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Non-Compartmental PK Modelling “model independent” Distributed models
Transit/Residence distributions
Michael Weiss
Martin Luther University
Halle-Wittenberg
)()(
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.