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

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

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

4. 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

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

6. 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.

7. The Mean Residence Time

8. 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

9. 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

10. 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

11. 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 =

12. 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

13. 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

14. Mean Residence Time

15. 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.

16. 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

17. 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

18. Computational methods
Area calculations by numerical integration
Linear trapezoidal
AUC
AUMC

19. 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

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

22. 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

23. Computational methods
Extrapolation to infinity
Assumes log-linear decline

24. 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

25. 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

26. 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

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

28. Fitting with a poly-exponential equation
For two compartments :

29. 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

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

31. 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

32. 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

33. 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

34. The Mean Absorption Time (MAT)

35. 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%

36. 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 :

37. 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

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

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

40. The Mean Transit Time (MTT)

41. 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.

42. The Mean Residence Number (MRN)

43. 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

44. 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

45. 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

46. 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

47. 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