non cpt analysis - 1 non compartmental analysis update: 13/08/2010

Non cpt analysis - 1

Non compartmental analysisNon compartmental analysis

Update: 13/08/2010


Stochastic interpretation

Statistical Moment ApproachStatistical 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


Synonymous

•Model-independent approach

•Non-compartmental analysis

Statistical Moment ApproachStatistical Moment Approach

!


• Clearance = Dose / AUC

• Vss =

• MRT = Vss / Cl = AUMC / AUC

• F% = AUC EV / AUC IV DEV = DIV

Dose x AUMCAUC2

The Main Non-compartmental Parameters

The Main Non-compartmental Parameters


(MRT system)

The Mean Residence TimeThe Mean Residence Time


• To measure the time each molecule 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

Exit

Non-compartmental analysisNon-compartmental analysis


• Clearance = flow = 2 balls/second

• MRT = t = (t1 + t2... t6)/n = (0.5 + 1 + 1.5 +…+6)/6 = 3

• Vss = Clearance x MRT = 6 balls• Tube volume x R2 x L = x R2 x 12R• Ball volume (6 x 4R3)/3

• Ratio Vballe/ Vtube = 0.67 = partition coefficient between balls and tube

Principle of the method

2 balls / s2 balls / s

rate of absorption



• 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 TimeMean Residence Time


• 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(t1) x t

AUCtot

C

(t)

C1

t1


x No


Cumulated sojourn times of molecule which has been eliminated during t at :


C

(t)

C1

t1

t1 : t1 x x No

tn : tn x x No

MRT= t1xtn x NoC1 x t x No Cn x t x No

AUCTOT AUCTOT

MRT = ti x Ci x t / AUCTOT = t C(t) t / C(t) t

tn

CnC1 x t AUCTOT

Cn x t AUCTOT




Requirements to compute MRT


Only one exit from the measurement compartment

First-order elimination : linearity

Principle of the method: (5)Entry (exogenous, endogenous)

Exit (single) : excretion, metabolism

recirculationexchanges

Central compartment

(measure)



• 2 exit sites

• MRT is not computable by statistical moments applied to plasma concentration



1 2


Computation MethodComputation Method

• Non-compartmental analysis• Trapezes

• Fitting to a polyexponential equation•

Equation parameters : Yi, i

• Assuming a compartmental model

• Model parameters : kij


• 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

Computation method (1)

2 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

0

0

0 0

0




• S0 by the arithmetic trapezoidal rule

C0

C1

C2

C3

t0 t1 t2 t3

extrapolation area

AUC1 = x (t1 - t0)2

C0 +C1

AUCTOT = S1 = AUC1 + AUC2 ... AUCn + extrapolation area

AUC1AUC2

AUC3




• Computation of S1 = AUMC with the arithmetic trapezoidal rule

AUMC1 = x (t1-t0) t0 x C0 + t1 x C1

2C0

t0 t1 t2 t3

AUMC1AUMC2

AUMC3

area to extrapolate

AUMCTOT = S2 = AUMC1 + AUMC2 +... AUMC extrapolated

C1

C2C3




• 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 3 2

2

2

3




• From the parameters of a given model

S0 = Yi / i

S1 = Yi /i

S2 = 2Yi /i

n

n

n

i =1

i =1

i =1

2

3




• Bicompartmental model :

C(t) = Y1 exp(-1t) + Y2 exp(-2t)

MRTsystem =Y1/1 + Y2 / 2

Y1/1 + Y2 / 2

2 2




• 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



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 = 0.693 MRT

MRT system: interpretationMRT system: interpretation


Multicompartmental model

terminal half-life vs MRTC

on

cen

trat

ion

MRT = 16 h

MRT = 4 h

t1/2 = 12 h 24 temps (h)

MRT system: interpretationMRT system: interpretation


• Comparison of published results • 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

Y1/1 + Y2 / 2

22 Y1/1 + Y2 / 2 + Y3 / 3

Y1/1 + Y2 / 2 + Y3 / 3

2 2 2

MRT systemMRT system


The Mean Absorption Time(MAT)

The Mean Absorption Time(MAT)


Definition : mean time for the arrival of bioavailable drug

MATKa

F = 100%

K10

MAT = 1

Ka

Administration

The MATThe MAT


1- IV administration

MRTIV = 1 / K10

2- Oral administration

MRToral longer than MRTIV

MRToral = 1 / K10 + 1 / Ka

MAT = MRToral - MRTIV = 1 / Ka

How to evaluate the MAT

KaK10

IVPo

The MATThe MAT


The MATThe MAT

MAT and bioavailability

• The MAT measures the MRT at the administration site and not the "rate" of drug arrival in the central compartment


The MATThe MAT


• Actually, the MAT is the MRT at the injection site

• MAT does not provide information about the absorption process unless F = 100%



MATKa1

K10

Ka2F = Ka1 / (Ka1 +Ka2)

MRT oral = + = +1

Ka1 + Ka2

1 1 1

K10 K10Ka

MAT is influenced by all processes of elimination (absorption, degradation,…) located at the administration site

!

The MATThe MAT


Conclusion : by measuring (AUMC/AUC), the same MAT will be obtainedThis does not mean that the absorption processes towards the central compartment are equivalent

MAT and bioavailability1 1.5 2

1 0.5 0

MAT = 1/(1+1) = 0.5h MAT= 1/(1.5+0.5)= 0.5h MAT=1/(0+2)=0.5h

!

The MATThe MAT

K10K10

K10


MATB < MATA

but

Absorption clearance of B is lower than that of A !

MAT and bioavailability1 0.5

1 4MATA = = 0.5 h

1

(1 + 1)MATB = = 0.28 h

1

(4 + 0.5)

A B

!

The MATThe MAT


the MATthe 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 x Vabs

Ka1

Clabs

Vabs


The MATThe 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

Vabs = 1 LKa1 = 0.2 h-1

Vabs = 1 L0.1 h-1

0.1 h-1


The Mean Dissolution Time(MDT)

The Mean Dissolution Time(MDT)


• in vitro measurement:

• dissolution test

• statistical moments approach

• modelling approach (Weibull)

The MDTThe MDT


• in vivo measurement (1) :

blood

elimination

solution

digestive tract

tablet

absorption

MRTtotal = MRTdissolution + MRTabsorption + MRT elimination

What is the dissolution rate of the pellet in the digestive tract ?

dissolution

The MDTThe MDT


IV administration

MRTIV = 6 h


IV

The MDTThe MDT


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


blood

eliminationadministrationof an oral solution

digestive tract

The MDTThe MDT


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


solution

administration

The MDTThe MDT


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

compartment (MRTT)

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

compartment (MRTT)


• 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

MRTcentral and MRTtissueMRTcentral and MRTtissue


Computations

• MRTC = AUC / Co = =

• MRTT = MRTsystem - MRTC

• MRTT = -

1

K10

Vc

Cl

AUMC

AUC

AUC

Co

N.B. : necessary to know Co accurately



Relationship with the extent of distribution

MRTsystem Vss

MRTcentral Vc

This ratio measures the affinity for the peripheral compartment

=



The Mean Transit Time(MTT)

The Mean Transit Time(MTT)


•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)The Mean Transit Times (MTT)


The Mean Transit Time in the measurement (central)

compartment (MTTcentral)

The Mean Transit Time in the measurement (central)

compartment (MTTcentral)


Calculation :

MTTC = - C(o) dCp/dt for t = 0

MTTC = - C(o) C'(o)

MTTC = Yi Yi i

N.B. : necessary to know Co accurately

i =1 i =1

n n

The MTTcentralThe MTTcentral


Computation : example for a bicompartmental model

• C(t) = 5 exp(-0.7t) + 2 exp(-0.07t)

• MTTC = (5 + 2) / (5 x 0.7 + 2 x 0.07) = 1.428 h

The MTTcentralThe MTTcentral


• 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 = - 1MRTC

MTTC

The MTTcentral and number of visitsThe MTTcentral and number of visits


= R + 1MRTC

MTTC

When there is no recycling (monocompartmental model) R = 0 and :

MRTC

MTTC

= 1 MRTC = MTTC



Bicompartmental model

Vc

K10

K12

K21

MTTC = 1 / (K10 + K12)MTTC = 1 / (Cl + Cld)R = K12 / K10 R = Cld / Cl

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.



The Mean Transit Time in the peripheral (tissue)

compartment (MTTtissue)

The Mean Transit Time in the peripheral (tissue)

compartment (MTTtissue)


Computation

• MTTT = MRTtissue / R

• MTTT =

MRTsystem - MRTcentral

R (visit)

The MTTtissue (MTTT)The MTTtissue (MTTT)


Computation : bicompartmental model

MTTT = = =1K21

Vss - Vc

Cld

Vt

Cld

MTTT : - does not rely on clearance - measures drug affinity for peripheral tissues

K12

K21

K10

Jusko.J.Pharm.Sci 1988.7: 157

The MTTtissueThe MTTtissue


• Interpretation of drug kinetics (1)

Gentamicin5600e-0.218t + 94.9e-0.012t

Digoxin21.4e-1.99t + 0.881e-0.017t

Clearance (L/h) 2.39

Cld (L/h) 0.632

Vss (L) 54.8

Vc (L) 14.0

VT (L) 40.8

12.0

52.4

585

33.7

551.0

time : h concentration : mg l-1


Application of the MRT conceptApplication of the MRT concept

Non cpt analysis - 58Jusko.J.Pharm.Sci 1988.7: 157

Gentamicin Digoxin

K12 (h-1) 0.045

K21 (h-1) 0.016

K10 (h-1) 0.170

R 0.265

1.56

0.095

0.338

4.37


• Interpretation of drug kinetics (2)


MTTcentral (transit time. central comp)

MRTC (residence time. central comp.)

MTTtissue (transit time peripheral comp.)

MRTtissue (residence time peripheral comp.)

MRTsystem (total)

Interpretation of the mean times


Gentamicin Digoxin

4.65

5.88

64.5

17.1

23.0

0.532

2.81

10.5

46.0

48.8



Stochastic interpretation of a kinetic relationship

Stochastic interpretation of a kinetic relationship

MRTC

(all the visits)MTTC

(for a single visit)

MRTT

(for all the visits)MTTT

(for a single visit)

Cldistribution

Rnumber de visits

Clelimination

MRTsystem = MRTC + MRTT

Clredistribution


Interpretation of a compartmental model 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 hDeterminist

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


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 hDeterminist

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 Interpretation of a compartmental model


Interpretation determinist vs stochasticInterpretation determinist vs stochastic

Gentamicin vs digoxinDeterministGentamicin Digoxin

Vc = 14 L VT = 40.8 L

0.17 h

0.045 h-1

0.016 h-1

t1/2 distribution : 3ht1/2 : 57 h

Cld:0.65 L/h0.26

0.65 L/h

MTTC: 4.65hMRTC: 5.88hVc = 14 L

MTTT: 64.5hMRTT:17.1hVT : 40.8 h

Cl = 2.39 L/hMR system: 23 h

Vc = 34 L VT = 551 L

0.338 h-1

0.56 h-1

0.095 h-1

t1/2 distribution : 0.3ht1/2 : 4 h

Cld:52 L/h4.4

ClR:52 L/h

MTTC: 0.5hMRTC: 2.81hVc = 34 L

MTTT: 10.5hMRTT:46hVT : 551 h

Cl = 12 L/hMR system: 48.8 h


Computation

• Statistical moments

• Parameters from compartmental model

MRTsystemMRTsystem


t1/2

MRT

0.693 Varea

Vss=


non cpt analysis - 1 non compartmental analysis update: 13/08/2010

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