chapt iv : modeling pkschapt iv : modeling pks · chapt iv : modeling pkschapt iv : modeling pks...

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.


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


The contextThe context

Pharmacology

PK

Real processReal process

Process - 2Process - 2Process - 1Process - 1

Drugadministration

Pharmacologicaleffects

Time

PDConcentrationConcentration


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


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


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 −⋅⋅−=⋅=⋅−

⎭⎬⎫


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


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


... 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


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

+ +


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


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

...


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


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


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=


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


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


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


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


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


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


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


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

+

+

←

⎡ ⎤ ⎡ ⎤<⎣ ⎦ ⎣ ⎦


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

⎧⎪⎨⎪⎩


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


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


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]


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]


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 → ∞ ≡


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


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


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


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

chapt iv : modeling pkschapt iv : modeling pks · chapt iv : modeling pkschapt iv : modeling pks...

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