged and solved for time t: Using Eq no 5, the time for a single 400-mg dose given by IV bolus injection to decline to 20 mg was calculated for a drug with a K M of 38 mg/L and a V max that varied from 200 to 100 mg/hr . With a V max of 200 mg/hr, the time for the 400-mg dose to decline to 20 mg in the body is 2.46 hours, whereas when the V max is decreased to 100 mg/hr, the time for the 400-mg dose to decrease to 20 mg is increased to 4.93 hours . Thus, there is an inverse relationship between the time for the dose to decline to a certain amount of drug in the body and the V max . Km=76 Km=38

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Thus, an increase in K M (with no change in V max) will increase the time for the drug to be eliminated from the body.

Determination of K M and V max : 

Determination of K M and V max In enzymatic kinetic work, the classic Michaelis-Menton equation can be written as

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Slope = 1/vmax Intercept = km/vmax

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Slope =1/vmax Intercept =km/vmax Slope = -km Intercept =vmax

Determination of K M and V max in Patients : 

Determination of K M and V max in Patients EXAMPLE Phenytoin was administered to a patient at dosing rates of 150 and 300 mg/day, respectively. The steady-state plasma drug concentrations were 8.6 and 25.1 mg/L, respectively. Find the K M and V max of this patient. What dose is needed to achieve a steady-state concentration of 11.3 mg/L?

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Method A Inverting eq no 10 on both sides Multiply both sides by cssvmax Rearrange

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Slope =630mg/day Intercept =27.6mg/L

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METHOD B From eq no 10 Dividing both sides by css yields

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METHOD C From eq no 6

Determination of K M and V max by Direct Method : 

Determination of K M and V max by Direct Method Combining the two equations yields

Dependence of Clearance on Dose : 

Dependence of Clearance on Dose Inverting and rearranging the equation yields Integration of Eq no 15 from time 0 to infinity gives

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Because the dose D0 = C 0 pV D, Eq no 16 may be expressed as To obtain mean body clearance, Cl av is then calculated from the dose and the AUC. Dividing the basic equation by cp gives

Equations for Drugs Distributed as One-Compartment Model and Eliminated by Nonlinear Pharmacokinetics : 

Equations for Drugs Distributed as One-Compartment Model and Eliminated by Nonlinear Pharmacokinetics MIXED DRUG ELIMINATION Example : sodium salicylate The equation that describes a drug that is eliminated by both first-order and Michaelis-Menten kinetics after IV bolus injection is given by

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Chronopharmacokinetics and Time-Dependent Pharmacokinetics : 

Chronopharmacokinetics and Time-Dependent Pharmacokinetics Chrono pharmacokinetics describe the changes in drug absorption, elimination due to normal physiological circadian rhythms (rhythmic or cyclical changes in plasma drug concentration that may occur daily due to normal changes in body functions). EXAMPLE : Prins et., al, reported that nephrotoxicity of aminoglycosides (Tobramycin) occurred more frequently when given during the rest period (midnight to 7:30 am) compared to other periods of the day due to significant increase in plasma drug concentration.

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The symptons of hypoxemia were found to be aggravated in some COPD patients due to changes in respiration during sleep cycle. Time-dependent pharmacokinetics generally refers to a noncyclical change in the drug absorption or drug elimination rate process over a period of time which may lead to nonlinear pharmacokinetics. This may be the result of alteration in the physiology or biochemistry in an organ or a region in the body that influences drug disposition. Time-dependant pharmacokinetics may be due to Auto-induction or Auto-inhibition of biotransformation enzymes.

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EXAMPLE : Pitlick and Levy reported that repeated doses of carbamazepine induce the enzymes responsible for its elimination (Auto-induction), there by increasing the clearance of the drug. Perrier et al., reported the Auto-inhibition may occur during the course of the metabolism of certain drugs, the metabolites formed increase in concentration and further inhibit metabolism of parent drug.

Nonlinear Pharmacokinetics Due to Drug–Protein Binding : 

Nonlinear Pharmacokinetics Due to Drug–Protein Binding Plasma curve comparing the elimination of two drugs given in equal IV doses. Curve A represents a drug 90% bound to plasma protein. Curve B represents a drug not bound to plasma protein.

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The concentration of free drug, cf, can be calculated at any time, as follows. cf =cp (1-fraction bound) One-Compartment Model Drug with Protein Binding

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Assuming a saturable and instantly reversible drug binding process where p = protein concentration in plasma cf = plasma concentration of free drug kd =k2/k1= dissociation constant of the protein drug complex. cb = plasma concentration of bound drug cp = total plasma drug concentration.

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Rearranging the above equation gives Solving for cf Rate of drug elimination is dcp/dt This differential equation describes the relationship of changing plasma drug concentration during elimination.


CONCLUSION Dose and time of drug administration significantly influence the pharmacokinetics of the drug clinically, the most dramatic source of dose dependence is capacity limited metabolism. Small changes in bioavailability may produce large changes in steady state concentration . Under these conditions, careful attention of an individual patients dosage requirement is needed, especially if the drug has a narrow therapeutic index.


REFERENCES Coffey J, Bullock FJ, Schoenemann PT: Numerical solution of nonlinear pharmacokinetic equations: Effect of plasma protein binding on drug distribution and elimination. J Pharm Sci 60:1623, 1971 [PMID: 5133908]Evans WE, Schentag JJ, Jusko WJ: Applied Pharmacokinetics—Principles of Therapeutic Drug Monitoring, 3rd ed. Applied Therapeutics, 1992, pp 2–33 Gibaldi M: Pharmacokinetic aspects of drug metabolism. Ann NY State Acad Sci 179:19, 1971 [PMID: 4936775]

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Hashimoto Y, Odani A, Tanigawara Y, Yasuhara M, Okuno T, Hori R: Population analysis of the dose-dependent pharmacokinetics of zonisamide in epileptic patients. Biol Pharm Bull17:323–326, 1994 [PMID: 8205132]Hashimoto Y, Koue T, Otsuki Y, Yasuhara M, Hori R, Inui K: Simulation for population analysis of Michaelis–Menton kinetics. J Pharmacokinet Biopharmacol 23:205–216, 1995 [PMID: 8719237]Hsu F, Prueksaritanont T, Lee MG, Chiou WL: The phenomenon and cause of the dose-dependent oral absorption of chlorothiazide in rats: Extrapolation to human data based on the body surface area concept. J Pharmacokinet Biopharmacol 15:369–386, 1987 [PMID: 3681665]

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