The half-life OCT 2010

The plasma half-life Synonymous half-life of elimination half-life of the terminal phase

Half-life has the apparent advantage of being a familiar term The half-life Half-life has the apparent advantage of being a familiar term

The half-life Definition ! Time required to divide the plasma concentration by two after reaching pseudo-equilibrium distribution Not the time necessary for the amount of administered drug in the body to fall by one-half !

The half-life Meaning in terms of drug elimination (half-life vs mean residence time (MRT)) MRT = 16 h Concentration MRT = 4 h t1/2 = 12 h 24 time (h)

The half-life Meaning in terms of drug elimination monocompartmental model t1/2 = time to eliminate half the dose t1/2 = time to eliminate half the remaining dose

Pharmacokinetic meaning of half-life

Half-life vs. Clearance Clearance : ability to eliminate Half-life : overall elimination during the terminal phase which depends on both clearance and distribution

Clearance vs. half-life Amiodarone Clearance: 1.9 ml/kg/min Half-life: 25 days Amikacine Clearance: 1.3 ml/kg/min Half-life: 2.3 hours

Meaning in terms of elimination and distribution The half-life Meaning in terms of elimination and distribution

The half-life is a hybrid parameter Distribution large small Clearance high low Half-life equal

Plasma clearance vs plasma half-life

Half-life (HL), Clearance (Cl) & volume of distribution (Vd) of Amiodarone vs. amikacin Drugs HL (h) Cl ml/kg/min Vd L/kg amiodarone 600 1.9 95 amikacin 2.5 1.3 0.27

The half-life is a hybrid parameter K12 t1/2 K21 K10 t1/2 = function of K10, K12 and K21

The half-life Hybrid parameter t 1/ 2 = t 1/ 2 = 2 k12 k21 k10 0.693 1/2 ( k12 + k21 + k10 ) - ( k12 + k21 + k10 ) 2 - 4 k21 k10 0.693 t 1/ 2 = 2

Half-life is a hybrid parameter reflecting both clearance and volume of distribution 0.693 x Volume of distribution t 1/ 2 = Clearance

If half-life increases it is because Volume of distribution  or Clearance 

The half-life t1/2 Terminal half-life is a hybrid parameter : Q Clint ° Q Vmax Km Pt Ka fup Clint Clearance t1/2 fuB VB Vt fub fuT Volume of Distribution fup : fraction of the total number of sites of free fixation Ka : affinity constant Km : Michaelis constant

Time parameters In pharmacokinetics, all the time parameters are hybrid parameters

Hybrid properties of time parameters Hybrid parameters k12 Vc Vp k21 k12 = distribution clearance / Vc k10 = plasma clearance / Vc k 21 = redistribution clearance / Vp t 1/2 vie = 0.693 Varea / plasma clearance MRT : Vss / plasma clearance k10

Consequences of the hybrid properties of time parameters Interpretation of rate constant

Interpretation of rate constant Interpretation of K10 k10 = Cl / Vc dependent variable 2 independent variables ! Wrong interpretation of clearance Cl = K10 x Vc (computation technique)

The half-life Consequence of its hybrid property a poor parameter to evaluate the influence of a pathology (e.g.: renal failure) in drug disposition dosage adaptation should be based on clearance not half-life

Why calculate a half-life ? Dosage regimen Dose Interval of administration Half-life Clearance

Half-life Volume of Clearance distribution Systemic exposure Absorption Volume of distribution Clearance bioavailability Half-life Systemic exposure Dosage regimen How much Dosing regimen How often?

Half-life defines the dosing interval If half-life is short Problem in maintaining steady-state drug concentration requires dosage form with a low input rate

Half-life defines the dosage interval If half-life is long drug accumulation long delay to reach steady state conditions requires a loading dose

Why calculate a half-life? In case of multiple administration : to predict drug accumulation to predict time of steady state

Why calculate a half-life ? To predict drug accumulation To predict the time of steady-state To determine the dosage interval Essential to develop a new drug with respect to compliance (eg: antibiotic)

Why calculate a half-life ? To predict drug accumulation R = 1 1- e - x t 0.693 t1/2 AUCss AUC1 Accumulation index AUCss AUC1 Half-life Interval of administration R= Mono-compartmental model or if drug is administered in the post distribution phase for a x-compartmental model

Why calculate a half-life To predict the time of steady state it is independent of the dosing interval it is only a function of the terminal half-life (3-4 times)

Why calculate a half-life Delay to reach steady state conditions if a drug is administered daily, the steady state will be reached after the 2nd or 3rd administration for all the drugs having a terminal half-life < 12 h

Half-life and delay to reach steady state conditions Monocompartmental 50% = half-life 90% = 3.3 half-life

Half-life, accumulation and steady-state concentration Hypothetical drug: half-life : 24h dosage interval :  = 24h maintenance dose = 50 R = = = 2 1 1- e -0.693/24h * 24h 1 1 - 0.5

Half-life, accumulation and steady-state concentration Days Just after dosing 24h later 1 50 25 2 75 37.5 3 87.5 43.75 4 93.75 46.88 5 96.88 48.44 6 98.44 49.22 7 99.22 49.61 8 99.61 49.80 infinity 100 50 Ratio = 2

Half-life, accumulation and steady-state concentration Hypothetical drug: half-life : 24h dosage interval :  = 12h maintenance dose = 25 R = = 3.41 1 1- e -0.693/24h * 12h

Half-life, accumulation and steady-state concentration Days Hours Just after dosing 12h later 1 0 25.00 17.68 1 12 42.68 30.18 2 24 55.18 39.02 2 36 64.02 45.27 3 48 70.27 49.69 3 60 74.69 52.82 4 72 77.82 55.03 4 84 80.03 56.60 5 96 81.60 57.70 5 108 82.70 58.48 6 120 83.48 59.04 6 132 84.04 59.43 7 144 84.43 59.70 7 156 84.70 59.90 8 168 84.90 60.04 8 180 85.04 60.13 infinity 85.37 60.37 Ratio= 3.42

Half-life Bicompartmental model

Half-life accumulation and delay to reach steady state conditions Bicompartmental Function of elimination fraction during initial and terminal phase t1/2 l2 control drug accumulation if at least 50% of the drug is eliminated during the terminal phase

Half-life and delay to reach steady state conditions Bicompartmental model : plasma concentrations 100 t1/2 = 5h 100 t1/2 = 5h 80 t1/2 = 48h t1/2 = 48h 20 R = +++ day R = + day 2 4 8 2 4 8 Apparent steady state is more rapidly reached when most of the drug is eliminated during the distribution phase t1/2 l2 control drug accumulation if at least 50% of the drug is eliminated during the terminal phase

Half-life and delay to reach steady state conditions Peripheral compartment delay to reach steady state conditions in the deepest tissular compartment is always controlled by t1/2 lz (residues,doping). a "pseudo-plateau" can be reached earlier in plasma and shallow tissues than in deep tissues compartment rate of accumulation is associated with Kz1 and Kz1 is a major determinant of lz

Half-life and delay to reach steady state conditions Plasma vs shallow and deep compartment 0.137 0.0869 shallow (2) 1 deep (3) 0.518 0.0479 3 0.0558 Amount 1 2 Time

Loading dose (LD) LD = Vss x Css Aim : to immediately reach the steady state conditions (Css) LD = Vss x Css

Loading dose (LD) (first dose=LD) maintenance dose 1- e - x t LD = 0.693 l1 or LD = maintenance dose x accumulation index LD for the first example =50x2=100 LD for the second example=25X3.42=85.5

Technical considerations for the calculation of half-life

Half-life Estimation Linear regression Non-linear regression Peeling method (residuals) Non-linear regression

Half-life How to calculate it t1/2 = 0.693 / z a semilogarithmic plot representation 100 z 10 Yz 1

Weighing factor and terminal half-life

Rate constant Rate of elimination Amount in body Cl V Kel = = Fractional rate of drug elimination can be viewed as the fraction of the volume of distribution from which drug is removed by unit of time e.g.: Kel = 0.01 h-1  1% per h

Sampling conditions for the appropriate estimation of t1/2 At least 3 times the expected t1/2 if 24 hours = sampling are over at least 3 days

Half-life and sampling times 100 t1/2= 48.7 hours -20% +20% Concentrations t1/2= 28 hours t1/2=20 hours D = +20% 10 10 20 30 40 50 60 70 Time (hours)

Half-life and level of quantification (LOQ)

Half-life and the level of quantification (LOQ) of the analytical technique C(t) = 100 e-0.139 t + 10 e-0.0139t + 1 e-0.00139t (ng/ml) half-life AUC(%) 10 5 h 33.3 2 days 66.6 1.0 20 days 100 0.1 20 50 100 Time (days)

Terminal half-life of gentamicin The very long terminal HL of gentamicin is due to its slow release from tissues and account for urinary excretion for 3 weeks after a dose in man and long withdrawal times in food producing animals

Variability of half-life Generally an analytical artefact Concentration LOQ time (h)

Bias in the estimation of PK parameters with respect to the LOQ C(t) = 100 e -0.139t + 10 e-0.0139t + 1 e-0.00139t Time (h), Dose = 100 LOQ (ng/ml) Ratio 1 0.0695 2.74 39.6 5.0 49.85 0.1 0.0464 12.4 267 33.4 498.5 1/0.66 1/4.466 1/6.7 1/6.6 1/10 Clearance Vss MRT (day) Varea (L) t1/2 (h)

Half-life and the LOQ of the analytical technique Where to stop ? (1) Answer : calculate the AUC associated with each phase Y(t) = Y1 exp(-1t) + Y2 exp(-2t) AUC1 = Y1 / 1 AUC2 = Y2 / 2 1 Y1 Y2 2

Half-life and the LOQ of the analytical technique Where to stop ? (2) Examples : Y(t) = 1000 exp(-1t) + 500 exp(-0.1t) + 1exp(-0.01t) AUC = 1000 + 5000 + 100 (16.4%) (82%) (1.6%) Y(t) = 1000exp(-1t) + 500exp(-0.1t) + 100exp(-0.01t) AUC = 1000 + 5000 + 10000 (6.8%) (31.3%) (62.5%)

Half-life and the LOQ of the analytical technique Where to stop ? (3) example of gentamicin phase 1 : t1/2 = 5 min phase 2 : t1/2 2 = 2 h phase 3 : t1/2 3 = 24 h AUC phase 1 and 2 = 98% Conclusion : 98% of gentamicin has already been eliminated when the equilibrium of pseudo-distribution occurs

Le cas des « Very late terminal phases » LOQ Benchmark concentration

The half-time for extravascular route of administration

Prednisolone (0.6mg/kg) Pred. sodium succinate Pred. acetate HL=3.7h 104 HL=3.7h 20 HL=48h 15 103 10 IV Plasma Concentration (ng / ml) IM 6 102 IM 10 2 2 4 6 8 10 24 48 72 96 144 Hours Toutain et al. Am.J.Vet.Res 1985, 46:719-725

Half-absorption or half-elimination ? What is the meaning of the terminal half-life after an extravascular drug administration? Half-absorption or half-elimination ? • a rate-limited absorption (flip-flop) must be recognized 100 (C) 10 EV: rate of absorption IV EV: rate of elimination 1 0.1 time 5 10 15 20 25 30

Terminal half-life and the flip-flop case Slow process of absorption K12 Ka1 K21 Ka2 negligible K10 (ng/ml) 100 Ka=Ka1+Ka2 # Ka1 = flip-flop 10 elimination 1 Ka1 Ka1 + Ka2 F% =  100% 0.1 Time 5 10 15 20 25 30

Flip-flop: a pictural view Baignoire avec réserve d’eau (dose), un robinet (ka) et une bonde (K10, clairance) (Ka) (K10)

Flip-flop: a pictural view (Ka) (K10)

What is the meaning of the terminal half-life after an extravascular drug administration? To measure the absorption rate using some special methods IM route k a1 k 10 k a  k a2  ? ? k a1 k 12 k a k 21 k a2 k10 k10

Half-life: summary Definition Interpretation Usefulness elimination distribution Usefulness single dose multiple dose

Drug elimination expressed in terms of amount in the body - dA/dt A dA/dt = - l  A or l = Rate of elimination Amount in the body l = Elimination rate constant is regarded as the fractional rate of drug removal