Non compartmental analysis Update: 13/08/2010

Statistical Moment Approach Stochastic interpretation Individual particles are assumed to move independently among kinetic spaces according to fixed transfert probabilities The behaviour of drug particles is described by the statistical moments

Statistical Moment Approach ! Synonymous Model-independent approach Non-compartmental analysis

The Main Non-compartmental Parameters Clearance = Dose / AUC Vss = MRT = Vss / Cl = AUMC / AUC F% = AUC EV / AUC IV DEV = DIV Dose x AUMC AUC2

The Mean Residence Time (MRT system)

Non-compartmental analysis Principle of the method: (1) Entry To measure the time each molecule stays in the system: t1, t2, t3...tn MRT = mean of the different times MRT = t1 + t2 + t3 +...tn n Exit

Non-compartmental analysis Principle of the method rate of absorption 2 balls / s 2 balls / s Clearance = flow = 2 balls/second MRT = t = (t1 + t2... t6)/n = (0.5 + 1 + 1.5 +…+6)/6 = 3 Vss = Clearance x MRT = 6 balls Tube volume  x R2 x L =  x R2 x 12R Ball volume (6 x 4R3)/3 Ratio Vballe/ Vtube = 0.67 = partition coefficient between balls and tube

Mean Residence Time Principle of the method : (2) The random variable (RV) is the presence time in the system This random variable is characterized by its mean (MRT) and its variance (VRT) The plasma concentration curve provides this information under minimal assumptions

Non-compartmental analysis Principle of the method: (3) Administration of No molecules at t=0 AUCtot will be proportional to No The molecules eliminated at t1 had a sojourn time of t1 in the system Number of molecules eliminated at t1 : C C1 C(t1) x t AUCtot t1 (t) x No

Non-compartmental analysis Principle of the method: (4) Cumulated sojourn times of molecule which has been eliminated during t at : C C1 C1 x t AUCTOT t1 : t1 x x No tn : tn x x No Cn Cn x t AUCTOT (t) t1 tn C1 x t x No Cn x t x No MRT=  t1x   tn x No AUCTOT AUCTOT MRT =  ti x Ci x t / AUCTOT =  t C(t) t /  C(t) t

Non-compartmental analysis Requirements to compute MRT

Central compartment (measure) Mean Residence Time Principle of the method: (5) Entry (exogenous, endogenous) Central compartment (measure) recirculation exchanges Exit (single) : excretion, metabolism Only one exit from the measurement compartment First-order elimination : linearity

Non-compartmental analysis Principle of the method: (6) 1 2 2 exit sites MRT is not computable by statistical moments applied to plasma concentration

Computation Method Non-compartmental analysis Trapezes Fitting to a polyexponential equation Equation parameters : Yi, li Assuming a compartmental model Model parameters : kij

Non-compartmental analysis Computation method (1) The 3 statistical moments S0 = (ti - ti-1) (Ci + Ci-1) / 2 = AUC S1 = (ti - ti-1) (Ci x ti + Ci x ti -1) / 2 = AUMC S2 = (ti - ti-1) (Ci x ti + Ci x ti -1) / 2 = AUMMC AUC = S0 MRT = S1 / S0 VRT = S2 / S0 - (S1 - S0)2 2 2

Non-compartmental analysis Computation method (2) The 3 centered moments (normalized in relation to the origin) AUC =  C(t) x dt MRT = t x C(t) x dt / C(t) x dt VRT = (t - MRT)2 x C(t) x dt /  C(t) x dt     

Non-compartmental analysis Computation method (3) S0 by the arithmetic trapezoidal rule C0 +C1 C0 AUC1 = x (t1 - t0) 2 C1 extrapolation area C2 C3 AUC1 AUC2 AUC3 t0 t1 t2 t3 AUCTOT = S1 =  AUC1 + AUC2 ... AUCn + extrapolation area

Non-compartmental analysis Computation method (4) Computation of S1 = AUMC with the arithmetic trapezoidal rule t0 x C0 + t1 x C1 C0 AUMC1 = x (t1-t0) 2 C1 area to extrapolate C2 C3 AUMC1 AUMC2 AUMC3 t0 t1 t2 t3 AUMCTOT = S2 = AUMC1 + AUMC2 +... AUMC extrapolated

Non-compartmental analysis Computation method (5) How to extrapolate S0 : Cz / 2 S1 : tz x Cz / z + Cz / 2 S2 : t2z Cz / z + 2tz Cz / z + 2Cz/z Cz : the last measured concentration at tz Problem with z et z 2 2 3 2 3

Non-compartmental analysis Computation method (6) From the parameters of a given model S0 =  Yi / i S1 =  Yi /i S2 =  2Yi /i n i =1 n 2 i =1 n 3 i =1

Non-compartmental analysis Computation method (7) Bicompartmental model : C(t) = Y1 exp(-1t) + Y2 exp(-2t) MRTsystem = Y1/1 + Y2 / 2 2 2 Y1/1 + Y2 / 2

Non-compartmental analysis Principle of the method: MRT =  t x C(t) x t  C(t) x t MRT =  t C(t) dt  C(t) dt  

MRT system: interpretation Monocompartmental model (IV) t1/2 : time to eliminate 50% of the molecules MRT : time to eliminate 63.2% of the molecules MRT = 1/ K10 t1/2 = 0.693 MRT

MRT system: interpretation Multicompartmental model terminal half-life vs MRT MRT = 16 h Concentration MRT = 4 h t1/2 = 12 h 24 temps (h)

MRT system Comparison of published results Y1/1 + Y2 / 2 2 2 Author 1 : bicompartmental model: t1/2 = 6h Author 2 : tricompartmental model: t1/2 = 18h Solution : a posteriori computation of MRTsystem MRT bicompartmental MRT tricompartmental Y1/1 + Y2 / 2 2 2 Y1/1 + Y2 / 2 + Y3 / 3 2 2 2 ? = Y1/1 + Y2 / 2 Y1/1 + Y2 / 2 + Y3 / 3

The Mean Absorption Time (MAT)

The MAT Definition : mean time for the arrival of bioavailable drug 1 Ka MAT K10 F = 100% Administration 1 MAT = Ka

The MAT How to evaluate the MAT 1- IV administration MRTIV = 1 / K10 Po IV Ka K10 1- IV administration MRTIV = 1 / K10 2- Oral administration MRToral longer than MRTIV MRToral = 1 / K10 + 1 / Ka MAT = MRToral - MRTIV = 1 / Ka

MAT and bioavailability The MAT MAT and bioavailability The MAT measures the MRT at the administration site and not the "rate" of drug arrival in the central compartment

MAT and bioavailability The MAT MAT and bioavailability Actually, the MAT is the MRT at the injection site MAT does not provide information about the absorption process unless F = 100%

The MAT MAT and bioavailability F = Ka1 / (Ka1 +Ka2) MRT oral = + = + ! MAT is influenced by all processes of elimination (absorption, degradation,…) located at the administration site

The MAT MAT and bioavailability 1 1.5 2 K10 1 K10 0.5 K10 MAT = 1/(1+1) = 0.5h MAT= 1/(1.5+0.5)= 0.5h MAT=1/(0+2)=0.5h Conclusion : by measuring (AUMC/AUC), the same MAT will be obtained This does not mean that the absorption processes towards the central compartment are equivalent !

The MAT MAT and bioavailability MATB < MATA but 1 0.5 A B 4 1 1 1 MATA = = 0.5 h MATB = = 0.28 h (1 + 1) (4 + 0.5) MATB < MATA but Absorption clearance of B is lower than that of A ! !

the MAT To accurately interpret the MAT in physiological terms it is necessary to: express the rate of absorption using the clearance concept Clabs = Vabs is unknown but this approach provides a meaning to the comparison of 2 MAT when the bioavailability is known Ka1 Vabs Clabs Ka1 x Vabs !

MAT and bioavailability The MAT MAT and bioavailability Given a MAT of 5 h with F = 100% Clabs = Ka1 x Vabs = 0.2 L/h Given a MAT of 5 h with F = 50% Clabs = Ka1 x Vabs = 0.1 L/h Ka1 = 0.2 h-1 Vabs = 1 L 0.1 h-1 Vabs = 1 L 0.1 h-1

The Mean Dissolution Time (MDT)

The MDT in vitro measurement: dissolution test statistical moments approach modelling approach (Weibull)

The MDT in vivo measurement (1) : solution tablet digestive tract blood dissolution absorption elimination MRTtotal = MRTdissolution + MRTabsorption + MRT elimination What is the dissolution rate of the pellet in the digestive tract ?

The MDT in vivo measurement (2) : IV IV administration MRTIV = 6 h

The MDT in vivo measurement (3) : oral administration of the drug of an oral solution elimination digestive tract blood oral administration of the drug Computation of MRTpo, solution from plasma concentrations MRToral, solution = MRTabsorption + MRTelimination = 8 h MAT = MRTpo - MRTIV MAT = 8h - 6h = 2h

Tablet administration The MDT in vivo measurement (4) : administration solution Tablet administration computation of MRToral,tablet from plasma concentrations MRToral,tablet=MRTdissolution + MAT + MRTelimination=18 h MRT dissolution= MRToral,tablet - (MAT + MRT IV) MRT dissolution= 18 - (2+6) = 10h

Mean residence time in the central compartment (MRTc) and in the peripheral (tissue) compartment (MRTT)

MRTcentral and MRTtissue Definition : mean time for the analyte within the measured compartment (MRTC) or outside the compartment (MRTT) MRTC MRTT MRTsystem = MRTC + MRTT The MRT are additive

MRTcentral and MRTtissue Computations MRTC = AUC / Co = = MRTT = MRTsystem - MRTC MRTT = - 1 Vc K10 Cl AUMC AUC AUC Co N.B. : necessary to know Co accurately

MRTcentral and MRTtissue Relationship with the extent of distribution MRTsystem Vss MRTcentral Vc This ratio measures the affinity for the peripheral compartment =

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 central compartment to its next exit

The Mean Transit Time in the measurement (central) compartment (MTTcentral)

The MTTcentral Calculation : MTTC = - C(o) dCp/dt for t = 0 MTTC = - C(o) C'(o) MTTC =  Yi  Yi i n n i =1 i =1 N.B. : necessary to know Co accurately

The MTTcentral Computation : example for a bicompartmental model C(t) = 5 exp(-0.7t) + 2 exp(-0.07t) MTTC = (5 + 2) / (5 x 0.7 + 2 x 0.07) = 1.428 h

The MTTcentral and number of visits Definition : The analyte "traveled" several times between the central and peripheral compartment R is the average number of times the drug molecule returns to the central compartment after passage through it MRTC R = - 1 MTTC

The MTTcentral and number of visits MRTC = R + 1 MTTC When there is no recycling (monocompartmental model) R = 0 and : MRTC = 1 MRTC = MTTC MTTC

The MTTcentral and number of visits Bicompartmental model K12 MTTC = 1 / (K10 + K12) MTTC = 1 / (Cl + Cld) R = K12 / K10 R = Cld / Cl Vc K21 K10 MTTC describes the first pass of the analyte in the central compartment and does not take into account the recirculating process of the distributed fraction.

The Mean Transit Time in the peripheral (tissue) compartment (MTTtissue)

The MTTtissue (MTTT) Computation MTTT = MRTtissue / R MTTT = MRTsystem - MRTcentral R (visit)

The MTTtissue Computation : bicompartmental model Vss - Vc Vt 1 K21 Cld Cld K12 K21 K10 MTTT : - does not rely on clearance - measures drug affinity for peripheral tissues Jusko.J.Pharm.Sci 1988.7: 157

Application of the MRT concept Interpretation of drug kinetics (1) Digoxin 21.4e-1.99t + 0.881e-0.017t Gentamicin 5600e-0.218t + 94.9e-0.012t Clearance (L/h) 2.39 Cld (L/h) 0.632 Vss (L) 54.8 Vc (L) 14.0 VT (L) 40.8 12.0 52.4 585 33.7 551.0 time : h concentration : mg l-1 Jusko.J.Pharm.Sci 1988.7: 157

Application of the MRT concept Interpretation of drug kinetics (2) Gentamicin Digoxin K12 (h-1) 0.045 K21 (h-1) 0.016 K10 (h-1) 0.170 R 0.265 1.56 0.095 0.338 4.37 Jusko.J.Pharm.Sci 1988.7: 157

Application of the MRT concept Interpretation of the mean times Gentamicin Digoxin MTTcentral (transit time. central comp) MRTC (residence time. central comp.) MTTtissue (transit time peripheral comp.) MRTtissue (residence time peripheral comp.) MRTsystem (total) 4.65 5.88 64.5 17.1 23.0 0.532 2.81 10.5 46.0 48.8 Jusko.J.Pharm.Sci 1988.7: 157

Stochastic interpretation of a kinetic relationship Cldistribution MRTC (all the visits) MTTC (for a single visit) MRTT (for all the visits) MTTT (for a single visit) R number de visits Clredistribution Clelimination MRTsystem = MRTC + MRTT

Interpretation of a compartmental model Determinist vs stochastic Digoxin 21.4 e-1.99t + 0.881 e-0.017t Cld = 52 L/h 0.3 h MTTT : 10.5h MRTT : 46h VT : 551 L MTTC : 0.5h MRTC : 2.81h Vc 34 L 4.4 41 h ClR = 52 L/h stochastic Cl = 12 L/h Determinist MRTsystem = 48.8 h 1.56 h-1 Vc : 33.7 L VT : 551L 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 MTTT : 64.5h MRTT : 17.1h VT : 40.8 L MTTC : 4.65h MRTC : 5.88h Vc : 14 L 0.265 t1/2 =57h ClR = 0.65 L/h stochastic Clélimination = 2.39 L/h Determinist MRTsystem = 23 h 0.045 h-1 Vc : 14 L VT : 40.8L 0.016 h-1 0.17 h-1 t1/2 = 57 h

Interpretation determinist vs stochastic Gentamicin vs digoxin Determinist Digoxin Gentamicin 0.56 h-1 0.045 h-1 Vc = 34 L VT = 551 L Vc = 14 L VT = 40.8 L 0.095 h-1 0.016 h-1 0.338 h-1 0.17 h t1/2 distribution : 0.3h t1/2 : 4 h t1/2 distribution : 3h t1/2 : 57 h MTTT: 10.5h MRTT:46h VT : 551 h Cld:0.65 L/h 0.26 0.65 L/h MTTT: 64.5h MRTT:17.1h VT : 40.8 h MTTC: 0.5h MRTC: 2.81h Vc = 34 L Cld:52 L/h 4.4 ClR:52 L/h MTTC: 4.65h MRTC: 5.88h Vc = 14 L Cl = 12 L/h Cl = 2.39 L/h MR system: 23 h MR system: 48.8 h

MRTsystem Computation Statistical moments Parameters from compartmental model

Mean Residence Time t1/2 MRT 0.693 Varea Vss =