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

2. 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

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

4. 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

5. The Mean Residence Time
(MRT system)

6. 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

7. 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 = ( …+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

8. 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

9. 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

10. 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

11. Non-compartmental analysis
Requirements to compute MRT

12. 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

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

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

15. 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

16. 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     

17. 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

18. 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 + AUMC AUMC extrapolated

19. 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

20. 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

21. 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

22. 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  

23. 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 = MRT

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

25. 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

26. The Mean Absorption Time (MAT)

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

28. The MAT How to evaluate the MAT 1- IV administration MRTIV = 1 / K10Po IV Ka
K10
1- IV administration
MRTIV = 1 / K10
2- Oral administration
MRToral longer than MRTIV
MRToral = 1 / K / Ka
MAT = MRToral - MRTIV = 1 / Ka

29. 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

30. 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%

31. 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

32. The MAT MAT and bioavailability 1 1.5 2 K10 1 K10 0.5 K10
MAT = 1/(1+1) = 0.5h MAT= 1/( )= 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 !

33. The MAT MAT and bioavailability MATB < MATA but1
0.5 A B 4 1 1 1
MATA = = 0.5 h
MATB = = 0.28 h
(1 + 1)
( )
MATB < MATA
but
Absorption clearance of B is lower than that of A ! !

34. 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 !

35. 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

36. The Mean Dissolution Time (MDT)

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

38. 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 ?

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

40. 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

41. 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

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

43. 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

44. 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

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

46. The Mean Transit Time (MTT)

47. 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

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

49. 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

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

51. 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 =
MTTC

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

53. 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.

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

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

56. The MTTtissue Computation : bicompartmental model Vss - Vc Vt1
K21
Cld
Cld
K12
K21
K10
MTTT : - does not rely on clearance
- measures drug affinity for peripheral tissues
Jusko.J.Pharm.Sci : 157

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

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

59. 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 : 157

60. 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

61. Interpretation of a compartmental model
Determinist vs stochastic
Digoxin
21.4 e-1.99t 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

62. 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
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

63. 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

64. MRTsystem Computation Statistical moments
Parameters from compartmental model

65. Mean Residence Time
t1/2
MRT
0.693 Varea
Vss =