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Non cpt analysis - 1 Non compartmental analysis Update: 13/08/2010.

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Presentation on theme: "Non cpt analysis - 1 Non compartmental analysis Update: 13/08/2010."— Presentation transcript:

1 Non cpt analysis - 1 Non compartmental analysis Update: 13/08/2010

2 Non cpt analysis - 2 Stochastic interpretation Statistical Moment Approach 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

3 Non cpt analysis - 3 Synonymous Model-independent approach Non-compartmental analysis Statistical Moment Approach !

4 Non cpt analysis - 4 Clearance = Dose / AUC V ss = MRT = V ss / Cl = AUMC / AUC F% = AUC EV / AUC IV D EV = D IV Dose x AUMC AUC 2 The Main Non-compartmental Parameters

5 Non cpt analysis - 5 (MRT system) The Mean Residence Time

6 Non cpt analysis - 6 To measure the time each molecule stays in the system: t 1, t 2, t 3...tn MRT = mean of the different times MRT = n t 1 + t 2 + t 3 +...tn Principle of the method: (1) Entry Exit Non-compartmental analysis

7 Non cpt analysis - 7 Clearance = flow = 2 balls/second MRT = t = (t 1 + t 2... t 6 )/n = (0.5 + 1 + 1.5 +…+6)/6 = 3 Vss = Clearance x MRT = 6 balls Tube volume  x R 2 x L =  x R 2 x 12R Ball volume (6 x 4  R 3 )/3 Ratio V balle / V tube = 0.67 = partition coefficient between balls and tube Principle of the method 2 balls / s rate of absorption Non-compartmental analysis

8 Non cpt analysis - 8 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 Principle of the method : (2) Mean Residence Time

9 Non cpt analysis - 9 Administration of No molecules at t=0 AUC tot will be proportional to No The molecules eliminated at t 1 had a sojourn time of t 1 in the system Number of molecules eliminated at t 1 : Principle of the method: (3) C(t 1 ) x  t AUC tot C (t) C1C1 t1t1 Non-compartmental analysis x No

10 Non cpt analysis - 10 Cumulated sojourn times of molecule which has been eliminated during  t at : Principle of the method: (4) C (t) C1C1 t1t1 t 1 : t 1 x x No tn : t n x x No MRT=  t 1x   t n x  No C 1 x  t x No C n x  t x No AUC TOT MRT =  ti x Ci x  t / AUC TOT =  t C(t)  t /  C(t)  t tntn CnCn C 1 x  t AUC TOT C n x  t AUC TOT Non-compartmental analysis

11 Non cpt analysis - 11 Non-compartmental analysis Requirements to compute MRT

12 Non cpt analysis - 12 Only one exit from the measurement compartment First-order elimination : linearity Principle of the method: (5) Entry (exogenous, endogenous) Exit (single) : excretion, metabolism recirculation exchanges Central compartment (measure) Mean Residence Time

13 Non cpt analysis - 13 2 exit sites MRT is not computable by statistical moments applied to plasma concentration Principle of the method: (6) Non-compartmental analysis 1 2

14 Non cpt analysis - 14 Computation Method Non-compartmental analysis Trapezes Fitting to a polyexponential equation Equation parameters : Y i, i Assuming a compartmental model Model parameters : k ij

15 Non cpt analysis - 15 The 3 statistical moments S 0 =  (t i - t i -1 ) (C i + C i -1 ) / 2 = AUC S 1 =  (t i - t i -1 ) (C i x t i + C i x t i -1 ) / 2 = AUMC S 2 =  (t i - t i -1 ) (C i x t i + C i x t i -1 ) / 2 = AUMMC AUC = S 0 MRT = S 1 / S 0 VRT = S 2 / S 0 - (S 1 - S 0 ) 2 Computation method (1) 2 2 Non-compartmental analysis

16 Non cpt analysis - 16 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 0 0 0 0 0      Computation method (2) Non-compartmental analysis

17 Non cpt analysis - 17 S 0 by the arithmetic trapezoidal rule C0C0 C1C1 C2C2 C3C3 t0t0 t1t1 t2t2 t3t3 extrapolation area AUC 1 = x (t 1 - t 0 ) 2 C 0 +C 1 AUC TOT = S 1 =  AUC 1 + AUC 2... AUCn + extrapolation area AUC 1 AUC 2 AUC 3 Computation method (3) Non-compartmental analysis

18 Non cpt analysis - 18 Computation of S 1 = AUMC with the arithmetic trapezoidal rule AUMC 1 = x (t 1 -t 0 ) t 0 x C 0 + t 1 x C 1 2 C0C0 t0t0 t1t1 t2t2 t3t3 AUMC 1 AUMC 2 AUMC 3 area to extrapolate AUMC TOT = S 2 = AUMC 1 + AUMC 2 +... AUMC extrapolated C1C1 C2C2 C3C3 Computation method (4) Non-compartmental analysis

19 Non cpt analysis - 19 How to extrapolate S 0 : C z / 2 S 1 : t z x C z / z + C z / 2 S 2 : t 2 z C z / z + 2t z C z / z + 2C z / z C z : the last measured concentration at t z Problem with z et z 3 2 2 2 3 Computation method (5) Non-compartmental analysis

20 Non cpt analysis - 20 From the parameters of a given model S 0 =  Y i / i S 1 =  Y i /  i S 2 =  2Y i /  i n n n i =1 2 3 Computation method (6) Non-compartmental analysis

21 Non cpt analysis - 21 Bicompartmental model : C(t) = Y 1 exp(- 1 t) + Y 2 exp(- 2 t) MRT system = Y 1 / 1 + Y 2 / 2 2 2 Computation method (7) Non-compartmental analysis

22 Non cpt analysis - 22 MRT =  t x C(t) x  t  C(t) x  t MRT =  t C(t) dt  C(t) dt Principle of the method:  0 0  Non-compartmental analysis

23 Non cpt analysis - 23 Monocompartmental model (IV) t1/2 : time to eliminate 50% of the molecules MRT : time to eliminate 63.2% of the molecules MRT = 1/ K 10 t1/2 = 0.693 MRT MRT system: interpretation

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

25 Non cpt analysis - 25 Comparison of published results Author 1 : bicompartmental model: t1/2 = 6h Author 2 : tricompartmental model: t1/2 = 18h Solution : a posteriori computation of MRT system MRT bicompartmental MRT tricompartmental ?=?= Y 1 / 1 + Y 2 / 2 2 2 Y 1 / 1 + Y 2 / 2 + Y 3 / 3 2 2 2 MRT system

26 Non cpt analysis - 26 The Mean Absorption Time (MAT)

27 Non cpt analysis - 27 Definition : mean time for the arrival of bioavailable drug MAT Ka F = 100% K 10 MAT = 1 Ka Administration The MAT

28 Non cpt analysis - 28 1- IV administration MRT IV = 1 / K 10 2- Oral administration MRT oral longer than MRT IV MRT oral = 1 / K 10 + 1 / Ka MAT = MRT oral - MRT IV = 1 / Ka How to evaluate the MAT Ka K 10 IVPo The MAT

29 Non cpt analysis - 29 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

30 Non cpt analysis - 30 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%

31 Non cpt analysis - 31 MAT and bioavailability MAT Ka 1 K 10 Ka 2 F = Ka 1 / (Ka 1 +Ka 2 ) MRT oral = + = + 1 Ka 1 + Ka 2 1 1 1 K 10 Ka MAT is influenced by all processes of elimination (absorption, degradation,…) located at the administration site ! The MAT

32 Non cpt analysis - 32 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 MAT and bioavailability 1 1.5 2 10.50 MAT = 1/(1+1) = 0.5h MAT= 1/(1.5+0.5)= 0.5h MAT=1/(0+2)=0.5h ! The MAT K 10

33 Non cpt analysis - 33 MAT B < MAT A but Absorption clearance of B is lower than that of A ! MAT and bioavailability 1 0.5 1 4 MAT A = = 0.5 h 1 (1 + 1) MAT B = = 0.28 h 1 (4 + 0.5) A B ! The MAT

34 Non cpt analysis - 34 the MAT To accurately interpret the MAT in physiological terms it is necessary to: express the rate of absorption using the clearance concept Cl abs = V abs is unknown but this approach provides a meaning to the comparison of 2 MAT when the bioavailability is known ! Ka 1 x V abs Ka 1 Cl abs V abs

35 Non cpt analysis - 35 The MAT MAT and bioavailability Given a MAT of 5 h with F = 100% Cl abs = Ka 1 x V abs = 0.2 L/h Given a MAT of 5 h with F = 50% Cl abs = Ka 1 x V abs = 0.1 L/h V abs = 1 L Ka 1 = 0.2 h -1 V abs = 1 L 0.1 h -1

36 Non cpt analysis - 36 The Mean Dissolution Time (MDT)

37 Non cpt analysis - 37 in vitro measurement: dissolution test statistical moments approach modelling approach (Weibull) The MDT

38 Non cpt analysis - 38 in vivo measurement (1) : blood elimination solution digestive tract tablet absorption MRT total = MRT dissolution + MRT absorption + MRT elimination What is the dissolution rate of the pellet in the digestive tract ? dissolution The MDT

39 Non cpt analysis - 39 IV administration MRT IV = 6 h in vivo measurement (2) : IV The MDT

40 Non cpt analysis - 40 oral administration of the drug Computation of MRT po, solution from plasma concentrations MRT oral, solution = MRT absorption + MRT elimination = 8 h MAT = MRT po - MRT IV MAT = 8h - 6h = 2h in vivo measurement (3) : blood elimination administration of an oral solution digestive tract The MDT

41 Non cpt analysis - 41 Tablet administration computation of MRT oral,tablet from plasma concentrations MRT oral,tablet =MRT dissolution + MAT + MRT elimination =18 h MRT dissolution = MRT oral,tablet - (MAT + MRT IV ) MRT dissolution = 18 - (2+6) = 10h in vivo measurement (4) : solution administration The MDT

42 Non cpt analysis - 42 Mean residence time in the central compartment (MRTc) and in the peripheral (tissue) compartment (MRT T )

43 Non cpt analysis - 43 Definition : mean time for the analyte within the measured compartment (MRT C ) or outside the compartment (MRT T ) MRT C MRT T MRT system = MRT C + MRT T The MRT are additive MRT central and MRT tissue

44 Non cpt analysis - 44 Computations MRT C = AUC / Co = = MRT T = MRT system - MRT C MRT T = - 1 K 10 Vc Cl AUMC AUC Co N.B. : necessary to know Co accurately MRT central and MRT tissue

45 Non cpt analysis - 45 Relationship with the extent of distribution MRTsystem Vss MRTcentral Vc This ratio measures the affinity for the peripheral compartment = MRT central and MRT tissue

46 Non cpt analysis - 46 The Mean Transit Time (MTT)

47 Non cpt analysis - 47 Definition : Average interval of time spent by a drug particle from its entry into the central compartment to its next exit The Mean Transit Times (MTT)

48 Non cpt analysis - 48 The Mean Transit Time in the measurement (central) compartment (MTT central )

49 Non cpt analysis - 49 Calculation : MTT C = - C(o) dCp/dt for t = 0 MTT C = - C(o) C'(o) MTT C =  Y i  Y i i N.B. : necessary to know Co accurately i =1 nn The MTT central

50 Non cpt analysis - 50 Computation : example for a bicompartmental model C(t) = 5 exp(-0.7t) + 2 exp(-0.07t) MTT C = (5 + 2) / (5 x 0.7 + 2 x 0.07) = 1.428 h The MTT central

51 Non cpt analysis - 51 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 R = - 1 MRT C MTT C The MTT central and number of visits

52 Non cpt analysis - 52 = R + 1 MRT C MTT C When there is no recycling (monocompartmental model) R = 0 and : MRT C MTT C = 1 MRT C = MTT C The MTT central and number of visits

53 Non cpt analysis - 53 Bicompartmental model Vc K 10 K 12 K 21 MTT C = 1 / (K 10 + K 12 ) MTT C = 1 / (Cl + Cl d ) R = K 12 / K 10 R = Cl d / Cl MTT C 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 MTT central and number of visits

54 Non cpt analysis - 54 The Mean Transit Time in the peripheral (tissue) compartment (MTT tissue )

55 Non cpt analysis - 55 Computation MTT T = MRT tissue / R MTT T = MRT system - MRT central R (visit) The MTT tissue (MTT T )

56 Non cpt analysis - 56 Computation : bicompartmental model MTT T = = = 1 K 21 Vss - Vc Cl d Vt Cl d MTT T : - does not rely on clearance - measures drug affinity for peripheral tissues K 12 K 21 K 10 Jusko.J.Pharm.Sci 1988.7: 157 The MTT tissue

57 Non cpt analysis - 57 Interpretation of drug kinetics (1) Gentamicin 5600e -0.218t + 94.9e -0.012t Digoxin 21.4e -1.99t + 0.881e -0.017t Clearance (L/h) 2.39 Cl d (L/h) 0.632 Vss (L) 54.8 Vc (L) 14.0 V T (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

58 Non cpt analysis - 58 Jusko.J.Pharm.Sci 1988.7: 157 GentamicinDigoxin K 12 (h -1 ) 0.045 K 21 (h -1 ) 0.016 K 10 (h -1 ) 0.170 R 0.265 1.56 0.095 0.338 4.37 Application of the MRT concept Interpretation of drug kinetics (2)

59 Non cpt analysis - 59 MTT central (transit time. central comp) MRT C (residence time. central comp.) MTT tissue (transit time peripheral comp.) MRT tissue (residence time peripheral comp.) MRT system (total) Interpretation of the mean times Jusko.J.Pharm.Sci 1988.7: 157 Gentamicin Digoxin 4.65 5.88 64.5 17.1 23.0 0.532 2.81 10.5 46.0 48.8 Application of the MRT concept

60 Non cpt analysis - 60 Stochastic interpretation of a kinetic relationship 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 de visits Cl elimination MRT system = MRT C + MRT T Cl redistribution

61 Non cpt analysis - 61 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

62 Non cpt analysis - 62 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

63 Non cpt analysis - 63 Interpretation determinist vs stochastic Gentamicin vs digoxin Determinist Gentamicin Digoxin Vc = 14 L V T = 40.8 L 0.17 h 0.045 h -1 0.016 h -1 t 1/2 distribution : 3h t 1/2 : 57 h Cl d :0.65 L/h 0.26 0.65 L/h MTT C : 4.65h MRT C : 5.88h Vc = 14 L MTT T : 64.5h MRT T :17.1h V T : 40.8 h Cl = 2.39 L/h MR system : 23 h Vc = 34 L V T = 551 L 0.338 h -1 0.56 h -1 0.095 h -1 t 1/2 distribution : 0.3h t 1/2 : 4 h Cl d :52 L/h 4.4 Cl R :52 L/h MTT C : 0.5h MRT C : 2.81h Vc = 34 L MTT T : 10.5h MRT T :46h V T : 551 h Cl = 12 L/h MR system : 48.8 h

64 Non cpt analysis - 64 Computation Statistical moments Parameters from compartmental model MRT system

65 Non cpt analysis - 65 t 1/2 MRT 0.693 Varea Vss = Mean Residence Time


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