Glossary
Page 1 of 24
Collection of terms, symbols, equations, and
explanations of common pharmacokinetic and
pharmacodynamic parameters and some
statistical functions
Version: 16 Februar 2004
Authors: AGAH working group PHARMACOKINETICS
...\PK-glossary_PK_working_group_2004.pdf
Glossary
Page 2 of 24
Collection of terms, symbols, equations, and explanations of common
pharmacokinetic and pharmacodynamic parameters and some
statistical functions
TABLE OF CONTENTS
Page
TABLE OF CONTENTS .................................................................................................................2
1
Pharmacokinetic Parameters from noncompartmental analysis (NCA) ...............................3
1.1 Parameters obtained from concentrations in plasma or serum...........................................3
1.1.1 Parameters after single dosing..................................................................................3
1.1.2 Parameters after multiple dosing (at steady state)....................................................6
1.2 Parameters obtained from urine ..........................................................................................7
2
Pharmacokinetic parameters obtained from compartmental modeling................................8
2.1 Calculation of concentration-time curves.............................................................................9
2.2 Pharmacokinetic Equations - Collection of Equations for Compartmental Analysis .........10
3
Pharmacodynamic GLOSSARY ................................................................................................20
3.1 Definitions ..........................................................................................................................20
3.2 Equations: PK/PD Models..................................................................................................21
4
Statistical parameters................................................................................................................22
4.1 Definitions ..........................................................................................................................22
4.2 Characterisation of log-normally distributed data ..............................................................23
...\PK-glossary_PK_working_group_2004.pdf
AGAH Working group PK/PD modelling
1
Page 3 of 24
PHARMACOKINETIC PARAMETERS FROM NONCOMPARTMENTAL
ANALYSIS (NCA)
1.1
Parameters obtained from concentrations in plasma or serum
1.1.1
Parameters after single dosing
Symbol
Unit /
Dimensio
n/
Definition
Calculation
Dimension
AUC
AUC(0-∞)
Amount·time/ Area under the concentration-time
volume
curve from zero up to ∞ with
extrapolation of the terminal phase
Cz
AUC=AUC (0 - t z ) +
λz
, Cz may be
measured (Cz,obs) or calculated (Cz,calc)
AUC(0-t),
AUCt
Amount·time/ Area under the concentration-time
According to the linear trapezoidal rule:
volume
curve from zero up to a definite time t
n- 1
AUC (0 - t) = ∑ AUC (t i -t i +1 )
wit
i= 1
h t1=0 and tn=t,
Concentrations Ci measured at times
ti, i=1,…,n.
AUC (t i
− t i +1 )
= 12 ⋅ (C i + C i+1 ) ⋅ (t i+1 − t i )
or according to the log-linear trapezoidal rule:
AUC(t i
− t i +1 )
=
(C i − C i +1 ) ⋅ (t i +1 − t i )
(ln C i − ln C i +1 )
(the logarithmic trapezoidal rule is used for the
descending part of the concentration-time curve, i.e. if
Ci>1.001*Ci+1>0)
AUC(0-tz)
AUCextrap %
AUMC
AUMC(0-t)
Amount·time/ Area under the concentration-time
volume
curve from zero up to the last
%
Amount·
2
(time) /
volume
Amount·
2
(time) /
volume
concentration ≥LOQ (Cz)
Area under the concentration-time
curve extrapolated from tz to ∞ in %
of the total AUC
Area under the first moment of the
concentration-time curve from zero
up to ∞ with extrapolation of the
terminal phase
Area under the first moment of the
concentration-time curve from zero
up to a definite time t
n- 1
AUMC(0 - t) = ∑ AUMC(t i -t i +1 )
i= 1
with t1=0 and tn=t. Concentrations Ci
measured at times ti, i=1,…,n.
See AUC(0-t)
AUC extrap % =
AUC-AUC (0-t z )
⋅ 100
AUC
AUMC = AUMC (0 -
tz )
+
t z ⋅ Cz
λz
+
Cz
2
λz
,
Cz may be measured (Cz,obs) or calculated (Cz,calc)
AUMC (t i
=
1
6
− t i +1 )
(t i+1 − t i )(t i +1 (C i + 2C i+1 ) + t i (2C i + C i +1 ))
(linear trapezoidal rule)
=
C i t i − C i +1t i +1 C i − C i +1
ln C i − ln C i +1
+
with B =
B
t i +1 − t i
B2
(log-linear trapezoidal rule)
AUMC(0-tz)
AUMCextrap %
Amount·
2
(time) /
volume
%
See AUMC(0-t)
Area under the first moment of the
concentration-time curve from zero to
the last quantifiable concentration
Area under the first moment of the
AUMC-AUMC
concentration-time curve
AUMC extrap % =
AUMC
extrapolated from tz to ∞ in % of the
total AUMC
...\PK-glossary_PK_working_group_2004.pdf
(0-t z )
⋅ 100
AGAH Working group PK/PD modelling
Symbol
Unit /
Dimension
Page 4 of 24
Definition
Cp or C
Amount/ volume Plasma concentration
Cs or C
Amount/ volume Serum concentration
Cu
Amount/ volume Unbound plasma concentration
CL
Volume/ time or Total plasma, serum or blood
volume/ time/ kg clearance of drug after intravenous
CL / f
Volume/ time or Apparent total plasma or serum
volume/ time/ kg clearance of drug after oral
Calculation
CL =
administration
CLint
CL / f =
administration
Volume/ time or Intrinsic clearance – maximum
volume/ time/ kg elimination capacity of the liver
CLH,b
Volume/ time or Hepatic blood clearance, product of
volume/ time/ kg hepatic blood flow and extraction
CLCR
Volume/ time or Creatinine clearance
volume/ time/ kg
D iv
AUC
D po
AUC
CLH = QH·EH
ratio
CLm
Volume/ time
Metabolic clearance
Cz, calc
Amount/ volume Predicted last plasma or serum
Cz or Cz, obs
Amount/ volume Last analytically quantifiable plasma
Cmax
D
f
concentration
or serum concentration above LOQ
Amount/ volume Observed maximum plasma or
serum concentration after
administration
Dose administered
Amount
Fraction of the administered dose
systemically available
-
Measured or Cockcroft & Gault formula
Calculated from a log-linear regression through
the terminal part of the curve
directly taken from analytical data
directly taken from analytical data
f =
AUC po ⋅ D iv
AUC iv ⋅ D po
Absolute bioavailability, systemic
availability in %
Fraction of the administered dose in
comparison to a standard (not iv)
F = f ⋅ 100
%
Relative bioavailability in %
Frel = frel ⋅ 100
fa
-
For orally administered drugs: f = fa*(1-EH)
fm
-
fu
-
Fraction of the extravascularly
administered dose actually absorbed
Fraction of the bioavailable dose
which is metabolized
Fraction of unbound (not proteinbound or free) drug in plasma or
serum
Half-value duration (time interval
during which concentrations exceed
50% of Cmax)
Terminal rate constant (slowest rate
constant of the disposition)
F
%
frel
-
Frel
HVD
Time
-1
λz
(Time)
ke or kel
(Time)
LOQ
-1
Elimination rate constant from the
central compartment
Amount/ volume Lower limit of quantification
...\PK-glossary_PK_working_group_2004.pdf
f rel =
AUC ⋅ DSTD
AUCSTD ⋅ D
STD = Standard
fu = Cu /C
negative of the slope of a ln-linear regression
of the unweighted data considering the last
concentration-time points ≥ LOQ
calculated from parameters of the
multiexponential fit
AGAH Working group PK/PD modelling
Symbol
MAT
Unit /
Dimension
Time
Page 5 of 24
Definition
Mean absorption time
Calculation
MAT = MRTev - MRTiv
(ev = extravasal, e.g. im, sc, po)
MDT
Time
Mean dissolution time
MRT
Time
Mean residence time (of the
AUMC
MRT =
unchanged drug in the systemic
AUC
circulation)
Metabolic ratio of parent drug AUC and
AUC parent
MR =
metabolite AUC
AUC metabolite
MR
-
t1/2
Time
Terminal half-life
tlag
Time
tz
Time
tmax
Time
Lag-time (time delay between drug
administration and first observed
concentration above LOQ in plasma)
Time p.a. of last analytically
quantifiable concentration
Time to reach Cmax
Vss
Vz
Apparent volume of distribution at
Volume
or volume/kg equilibrium determined after
intravenous administration
Volume
or volume/kg
Vss / f
Volume
or volume/kg
Vz / f
Volume
or volume/kg
t 1/ 2
=
ln 2
λz
directly taken from analytical data
directly taken from analytical data
directly taken from analytical data
Vss = CL ⋅ MRT =
D ⋅ AUMC
(AUC) 2
Volume of distribution during terminal
phase after intravenous administration
Vz=
Apparent volume of distribution at
equilibrium after oral administration
Vss /f = CL ⋅ MRT =
Apparent volume of distribution during
terminal phase after oral /
extravascular administration
Vz /f =
...\PK-glossary_PK_working_group_2004.pdf
Div
AUC ⋅ λ z
D ⋅ AUMC
(AUC) 2
D po
AUC ⋅ λz
po instead of iv !
AGAH Working group PK/PD modelling
1.1.2
Parameters after multiple dosing (at steady state)
Symbol
Aave
AUCτ,ss
Unit /
Dimension
Amount
Average amount in the body at
steady state
Calculation
A ave =
f ⋅ DM
λz ⋅ τ
by trapezoidal rule
steady state
%
Cav,ss
Amount
/volume
Cmax,ss
Amount
/volume
Cmin,ss
Amount
/volume
Ctrough
Amount
/volume
DM
Amount
LF
-
PTF %
Definition
Amount·time/ Area under the concentration-time
volume
curve during a dosing interval at
AUCss
AUCF%
Page 6 of 24
%
Percent fluctuation of the
concentrations determined from areas AUCF% = 100 ⋅ AUC(above C ave ) + AUC(below C ave )
AUC
under the curve
Average plasma or serum
AUCτ , ss
C av,ss =
concentration at steady state
τ
Maximum observed plasma or serum directly taken from analytical data
concentration during a dosing interval
at steady state
Minimum observed plasma or serum directly taken from analytical data
concentration during a dosing interval
at steady state
Measured concentration at the end of directly taken from analytical data
a dosing interval at steady state (taken
directly before next administration)
Maintenance dose
design parameter
Linearity factor of pharmacokinetics
after repeated administration
Peak trough fluctuation over one
dosing interval at steady state
Accumulation ratio calculated from
AUCτ,ss at steady state and AUCτ
after single dosing
Accumulation ratio calculated from
Cmax,ss at steady state and Cmax after
single dosing
Accumulation ratio calculated from
Cmin,ss at steady state and from
concentration at t=τ after single dose
RA (AUC)
RA (Cmax)
RA (Cmin)
Theoretical accumulation ratio
Rtheor
TCave
Time
tmax,ss
Time
τ
Time
Time period during which plasma
concentrations are above Cav,ss
Time to reach the observed
maximum (peak) concentration at
steady state
Dosing interval
...\PK-glossary_PK_working_group_2004.pdf
LF =
AUC τ, ss
sd = single dose
AUC sd
PTF % = 100 ⋅
RA
(AUC)
RA
(Cmax) =
RA
=
C ss,av
AUC τ,ss
=
AUC τ,sd
C max,ss
C max,sd
(Cmin) =
R theor
C ss,max - C ss,min
1
1- 2 - ε
C min, ss
Cτ ,sd
=
1
1− e − λ z τ
sd = single dose
sd = single dose
, ε=
τ
t1 / 2
derived from analytical data by linear interpolation
directly taken from analytical data
directly taken from study design
AGAH Working group PK/PD modelling
1.2
Page 7 of 24
Parameters obtained from urine
Unit /
Dimension
Definition
Ae(t1-t2)
Amount
Amount of unchanged drug excreted
into urine within time span from t1 to t2.
Cur * Vur
Ae(0-
Amount
Cumulative amount (of unchanged
drug) excreted into urine up to infinity
after single dosing
(can commonly not be determined)
Amount
Amount (of unchanged drug) excreted
into the urine during a dosing interval
(τ) at steady state
Amount/
Drug concentration in urine
Symbol
Aeτ,ss
)
Aess
Cur
Calculation
volume
CLR
Volume/ time Renal clearance
or volume/
time/ amount
CLR =
Ae(0 − ∞)
Ae(0 − τ )
≈
AUC
AUC (0 − τ )
after multiple dose CLR =
fe
-
Fraction of intravenous administered
drug that is excreted unchanged in
urine
fe =
fe/f
-
Fraction of orally administered drug
excreted into urine
fe / f =
Fe
%
Total urinary recovery after intravenous
administration = fraction of drug
excreted into urine in %
tmid
Time
Vur
Volume
Ae(0 − τ )
AUCτ , ss
Ae
Div
Ae
Dpo
Fe = fe ⋅ 100
Mid time point of a collection interval
Volume of urine excreted
...\PK-glossary_PK_working_group_2004.pdf
directly taken from measured lab data
AGAH Working group PK/PD modelling
2
Page 8 of 24
PHARMACOKINETIC PARAMETERS OBTAINED FROM COMPARTMENTAL
MODELING
Symbol
A,B,C or
Ci, i=1,...,n
Unit /
Dimensions
Amount/ volume Coefficients of the polyexponential
Calculation
by multiexponential fitting
equation
-1
Exponents of the polyexponential
equation (slope factor)
-1
Exponent of the i (descending)
exponential term of a polyexponential
equation
α, β, γ
(Time)
λI
(Time)
AUC
Definition
th
Amount·time/
volume
Area under the curve (model)
by multiexponential fitting
by multiexponential fitting
n
⎡C ⎤
AUC = ∑ ⎢ i ⎥
i=1 ⎣ λ i ⎦
extravascular :
iv :
n ⎡
ka
AUC = ∑ ⎢Ci ⋅
k a − λi
i=1 ⎣
⎛ 1 1 ⎞⎤
⋅ ⎜⎜ − ⎟⎟⎥
⎝ λ i k a ⎠⎦
Note: Ci is the linear coefficient of the
polyexponential equation
AUMC
2
Amount·(time) / Area under the first moment curve
volume
n ⎡
C ⎤
AUMC = ∑ ⎢ 2i ⎥
i =1 ⎣ λ i ⎦
extravascular :
iv :
n ⎡
ka
AUMC = ∑ ⎢Ci ⋅
k
i =1 ⎣
a − λi
⎛ 1 1 ⎞⎤
⋅ ⎜⎜ 2 − 2 ⎟⎟⎥
⎝ λ i k a ⎠⎦
Note: Ci is the linear coefficient of the
polyexponential equation
C(0)
Amount/ volume Initial or back-extrapolated drug
concentration following rapid
intravenous injection
n
C (0) =
∑Ci
i =1
Note: Ci is the linear coefficient of the
polyexponential equation
C(t)
CL
fi
Amount/ volume Drug concentration at time point t
Volume/ time
-
Fractional area, area under the various
phases of disposition (λi) in the plasma
concentration-time curve after iv
dosing
f ⋅ Dose
AUC
iv: f=1
Ci
fi =
λi
AUC
with
n
∑f
i =1
i
=1
-1
Zero order rate constant
Design parameter or determined by
multiexponential fitting
-1
Elimination rate constant from the
central compartment
calculated from parameters of the
multiexponential fit
-1
Absorption rate constant
by multiexponential fitting
k0
(Time)
ke or kel
(Time)
ka or kabs
(Time)
Km
CL =
Number of compartments in a multicompartmental model
i
kij
Clearance
See 2.2
Transfer rate between compartment i
and j in a multi-compartmental model
Amount/ volume Michaelis –Menten constant
-1
(Time)
...\PK-glossary_PK_working_group_2004.pdf
by multiexponential fitting
by nonlinear fitting
AGAH Working group PK/PD modelling
Symbol
MRT
Unit /
Dimensions
Time
Page 9 of 24
Definition
Calculation
Mean residence time
iv: MRT =
AUMC
AUC
extravascular: MRT =
Qi
Amount/Time
Intercompartmental clearance between
central compartment and compartment
i
k0
Amount/Time
Zero order infusion rate
design parameter
th
t 1/ 2, λi
Time
Half-life associated with the i
ln2
exponent of a polyexponential equation t1/ 2, λi = λ
i
τ
Time
Infusion duration
t
Time
Time after drug administration
Vc
Volume or
Apparent volume of the central or
Volume /amount plasma or serum compartment
design parameter
Vc =
f ⋅ Dose
n
∑C
i=1
iv: f=1
Vmax
Amount/Time
Maximum metabolic rate
...\PK-glossary_PK_working_group_2004.pdf
i
AUMC
1
− (tlag + )
AUC
ka
AGAH Working group PK/PD modelling
2.1
Page 10 of 24
Calculation of concentration-time curves
Application
iv bolus
Parameter
concentration after bolus administration
Calculation
n
[
Cp ( t ) = ∑ B i ⋅ e −λ ⋅t
i
]
i=1
short-term iv
infusion
concentration during infusion
n ⎡
⎤
B
Cp ( t < T ) = ∑ ⎢ i ⋅ (1 − e −λ ⋅t )⎥
i=1 ⎣ λ i
⎦
i
peak level
n ⎡
⎤
B
Cmax = ∑ ⎢ i ⋅ (1 − e − λ ⋅ T )⎥
λ
i =1 ⎣
⎦
i
i
concentration after infusion
C p (t ) =
k0
Vc
⎡ Bi ⎛ λi t * ⎞ − λi t ⎤
⋅ ⎜ e − 1⎟ ⋅ e
⎥
⎝
⎠
i
⎦
i =1
n
∑ ⎢⎣ λ
with t*=min(t,T)
continuous iv
infusion
concentration at steady state
extravascular
C ss =
Ro
CL
n ⎡
⎤
ka
⋅ (e −λ ⋅tl − e −k ⋅tl )⎥
Cp ( t ) = ∑ ⎢B i ⋅
k a − λi
i=1 ⎣
⎦
i
tl = t – tlag
...\PK-glossary_PK_working_group_2004.pdf
a
AGAH Working group PK/PD modelling
2.2
Page 11 of 24
Pharmacokinetic Equations - Collection of Equations for Compartmental Analysis
One Compartment Model, IV bolus, single dose, one elimination pathway only
(assumed to be urinary excretion)
D ⎯⎯→ X ⎯⎯e → U
U - drug amount in urine
dX
= −k e ⋅ X (t )
dt
ke= elimination rate constant
X = drug amount in the body
U =drug amount in the urine
i .v .
k
dU
= k e ⋅ X (t )
dt
D = X ( 0) = X (t ) + U (t ) = U ( ∞ )
X (t ) = X (0) ⋅ e − k ⋅te
D = dose administered
X(t) = amount in plasma at time t after
administration
U(t) = amount in urine at time t
C p (t ) =
Vc =
X (t )
; C p (t ) = C p (0) ⋅ e − k e ⋅t
Vc
X (0)
D
=
C p (0) C p (0)
C p (t ) =
D − ke ⋅t
e
Vd
;
t1 / 2 =
ln( 2)
ke
Cp= Conc. in plasma after single dose
ke= negative slope of concentration-time
plot in ln-linear scaling
Cp (0)= intercept with y axis
Cp(t) - plasma conc at any time
Urinary excretion
U (t ) = U (∞ )(1 − e − ket ) ;
ln(U (∞ ) − U (t )) = ln U (∞ ) − k e ⋅ t
„Sigma-Minus Plot“ (page 21)
Calc. of ke from urine data based on lnlinear plot of (U (∞) − U (t )) versus t, ke is
the negative slope, but you need total
amount U(∞) of drug excreted into urine,
which frequently is not identical to the
dose administered, in contrast to the
assumptions of the model
dU
= k e ⋅ X ( 0 ) ⋅ e − k e ⋅t ;
dt
Other method based on urinary excretion
rate (total amount of drug need not be
known) ∆U/∆t -sampling intervals
tmid - mean time point of the sampling
interval
ln
∆U
= ln(k e X (0)) − k e ⋅ t mid
∆t
dU
CLR = dt
C p (t )
; CLR = k e ⋅ Vc
∆U
= CLR ⋅ Cp (t mid )
∆t
U t = CL R ⋅ AUC ( 0 − t )
Cp (0)
(1 − e −ket t )
AUC(0 − t ) =
ke
...\PK-glossary_PK_working_group_2004.pdf
Urinary excretion rate -described by renal
clearance CLR
Cp(tmid) = conc. in plasma at the mean
time point of the urine collection interval,
measured or derived by log-linear
interpolation
CLR = slope of a plot ∆U/∆t versus
Cp(tmid)
AGAH Working group PK/PD modelling
Page 12 of 24
One Compartment Model, IV Inj. and Parallel Elimination Pathways (renal, biliary,
metabolic), single dose
k e = k ren + k bil + k met
kren = rate constant of renal
elimination
kbil = rate constant of biliary
elimination
kmet = rate constant of metabolic
elimination
dX
dU
dB
= −k e X (t ) ;
= k ren X (t ) ;
= k bil X (t ) ;
dt
dt
dt
X = amount in plasma
U = amount in urine
B = amount in bile
M = amount of metabolites in
plasma
dM
= k met X (t )
dt
D = X ( 0 ) = X ( t ) + U ( t ) + B( t ) + M ( t ) = U ( ∞ ) + B ( ∞ ) + M ( ∞ )
Plasma concentration
C p (t ) = C p ( 0) ⋅ e − k e t
Drug amount in urine
k ren
⋅ D ⋅ (1 − e −ket )
ke
U (t ) =
Up to infinite time (t = ∝)
k ren
U (∞ ) k ren
D ;
=
;
ke
D
ke
ln(U (∞) − U (t )) = ln U (∞) − ke ⋅ t
U (∞ ) =
ke - slope can calc.from the
Sigma Minus Plot (U(∞)-U(t) vs t
fb – fraction of bound drug
CLR = k ren⋅ ⋅ Vc
CLR
u
; CLR =
(1 − f b )
B( t ) =
k bil
⋅ D ⋅ (1 − e − ket )
ke
M (t ) =
k met
⋅ D ⋅ (1 − e − ket ) ; CLmet = k met ⋅ ⋅ Vc
ke
dM p
dt
; CLbil
= k bil ⋅ ⋅ Vc
= k met X (t ) − k eM M p (t )
C M (t ) =
CLtot =
M
k met D
(e − k e t − e − k e t )
M
V (k e − k e )
M
c
Biliary excretion can be calc. In
analogous fashion assuming no
reabsorption
Total amounts of metabolites
including further excretion of
M
metabolite into urine ( k e ).
M
C (t) = concentration of the
metabolite in the central
circulation
D
= k e ⋅ Vc ; CLtot = CLR + CLbil + CLmet
AUC
D : U(∞) : B(∞) : M(∞) = ke : kren : kbil : kmet = CLtot : CLR: CLbil: CLmet
...\PK-glossary_PK_working_group_2004.pdf
after the end of all elimination
into the different compartments
AGAH Working group PK/PD modelling
Page 13 of 24
One Compartment, multiple IV injection (i intervals τ)
Cn- concentration after nth administration
every τ hours
⎛ (1 − e − nkτ ) ⎞
D
⎟
Cn (t ) = e − ket ⋅ ⎜⎜
− kτ ⎟
Vc
⎝ (1 − e ) ⎠
C ss (t) = C 0 ⋅
e −k e t
(1 − e
C ss ,max = C0 ⋅ R =
− kτ
)
Fluc. =
Css ,max
Css ,min
= Peak
D
1
⋅
Vc 1 − e − keτ
C ss ,min = C0 ⋅ R ⋅ e − keτ =
% Fluctuation =
During steady-state conditions (n=∞),
C0=concentration immediately after initial
(first) injection = D/Vc
1
R =
1 − e − k eτ
= C0 ⋅ R ⋅ e −k e t
D e − keτ
⋅
= C ss ,max ⋅ e − keτ
Vc 1 − e − ket
C ss ,max − C ss ,min
Css ,max
= Trough
Fluctuation depends on the relation
between ke (or t1/2) and τ, not on the dose
⋅100
= e keτ
⎞
⎛C
ln⎜ ss ,max ⎟
⎜ C ss ,min ⎟
⎠
τ= ⎝
ke
−
C
ss
=
AUC
C ss,max =
τ
=
Useful for calculation of the maintenance
dose
C ss -average ss conc., weighted mean,
value between Cmax and Cmin ;
includes no inform. about fluctuations in
plasma levels + no inform. about
magnitude of Cmax or Cmin
D
CL ⋅ τ
DM
D
1
⋅
= L ;
− k eτ
Vc 1 − e
Vc
...\PK-glossary_PK_working_group_2004.pdf
DL =
DM
1 − e − k eτ
DL = loading dose required to immediately
achieve the same maximum concentration
as at steady state with a maintenance
dose
DM every τ hours
AGAH Working group PK/PD modelling
Page 14 of 24
One Compartment Model, IV Infusion, Zero Order Kinetics
k0
ke
D ⎯⎯→
X ⎯⎯→
E
k0- constant infusion rate
dX
= k 0 − k e ⋅ X (t )
dt
C ss =
during constant rate infusion
k0
⋅ (1 − e − ket )
k e ⋅ Vd
C (t ) =
ss - t = ∞ , infusion equilibrium, like ss
k0
k
= 0
k e ⋅ Vd CLtot
R0 =Css.CL ; Cltot =ke Vc ;
R 0T
D
=
CL tot =
AUC ( 0 − T )
AUC
Plasma concentr. at SS , CL at SS
proportional to Css at SS
C ss =
R0
CL
C (t ) =
R0
(1 − e −ket )
CL
C max =
; C(t ) = C ss (1 − e
− ket
)
for example: time to reach 90% SS ?
(ln 0.1)
C (t )
= 0.90 = (1 − e − k e t ) ; t =
− ke
Css
Cmax -occurs at the end of infusion,
setting t=τ (total time of infusion)
R0
⋅ (1 − e −k eT )
k e ⋅ Vd
After End of Infusion:
Plasma level after end of infusion with
C(t ) = C max ⋅ e − k e ( t −T )
t = time after start of the infusion
Short term Infusion:
k
LD = C ss ⋅ Vc = 0
ke
Loading dose
Incrementa l LD = Vc ⋅ (C desired − C initial )
Plasma level depends on infusion duration
(τ) and t1/2:
C (t )
⋅100 = (1 − e − ke ) ⋅100
Css
C
1 < t1/2 < τ: C (t1 / 2 ) = ss
2
One Compartment Model, Short Term Infusion, Zero Order, multiple dose
C n (t ) = C n −1 (τ ) ⋅ e − ket +
C n (τ ) =
Cn(t) = concentration after nth infusion in
intervals of τ
k0
(1 − e − ket )
k e ⋅ Vd
⎛ 1 − e − nkeτ
−ke (τ −T )
k0
(e
− e − keτ ) ⋅ ⎜
⎜ 1 − e − keτ
k e ⋅ Vd
⎝
...\PK-glossary_PK_working_group_2004.pdf
⎞
⎟
⎟
⎠
n = number of doses
AGAH Working group PK/PD modelling
Page 15 of 24
One Compartment Model, Oral Administration With Resorption First Order, single dose
D → A ⎯⎯a → X ⎯⎯e → E
k
dA
= −k a A
dt
k
dX
dE
= ka A − ke X ;
= ke X
dt
dt
;
f D = A(t) + X(t) + E(t)= E(∞)
C (t ) =
In most cases: k a > k e , this means that
f ⋅ D ⋅ ka
⋅ (e − k e t )
Vd ⋅ ( k a − k e )
ln Cterm (t ) =
e − ka t approches zero much faster
−k t
than e e - calc. of ke from slope of
terminal phase
k a < k e - Flip-Flop, but you need an
additional iv administration to distinguish
this case
f ⋅ D ⋅ ka
− k e t ; C(t)->Cterm(t) for t->∞
Vd ⋅ ( k a − k e )
Cterm (t ) − C (t ) =
f ⋅ D ⋅ ka
⋅ e− k at
Vc ⋅ (ka − ke )
ln(Cterm (t ) − C (t )) =
t max
BATEMAN-Function
f ⋅ D ⋅ ka
⋅ (e − k e t − e − k a t )
Vd ⋅ ( k a − k e )
Cterm (t ) =
C (t ) =
F = fraction of dose available for
absorption
A(t ) = f ⋅ D ⋅ e − kat
;
ka - feathering-method (can reasonably be
used only if there are at least 4 data points
in the increasing part of the concentrationtime curve)
f ⋅ D ⋅ ka
− kat
Vd ⋅ ( k a − k e )
substraction of C from C’ (semilog. ∆(C’-C)
versus t - slope -ka)
with t0 - lag time
f ⋅ D ⋅ ka
⋅ (e − k e ( t − t 0 ) − e − k a ( t − t 0 ) )
Vd ⋅ ( k a − k e )
⎛k ⎞
ln⎜⎜ a ⎟⎟
ke
ln(k a ) − ln(k e )
= ⎝ ⎠=
ka − ke
k a − ke
C max =
A = unabsorbed drug available at
resorption place
E = sum of the excreted amount of drug
ka = absorp. rate constant
tmax does not depend on the bioavailability f
and, since ke commonly is substancedependent and not preparation-dependent,
reflects ka
,
f ⋅ D ⋅ k a − ket max
⋅e
Vd
One Compartment Model, Oral Administration With Resorption First Order, multiple
dose
C n (t ) =
Cn(t) = concentration after the nth consecutive
dosing in intervals τ; BATEMAN-Function
expanded by accumulation factor
f ⋅ D ⋅ ka
⋅ (re ⋅ e − ket − ra ⋅ e − k a t )
Vd ⋅ ( k a − k e )
⎛ e − k et
f ⋅ D ⋅ ka
e − k at
⋅⎜
−
C ss (t ) =
− k eτ
⎜
Vd ⋅ ( k a − k e ) ⎝ 1 − e
1 − e − k aτ
t ss ,max =
⎛ k (1 − e − k e τ ) ⎞
1
⎟
⋅ ln ⎜⎜ a
− k aτ ⎟
ka − ke
)⎠
⎝ k e (1 − e
...\PK-glossary_PK_working_group_2004.pdf
⎞
⎟
⎟
⎠
re =
1 − e − nkeτ
;
ra =
1 − e − nk aτ
; n = ∞ for steady
1 − e − k aτ
1 − e − keτ
state, in most cases ra ≈ 1
tss,max < tmax for ka > ke
AGAH Working group PK/PD modelling
Page 16 of 24
Two Compartment Model, IV Inj (without Resorption), single dose
iv
k10
D ⎯⎯→
X c ⎯ ⎯→
E
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ k12 ↓↑ k 21
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ X
p
Xc = amount in central compartment
Xp = amount in peripheral comp.
dX c
= k12 ⋅ X c + k10 ⋅ X c − k 21 ⋅ X p
dt
dX p
= k 12 ⋅ X c − k 21 ⋅ X T
dt
−
;
dE
= k10 ⋅ X c
dt
D = X (0) = X c (0) = X c (t ) + X p (t ) + E (t ) = E (∞)
E(∞) - Sum of drug eliminated
⎤
⎡ α − k 21
⋅ (eαt * − 1) ⋅ e −αt + ⎥
⎢
(
)
α
α
β
⋅
−
k
⎥
C (t ) = 0 ⎢
VC ⎢ k 21 − β
βt *
− βt ⎥
(
e
1
)
e
⋅
−
⋅
⎥
⎢ β ⋅ (α − β )
⎦
⎣
Concentration in plasma =
Conc. in central compartment
Aiv =
(α − k 21 ) D
⋅
(α − β ) V c
α =
1
k 12 + k 21 + k 10 +
2
β =
1
k 12 + k 21 + k 10 −
2
; Biv =
( k 21 − β ) D
⋅
(α − β ) Vc
Vc = volume of the central comp.,
α>k21>β
(
( k 12 + k 21 + k 10 ) 2 − 4 ⋅ k 21 ⋅ k 10
(
( k 12 + k 21 + k 10 ) 2 − 4 ⋅ k 21 ⋅ k 10
α ⋅ β = k21 ⋅ k10
α + β = k12 + k 21 + k10
;
Cterm (t ) = B ⋅ e − β ⋅t
A ⋅ β + B ⋅α
A+ B
→ ln C term (t ) = ln B − β ⋅ t
;
k10 =
k12 = α + β − k 21 − k10 =
AUC (0 − t ) =
A
α
α ⋅β
k 21
=
A+ B
A B
+
α
B
β
(1 − e − βt )
...\PK-glossary_PK_working_group_2004.pdf
disposition rate constants, equal for
iv and oral administration
β
AUC =
α>β, for elim. phase first term =0
A,B, α, β, can determined by
feathering method
Plot ln(Cterm(t)) vs t with slope β,
intercept ln(B)
Plot ln((C(t)-Cterm(t))) vs t with slope
α, intercept ln(A)
A,B iv ≠ A,B oral
k10=kren+ kmet (+kbil+ kother)
;
k ren U ∞
=
k10
E∞
AB( β − α ) 2
( A + B)( A ⋅ β + B ⋅ α )
(1 − e −αt ) +
)
α, β = Macro constants (or Hybrid
constants, independent of dose,
A+B proportional to dose
k 12 , k 21 , k10 - Micro constants
C (t ) − Cterm (t ) = A ⋅ e −αt → ln(C (t ) − C term (t )) = ln A − α ⋅ t
k 21 =
)
A
α
+
B
β
AUC - by integration of the general
equation for C
AGAH Working group PK/PD modelling
Page 17 of 24
Two Compartment Model, IV Inj, single dose
iv
k10
D ⎯⎯→
X c ⎯ ⎯→
E
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ k12 ↓↑ k 21
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ X
p
(
X p (t ) =
D ⋅ k12 −αt
e − e − βt
β −α
t max, p =
ln α − ln β
α −β
)
Xp = drug amount in the tissues
(peripheral compartment)
dX p
= 0 at tmax,p
dt
Most membranes central
compartment / tissue are crossed by
diffusion – by unbound drug only
fb = fraction bound (to protein)
Vc – volume of distribution in the
central compartment
Ccf (t max, p ) = C (t max, p ) ⋅ (1 − f b ) =
C p (t max, p ) ⋅ (1 − f b, p ) = C pf (t max, p )
Vc =
D
D
=
C (0) A + B
Xc + X p
X
X
=
= (assumed ) c
Vc
Vc + V p
Vc
Vd , ss = Vss =
Vss =
c
⎛ k ⎞
= ⎜⎜1 + 21 ⎟⎟ ⋅ Vc
⎝ k12 ⎠
Vss can also be calculated from macro
constants
A ⋅ β 2 + B ⋅α 2
⋅D
( A ⋅ β + B ⋅α ) 2
C max, p =
(
k 21 ⋅ D
−α ⋅t
− β ⋅t
⋅ e max, p − e max, p
Vc ⋅ ( β − α )
dE
dt
C (t )
=
AUC =
A
α
CL
β
=
+
In the strictest sense only true at
equilibrium
Xp
k 21
Vc ; C p =
Vp
k12
)
k10 ⋅ X c (t )
= k10 ⋅ Vc
C (t )
CL = ke ⋅ Vss ; ke =
Vz =
k 21
k12
Xc
Vc
C ss
V p = Vss − Vc =
CL =
(1 + )⋅ X
=
Xc + X p
Other “volume” terms are
proportionality factors assuming that
Cc = CT, they may take on
unphysiological values. Initially Xc and
Cc high with XT and Cp nearly 0. In the
end frequently CT > Cp. Vd = volume
of distribution of the total organism –
not constant in time!
Vss = volume of distribution at
equilibrium, when flows Xc ↔ XT
balance: k12·Xc = k21·XT
B
β
=
k10 ⋅ Vc
k ⋅k
= 10 21
Vss
k21 + k12
k 21
⋅
D
α ⋅ β Vc
;
This is the definition of ke for a twocompartment model
k ⋅k
D
= 21 10 Vc = k10 ⋅ Vc = CL
AUC
k 21
D
β ⋅ AUC
CL = k10 ⋅ Vc = ke ⋅ Vss = β ⋅ Vz =
D
AUC
...\PK-glossary_PK_working_group_2004.pdf
Vz – volume of distribution during
terminal phase, calculated based on
the rate constant
Vz > Vss > Vc – during terminal phase
XT > Xc
AGAH Working group PK/PD modelling
Page 18 of 24
Two compartment Model single dose infusion (or zero order resorption)
k0
k10
A ⎯⎯→
X c ⎯ ⎯→
E
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ k12 ↓↑ k 21
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ X
k0 =
p
D
T
Infusion of dose D during τ at
constant rate k0
⎤
⎡ α − k 21
⋅ (eαt * − 1) ⋅ e −αt + ⎥
k 0 ⎢⎢ α ⋅ (α − β )
⎥
C (t ) =
VC ⎢ k 21 − β
βt *
− βt ⎥
⋅ (e − 1) ⋅ e
⎥
⎢
⎦
⎣ β ⋅ (α − β )
General equation for calc. of C(t)
during and after infusion, t* =
min(τ,t)
⎤
k 0 ⎡ α − k 21
k −β
⋅ (1 − e −αt ) + 21
⋅ (1 − e − βt )⎥
⎢
β ⋅ (α − β )
VC ⎣ α ⋅ (α − β )
⎦
k0
k
k0 = k10·Ass = k10·Css·Vc; C ss =
= 0
k10 ⋅ Vc CL
during infusion, t*=t
(eλt*-1)e-λt becomes 1- e-λt
⎡ (α − k 21 ) ⋅ (1 − e −αT ) −α (t −T ) ⎤
⋅e
+⎥
⎢
α ⋅ (α − β )
k0 ⎢
⎥
C (t ) =
VC ⎢ (k 21 − β ) ⋅ (1 − e − βT ) − β (t −T ) ⎥
⎢
⎥
⋅e
β ⋅ (α − β )
⎢⎣
⎥⎦
after end of infusion,
t-τ = time after end
C (t ) =
...\PK-glossary_PK_working_group_2004.pdf
For a continuing infusion, τ→∞
AGAH Working group PK/PD modelling
Page 19 of 24
Two compartment Model, single dose with Resorption First Order
ka
k10
A ⎯⎯→
X c ⎯ ⎯→
E
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ k12 ↓↑ k 21
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ X
p
(k21 − α )
(k21 − β )
⎡
α⋅t
−β ⋅t ⎤
⎢ (β − α ) ⋅ (k − α ) ⋅ e + (β − α ) ⋅ (k − β ) ⋅ e ⎥
ka ⋅ F ⋅ D ⎢
a
a
⎥
⋅
C(t ) =
⎢
⎥
(k21 − ka )
Vc
−kat
⋅e
⎢+
⎥
⎣ (α − ka ) ⋅ (β − ka )
⎦
k 21 − α
k 21 − β
+
( β − α ) ⋅ ( ka − α ) (α − β ) ⋅ (k a − β )
=−
k21 − ka
(α − ka ) ⋅ ( β − k a )
C-central compartment
with micro constants
C(t) = A⋅ e−α⋅t + B ⋅ e−β ⋅t − ( A + B) ⋅ e−kat
C-central compartment
with macro constants
AIV =
D (k 21 − α )
⋅
Vc ( β − α )
Aoral =
ka ⋅ f
⋅ Aiv
(k a − α )
BIV =
;
; Boral =
D (k 21 − β )
⋅
Vc (α − β )
ka ⋅ f
⋅ Biv
(k a − β )
Vc
D
=
f
f ⋅ Aiv + f ⋅ Biv
C p (t ) =
B ⋅ k 21 − βt ( A + B) ⋅ k 21 − k a t
A ⋅ k 21
⋅e
e −
⋅ e −αt +
(k 21 − α )
(k 21 − β )
(k 21 − k a )
Without iv data only Vc/f can be
determined, but based on
knowledge of f⋅Aiv and f⋅Biv the
micro constants k10, k21, k12 may
be derived
CT-deep compartment
Two compartment Model, multiple dose with Resorption First Order
Cn (t x ) = A ⋅
− ( A + B)
(1 − e − nατ ) −αt x
1 − e − nβτ − βt x
⋅e
+B
e
−ατ
1− e
1 − e − βτ
1 − e −nkaτ
1 − e −kaτ
e −kat x
...\PK-glossary_PK_working_group_2004.pdf
Cn – concentration at
time tx after the nth
administration at interval
τ, time after first dosing =
n⋅τ
AGAH Working group PK/PD modelling
3
Page 20 of 24
PHARMACODYNAMIC GLOSSARY
3.1
Definitions
Symbol
AUEC
Unit / Dimension
Arbitrary units·time
Definition
Area under the effect curve
Ce
Amount/volume
Fictive ‘concentration’ in the effect compartment
Cp
Amount/volume
Drug concentration in the central compartment
E
(effect unit)
Effect
E0
(effect unit)
Baseline effect
Emax
(effect unit)
Maximum effect
EC50
Amount/volume
Imax
(effect unit)
I50
keo
Drug concentration producing 50% of maximum effect
Maximum inhibition
Drug concentration producing 50% of maximal inhibition
Amount/volume
(Time)
Rate constant for degradation of the effect compartment
-1
-1
Zero order constant for input or production of response
kin
(effect unit) (time)
kout
(time)
M50
Amount/volume
50% of maximum effect of the regulator
MEC
Amount/volume
Minimum effective concentration
-
Sigmoidicity factor (Hill exponent)
n
S
-1
(effect unit)/
First order rate constant for loss of response
Slope of the line relating the effect to the concentration
(amount/volume)
tMEC
Ve
Time
Volume
...\PK-glossary_PK_working_group_2004.pdf
Duration of the minimum (or optimum) effective
concentration
Fictive volume of the effect compartment
AGAH Working group PK/PD modelling
3.2
Equations: PK/PD Models
E=Efixed if C ≥ Cthreshold
E=
E=
Page 21 of 24
fixed effect model
E max ⋅ C
E50 + C
Emax model
Emax ⋅ C n
sigmoid Emax model
n
E50
+Cn
dR
= k in − k out ⋅ R
dt
Rate of change of the response over time with no
drug present
⎡
⎤
dR
C
= k in ⋅ ⎢1 −
⎥ − k out ⋅ R
dt
IC
+
C
50
⎣
⎦
Inhibition of build-up of response
⎡
I
⋅Cn ⎤
dR
= k in ⋅ ⎢1 − max
− k out ⋅ R
n⎥
dt
⎣⎢ IC50 + C ⎦⎥
⎡
⋅C ⎤
I
dR
= k in − k out ⋅ ⎢1 − max
⎥⋅R
+
dt
IC
50 C ⎦
⎣
Inhibition of loss of response
⎡ E ⋅C ⎤
dR
= k in ⋅ ⎢1 + max ⎥ − k out ⋅ R
dt
⎣ E50 + C ⎦
Stimulation of build-up of response
⎡ E ⋅C ⎤
dR
= k in − k out ⋅ ⎢1 + max ⎥ ⋅ R
dt
⎣ E50 + C ⎦
Stimulation of loss of response
...\PK-glossary_PK_working_group_2004.pdf
AGAH Working group PK/PD modelling
4
Page 22 of 24
STATISTICAL PARAMETERS
4.1
Definitions
Symbol
AIC
Definition
Akaike Information Criterion
(smaller positive values indicate a better fit)
CI
Confidence interval, e.g. 90%-CI
CV
Coefficient of variation in %
Calculation
AIC = n·ln(WSSR) +2p
n = number of observed (measured) concentrations,
p = number of parameters in the model
CI = x ± tn −1,α ⋅ SEM
CV = 100 ⋅
Median = ~
x
Median, value such that 50% of observed
values are below and 50% above
Mean = x
Arithmetic mean
SD
x
, SD = standard deviation
(n+1)st value if there are 2n+1 values or arithmetic
th
st
mean of n and (n+1) value if there are 2n
values
x=
1
n
n
∑x
i
i =1
MSC
Model selection criterion
AIC, SC, F-ratio test, Imbimbo criterion etc.
SC
Schwarz criterion
SC = n·ln(WSSR) +p·ln(n)
SD
Standard deviation
SD = Var
SEM
Standard error of mean
SSR
SS
Sum of the squared deviations between the
calculated values of the model and the
measured values
Sum of the squared deviations between the
measured values and the mean value C
SEM =
SD
n
n
SSR = ∑
i=1
n
SS = ∑
i=1
(
)
2
C i , obs - C i , calc
(
)
2
C i , obs - C
⎛ n
⎞
⎜ ∑ Ci ,obs ⎟
⎟
⎛ n
⎞ ⎜
⎠
SS = ⎜ ∑ Ci ,obs 2 ⎟ - ⎝ i =1
⎜
⎟
n
⎝ i =1
⎠
2
n = number of observed (measured) concentrations
use of the second formula is discouraged although
mathematically identical
WSS or WSSR Weighted sum of the squared deviations
between the calculated values of the model
and the measured values
n
WSSR = ∑ w i
i=1
(
C i , obs - C i , calc
)
2
Var
Variance
s² = SS/(n-1)
X25%
Lower quartile (25%- quantile), value such
that 25% of observed values are below and
75% above
may be calculated as median of values between
minimum and the overall median
X75%
Upper quartile (75%- quantile)
may be calculated as median of values between
the overall median and the maximum
...\PK-glossary_PK_working_group_2004.pdf
AGAH Working group PK/PD modelling
4.2
Page 23 of 24
Characterisation of log-normally distributed data
Symbol
Definition
Xg
Geometric mean of log-normally distributed
data
sdl
Standard deviation to the log-transformed
data
Calculation
⎡1 n
⎤
X g = exp ⎢ * ∑ ln(x i )⎥
n
i =1
⎣
⎦
sd l =
2
⎧ n
n
⎤ ⎫⎪
1 ⎪
1⎡
⋅ ⎨ ln( xi ) 2 − ⎢ ln( xi )⎥ ⎬
n − 1 ⎪ i =1
n ⎣⎢ i =1
⎦⎥ ⎪⎭
⎩
∑
∑
Scatter
Scatter-Factor
Scatter = e sd
CIg
Confidence interval of log-normally
distributed data
⎡1 n
⎤
CI g = exp ⎢ ⋅ ln( xi ) ± tn −1,0.05 ⋅ SEM ln ⎥
⎣⎢ n i =1
⎦⎥
CVg
Geometric coefficient of variation in %
Per16%
16% percentile of log-normally distributed
data
Per84%
84% percentile of log-normally distributed
data
...\PK-glossary_PK_working_group_2004.pdf
l
∑
CVg = 100 ⋅ eVarln − 1 [%]
Per16% =
Xg
Scatter
Per 84% = X g ⋅ Scatter