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CHAPT : IV A. ILIADIS1
CHAPT IV : Modeling PKsCHAPT IV : Modeling PKs
Modeling the mass-balance Unit processes, order of the process. Linear and nonlinear systems of differential equations. Superimposition principle.
Compartmental analysis Biological space vs. compartment. Compartmental modeling. Elementary mathematical description.Volumes of distribution and transfer rates. Open compartment mammillary models. Fundamental properties. Clearance definitions. Steady state and equilibrium state.
Modeling administration protocols Routes of administration: intra- and extra-vascular. Extravascular administrations: bioavailabilty and absorption rate. Time schedule and drug amounts. Periodic and irregular repeated administrations.
General closed form solution Sum-of-exponentials form. Compartments vs. exponential phases, micro- vs. Macro-rates. Computing macro-exponents and coefficients. The functional methodology scheme.
Simulations Phase filtering for intra- and extra-vascular routes (flip-flop). Fluctuations in repeated administrations. Sensitivity of kinetic profiles to the model parameters. Initial conditions. Need for a functional modeling.
CHAPT : IV A. ILIADIS2
Functional scheme - Chapt IVFunctional scheme - Chapt IV
Measurementnoise
Measurementnoise
PK modelPK model
Equivalence criterion
Equivalence criterion
NonlinearprogrammingNonlinear
programming
Administrationprotocol
Administrationprotocol
Priorinformation
Priorinformation
Observation
Prediction
PK processPK process
CHAPT : IV A. ILIADIS3
The contextThe context
Pharmacology
PK
Real processReal process
Process - 2Process - 2Process - 1Process - 1
Drugadministration
Pharmacologicaleffects
Time
PDConcentrationConcentration
CHAPT : IV A. ILIADIS4
States and unit processesStates and unit processes
can always be written as a sum of terms each of which represents a single or unit process which occurs in the system.
Systems are monitored by means ofquantifiable measures
( temperature, concentration, etc )
Systems are monitored by means ofquantifiable measures
( temperature, concentration, etc )
Basic laws govern changes in the real system
Basic laws govern changes in the real system
Mathematical descriptions expressed by differential equations :How the rate of change of a state variable depends on the current value of the others
nifi :1=
Mass-balance equations
( )dydt
f y y yii n= 1 2, ,...,
Use models todescribe quantifiable measures
by means of the state variables
Use models todescribe quantifiable measures
by means of the state variables iy
CHAPT : IV A. ILIADIS5
The order of the processThe order of the process
Ex : Bimolecular reaction Material Conc. State Var.
Forward reaction rate
Backward reaction rate
The differential equations :
The order of a unit process
A B
k
k
AB+→←
1
2
ABAB
y1y2y3
u k y yf = ⋅ ⋅1 1 2
u k yb = ⋅2 3
dydt
dydt
dydt
u u k y k y yb f1 2 3
2 3 1 1 2= = − = − = ⋅ − ⋅ ⋅
It is the sum of the exponents of each of the state variables in the term describing the process
CHAPT : IV A. ILIADIS6
Linear and nonlinear modelsLinear and nonlinear models
Definition : If the rates of change of all of the state variables of a model can be written as sums of processes of order 0 or 1, the model is linear otherwise nonlinear. These models are described by a set of linear or nonlinear differential equations, respectively.
Ex : Linear models Radioactive decay : The number of atoms decaying per unit of time
is proportional to the number of atoms present at that time
Diffusion Fick’s law : The flux or “the rate of transfer of material per unit area” crossing a membrane with section is proportional to the concentration gradient
with
the diffusion coefficient,
the permeability constant,
dNdt
N⎧⎨⎩
= − ⋅λ
S
DP
[ ] [ ] [ ]surface timeD =[ ] [ ] [ ]length timeP =
!ais CLSP ⋅
1 dq yDS dt x
∂∂
⋅ = − ⋅ ( )122
21
1 yySPdtdyV
dtdyV −⋅⋅−=⋅=⋅−
⎭⎬⎫
CHAPT : IV A. ILIADIS7
Space and compartmentsSpace and compartments
Biological space : may be well-stirred ( homogeneous, ex. blood stream ) or under-stirred ( heterogeneous, ex. intracellular space ).
Compartment : is a biological space that acts kinetically like a distinct, homogeneous, well-stirred amount of material : spaces × chemical forms.
Compartmental model : is a model which is made up of a finite number of compartments, and the compartments interact by exchanging material.
is a model for the administration rate or input function.( )u tk3
k12
k23
( )u t
k13
k2
k321
2
3
( )tu 1k 2k1 2
CHAPT : IV A. ILIADIS8
Linear modeling and ...Linear modeling and ...
Draw and connect compartmentsVolumes of distribution characterizes the size of the cpt ( units : volume ).
Transfer rate constants connect cpts among them ( units : time-1 ).
Elementary model for mathematical descriptionModeling each connection pathway by first-order unit processes :
Transfer rate constants are involved in a first-order ( linear ) unit process.
For the sampled compartments introduce the volumes of distribution .
Vi
kij
[ ]
ij
ij
dq k qdtdq q
kdt
⎧ = − ⋅⎪⎪⎨⎪ = −⎪⎩
The elimination rate from a given compartment is proportional to the amount of material in this compartment
The relative decrease of material from a given compartment per unit time is constant
kij
Vi
CHAPT : IV A. ILIADIS9
... compartmental structure... compartmental structure
For a given compartmental structure :Express the rate of variation of material in each compartment, by assembling unit processes and establishing a system of linear ordinary differential equations ( ODE ).
Ex :
and are the parameters ( called micro-rates, µrates ).
ODE are linear : is linear with respect to the administered amount, .
Due to this linearity, the superimposition principle holds :
Vi kij
D( )y t
( ) ( )
( )
dqdt
k q u t q
dqdt
k q k q q
11 1 1
21 1 2 2 2
0 0
0 0
= − ⋅ + =
= ⋅ − ⋅ =
The resulting kinetic after various administration protocols could be evaluated bylinearly combining the elementary kinetics obtained after each protocol
( )tu 1k 2k1 2
CHAPT : IV A. ILIADIS10
Superimposition principleSuperimposition principle
0 12 24 36 48 6010
0
101
102
t [h]
y[µg
/mL]
0 12 24 36 48 6010
0
101
102
t [h]
y[µg
/mL]
0 12 24 36 48 6010
0
101
102
t [h]
y[µg
/mL]
0 12 24 36 48 6010
0
101
102
t [h]
y[µg
/mL]
+ ⋅12 2h, D
+4 2h, D
+ +
CHAPT : IV A. ILIADIS11
Fick’s diffusion law central vs peripheral inter-cpt clearance.
central vs environment total clearance.
First order process
Compartmental modelingCompartmental modeling
dd SPQ ⋅=
ee SPCL ⋅=
( ) ( )
( )212
2
1211
1
yyQdtdyV
tuyyQyCLdtdyV
−⋅=
+−⋅+⋅−=
ekVCLVkVkQ
⋅=⋅=⋅=
1
221112
( ) ( )
22111122
12
1
21112
1
qkyVkdt
dqV
tuqVkykk
dtdy
e
⋅−⋅⋅=
+⋅+⋅+−=
AnatomicDiffusion
MetabolicReaction
11yV22 yV( )tuQ
CL
Correspondence
11yVk12
2q
ek
21k( )tu
CHAPT : IV A. ILIADIS12
Compartment mammillary modelsCompartment mammillary models
The commonly used structureCentral cpt :
The material is administered in the central cpt.
# distributed and eliminated from the central cpt.
Observations are made in the central cpt.
Peripheral cpt :
No exchanges among peripheral cpts.
The nbr of parameters in DE is twice :
, and pairs.
Mammillary cpt models are structurally identifiable.
k ki i1 1−V1
L
ek( )u t 1 1 V
k12
2
ek
21kL
Lk1
1Lk
...
CHAPT : IV A. ILIADIS13
Additional parametersAdditional parameters
Clearance definitionsInternal The capacity of a live organism to eliminate per time-unit the
volume in which the drug is distributed :
External The proportionality constant between and its image at output, the area under the time-concentration curve :
Volumes of distributionIf peripheral cpts are not sampled, corresponding cannot be evaluated.
To assess a fictitious volume, assume flux equality ( or cpt incompressibility )
Then and the total
CL V ke= ⋅1
( )CL D AUC=
( ) LiVkkV iii :2 111 =⋅= ( )1 1 12
1L
T i ii
V V k k=
⎡ ⎤= ⋅ +⎢ ⎥⎣ ⎦∑
LiVi :2=
LiVkVkQ iiii :2111 =⋅=⋅=
D
CHAPT : IV A. ILIADIS14
Equilibrium stateEquilibrium state
Definition
Pseudo-equilibrium, for some of cpts.
System equilibrium, for all # cpts.
Pseudo-equilibrium for any peripheral compartment
System equilibrium ( only for a long-time infusion )
ii yy
dtdy
=→= 10
ooeo yCLykVudtdy
1111 0 ⋅=⋅⋅=→=
( ) 0=dt
tdyi
Li :1=
The intersection of time-concentration profilesbehaves at the maximum concentration of iy
Reach at the equilibrium statedepends simply on the
1oyCL
CHAPT : IV A. ILIADIS15
Steady stateSteady state
Stationary state obtained after repeated administrationsAdministration protocol
States
is the period.
Obtain a target average level by periodic administrations of every
Input-output balance over a period :
Define : over a period for repeated doses and
to infinity for a single dose.
If then
aveC
aveR
o CCLDu ⋅=≡τ
RD τ
CLDCAUC R
aveR =⋅≡ τ
CLDAUC s
S ≡
SR DD =
( ) ( )( ) ( ) Lityty
tutu
ii :1=+=+=
ττ
τ
R SAUC AUC=
CHAPT : IV A. ILIADIS16
Administration characteristicsAdministration characteristics
Administration protocolsIntra- ( bolus IV or infusion ) and extravascular ( oral or intramuscular ) routes have been considered in single or repeated dose protocols.
Set factors defining an :
Modeling extravascular routesThe drug goes through an "absorption compartment" before reaching the central one.
Use two new parameters : bioavailability ( units : % ) ;
: absorption constant ( units : time-1 ).
can be evaluated when drug was given by both intra- and extravascular routes. Otherwise, assume complete bioavailability : apparent volumes of distribution and clearances are overestimated.
ka
aF
aF
Time schedule
Drug amounts
Administration protocol( )u t
CHAPT : IV A. ILIADIS17
Definition of terms : period : nbr. of components
: n° of present period : elapsed time from origin
: elapsed time from the beginning of the
-th period
: nbr. of components under the -th period
Complex administration protocolsComplex administration protocols
0 24 48 72 960
40
80
120
t [h]
Infu
sion
rate
[mg/
h]
...
...
′tr′
th( )1−b τr
st1 τr
t
( ) τ⋅−−=′ 1btt
rb t
′t
b
r′
τ
b
CHAPT : IV A. ILIADIS18
Practical issuesPractical issues
Use : - set for an irregular administration protocol ;
- the steady state is obtained for .
Ex : - single dose : and ;
- single dose repeated daily : and .
Form of the mathematical model : Instead of solving DE for each particular application, develop a closed form solution from which the particular cases may be obtained by using a system of control indexes.
For practical reasons, we will assume in next a null initial state.
Initial time : First experimental intervention on the real process :
Administration ( SD ), or
Sampling ( RD )
∞→τ∞→b1'== rr
h 24=τ 1'== rr∞→τ
CHAPT : IV A. ILIADIS19
General closed form solutionGeneral closed form solution
Generic model
Indexes : associated to the n° of cpt in which is predicted.
associated to the n° of exponential term.
The number of exponential terms is equal to the number of compartments.
Coefficients , and exponents are called macro-rates, Mrates. Mrates are the parameters to be estimated, components of with .
Algebraic relations make it possible to express Mrates as function of µrates and inversely.
Parametric identifiabilityHow many compartments are they seen through the observed data?
yMi
( )x p × 1 p L= ⋅2
( ) ( ) ( ) ( )1 exp,,,,,1
×′⋅−⋅⋅′⋅=′ ∑=
rDtaArbrahDxtyL
jjijj
TMi τ
ijA ja
ij
CHAPT : IV A. ILIADIS20
0 1 2 310
0
101
t [h]
y 1(t) [µ
g/m
L]
a2
a1
Differential and closed formsDifferential and closed forms
non
11yVk12
2y
ek
21k ( )q D1 0 =
Closed form, Mrates
( ) ( )( ) ( )
y t D A e A ey t D A e A e
a t a t
a t a t1 11 12
2 21 22
1 2
1 2
= ⋅ ⋅ + ⋅= ⋅ ⋅ + ⋅
− ⋅ − ⋅
− ⋅ − ⋅
Differential form, µrates
( ) ( )
( ) ( ) 00
0
221212
1112121
1
=−⋅=
=−+⋅−=
yyykdtdy
VDyyykyk
dtdy
e
CHAPT : IV A. ILIADIS21
Numbering compartments and phasesNumbering compartments and phases
0 2 4 6 8 10 1210
1
102
103
104
t [h]
y 1(t) o
r y2(t)
or y
3(t) [n
g/m
L]
1
23
-1 -11 12 13
-1 -1 -121 31
7.4 L 1.06 h 1.26 h5.2 h 1.51 h 0.084 he
V k kk k k
= = == = =
The slower profile, the deeper compartment
Order in decreasing order : ja 1 j La a a> >
Docetaxel PKs
CHAPT : IV A. ILIADIS22
Summary, notesSummary, notes
PK linearity was used to elaborate the previous formulas.
The estimated parameters are Mrates in the identification task.
Complex formula to predict used :inside the criterion function ;
for simulation and dosage regimen applications.
Main advantage : very general conditionsmulti-compartment configuration ;
periodic or irregular protocols ;
all possible routes of administration ;
closed form solution (no DE, fast computing).
( )y t xMi ,( )xJ
CHAPT : IV A. ILIADIS23
Functional methodologyFunctional methodology
( ) ( ) ( )( )
2
1
,,
mi M i
i M i
y y t xJ x SE x
y t xα=
⎡ ⎤−∝ = ⎢ ⎥
⎣ ⎦∑
Observation
Prediction
PK processPK process
( )y t xMi ,
miyt ii :1, =
( ) ( )rbrahDtu T ′⋅ ,,,,: τ
α,K
( )f x
( ) ( )
( ) ( )
1
1
k k
k k
x x
J x J x
+
+
←
⎡ ⎤ ⎡ ⎤<⎣ ⎦ ⎣ ⎦
CHAPT : IV A. ILIADIS24
The reference profile – model The reference profile – model
Intravascular route : 100 mg by bolus.
Proposed µ-rates :
Converted M-rates :
1-2
1-1
1-212
1-211
h 304.0h 723.3
L 10772.3L 10144.9
==
⋅=⋅=
−
−
aa
AA
1-21
1-12
1-1
h 302.1h 856.1h 868.0L 742.7
====
kkkV
e
0 1 2 3 4 5 610
-1
100
101
102
t [h]
y 1(t) [µ
g/m
L]
compartments2 : exp terms
phases
⎧⎪⎨⎪⎩
CHAPT : IV A. ILIADIS25
Conversion of µ-rates to M-ratesConversion of µ-rates to M-rates
Express as functions of .
Express as functions of .
Application :
2121 aaAA 21121 kkkV e
2121 aaAA21121 kkkV e
21
221
122121
12
121
11211221
1
1
aaak
VAkkaa
aaak
VAkkkaa
e
e
−−⋅=⋅=⋅
−−⋅=++=+
( )12
121
11212112
21
21
12
211221
1aaak
AVkkaak
kaak
AAaAaAk ee −
−⋅=+−+=
⋅=
+⋅+⋅
=
1-2
1-1
1-22
1-21 h 304.0h 723.3L 10772.3L 10144.9 ==⋅=⋅= −− aaAA
1-21
1-12
1-1 h 302.1h 856.1h 868.0L 742.7 ==== kkkV e
CHAPT : IV A. ILIADIS26
The "flip-flop"The "flip-flop"
ka
kaExtravascular case : The influence of absorption rate constant, .
Reference µ-rates.
Complete bioavailability.
Vary :
ka = 5 h-1
ka = 2 h-1
ka = 0 3. h-1
a1
a2
0 1 2 3 4 5 610
-1
100
101
t [h]
y 1(t) [µ
g/m
L]
ka = 0.3 h-1
ka = 5 h-1
ka = 2 h-1
CHAPT : IV A. ILIADIS27
Filtering for IV administrationFiltering for IV administration
Intravascular case : The influence of infusion time .
Reference µrates.
Vary :
T
T
2
2ln5a
⋅
h 5.0=T
h 2=T
h 12=T
The infusion rate controls the number of decreasing phases 0 2 4 6 8 10 12 14 16 18
10-1
100
101
t [h]
y 1(t) [µ
g/m
L]
CHAPT : IV A. ILIADIS28
Intravascular case : The influence of the period .
Adm. unit : 100 mg / 0.05 h.
with increasing :
and decrease,
increases,
.
Repeated adms, same unit doseRepeated adms, same unit dose
miny maxyτ
h22/1 ≈t
τ
τ miny maxy1 10.94 22.82 2 4.54 16.56 4 1.60 13.66 8 0.36 12.44
max min/y ymax miny y cst− ≈ 0 6 12 18 24
10-1
100
101
102
t [h]
y 1(t) [µ
g/m
L]
CHAPT : IV A. ILIADIS29
Repeated adms, same daily doseRepeated adms, same daily dose
Intravascular case : The influence of the number of daily administrations.
Constant daily dose :
100 mg / 0.05 h.
with increasing :
decreases,increases,
decreases,fluctuations are reduced.
r
r
r miny maxy 2 0.051 6.093 6 0.267 2.277 12 0.379 1.369
minymaxymax miny y−
0 6 12 18 2410
-1
100
101
t [h]
y 1(t)
[µg/
mL
]
[ ]Steady state / Equilibrium stater → ∞ ≡
CHAPT : IV A. ILIADIS30
Sensitivity to the initial conditionsSensitivity to the initial conditions
Configuration with initial conditions : ( the same model as above )
Central cpt :
Peripheral cpt :
for the quick decline disappears
( )y1 0 10= ⋅ g mL-1µ( )y2 0 0 10 100= ⋅ g mL-1µ( ) ( )1 20 0y y≈
0 1 2 3 4 5 610
-1
100
101
102
t [h]
y 1(t) [µ
g/m
L]
0 1 2 3 4 5 610
-1
100
101
102
t [h]
y 2(t) [µ
g/m
L]
?
Central
!
Peripheral
CHAPT : IV A. ILIADIS31
Profile = model + adm.prot.Profile = model + adm.prot.
What is the administration protocol revealing all the model features from the observed kinetic profile? The pulse input IV bolus !
Warning : Interpretation of observed kinetic profiles
CHAPT : IV A. ILIADIS32
Sensitivity to the µ-rate k12Sensitivity to the µ-rate k12
Intravascular case : 75 mg / 0.05 hthe same model as above :
variable
for the kinetic becomes mono-phasic
[ ] -112 0.1 6 hk ∈ −
12 21k k<<
k21 1302= . h-1
0 1 2 3 4 5 60
2
4
6
8
k12 [h-1 ]
a 1, a2 [h
-1 ]
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
A 1, A2
0 1 2 3 4 5 610
-1
100
101
t [h]
y 1(t) [µ
g/m
L]↓ 12k
1a
2a
1A
2A
CHAPT : IV A. ILIADIS33
Sensitivity to the µ-rate k21Sensitivity to the µ-rate k21
Intravascular case : 75 mg / 0.05 hthe same model as above :
variable
decreasing decelerates the slopes of phases
[ ]k21 0 1 6∈ −. h-1
21k
k12 1856= . h-1
0 1 2 3 4 5 60
2
4
6
8
k12 [h-1 ]
a 1, a2 [h
-1 ]
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
A 1, A2
0 1 2 3 4 5 610
-1
100
101
t [h]
y 1(t) [µ
g/m
L]↓ 21k
1a
2a
1A
2A