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

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Non-compartmental analysisand

The Mean Residence Time approach

A Bousquet-Mélou

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Mean Residence Time approach

Statistical Moment Approach

Non-compartmental analysis

Synonymous

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Statistical Moments

Mean

• Describe the distribution of a random variable :• location, dispersion, shape ...

Standard deviation

Random variable values

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Stochastic interpretation of drug disposition

Statistical Moment Approach

• The statistical moments are used to describe the distribution

of this random variable, and more generally the behaviour of

drug particules in the system

• 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

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0

dttCt n

AUCdttCdttCt

00

0

AUMCdttCt

0

• n-order statistical moment

• zero-order :

• one-order :

Statistical Moment Approach

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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.

Statistical Moment Approach

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The Mean Residence Time

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• Evaluation of the time each molecule of a dose stays in the system: t1, t2, t3…tN

• MRT = mean of the different times

MRT = N

t1 + t2 + t3 +...tN

Principle of the method: (1)Entry : time = 0, N molecules

Exit : times t1, t2, …,tN

Mean Residence Time

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• 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 TimeMean Residence Time

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Only one exit from the measurement compartment

First-order elimination : linearity

Principle of the method: (3)Entry (exogenous, endogenous)

Exit (single) : excretion, metabolism

recirculationexchanges

Central compartment

(measure)

Mean Residence TimeMean Residence Time

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Consequence of linearity

• AUCtot is proportional to N

• Number n1 of molecules eliminated at t1+ t is proportional to AUCt:

Principle of the method: (4)

C

(t)

C1

t1

Mean Residence TimeMean Residence Time

C(t1) x t

AUCtot

X Nn1 =AUCt

AUCtot

X N =

• N molecules administered in the system at t=0

• The molecules eliminated at t1 have a residence time in the system equal to t1

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Cumulated residence times of molecules eliminated during t at :

Principle of the method: (5)

C

(t)

C1

t1

t1 : t1 x x N

tn : tn x x N

MRT = t1xtn x

N

C1 x t x N Cn x t x N

AUCTOT AUCTOT

tn

CnC(1) x t AUCTOT

C(n) x t AUCTOT

Mean Residence TimeMean Residence Time

n1

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Principle of the method: (5)

MRT = t1xtn x

N

C1 x t x N Cn x t x N

AUCTOT AUCTOT

MRT = =

Mean Residence TimeMean Residence Time

MRT = t1xC1 x ttn x Cn x t AUCTOT

t C(t) t

C(t) t

ti x Ci x t

AUCTOT

AUMC

AUC=

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Mean Residence TimeMean Residence Time

0

dttCAUC

0

dttCtAUMC

AUC

AUMCMRT

15From: Rowland M, Tozer TN. Clinical Pharmacokinetics – Concepts and Applications, 3rd edition, Williams and Wilkins, 1995, p. 487.

AUC

AUMC

• AUC = Area Under the zero-order moment Curve

• AUMC = Area Under the first-order Moment Curve

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• 2 exit sites• Statistical moments obtained from plasma concentration

inform only on molecules eliminated by the central compartment

Limits of the method:

Mean Residence TimeMean Residence Time

Centralcompartment

(measure)

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• Non-compartmental analysis

Trapezes

• Fitting with a poly-exponential equation

Equation parameters : Yi, i

• Analysis with a compartmental model Model parameters : kij

Computational methodsComputational methods

Areacalculations

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1. Linear trapezoidal

CC

tt

1

2

1

2

Concentration

Time

Computational methodsComputational methods

Area calculations by numerical integration

2

CCtt 1ii

i1ii

2

CtCttt 1i1iii

i1ii

AUC

AUMC

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1. Linear trapezoidal

Computational methodsComputational methods

Area calculations by numerical integration

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

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2. Log-linear trapezoidalC

C

tt

1

2

1

2

Concentration

Time

Computational methodsComputational methods

Area calculations by numerical integration

1i

i

1ii

i1ii

CC

CCtt

LnAUC

AUMC

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2. Log-linear trapezoidal

Computational methodsComputational methods

Area calculations by numerical integration

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

< Linear trapezoidal

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Computational methodsComputational methods

Extrapolation to infinity

last

lastt z

lastt

CdttCAUC

Assumes log-linear decline

z

lastlast

z

lastt

CtCAUMC

last

2

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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

AUC Determination

Area (mg.hr/L)-2.2753.131.831.130.9450.900

Total 10.21

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

Computational methodsComputational methods

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• MRT = AUMC / AUC

• Clearance = Dose / AUC

• Vss = Cl x MRT =

• F% = AUC EV / AUC IV DEV = DIV

Dose x AUMCAUC2

The Main PK parameters can be calculated using non-compartmental analysis

Non-compartmental analysisNon-compartmental analysis

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• Non-compartmental analysis

Trapezes

• Fitting with a poly-exponential equation

Equation parameters : Yi, i

• Analysis with a compartmental model Model parameters : kij

Computational methodsComputational methods

Areacalculations

Areacalculations

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Fitting with a poly-exponential equationFitting with a poly-exponential equation

Area calculations by mathematical integration

n

1i

tλi

ieC(t) Y

n

1i i

i

λ

YAUC

n

1i2i

i

λ

YAUMC

For one compartment :

10

0

k

CAUC

210

0

k

CAUMC

10k

1MRT

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Fitting with a poly-exponential equationFitting with a poly-exponential equation

For two compartments :

2

2

1

1 YY

AUCY

AUCi

i

22

221

12

YY

AUMCY

AUMCi

i

tλ2

tλ1

21 eeC(t) YY

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• Non-compartmental analysis

Trapezes

• Fitting with a poly-exponential equation

Equation parameters : Yi, i

• Analysis with a compartmental model Model parameters : kij

Computational methodsComputational methods

Direct MRTcalculations

Areacalculations

Areacalculations

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Example : Two-compartments model

Analysis with a compartmental modelAnalysis with a compartmental model

1

k12

k21

k10

2

dt

dX1 11210 Xkk 221 Xk

dt

dX2112 Xk 221 Xk-

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dX1/dt

X1 X2

Example : Two-compartments model

K is the 2x2 matrix of the system of differential equations describing the drug transfer between compartments

Analysis with a compartmental modelAnalysis with a compartmental model

dX2/dtK =

1210 kk

12k

21k

21k-

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MRTcomp1

Dosing in 1

MRTcomp2

MRTcomp1

MRTcomp2

Analysis with a compartmental modelAnalysis with a compartmental model

(-K-1) =

Then the matrix (- K-1) gives the MRT in each compartment

Dosing in 2

Comp 1

Comp 2

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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 Residence Times

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The Mean Absorption Time(MAT)

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Definition : mean time required for the drug to reach the central compartment

1 A comp. systemEV

MRTAUC

AUMC

A 1

Ka

F = 100%

K10

The Mean Absorption TimeThe Mean Absorption Time

1 compIV

MRTAUC

AUMC

IVEV

IVEVA comp. AUC

AUMC

AUC

AUMCMRT

!MATMRT A comp.

because bioavailability = 100%

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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 :

!

The Mean Absorption TimeThe Mean Absorption Time

F

MRTMAT A comp.

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• In vivo measurement of the dissolution rate in the digestive tract

blood

solution

digestive tract

MDT = MRTtablet - MRTsolution

dissolution absorption

tablet solution

The Mean Dissolution TimeThe Mean Dissolution Time

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Mean Residence Time in the Central Compartment (MRTC) and in

the Peripheral (Tissues) Compartment (MRTT)

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MRTC MRTTMRTsystem = MRTC + MRTT

MRTcentral and MRTtissue

Entry

Exit (single) : excretion, metabolism

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The Mean Transit Time(MTT)

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• 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 Transit Times (MTT)

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The Mean Residence Number(MRN)

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•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 :

The Mean Residence Number (MRN)

MRN =MRT

MTT

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Stochastic interpretation of the drug disposition in the body

MRTC

(all the visits)MTTC

(for a single visit)

MRTT

(for all the visits)MTTT

(for a single visit)

Cldistribution

Rnumber

of cycles

Clelimination

Clredistribution

Mean numberof visits

RR+1IV

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Stochastic interpretation of the drug disposition in the body

Computation : intravenous administration

MRTsystem = AUMC / AUC

MRTC = AUC / C(0)

MTTC = - C(0) / C’(0) R + 1 = MRTC

MTTC

MRTT = MRTsystem- MRTC

MTTT =MRTT

R

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Interpretation of a Compartmental Model

Determinist vs stochasticDigoxin

stochastic

MTTC : 0.5hMRTC : 2.81hVc 34 L

Cld = 52 L/h

4.4

ClR = 52 L/h

MTTT : 10.5hMRTT : 46hVT : 551 L

Cl = 12 L/h

MRTsystem = 48.8 h

Determinist

Vc : 33.7 L1.56 h-1

VT : 551L0.095 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

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Determinist vs stochasticGentamicin

stochastic

MTTC : 4.65hMRTC : 5.88hVc : 14 L

Cld = 0.65 L/h

0.265

ClR = 0.65 L/h

MTTT : 64.5hMRTT : 17.1hVT : 40.8 L

Clélimination = 2.39 L/h

MRTsystem = 23 h

Determinist

Vc : 14 L0.045 h-1

VT : 40.8L0.016 h-1

0.17 h-1

t1/2 = 57 h

y =5600 e-0.281t + 94.9 e-0.012t

t1/2 =3h

t1/2 =57h

Interpretation of a Compartmental Model