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ompartment Models

Lecture#

Quantitative Analysis and Data Processing in Nuclear Medicine

Islamabad, Pakistan Da NangVietnam

Dome of the Rock,Jerusalem, Israel

Lecture#

Compartment Models

Construction

of

Mathematical

Models

●

Structural modeling is the process by which ones

●

Structural

modeling

is

the

process

by

which

ones

knowledge

and

assumptions

about

the

system

are

formalized

first

as

a

schematic

and

then

mathematically.

Compartment Models

Construction

of

Mathematical

Models

●

Parameter estimation is the process by which the

●

Parameter

estimation

is

the

process

by

which

the

parameters

characterizing

the

model

are

adjusted

so

as

to

obtain

a

best

fit

of

the

available

data.

●

For

any

hypothesized

structural

model,

parameter

estimation

provides

information

to

assess

the

adequacy

of

the model.the

model. Criteria

based

upon

goodness

‐ of

‐ fit,

precision

of

the

parameter

estimates,

parsimony,

and

plausibility

permit

an

investigator

to

judge the quality of the modeljudge

the

quality

of

the

model

.

Compartment Models

Construction

of

Mathematical

Models

●

Such model can be used for predictive purposes e g

●

Such

model

can

be

used

for

predictive

purposes

e

.g.

estimating

the

system

parameters

and

simulating

future

experiments.

●

The

model

will

either

correctly

predict

the

results

of

these

experiments

or

not.

If it does not then the model structure will have to be changedIf

it

does

not

,^

then

the

model

structure

will

have

to

be

changed

,

and

the

process

of

compatibility

with

previous

data

and

physiological

plausibility

reexamined.

7

Compartment Models

Mathematical

Modeling

●

There are many types of mathematical models that can be

●

There

are

many

types

of

mathematical

models

that

can

be

used

to

interpret

tracer

kinetic

data.

All

have

assumptions

associated

with

them.

These

assumptions

need

to

be

understood

in

order

to

apply

them

correctly.

In

addition,

Selection

of

an

appropriate

model

for

a

p

articular

situation

can

pp

p

p

depend

upon

the

information

that

is

needed.

8

Compartment Models

Need

for

Modeling

●

A model is a hypothesis using mathematical terms to

●

A

model

is

a

hypothesis

using

mathematical

terms

to

describe

quantitative

relationships

concisely.

●

Such

mathematical

models

can

be

devised

to

simulate

the

rate

processes

of

drug

absorption,

distribution,

and

elimination

to

describe

and

predict

drug

concentrations

in

the body as a function of time

the

body

as

a

function

of

time

Compartment Models

Need

for

Modeling

●

The

p

redictive

capability

of

a

model

lies

in

the

p

roper

p

p

y

p

p

selection

and

development

of

mathematical

function(s)

that

parameterize

the

essential

factors

governing

the

kinetic

p

rocess. p

●

The

key

parameters

in

a

process

are

commonly

estimated

by

fitting

the

model

to

the

experimental

data,

known

as

variablesvariables

●

A

pharmacokinetic

parameter

is

a

constant

for

the

drug

that

is

estimated

from

the

experimental

data.

For

example,

estimated

pharmacokinetic

parameters

such

as

k

depend

on

the

method

of

tissue

sampling,

the

timing

of

the

sample,

drug

analysis,

and

the

predictive

model

selected.

Compartment Models

Pharmacokinetic

Models

●

Pharmacokinetic

models

are

used

to:

Predict

plasma,

tissue,

and

urine

drug

levels

with

any

dosage

regimen 2

. Calculate the optimum dosage regimen for each patient 2 .^

Calculate

the

optimum

dosage

regimen

for

each

patient

individually

Estimate

the

possible

accumulation

of

drugs

and/or

metabolites

4

Correlate drug concentrations with pharmacologic or toxicologic

4

.^

Correlate

drug

concentrations

with

pharmacologic

or

toxicologic

activity

Evaluate

differences

in

the

rate

or

extent

of

availability

between

formulations (bioequivalence)formulations

(bioequivalence)

Describe

how

changes

in

physiology

or

disease

affect

the

absorption,

distribution,

or

elimination

of

the

drug

7

Explain drug interactions

7

.^

Explain

drug

interactions

Compartment Models

Pharmacokinetic

Models

●

Simplifying assumptions are made in pharmacokinetic

●

Simplifying

assumptions

are

made

in

pharmacokinetic

models

to

describe

a

complex

biologic

system

concerning

the

movement

of

drugs

within

the

body.

For

example,

most

pharmacokinetic

models

assume

that

the

plasma

drug

concentration

reflects

drug

concentrations

globally

within

the

body.

Compartment Models

Empirically

based

Models

●

These models use an empirical formula to estimate drug

●

These

models

use

an

empirical

formula

to

estimate

drug

level

over

time

and

it

is

justified

when

limited

information

is

available.

●

Empirical

models

are

practical

but

not

very

useful

in

explaining

the

mechanism

of

the

actual

process

by

which

the drug is absorbed, distributed, and eliminated in thethe

drug

is

absorbed,

distributed,

and

eliminated

in

the

body.

Compartment Models

Physiologically

based

Models

●

If the tissue drug concentrations and binding are known

●

If

the

tissue

drug

concentrations

and

binding

are

known

physiologic

pharmacokinetic

models,

which

are

based

on

actual

tissues

and

their

respective

blood

flow,

describe

the

d t

li ti

ll

d

ata

realistically.

●

Physiologic

pharmacokinetic

models

are

frequently

used

in

describing drug distribution in animals, because tissuedescribing

drug

distribution

in

animals,

because

tissue

samples

are

easily

available

for

assay.

On

the

other

hand,

tissue

samples

are

often

not

available

for

h

bj

t

t

h

i l

i^

l^

d l

h

uman

subjects,

so

most

physiological

models

assume

an

average

set

of

blood

flow

for

individual

subjects.

Compartment Models

Mammillary Model

●

The

mammillary

model is the most common compartment

●

The

mammillary

model

is

the

most

common

compartment

model

used

in

pharmacokinetics.

●

The

mammillary model

is

a

strongly

connected

system,

because

one

can

estimate

the

amount

of

drug

in

any

compartment

of

the

system

after

drug

is

introduced

into

a

given compartment.given

compartment.

Compartment Models

Mammillary Model

●

In the one

compartment model drug is both added to and

●

In

the

one

compartment

model

drug

is

both

added

to

and

eliminated

from

a

central

compartment.

The

central

compartment

is

assigned

to

represent

plasma

and

highly

f

d

ti

th t

idl

ilib

t

ith d

perfused tissues

th

at

rapidly

equilibrate

with

d

rug.

When

an

intravenous

dose

of

drug

is

given,

the

drug

enters

directly

into

the

central

compartment.

Elimination

of

drug

occurs

from

the

central

compartment

because

the

organs

involved

in

drug

elimination,

primarily

kidney

and

liver,

are

well

‐ perfused

tissues.