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Non-compartmental analysis and The 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.

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Non-compartmental analysis and The Mean Residence Time approach

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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.


The Mean Residence Time


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


Mean Residence Time


  • 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(MAT)


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 Time(MTT)


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.


The Mean Residence Number(MRN)


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


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