Non-compartmental analysisandThe Mean Residence Time approach A Bousquet-Mélou
Synonymous Mean Residence Time approach Statistical Moment Approach Non-compartmental analysis
Standard deviation Random variable values Mean Statistical Moments • Describe the distribution of a random variable : • location, dispersion, shape ...
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. 1978 Dec;6(6):547-58. Yamaoka K, Nakagawa T, Uno T. Statistical moments in pharmacokinetics: models and assumptions. J Pharm Pharmacol. 1993 Oct;45(10):871-5. Dunne A.
Mean Residence Time Principle of the method: (1) • Evaluation of the time each molecule of a dose stays in the system: t1, t2, t3…tN • MRT = mean of the different times MRT = Entry : time = 0, N molecules t1 + t2 + t3 +...tN N Exit : times t1, t2, …,tN
Mean Residence Time • 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 Principle of the method: (3) Only one exit from the measurement compartment First-order elimination : linearity Entry (exogenous, endogenous) Central compartment (measure) recirculation exchanges Exit (single) : excretion, metabolism
C C1 AUCDt C(t1) x t (t) t1 X N X N n1 = = AUCtot AUCtot Mean Residence Time Principle of the method: (4) Consequence of linearity • AUCtot is proportional to N • Number n1 of molecules eliminated at t1+ t is proportional to AUCDt: • N molecules administered in the system at t=0 • The molecules eliminated at t1 have a residence time in the system equal to t1
Mean Residence Time Principle of the method: (5) Cumulated residence times of molecules eliminated during t at : C C1 C(1) x t AUCTOT t1 : t1 x x N tn : tn x x N Cn n1 C(n) x t AUCTOT (t) tn t1 Cn x t x N C1 x t x N MRT = t1x tn x N AUCTOT AUCTOT
Mean Residence Time Principle of the method: (5) Cn x t x N C1 x t x N MRT = t1x +tn x N AUCTOT AUCTOT MRT = t1xC1 x t +tn x Cn x t AUCTOT ti x Ci x t t C(t) t AUMC MRT = = = AUC AUCTOT C(t) t
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.
Mean Residence Time Limits of the method: • 2 exit sites • Statistical moments obtained from plasma concentration inform only on molecules eliminated by the central compartment Central compartment (measure)
Computational methods • Non-compartmental analysis Trapezes Area calculations • Fitting with a poly-exponential equation Equation parameters : Yi, li • Analysis with a compartmental model Model parameters : kij
Computational methods Area calculations by numericalintegration • Linear trapezoidal AUC AUMC
Computational methods Area calculations by numericalintegration • 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 numericalintegration 2. Log-linear trapezoidal AUC AUMC
Computational methods Area calculations by numericalintegration 2. Log-linear trapezoidal < Linear trapezoidal • Disadvantages: • Produces large errors on an ascending curve, near the peak, or steeply declining polyexponential curve • Advantages: • Hand calculator • Very accurate for mono-exponential • Very accurate in late time points where interval between points is substantially increased
Computational methods Extrapolation to infinity Assumes log-linear decline
Computational methods AUC Determination AUMC Determination C x t (mg/L)(hr) 0 2.00 3.39 3.50 3.01 2.00 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) 0 2.55 1 2.00 3 1.13 5 0.70 7 0.43 10 0.20 18 0.025 Area (mg.hr/L) - 2.275 3.13 1.83 1.13 0.945 0.900 Total 10.21
The Main PK parameters can be calculated using non-compartmental analysis Non-compartmental analysis • MRT = AUMC / AUC • Clearance = Dose / AUC • Vss = Cl x MRT = • F% = AUC EV / AUC IVDEV = DIV Dose x AUMC AUC2
Computational methods • Non-compartmental analysis Trapezes Area calculations • 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 mathematicalintegration For one compartment :
Fitting with a poly-exponential equation For two compartments :
Computational methods • Non-compartmental analysis Trapezes Area calculations • 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 Definition : mean time required for the drug to reach the central compartment IV EV Ka 1 A 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 : !
solution tablet solution The Mean Dissolution Time • In vivo measurement of the dissolution rate in the digestive tract absorption dissolution digestive tract blood MDT = MRTtablet - MRTsolution
Mean Residence Time in the Central Compartment (MRTC) and in the Peripheral (Tissues) Compartment (MRTT)
MRTC MRTT MRTcentral and MRTtissue Entry MRTsystem = MRTC + MRTT Exit (single) : excretion, metabolism
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.
MRT MRN = MTT 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 :
Stochastic interpretation of the drug disposition in the body Mean number of visits R+1 R IV Cldistribution MRTT (for all the visits) MTTT (for a single visit) MRTC (all the visits) MTTC (for a single visit) R number of cycles Clredistribution Clelimination
MRTC MRTT MTTT = MTTC R R + 1 = Stochastic interpretation of the drug disposition in the body Computation : intravenous administration MRTsystem = AUMC / AUC MRTC = AUC / C(0) MRTT = MRTsystem- MRTC MTTC = - C(0) / C’(0)
Interpretation of a Compartmental Model Digoxin Determinist vs stochastic 21.4 e-1.99t + 0.881 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 VT : 551L Vc : 33.7 L 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 + 94.9 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 MRTsystem = 23 h VT : 40.8L Vc : 14 L 0.016 h-1 0.17 h-1 t1/2 = 57 h