Concepts of Pharmacology - Half Life Calculation -

1. Concepts of Pharmacology- Half Life Calculation - C. M. Prada, MD July 12, 2006

2. Pharmacokinetics = availability - dosage and rate of administration - modes of transport – across biologic membranes; bound to proteins from plasma and tissues - blood flow to the site of action - extent and speed of the metabolic process of the drug - rate of the removal of the drug (and its metabolic products) Pharmacodynamics = pharmacologic effect (in relation with the plasma drug concentration) - cellular mechanisms of drug action - clinical evaluation of drug effects - biologic variability Drug Pharmacology

3. Pharmacokinetics Definition: quantitative study of absorption, distribution, metabolism, and elimination of chemicals in the body, as well as the time course of these effects. Summary: - absorption - distribution - metabolism - elimination

4. Concentration of a drug at its site of action is a fundamental determinant of its pharmacologic effects. Drugs are transported to and from their sites of action in the blood – because of that: the concentration at the active site is a function of the concentration in the blood. The change in drug concentration over time in the blood, at the site of action, and in other tissues is a result of complex interactions of multiple biologic factors with the physicochemical characteristics of the drug. Pharmacokinetics Principles

5. Etymology medicine pharmacon = movement kinesis =

6. Pharmacokinetic Concepts: Rate Constants and Half-Lives • Disposition of most drugs follows first-orderkinetics – a constant fraction of the drug is removed during a finite period of time. • This fraction is equivalent to the rateconstant of the process. • Rate constants: k; min-1 or h-1 • The absolute amount of drug removed is proportional to the concentration of the drug • In first-order kinetics, the rate of change of the concentration at any given time is proportional to the concentration present at that time. • When the concentration is high, it will fall faster than when it is low. • First-order kinetics apply not only to elimination, but also to absorption and distribution.

7. Half-Lives • The rapidity of pharmacokinetic processes is often described with half-lives • Half-Life = the time required for the concentration to change by a factor of 2. • Half-Life = the period of time required for the concentration or amount of drug in the body to be reduced to exactly one-half of a given concentration or amount. Half-Life =the time required for half the quantity of a drug or other substance deposited in a living organism to be metabolized or eliminated by normal biological processes. Also called biological half-life.

8. Zero-Order Elimination approximately constant rate of elimination

10. First-Order Elimination Ct is concentration after time t C0is the initial concentration (t=0) k is the elimination rate constant - first-order logarithmic process - that is, a constant proportion of the agent is eliminated per unit time (Birkett, 2002) Birkett DJ (2002). Pharmacokinetics Made Easy (Revised Edition). Sydney: McGraw-Hill Australia. ISBN 0-07-471072-9.

14. First-Order Elimination (cont.) dC 1). ------ = kC dt dC 2). ------ = k dt C 3).ln C = - kt + D 4). C = eD e-kt 5). At time t = 0: C = eD 6).C0 = eD 7). Ct = C0 e-kt

15. At the time t = t1/2: Ct = C0 e-kt C(1/2) = C0 x 1/2 C0 x 1/2 = C0 e-kt1/2 e-kt1/2 = 1/2 - kt1/2 = ln 1/2 = - ln 2 ln 2 t1/2 = ------- k

16. The reduction of the quantity in terms of the number of half-lives elapsed: ln 2 t1/2 = ------- k

17. First-Order Elimination (cont.) Relationship between the elimination rate constant and half-life: ln(Cpeak) - ln(Ctrough) kelim = --------------------------------- t interval t½ = 0.693 / kelim Half-life is determined by clearance(CL) and volume of distribution (VD): Only for IV

18. Half-Life Calculation • Directly from the corresponding rateconstants: (ln 2) 0.693 t1/2 = ---------- = ----------- k k Ex.: rate constant of 0.1 min-1 translates into a half life of 6.93 min.

19. Half – Life (cont.) • Half-Life of any first-order kinetic process can be calculated – ex.: drug absorption, distribution, elimination • First order processes – asymptotically approach completion, because a constant fraction of the drug is removed per unit of time (not an absolute amount). • The process will be almost complete after 5 (97%) …half-lives:

21. Half-Life Elimination (cont.) • Repeated equal doses of a drug morefrequently than 5 elimination half-times: result in drug being administered at a rate greater than its plasma clearance – accumulation in plasma.

22. Equations for Half Lives For a zero order reactionA products   ,    rate =  k: t½ =  [Ao] / 2k For a first order reaction     A products  ,   rate =  k[A]: t½ = 0.693 / k For a second order reaction   2A products   or   A + B products   (when [A] = [B]), rate = k[A]2: t½ = 1 / k [Ao] http://www.chem.purdue.edu/gchelp/howtosolveit/Kinetics/Halflife.html

23. For a zero order reactionA products   ,    rate =  k:

24. For a first order reaction      A products  ,   rate =  k[A]:

25. For a second order reaction   2A products   or   A + B products   (when [A] = [B]),   rate = k[A]2:

26. The Elimination Half-Time Limits • Only in single-compartment models does it actually represent the time required for a drug to reach half of its initial concentration after administration • This is because in a single-compartment model elimination is the only process that can alter drug concentration • Intercompartmental distribution cannot occur because there are no other compartments for the drug to be distributed to and from • Most drugs in Anesthesia: lipophilic – therefore are more suited to multicompartmental model • In multicompartmental models, the metabolism and excretion of some intravenous anesthetic drugs may have only minor contribution to changes in plasma concentration when compared with the effects of intercompartmental distribution

27. Drug Elimination • Elimination = all the various processes that terminate the presence of a drug in the body. • Processes: - metabolism - renal excretion - hepato-biliary excretion - pulmonary excretion (inhaled anesthetics mainly) - other: saliva, sweat, breast milk, tears.

28. Renal Excretion • Both metabolically changed and unchanged drugs • LMW substances: filtered from blood through the Bowman membrane of the capsule • Some: actively secreted • Reabsorption in the tubule: depending on the lipid solubility, degree of ionization, molecular shape, carrier mechanism (for some). • Weak acid: best reabsorbed from an acidic urine. • Important to know if the drug is dependent on renal function or excretion.

29. Hepatobiliary Excretion • Drugs metabolites – excreted in the intestinal tract with the bile. • Majority: reabsorbed into the blood and excreted through urine. (enterohepatic cycle). • Poorly lipid-soluble organic compounds – at least three active transport mechanisms

30. Pulmonary Excretion • Volatile anesthetics and anesthetic gases: in large part eliminated unchanged through the lung • The factors that determine uptake operate in reverse manner

31. Multicompartment Pharmacokinetics • Instead of a single exponential process with one half-time, the pharmacokinetics are described by 2 or more exponentialprocesses – can calculate a half-time for each process: l1, l2, l3, etc. (referred to as: a, b, g). • The time for the concentration to decrease by 50% is dependent on the preceding dosinghistory and can vary with the duration of drugadministration. • The time to decrease plasma concentration by half is not equal with the time to remove half of the drug from the body - terminal half-life.

32. Multicompartment Models • 3 compartments: • Central = vascular bed • 2nd = rapidly equilibrating, high perfusion (muscle) • 3rd = large compartment – slow equilibrating, low perfusion (fat). • 5 compartments: (isoflurane and sevoflurane measurements) • Central • Vessel-rich • Muscle group • 4th compartment • The fat group

33. Context-Sensitive Half-Time • Improved our understanding of anesthetic drug disposition; is clinically applicable. • Effect of distribution on plasma drugconc. varies in magnitude and direction over time - depends on the drugconcentration gradients between various compartments. - ex.: early part of the infusion of a lipophilic drug, the distributive factor decrease its plasma conc. as the drug is transported to the unsaturated peripheral tissues – later, after the infusion is discontinued: drug will re-enter in the central circulation

34. Context-Sensitive Half-Time (cont.) • Def.: context-sensitive half-time describes the time required for the plasma drug concentration to decline by 50% after terminating an infusion of a particular drug. • Calculated by using computer simulation of multicompartmental models of drug disposition:

36. Context-Sensitive Half-Time (cont.) • Reflects the combined effects of distribution and metabolism on drug disposition • The data confirm the clinical impression that as the infusion duration increases, the context sensitive half-time of all drugs increases. This phenomenon is not described by the elimination half-life. • Norelationship with the Half-Life. • N.B.: For a one-compartment model: context-sensitive half-life = elimination half-life.

37. Context-Sensitive Half-Time (cont.) • Compare fentanyl (half life 462 min.) and sufentanil (half-life 577 min.) – storage and release of fentanyl from the peripheral binding sites: delay the declines of plasma concentration. • Compare propofol and thiopental – comparable c-s h-t following a brief infusion only. • Because of : 1). high metabolic clearance of propofol and 2). relatively slow rate of return to plasma of propofol from peripheral compartments.

38. Context-Sensitive Half-Time (cont.) • Alfentanil – studied for ambulatory techniques • Elimination half – life: 111 min. • Small distribution volume – not significant in plasma decay after infusion • Sufentanil – elimin. half – life: 577 min. • Less context-sensitive half-time (for infusions up to 8 hours); large volume of distribution for sufentail • After termination of its infusion, the decay in plasma drug conc. is accelerated not only by elimination, but also by continuous redistribution into the peripheral compartments.

39. Context-Sensitive Half-Time and Time to Recovery • Time to recovery depends on other additional factors: • Plasma concentration below which awakening can be expected • Awakening from anesthesia is a function of effect-site concentration decay • Effect-site equilibration – half-time of equilibration between drug concentration in blood and the drug effect can be determined: t1/2 ke0

41. 1). Zero order pharmacokinetics If there is one thing that separates pharmacology from other medical subjects, it is zero order pharmacokinetics! Salicylic acid is an example of a drug that behaves this way. What is drug elimination according to zero order kinetics? A constant amount of drug is removed per unit of time. This makes the rate of metabolism saturable, so that small changes in dose, can give dramatic changes in plasma concentration

42. 2). A drug was given intravenously at a dose of 200mg. The initial concencentration in plasma (Co) was 10 µg/ml, and the Kel (elimination constant) was 0.02/h. Determine the plasma clearance (Clpl) and t1/2 for this drug. Distribution volume: Vd = dose/ Co = 200 mg / 10 mg/L = 20 L Cl pl = Vd x Kel = 20 L x 0.02/h = 0.4 L/h T1/2 = ln 2 / kel = 0.693 x 0.02 = 35 hours

43. The kinetic order of a reaction is determined by the exponent of the rate equation, n, in dc/dt = K Cn:

44. Saturable kinetics usually follow Michaelis-Menten equation: dC/dt = [(dC/dt)m.C] / (Km + C) where (dc/dt)m is the maximum rate that a reaction can reach and Km is the Michaelis-Menton constant that corresponds to a concentartion at which the rate is 1/2 of the maximum. When C is very small it can be ignored from denominator and reduces the above equation to: dC/dt = [(dC/dt)m.C] / Km or dC/dt = (dC/dt)m/Km.C. Hence the rate becomes dependent upon a constant, [(dC/dt)m / Km], and C, and the kinetics change to a first order. On the other hand during the time when C is very high, Km becomes negligible and the equation reduces to: dC/dt = (dC/dt)m.C / C Now C can be cancelled from both nominator and denominator to give: dC/dt = (dC/dt)m i.e., the rate depends only upon a constant but not upon concentration. From then on the reaction proceeds according to zero order kinetics. Between these two extremes the order of the reaction is a mixture of first and zero kinetics.

45. Concentration dependency of the kinetic order of saturable reaction. www.pharmacy.ualberta.ca/pharm415/orderof.htm

46. Michaelis Menten Equation The Michaelis Menten process is somewhat more complicated with a maximum rate (velocity, Vm) and a Michaelis constant (Km) and the amount or concentration remaining.

47. Half-Lives for Some Common Drugs • Narcan (Naloxone): plasma half-life: - adult: 64±12 min; neonate: 3.1 ± 0.5 hours • Fentanyl: half-life: 2 – 9 hours • Morphine: terminal half-life: 1.5 – 4.5 hours • Midazolam: elimination half-life: 1.8 – 6.4 hours • Flumazenil (Romazicon): distribution half-life: 7 – 15 min.; terminal half–life: 41 – 79 min

48. http://www.globalrph.com/narcoticonv.htm

49. e = 1 + 1/1! + 1/2! + 1/3! +…… e is the limit of (1 + 1/n)n as n tends to infinity e = 2.718281828459045235 Euler

50. Disclaimer • This is a compilation intended only for the personal use of the MHMC residents, and not for publication • For the bibliographic sources, please send an e-mail to: cprada@metrohealth.org