Nuclear Imaging h20

7/30/2019 Nuclear Imaging h20

1/36

Tracer kinetic modelling

Department of Nuclear Medicine & Molecular Imaging

Chapter 20


2/36

Dept. Of Nuclear Medicine & Molecular ImagingChapter 20

Tracer Kinetic Modelling

The spatial distribution of a radiotracer in the body is determined

by the characteristics of the tracer.

by the characteristics of the tissue under investigation.

This distribution is time-varying.

So study of the tracer kinetics may give insight into the

underlying processes.

Perfusion/transport

Metabolism

Binding


3/36



As an example, consider a tracer which is injected at point A and

measured at point B.

The time-activity curve will depend on the flow in the tube.


4/36


Kinetic modeling in a nutshell

Tissue distributionInjection

Tissue


5/36



Model


Tissue


6/36



Plasma sampling PET MeasurementModel


Tissue


7/36



Plasma sampling PET MeasurementModel


Tissue

Model fit

Model parameters


8/36


Tracers

A tracer is a substance that follows (traces) a physiologic or

biochemical process. Some characteristic of an ideal tracer are:

Its behavior should be identical or at least directly related to a known

natural substance.

Its mass should not influence the process which is being studied. This

requires a high specific activity (MBq/mmol).

It should not exhibit an isotope effect, or this effect must be

predictable.


9/36


Isotopes

Single photon isotopes such as 99mTc, 67Ga, 111In or 123I and

positron emitters such 18F or 89Zr are not normally present in the

body.

This is not a major problem with distribution, transport or

excretion measurements.

However, a biochemical reaction is much more selective and may

not work, or work differently, when one of these isotopes is used.


10/36


Isotopes

The positron emitters 11C, 13N and 15O are common in biological

substances.

So their use is advantageous since they will have no isotope

effect. (The mass effect can be neglected.)

However, their use is not simple considering their half-lives of

only 20, 10 and 2 min.

For PET, 11C is probably the best tracer in principle, but often 18F

is used due to its longer half-live.


11/36


Compartments

Tracer kinetic modeling uses the concept of compartments.

The changes in tracer concentration are described by transports

between compartments.

Compartments may have a physical analog such as the

intracellular fluid but they can also represent a transport from

one chemical form to another of from a free tracer to a bound

tracer.


12/36


Compartments


13/36


Linearity of transport

Passive diffusion is linear with the concentration.

However, since we have a finite number of transporters, enzymes or

receptors, their associated processes are non-linear by definition.

For example the Michaelis-Menten equation for tracer transport

when there is competition between the tracer and an endogenous

compound is given by:

)()(

1

)(max

tCKK

tC

tCTTtt

me

m

e

ttt

+

+

=


14/36


Linearity of transport

Now devise the experiment in such a way that:

The endogenous concentration is constant. (Steady-state condition)

The tracer concentration is negligible. (Tracer condition)

The tracer transport can then be approximated by:

Thus under these conditions the tracer transport is indeed linear with its

concentration.

)()(

1)()(

1

)( maxmax tkCtC

KK

C

T

tCKK

tC

tCTT

tt

t

me

m

e

t

tt

me

m

e

ttt

=

+

=

+

+

=


15/36


Compartment Models: Linearity

Since the transport is linear with the tracer concentration, the above model is

described by:

C1

K1

k2

k3C2Cp

132

13211 )(

Ckdt

dC

CkkCK

dt

dCp

=

+=


16/36


Compartment Models: Linearity

Since the transport is linear with the tracer concentration, the above model is

described by:

2413

2

241321

1 )(

CkCkdt

dC

CkCkkCKdt

dCp

=

++=

C1

K1

k2

k3

C2Cpk4


17/36


Rate of metabolism

Let us assume that k3 represents some metabolic process.

Since this process is uni-directional, the transport from C1 to C2 represents

the net metabolism of the tissue.

This transport is given by: k3C1

Under steady state conditions, the concentrations Cp and C1 are constant.

Thus:

So we find that:

C1 C2Cp

ppC

kk

KCCkkCK

32

1

11321)(0

+

=+=

pC

kk

kKCk

dt

dCM

32

31

13

2

+

===

k3


18/36


Rate of metabolism

The total tissue signal is then given by:

and since Cp is assumed constant, the tissue-plasma ratio thus becomes:

C1 C2Cp

dCkk

kKC

kk

KCCC

t

ppt)(

032

31

32

1

21 +

+

+

=+=

tkk

kK

kk

K

C

dC

kk

kK

kk

K

C

C

p

t

p

p

t

32

31

32

1

0

32

31

32

1

)(

+

+

+

=

+

+

+

=


19/36


The distribution volume gives the ratio of tissue and plasma concentration

at equilibrium.

We also use the partition coefficient which uses a different concept but

gives exactly the same value.

This can be determined directly from the data if steady state conditions

are fulfilled.

Distribution volumeC1 C2Cp


20/36


Under steady state conditions all concentrations will be constant. Thus:

and the distribution volume thus equals:

p

pp

Ckk

kKC

k

kCCkCk

Ck

KCCkCkkCK

42

31

1

4

3

22413

2

1

1241321

0

)(0

===

=++=

Distribution volumeC1 C2Cp

+=+==

4

3

2

121 1kk

kK

CCC

CCDV

pp

t


21/36


The derivation of the metabolic rate and the distribution volume assumes

steady state for the tracer.

This is only possible if the underlying processes are also in steady state.

However, given the resulting equations, one can also calculate them from the

individual rate constants.

We can calculate the rate constants by fitting the measured data to the

model.

Thus, with this approach it is not essential that the tracer is in steady state.

However, it remains essential that the system is in steady state.

Steady state condition


22/36


Compartment models solutions

Plasma sampling PET Measurement

Tissue distributionInjectionTissue

Model fit

K1, k2

tA

t CkCKdt

dC21

=


23/36


Example: FDG measurement

FDG PET measurement and fit

Time [min]

0 10 20 30 40 50 60

Activity[Bq

/cc]

0

5000

10000

15000

20000

Target

Fitted

Free

Fixed

Blood

minccmlR

b

mink

mink

minccmlK

tissueplasma

v

1

1

tissueplasma

029.0

024.0

067.0

108.0

075.0

3

2

1

=

=

=

=

=

Free

K1

k2

k3

FixedPlasma


24/36


SA4503 PET measurement and fit

Time [min]

0 20 40 60 80

Activity[Bq

/cc]

0

2000

4000

6000

8000

10000

12000

14000

Target

Fitted

Free

Bound

Blood

tissueplasma

v

1

1

1

tissueplasma

ccmlDV

BP

b

mink

mink

mink

minccmlK

0.6

6.0

024.0

021.0

013.0

107.0

394.0

4

3

2

1

=

=

=

=

=

=

=

Example: Receptor measurement

Free

K1

k2

k3

k4

BoundPlasma


25/36


Extraction

The Renkin-Crone model describes the extraction

for a rigid tube with identical arterial/venous flow

and extraction of a surface S with permeability P.

This model can also be transformed into a

compartment model with perfusion F and a tissue

extraction PS.

( )F

PS

eE

=1

FPS

PSE

+

=


26/36


Extraction & Clearance

From the extraction we can calculate the

clearance i.e. the product of extraction

and flow.

For low flow, the extraction is ~1 i.e. the

clearance becomes dependent of flow.

For high flow, the extraction behaves

like 1/F. Thus the clearance becomes

independent of flow.


27/36


Tracer Tracee revisited

We showed the importance of the steady-state and tracer condition.

We also showed that the individual rate constants can be measured using

PET.

However, the rate constants are determined for the tracer.

How do they relate to the process under investigation?


28/36


Tracer Tracee revisited

Consider the model for glucose (tracee) and FDG (tracer) i.e. by

measuring the FDG kinetics we want to learn something about the glucose

consumption.


29/36


Tracer - Tracee

Shown are the tracer kinetic models for glucose and FDG.

Not that both are transported by the same transport systems and both are

metabolized by hexokinase.


30/36


Tracer - Tracee

We can measure

The total tissue activity as a function of time using PET.

The plasma activity by taking plasma samples.

We can then solve the tracer kinetic model using the measured input and

output to obtain the optimal parameter set K1*, k2

*, k3* and k4

*.

Generally, it is assumed that k4*=0. The rate of metabolism for FDG is

then given by:

*

3

*

2

*

3

*

1

kkkKR+

=


31/36


Tracer - Tracee

We can measure K1*, k2* and k3* and thus the rate of FDG metabolism.

However, FDG is not identical to glucose so:

This is incorporated into the lumped constant LC as:

So, under the model assumptions, the measurement of FDG kinetics gives

information about the steady-state glucose metabolism.

ppC

kk

kKC

kk

kKM

*

3

*

2

*

3

*

1

32

31

+

+

=

LC

C

kk

kKC

kk

kKM p

p *

3

*

2

*

3

*

1

32

31

+

=

+

=


32/36


Take home message

PET allows the quantitative measurement (in Bq/ml) of the

distribution of an injected tracer.

From blood samples, the plasma input curve can be obtained.

So given a specific I/O model, the model parameters can be

measured.

These model parameters give insight into the fate of the tracer in

terms of transport, metabolism and/or binding.


33/36


Take home message

These kinetic models require the system to be in steady-state.

(Steady-state condition.)

Generally, the tracer itself will fluctuate in time however its

concentration should be negligible. (Tracer condition.)

If the tracer has an endogenous analog, its model parameters will

be identical or at least related to those of the analog.

Thus, the tracer tells us something about this endogenous analog. For example 18FDG can be used to determine the glucose

metabolism.


34/36


Take home message

The individual rate constants can be of interest e.g. K1 can be

related to perfusion.

Often we need to combine the model parameters to get

meaningful results.

The most used are the rate of metabolism and the distribution

volume.


35/36



36/36



As an example, consider a tracer which is injected at point A and

measured at point B.

The time-activity curve will depend on the flow in the tube.

Documents

Nuclear Imaging h20