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**Non-compartmental analysis and The Mean Residence Time approach**

A Bousquet-Mélou

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

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**Statistical Moments Describe the distribution of a random variable :**

location, dispersion, shape ... Random variable values Mean Standard deviation

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**Statistical Moment Approach**

Stochastic interpretation of drug disposition 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 The statistical moments are used to describe the distribution of this random variable, and more generally the behaviour of drug particules in the system

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**Statistical Moment Approach**

n-order statistical moment zero-order : one-order :

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**Statistical Moment Approach**

Statistical moments in pharmacokinetics. J Pharmacokinet Biopharm Dec;6(6): Yamaoka K, Nakagawa T, Uno T. Statistical moments in pharmacokinetics: models and assumptions. J Pharm Pharmacol Oct;45(10):871-5. Dunne A.

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**The Mean Residence Time**

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**Principle of the method: (1)**

Mean Residence Time Principle of the method: (1) Entry : time = 0, N molecules Evaluation of the time each molecule of a dose stays in the system: t1, t2, t3…tN MRT = mean of the different times MRT = t1 + t2 + t3 +...tN N Exit : times t1, t2, …,tN

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**Principle of the method : (2)**

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

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**Central compartment (measure)**

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

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**Principle of the method: (4)**

Mean Residence Time Principle of the method: (4) N molecules administered in the system at t=0 The molecules eliminated at t1 have a residence time in the system equal to t1 Consequence of linearity AUCtot is proportional to N Number n1 of molecules eliminated at t1+ t is proportional to AUCDt: C (t) C1 t1 C(t1) x t AUCtot X N n1 = AUCDt =

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**Principle of the method: (5)**

Mean Residence Time Principle of the method: (5) Cumulated residence times of molecules eliminated during t at : C C1 C(1) x t AUCTOT Cn t1 : t1 x x N tn : tn x x N n1 C(n) x t AUCTOT tn (t) t1 C1 x t x N Cn x t x N MRT = t1x tn x N AUCTOT AUCTOT

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**Principle of the method: (5)**

Mean Residence Time Principle of the method: (5) C1 x t x N Cn x t x N MRT = t1x + tn x N AUCTOT AUCTOT MRT = t1x C1 x t + tn x Cn x t AUCTOT ti x Ci x t t C(t) t AUMC MRT = = = AUC AUCTOT C(t) t

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Mean Residence Time

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**AUC = Area Under the zero-order moment Curve**

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

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**compartment (measure)**

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

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**Computational methods**

Area calculations Non-compartmental analysis Trapezes Fitting with a poly-exponential equation Equation parameters : Yi, li Analysis with a compartmental model Model parameters : kij

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**Computational methods**

Area calculations by numerical integration Linear trapezoidal AUC AUMC

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**Computational methods**

Area calculations by numerical integration Linear trapezoidal 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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**Computational methods**

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

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**Computational methods**

Area calculations by numerical integration 2. Log-linear trapezoidal < Linear trapezoidal 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

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**Computational methods**

Extrapolation to infinity Assumes log-linear decline

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**Computational methods**

AUC Determination AUMC Determination C x t (mg/L)(hr) 2.00 3.39 3.50 3.01 0.45 Area (mg.hr2/L) - 1.00 5.39 6.89 6.51 7.52 9.80 37.11 Time (hr) C (mg/L) Area (mg.hr/L) - 2.275 3.13 1.83 1.13 0.945 0.900 Total

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**Non-compartmental analysis**

The Main PK parameters can be calculated using non-compartmental analysis MRT = AUMC / AUC Clearance = Dose / AUC Vss = Cl x MRT = F% = AUC EV / AUC IV DEV = DIV Dose x AUMC AUC2

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**Computational methods**

Area calculations Non-compartmental analysis Trapezes Fitting with a poly-exponential equation Equation parameters : Yi, li Area calculations Analysis with a compartmental model Model parameters : kij

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**Fitting with a poly-exponential equation**

Area calculations by mathematical integration For one compartment :

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**Fitting with a poly-exponential equation**

For two compartments :

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**Computational methods**

Area calculations Non-compartmental analysis Trapezes Fitting with a poly-exponential equation Equation parameters : Yi, li Area calculations Analysis with a compartmental model Model parameters : kij Direct MRT calculations

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**Analysis with a compartmental model**

Example : Two-compartments model k12 1 2 k21 k10

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**Analysis with a compartmental model**

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

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**Analysis with a compartmental model**

Then the matrix (- K-1) gives the MRT in each compartment Dosing in 1 Dosing in 2 MRTcomp1 MRTcomp1 Comp 1 (-K-1) = MRTcomp2 MRTcomp2 Comp 2

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**The Mean Residence Times**

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

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**The Mean Absorption Time (MAT)**

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**The Mean Absorption Time**

Definition : mean time required for the drug to reach the central compartment IV EV Ka A 1 K10 F = 100% ! because bioavailability = 100%

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**The Mean Absorption Time**

! 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 :

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**The Mean Dissolution Time**

In vivo measurement of the dissolution rate in the digestive tract tablet solution dissolution absorption solution digestive tract blood MDT = MRTtablet - MRTsolution

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Mean Residence Time in the Central Compartment (MRTC) and in the Peripheral (Tissues) Compartment (MRTT)

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**MRTcentral and MRTtissue**

Entry MRTC MRTT MRTsystem = MRTC + MRTT Exit (single) : excretion, metabolism

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**The Mean Transit Time (MTT)**

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**The Mean Transit Times (MTT)**

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.

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**The Mean Residence Number (MRN)**

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**The Mean Residence Number (MRN)**

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 : MRN = MRT MTT

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**Stochastic interpretation of the drug disposition in the body**

Mean number of visits R+1 R IV Cldistribution MRTC (all the visits) MTTC (for a single visit) MRTT (for all the visits) MTTT (for a single visit) R number of cycles Clredistribution Clelimination

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**Stochastic interpretation of the drug disposition in the body**

Computation : intravenous administration MRTsystem = AUMC / AUC MRTC = AUC / C(0) MRTT = MRTsystem- MRTC R + 1 = MRTC MTTC MTTC = - C(0) / C’(0) MTTT = MRTT R

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**Interpretation of a Compartmental Model**

Determinist vs stochastic Digoxin 21.4 e-1.99t e-0.017t Cld = 52 L/h 0.3 h MTTC : 0.5h MRTC : 2.81h Vc 34 L MTTT : 10.5h MRTT : 46h VT : 551 L 4.4 41 h ClR = 52 L/h stochastic Cl = 12 L/h Determinist 1.56 h-1 Vc : 33.7 L VT : 551L MRTsystem = 48.8 h 0.095 h-1 0.338 h-1 t1/2 = 41 h

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**Interpretation of a Compartmental Model**

Determinist vs stochastic Gentamicin y =5600 e-0.281t e-0.012t Cld = 0.65 L/h t1/2 =3h MTTC : 4.65h MRTC : 5.88h Vc : 14 L MTTT : 64.5h MRTT : 17.1h VT : 40.8 L 0.265 t1/2 =57h ClR = 0.65 L/h stochastic Clélimination = 2.39 L/h Determinist 0.045 h-1 VT : 40.8L MRTsystem = 23 h Vc : 14 L 0.016 h-1 0.17 h-1 t1/2 = 57 h

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