Non-Compartmental PK Modelling “model independent” Distributed models

Transit/Residence distributions

Michael Weiss

Martin Luther University

Halle-Wittenberg

michael.weiss@medizin.uni-halle.de

)()(

tCCLdt

tdAe

Basic Equation

Rate of drug elimination = Clearance x Plasma concentration

(1)

dttCCLAdt

tdAe

e )()(

00

AUCCLDiv

Note: ive DA )( (nothing remains in the body)

Well-mixed plasma

compartment !

“ model independent “ or noncompartmental analysis)

Estimation of Clearance (single dose)

AUC

DCL iv

AUC

C(t)

t

Single dose

Div dttCAUC

0

)(

!

Intravenous dose

Area Under the Curve

Estimation of Clearance (infusion)

ssCCLSteady state after continuous

i.v. infusion DR

Output (elimination rate) = Input (dose rate, infusion rate)

t

C(t)

Css

DR

ssC

DRCL

Elimination rate

Dose rate

Estimation of Bioavailability

dttCCLAe )(0

(cf. Eq. 1)

ivor

orivivor

AUCD

AUCDFDDif :

iv

or

iv

or

ive

ore

AUC

AUC

dttCCL

dttCCL

A

AF

)(

)(

0

0

,

, Assumption:

CL unchanged ! (13)

ncirculatiosystemicthereachesthatAmountAe

Determinants of Clearance

Organ

QCin QCout

E = 1-F (extraction)

Cout = F Cin

F (availability)

)( otherRH CLCLCLCLhepatic renal

organorganorgan EQCL (4)

N

i

iiEQCL1

Renal Clearance

2

1

21

)(

,

t

t

tteR

R

dttC

ACL

21

21

tt

tt

AUC

excretedamount

(5)

RH CLCLCLt1 t2

AUC

Relative Bioavailability

Conor

Coniv

Treativ

Treator

Con

Treat

AUC

AUC

AUC

AUC

F

F

,

,

,

,

D

tAtTtF e )(

Pr)(

If an amount of drug molecules (dose Div) is instantaneously injected intravenously,

each molecule will spend a random time T in the body until it is eliminated (the

disposition residence time of that molecule).

Residence Time Distribution

Residence time distribution, F(t), is defined by the fraction of molecules which have a

residence time less than t:

Ae(t) is the cumulative amount of drug

eliminated up to time t.

F(0) = 0 and F(∞) = 1

)()(

tCCLdt

tdAe

AUC

tC

dttC

tCtf

)(

)(

)()(

0

Density function

0 0 0

)](1[)()(][ dttFdttFdtttfTEMRT

AUC

dtttC

MRT 0

)(

)(

)]()([

,

0

,,

Re

ReRe

A

dttAA

MRT

)()(

tCLCdt

tdAe

Weiss, Eur J Clin Pharmacol , 1992

Mean Residence Time

][)( tTPtF Probability that residence time T of a molecule exceeds t.

Continuous infusion: Mass dA which entered the body in the time interval [t-dt,t] which

remains in the body at time t is given by:

dttFDR )(

dttFDRtA

t

0

)()( dttFDRAAss

0

)()(

MDRTDRAss

CLMDRTC

MDRTDR

C

AV

ssss

ssss

Weiss, J Pharm Sci , 1991

1. Disposition Curves (Bolus Injection)

Clearance CL

Volume of distribution at steady state Vss

Mean Disposition Residence Time MDRT

CL

VMDRT ss

(14)

Mean Disposition Residence Time

A(t)

t

Div

10 % of Div

t 90%

Bolus injection

t 90%

Continuous infusion

t

90 % of Css

Css

C(t)

2/1%90 4 ttMDRTt 7.3%90(15)

Weiss, J Pharmacokin Biopharm, 1986

Multiple Dosing

ssC

C(t)

Dor Dor Dor Dor Dor Dor Dor Dor

dosing interval

maintenance dose

average

concentration

dose rate orFD

Dor = Dm

C(t)

Dor Dor Dor Dor Dor Dor Dor Dor

Cmax

increasing

toxicity

decreasing

efficacy

Cmin

Therapeutic Drug Monitoring (TDM)

C(t)

Dor Dor Dor Dor Dor Dor Dor Dor

AUC

AUCss

AUCAUCss

MDRT

doseemaintenanc

statesteadyatbodyinamount

ssC

CL

FD

CL

DRC m

ss

CL determines maintenance dose Dm

V determines loading dose DL

MDRT determines time to steady state t90%

and dosing interval

Basic Pharmacokinetic Parameters

n

i

t

iivieBtC

1

)( fit to data, estimate Bi and i (i= 1..n)

n

i i

i

iv

B

DCL

1

n

i i

iBAUCdttC

10

)(

Parametric Curve Model

n

i i

i

n

i i

i

iv

iv

iv

iv

B

B

AUC

AUMC

dttC

dtttC

MDRT

1

12

0

0

)(

)(

MDRTCLVss

Mean Residence Time after Oral and Intravenous Administration

Absorption

Disposition

C(t)

Dissolution

Dor

Div

Mean Dissolution

Time

Mean Absorption

Time

Mean Disposition Residence

Time

Mean Input Time

Mean Body Residence Time

MBRT = MDT + MAT + MDRT

Mean Body Residence Time MBRT

or

or

or

or

AUC

AUMC

dttC

dtttC

MBRT

0

0

)(

)(

Mean Disposition Residence Time MDRT

iv

iv

iv

iv

AUC

AUMC

dttC

dtttC

MDRT

0

0

)(

)(

Mean Input Time MATMDTMIT

Oral Administration

Dissolution

Absorption

SolutionTablet MBRTMBRTMDT

MDRTMBRTMAT Solution

MAT

Dor

FDor

. . . . . . . . . . . . . . . . . . . . MDT

Systemic

Circulation

MDTin vitro

MDTin vivo

C(t) after extravascular (oral) administration

0

)( AUCdttC0

)( AUMCdtttCand

by numerical integration Trapezoidal rule

C(t)

ti ti+1

Ci

Ci+1

tN

,NtAUC

Trapezoidal rule

,,0

2111

1

1

)1

()(2

1

NN tt

zz

NNiiiiii

N

i

AUMCAUMC

tCttCtCtAUMC

,,0

11

1

1

)(2

1

NN tt

z

Niiii

N

i

AUCAUC

CttCCAUC

ti+1 ti

)(2

111 iiii ttCC

Reactor: Turbulent mixing

Transit time dispersion in microcirculatory network:

mixing

How to describe mixing/distribution kinetics ?

Steady-state → transient state

Circulation without dispersion: no mixing

Residence time sytem

Transit time sytem

Disposition

curve

Outflow

curve

Transit time dispersion Rate of distribution

Mean transit time Extent of distribution

Normalized (dimensionless) variance

Relative Dispersion of Disposition Residence Time Distribution

2

2

MDRT

VDRTRDD

n

i

t

iivieBtC

1

)(

n

ij

i

ij

j

BjdttCtm

11

0

!)(

2

0

1

0

2

m

m

m

mVDRT

0

1

m

mMDRT

t

Div Weiss, Pharm Res , 2007

Closed (noneliminating) system (CL = 0)

t

Div

Rate of Distribution: Mixing Clearance

21

2

DM

iv

MRD

CL

AUC

DCL)1(

2

1 2

DM RD

AUC

AUC

Well-mixed system (1-compartment model)

12

DRD Exponential distribution

dtCtCAUCM

0

)()(

ss

iv

V

DC )(

V

DC iv)0(

)()()(

0 CtCCLdt

tdCV M

AUCM: Circulatory Transit Time

AUCM

C( )

t

Div

Closed

(noneliminating)

system (CL = 0)

Weiss & Pang, J Pharmacokin Biopharm, 1992

)1(2

1 2

civ

M RDQ

DAUC

VCTMCT

)1(1 22

cD RDQ

CLRD

From Flow-to Diffusion-Limited Distribution Kinetics A Continuous Transition

Cardiac Output (l/min)

Antipyrine

Inulin

0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

Dis

trib

uti

on

Cle

aran

ce (

l/m

in) flow-limited

diffusion -limited

Deff~ 7*Dinulin Sorbitol

Thiopental

Weiss et al, Pharm Sci, 2007

5 6 7 8 9 10 11

0

1

2

3

CLM

(l/min)

Q (l/min)

Slope 0.26 ± 0.07, P < 0.05; R = 0.84

Distribution Kinetics of Alfentanil

Data from: Henthorn et al., Clin Pharmacol Ther, 1992

4

1

)(i

t

iivieBtC

4

11

!i

j

i

i

j

Bjm

Thiopental: heterogeneity of residence time distribution increases with obesity

Weiss , Pharmacokin Pharmacodyn, 2008

Brain

Heart

Kidney

Testes

Fat

Gut

Carcass

Vei

ns

Art

erie

s

Lung

Pancreas

Spleen

Skin

Liver

Muscle

Pulmonary

Circulation

Systemic

Circulation

Minimal Circulatory PK Model

Heterogeneous subsystems

Transit time distributions

© Weiss 2005

Why are they relevant?

Less than 0.1% of PK models used in literature are circulatory models

2) First-principles modeling of distribution kinetics

Role of cardiac output, convective dispersion and intratissue diffusion

(ICG, inulin, antipyrine, thiopental, rocuronium)

Modeling of slow tissue binding (digoxin)

Use of the multiple indicator approach in parameter estimation

1) Description of initial mixing kinetics (2 min after bolus injection)

Front-end kinetics of short acting iv anesthetics

Circulatory minimal model

Pulmonary Circulation

Systemic Circulation

Div

Cardiac

Output, Q

Arterial

Sampling

Transit Time

Density

MTT=V/Q, RD

Elimination

CL = EQ

tMTTRD

MTTt

tRD

MTTtf IG

2

)(exp

2)(

2

3

Extraction

E © Weiss 2005

Weiss et al., Br J Clin Pharmacol ,1996

)(ˆ)(ˆ)1(1

)(ˆ)(ˆ

sfsfE

sfsf

ps

p

circ

)(ˆ)( 1 sfQ

DLtC circ

Recirculatory PK Model

Numerical inverse Laplace Transformation

Q

CLE Extraction (probability of elimination in one passage through systemic

circulation)

Schalla & Weiss, Eur J Pharm Sci, 1999

Hemodynamic Influences on Sorbitol Kinetics in Humans Inverse Gaussian Transit Time Density

Pla

sma

So

rbit

ol

(µg

/ml)

0 5 10 15 20 25 30 35

Time (min)

0

50

100

150

200

Control

Orciprenaline

(10 µg/min)

Sorbitol

0.8 g, 1min

RDs + 27 %

CLM + 44 %

CL + 24 %

Weiss et al., Br J Clin Pharmacol,1996

Q + 53%

0

20

40

60

80

100

0 5 10 15 20 25 300

40

80

120

160

200

0 2 4 6 8 100 2 4 6 8 10

0

40

80

120

160

200

Physiological (recirculatory)

vs. Compartmental (biexponential)

Time (min)

simulated

fitted

Art

eria

l co

nce

ntr

atio

n

5 min

Infusion 1 min

simulated

predicted

Pul circ

Sys circ Central

Peripheral

0

40

80

120

160

200

0 2 4 6 8 100 2 4 6 8 10

0

40

80

120

160

200

0 2 4 6 8 10

0

40

80

120

160

200

0 2 4 6 8 10

0

40

80

120

160

200

0

40

80

120

160

200

0 2 4 6 8 100 2 4 6 8 10

0

40

80

120

160

200

0 2 4 6 8 10

0

40

80

120

160

200

0

10

20

30

40

50

0 10 20 30 40 50 600 10 20 30 40 50 60

0

10

20

30

40

50

Time (min)

Conce

ntr

atio

n

1 min 15 min

Arterial vs. peripheral venous sampling

CA(t)

CV(t)

First-principles modeling of distribution kinetics

Advective transport

Advective dispersion Vascular mixing

Permeation (Capillary uptake)

Diffusion (Extravascular)

Tissue Binding

Extraction, E

Pulmonary Circulation

Systemic Circulation

Div

Cardiac

Output, Q

Drug+vascular marker (ICG)

ICG (vascular marker)

Drug

fs(s)

fp(s)

Div

1. Simultaneous injection, ICG+drug

2. Fit of ICG data (IG model)

3. Fixing of ICG parameters

in drug model

4. Fit of drug data

Arterial Sampling

Q

VB,p VT,p

VB,s VT,s Intravenous

injection

C(t)

Arterial

sampling Dose

Cardiac output

Pulmonary blood and tissue volume

Systemic

Extravascular diffusion

CL

d

Intravascular

mixing

(vascular marker)

Microcirculatory network

Tissue

distribution

Microscopic volume element

vascular

tissue

phase

Dif

fusi

on

Vp

VT

Capillary

flow

Systemic circulation: Advection-diffusion model Stochastic model of transit time distribution

Weiss & Roberts, J Pharmacokin Biopharm, 1996

Extravascular Diffusion Kinetics Rocuronium (+ICG as vascular marker)

Systemic

Circulation

Q )(ˆ sf p

)(ˆ sfs

Vp 2

pRDQ

d VT,s

VB,s

vascular

ISF

Cell

extravascular

2

,sBRDVB,s

ssv

sfsf dd

d

sIGs tanhˆ)(ˆ

B

T

V

Vv

eff

dD

L2

)(ˆ, sf pIG

0 2 4 6 8 10

0

5

10

15

100 1017 8 2 3 4 5 6 7 8 2 3

100

101

78

2

3

4

5

6

78

2

3

ICG

conce

ntr

atio

n(µ

g/m

l)

Pre

dic

ted

co

nce

ntr

atio

n (

µg/m

l)

Time (min)

Observed concentration (µg/ml)

Vascular Mixing Kinetics Vascular Marker (ICG) in Patient

VB,p

VB,s

2

, pBRD

2

,sBRD

Q

E

Relative dispersion -> Intravascular mixing

)(ˆ, sf sIG

2

, pBRD 0.09 12

0.37 21

Population

Mean

Interpatient

CV(%)

Pulmonary circulation

Systemic circulation

Q (L/min) Cardiac output 3.52 20

2

,sBRD

TT dispersion

Time (min)

Weiss et al.,

J Pharmacokin

Pharmacodyn, 2011

101

102

103

104

105

0 50 100 150 200 2500 50 100 150 200 250

101

102

103

104

105

0 1 2

0

2

4

6

8

10

12

14

Ro

curo

niu

m c

on

cen

trat

ion

(n

g/m

l)

Time(min)

89 (37) 50(62)

2.66(97) 115 (61)

Distribution kinetics of rocuronium

Interstitial diffusion, time constant

Interstitial volumes

d (min)

VT,p (L))

VT,s (L)

Population Mean

(%RSE)

Interpatient %CV

(%RSE)

14.2 (30) 29 (96)

Individual estimates of ICG parameters were

used as fixed parameters in fitting rocuronium data.

Rocuronium Kinetcs in Patients

Weiss et al.,J Pharmacokin Pharmacodyn, 2011

2.5 3.0 3.5 4.0 4.5 5.0

0.25

0.30

0.35

0.40

0.45

Rel

ativ

e S

yst

emic

D

isp

ersi

on

Cardiac Output (L/min)

Fig. 3

Systemic transit time heterogeneity of ICG decreases linearly with cardiac output (P< 0.005)

2.5 3.0 3.5 4.0 4.5 5.0

1.4

1.6

1.8

2.0

2.2

2.4

Cardiac Output (L/min)

Pulm

onar

y B

lood V

olu

me,

VB

,p(L

)

Fig. 4

Central blood volume increases linearly with cardiac output (P<0.01)

Weiss et al.,J Pharmacokin Pharmacodyn, 2011

The validity of a model is determined

by the modeling objectives

Minimal PBPK models are relevant for explaining -Initial intravascular mixing (blood volumes, TT dispersion, role of the lungs) -Tissue distribution kinetics (permeation,diffusion,binding)

-Effect of obesity (highly lipid-soluble drugs) -Effect of cardiac output and hemorrhagic shock -Hemodynamic drug interactions -Hepatic function in vivo (ICG)

Model selection and experimental design are strongly interrelated: Frequent early blood sampling and multiple indicator method

Conclusions

References

Henthorn, T., T. Krejcie, et al. (2008). "Early drug distribution: a generally neglected aspect of

pharmacokinetics of particular relevance to intravenously administered anesthetic agents."

Clin Pharmacol Ther 84(1): 18-22.

Henthorn, T. K., T. C. Krejcie, et al. (1992). "The relationship between alfentanil distribution

kinetics and cardiac output." Clin Pharmacol Ther 52(2): 190-196.

Schalla, M. and M. Weiss (1999). "Pharmacokinetic curve fitting using numerical inverse Laplace

transformation." European journal of pharmaceutical sciences 7(4): 305-309.

Weiss, M. (1986). "Generalizations in linear pharmacokinetics using properties of certain classes of

residence time distributions. I. Log-convex drug disposition curves." Journal of

pharmacokinetics and biopharmaceutics 14(6): 635-657.

Weiss, M. (1991). "Nonidentity of the steady-state volumes of distribution of the eliminating and

noneliminating system." Journal of pharmaceutical sciences 80(9): 908-910.

Weiss, M. (1992). "The relevance of residence time theory to pharmacokinetics." European journal

of clinical pharmacology 43(6): 571-579.

Weiss, M. (2007). "Residence time dispersion as a general measure of drug distribution kinetics:

Estimation and physiological interpretation." Pharmaceutical research 24(11): 2025-2030.

Weiss, M. (2008). "How does obesity affect residence time dispersion and the shape of drug

disposition curves? Thiopental as an example." Journal of pharmacokinetics and

pharmacodynamics 35(3): 325-336.

Weiss, M., G. Hübner, et al. (1996). "Effects of cardiac output on disposition kinetics of sorbitol:

recirculatory modelling." British journal of clinical pharmacology 41(4): 261-268.

Weiss, M. and K. S. Pang (1992). "Dynamics of drug distribution. I. Role of the second and third

curve moments." Journal of pharmacokinetics and biopharmaceutics 20(3): 253-278.

Weiss, M., M. Reekers, et al. "Circulatory model of vascular and interstitial distribution kinetics of

rocuronium: a population analysis in patients." Journal of pharmacokinetics and

pharmacodynamics 38(2): 165-178.

Weiss, M. and M. S. Roberts (1996). "Tissue distribution kinetics as determinant of transit time

dispersion of drugs in organs: application of a stochastic model to the rat hindlimb." Journal

of pharmacokinetics and biopharmaceutics 24(2): 173-196.