Non-compartmental analysis and The Mean Residence Time approach

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

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

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

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

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

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.

The Mean Residence Time

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

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

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

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 =

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

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

Mean Residence Time

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.

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

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

Computational methods
Area calculations by numerical integration Linear trapezoidal AUC AUMC

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

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

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

Computational methods
Extrapolation to infinity Assumes log-linear decline

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

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

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

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

Fitting with a poly-exponential equation
For two compartments :

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

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

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

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

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

The Mean Absorption Time (MAT)

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%

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 :

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

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

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

The Mean Transit Time (MTT)

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.

The Mean Residence Number (MRN)

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

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

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

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

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