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1 Non-compartmental analysis and The Mean Residence Time approach A Bousquet-Mélou

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2 Mean Residence Time approach Statistical Moment Approach Non-compartmental analysis Synonymous

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3 Statistical Moments Mean Describe the distribution of a random variable : location, dispersion, shape... Standard deviation Random variable values

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4 Stochastic interpretation of drug disposition Statistical Moment Approach The statistical moments are used to describe the distribution of this random variable, and more generally the behaviour of drug particules in the system Individual particles are considered : they are assumed to move independently accross kinetic spaces according to fixed transfert probabilities The time spent in the system by each particule is considered as a random variable

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5 n-order statistical moment zero-order : one-order : Statistical Moment Approach

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6 Statistical moments in pharmacokinetics. J Pharmacokinet Biopharm. 1978 Dec;6(6):547-58. Yamaoka KYamaoka K, Nakagawa T, Uno T.Nakagawa TUno T Statistical moments in pharmacokinetics: models and assumptions. J Pharm Pharmacol. 1993 Oct;45(10):871-5. Dunne ADunne A. Statistical Moment Approach

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7 The Mean Residence Time

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8 Evaluation of the time each molecule of a dose stays in the system: t 1, t 2, t 3 …t N MRT = mean of the different times MRT = N t 1 + t 2 + t 3 +...t N Principle of the method: (1) Entry : time = 0, N molecules Exit : times t 1, t 2, …,t N Mean Residence Time

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9 Under minimal assumptions, the plasma concentration curve provides information on the time spent by the drug molecules in the body Principle of the method : (2) Mean Residence Time

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10 Only one exit from the measurement compartment First-order elimination : linearity Principle of the method: (3) Entry (exogenous, endogenous) Exit (single) : excretion, metabolism recirculation exchanges Central compartment (measure) Mean Residence Time

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11 Consequence of linearity AUC tot is proportional to N Number n1 of molecules eliminated at t1+ t is proportional to AUC t : Principle of the method: (4) C (t) C1C1 t1t1 Mean Residence Time C(t 1 ) x t AUC tot X N n1 = AUC t AUC tot X N = N molecules administered in the system at t=0 The molecules eliminated at t1 have a residence time in the system equal to t1

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12 Cumulated residence times of molecules eliminated during t at : Principle of the method: (5) C (t) C1C1 t1t1 t 1 : t 1 x x N tn : t n x x N MRT = t 1x t n x N C 1 x t x N C n x t x N AUC TOT tntn CnCn C(1) x t AUC TOT C(n) x t AUC TOT Mean Residence Time n1n1

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13 Principle of the method: (5) MRT = t 1x t n x N C 1 x t x N C n x t x N AUC TOT MRT = = Mean Residence Time MRT = t 1x C 1 x t t n x C n x t AUC TOT t C(t) t C(t) t ti x Ci x t AUC TOT AUMC AUC =

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14 Mean Residence Time

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15 From: Rowland M, Tozer TN. Clinical Pharmacokinetics – Concepts and Applications, 3 rd edition, Williams and Wilkins, 1995, p. 487. AUC AUMC AUC = Area Under the zero-order moment Curve AUMC = Area Under the first- order Moment Curve

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16 2 exit sites Statistical moments obtained from plasma concentration inform only on molecules eliminated by the central compartment Limits of the method: Mean Residence Time Central compartment (measure)

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17 Non-compartmental analysis Trapezes Fitting with a poly-exponential equation Equation parameters : Y i, i Analysis with a compartmental model Model parameters : k ij Computational methods Area calculations

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18 1.Linear trapezoidal Computational methods Area calculations by numerical integration AUC AUMC

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19 1.Linear trapezoidal Computational methods Area calculations by numerical integration Advantages: Simple (can calculate by hand) Disadvantages: Assumes straight line between data points If curve is steep, error may be large Under or over estimation, depending on whether the curve is ascending of descending

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21 2. Log-linear trapezoidal Computational methods Area calculations by numerical integration AUC AUMC

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22 2. Log-linear trapezoidal Computational methods Area calculations by numerical integration Advantages: Hand calculator Very accurate for mono- exponential Very accurate in late time points where interval between points is substantially increased Disadvantages: Produces large errors on an ascending curve, near the peak, or steeply declining polyexponential curve < Linear trapezoidal

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23 Computational methods Extrapolation to infinity Assumes log- linear decline

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24 Time (hr) C (mg/L) 0 2.55 1 2.00 3 1.13 5 0.70 7 0.43 10 0.20 18 0.025 AUC Determination Area (mg.hr/L) - 2.275 3.13 1.83 1.13 0.945 0.900 Total 10.21 AUMC Determination C x t (mg/L)(hr) 0 2.00 3.39 3.50 3.01 2.00 0.45 Area (mg.hr 2 /L) - 1.00 5.39 6.89 6.51 7.52 9.80 37.11 Computational methods

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25 MRT = AUMC / AUC Clearance = Dose / AUC V ss = Cl x MRT = F% = AUC EV / AUC IV D EV = D IV Dose x AUMC AUC 2 The Main PK parameters can be calculated using non-compartmental analysis Non-compartmental analysis

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26 Non-compartmental analysis Trapezes Fitting with a poly-exponential equation Equation parameters : Y i, i Analysis with a compartmental model Model parameters : k ij Computational methods Area calculations Area calculations

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27 Fitting with a poly-exponential equation Area calculations by mathematical integration For one compartment :

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28 Fitting with a poly-exponential equation For two compartments :

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29 Non-compartmental analysis Trapezes Fitting with a poly-exponential equation Equation parameters : Y i, i Analysis with a compartmental model Model parameters : k ij Computational methods Direct MRT calculations Area calculations Area calculations

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30 Example : Two-compartments model Analysis with a compartmental model 1 k12 k21 k10 2

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31 dX1/dt X1 X2 Example : Two-compartments model K is the 2x2 matrix of the system of differential equations describing the drug transfer between compartments Analysis with a compartmental model dX2/dt K =

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32 MRT comp1 Dosing in 1 MRT comp2 MRT comp1 MRT comp2 Analysis with a compartmental model (-K -1 ) = Then the matrix (- K -1 ) gives the MRT in each compartment Dosing in 2 Comp 1 Comp 2

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33 Fundamental property of MRT : ADDITIVITY The mean residence time in the system is the sum of the mean residence times in the compartments of the system Mean Absorption Time / Mean Dissolution Time MRT in central and peripheral compartments The Mean Residence Times

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34 The Mean Absorption Time (MAT)

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35 Definition : mean time required for the drug to reach the central compartment A 1 Ka F = 100% K 10 The Mean Absorption Time IV EV ! because bioavailability = 100%

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36 MAT and bioavailability Actually, the MAT calculated from plasma data is the MRT at the injection site This MAT does not provide information about the absorption process unless F = 100% Otherwise the real MAT is : ! The Mean Absorption Time

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37 In vivo measurement of the dissolution rate in the digestive tract blood solution digestive tract MDT = MRT tablet - MRT solution dissolution absorption tabletsolution The Mean Dissolution Time

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38 Mean Residence Time in the Central Compartment (MRT C ) and in the Peripheral (Tissues) Compartment (MRT T )

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39 MRT C MRT T MRT system = MRT C + MRT T MRT central and MRT tissue Entry Exit (single) : excretion, metabolism

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40 The Mean Transit Time (MTT)

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41 Definition : – Average interval of time spent by a drug particle from its entry into the compartment to its next exit – Average duration of one visit in the compartment Computation : – The MTT in the central compartment can be calculated for plasma concentrations after i.v. The Mean Transit Times (MTT)

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42 The Mean Residence Number (MRN)

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43 Definition : – Average number of times drug particles enter into a compartment after their injection into the kinetic system – Average number of visits in the compartment – For each compartment : The Mean Residence Number (MRN) MRN = MRT MTT

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44 Stochastic interpretation of the drug disposition in the body MRT C (all the visits) MTT C (for a single visit) MRT T (for all the visits) MTT T (for a single visit) Cl distribution R number of cycles Cl elimination Cl redistribution Mean number of visits RR+1 IV

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45 Stochastic interpretation of the drug disposition in the body Computation : intravenous administration MRT system = AUMC / AUC MRT C = AUC / C(0) MTT C = - C(0) / C(0) R + 1 = MRT C MTT C MRT T = MRT system - MRT C MTT T = MRT T R

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46 Interpretation of a Compartmental Model Determinist vs stochastic Digoxin stochastic MTT C : 0.5h MRT C : 2.81h Vc 34 L Cl d = 52 L/h 4.4 Cl R = 52 L/h MTT T : 10.5h MRT T : 46h V T : 551 L Cl = 12 L/h MRTsystem = 48.8 h Determinist Vc : 33.7 L 1.56 h -1 VT : 551L 0.095 h -1 0.338 h -1 t1/2 = 41 h 21.4 e -1.99t + 0.881 e -0.017t 0.3 h 41 h

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47 Determinist vs stochastic Gentamicin stochastic MTT C : 4.65h MRT C : 5.88h Vc : 14 L Cl d = 0.65 L/h 0.265 Cl R = 0.65 L/h MTT T : 64.5h MRT T : 17.1h V T : 40.8 L Cl élimination = 2.39 L/h MRTsystem = 23 h Determinist Vc : 14 L 0.045 h -1 V T : 40.8L 0.016 h -1 0.17 h -1 t 1/2 = 57 h y =5600 e -0.281t + 94.9 e -0.012t t 1/2 =3h t 1/2 =57h Interpretation of a Compartmental Model

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