MODELLING AND PARAMETER ESTIMATION IN PET Vesa Oikonen Turku PET Centre 2004-06-03.

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MODELLING AND PARAMETER ESTIMATION IN PET Vesa Oikonen Turku PET Centre 2004-06-03
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Transcript of MODELLING AND PARAMETER ESTIMATION IN PET Vesa Oikonen Turku PET Centre 2004-06-03.

MODELLING AND PARAMETER

ESTIMATION IN PET

Vesa OikonenTurku PET Centre2004-06-03

PET provides

• Quantitation of biochemical and physiological processes

• ... per organ volume• Noninvasive measurement• In vivo

PET provides

• Perfusion: ml blood / (min * 100 g tissue)• Glucose consumption: μmol glucose /

(min * 100 g tissue)• Oxygen consumption: ml O2 / (min * 100

g tissue)• Amino acid uptake• Fatty acid uptake• Concentration and affinity of receptor

(Bmax , KD)

PET also provides

• Change in perfusion (brain activation)

• Change in binding potential (receptor occupancy, endogenous ligand)

What PET actually provides

• Time course of the radioactivity concentration (TAC) in each image voxel (in units Bq/ml)

• Radioactivity concentration in blood plasma (input curve) is measured separately, or is replaced by reference region TAC from the image

Input curve

t t

Mixing in the heartExchange with interstitial volume

Exchange with intracellular volumeTime delay

Intravenous bolus infusion Measured arterial plasma TAC

Correction for labeled metabolite in plasma

0 15 30 450.0

0.2

0.4

0.6

0.8

1.0

Fra

ctio

n o

f au

then

tic

trac

er

Time (min)

0 15 30 45 600

10

20

30

40

50

60

70

80

90

100

Rad

ioac

tivi

ty c

on

c. (

kBq

/mL

)

Time (min)

Plasma TAC Authentic tracer Metabolized tracer

PET data

0 15 30 45 60 75 900

10

20

30

40

50

60

70

80

90

Co

nce

ntr

atio

n o

f au

then

tic

trac

er (

kBq

/mL

)

Time (min)

0 15 30 45 60 75 900

10

20

30

40

50

60

70

80

90

Co

nce

ntr

atio

n in

tis

sue

(kB

q/m

L)

Time (min)

”input” ”output”Authentic tracerconcentrationavailable in

arterial blood

Concentrationin tissue

measured byPET scannerPerfusion

Endothelial permeabilityVascular volume fraction

Transport across cell membranes

Specific binding to receptorsNon-specific binding

Enzyme activity

Dynamic processes

1. Translocation

2. Transformation

3. Binding

E

Translocation

• Delivery and removal by circulatory system

• Active and passive transport over membranes

• Vesicular transport inside cells

Transformation

• Enzyme-catalyzed reactions: (de)phosphorylation, (de)carboxylation, (de)hydroxylation, (de)hydrogenation, (de)amination, oxidation/reduction, isomerization

• Spontaneous reactions

Binding

• Binding to plasma proteins• Specific binding to receptors and

activation sites• Specific binding to DNA and RNA• Specific binding between antibody

and antigen• Non-specific binding

First-order kinetics

• Models that can be reasonably analyzed with standard mathematical methods assume first-order processes

• Process is of ”first-order”, when its speed depends on one concentration only

First-order kinetics

A Pk

For a first-order process A->P, the velocity vcan be expressed as

)()()(

tCkdt

tdC

dt

tdCv A

AP

, where k is a first-order rate constant;it is independent of concentration and time;its unit is sec-1 or min-1.

First-order kinetics -radioactive decay

eOF k 1818

dtktC

tdCdtk

tC

tdCtkC

dt

tdC

F

F

F

FF

F

)(

)(

)(

)()(

)(

18

18

18

1818

18

Integrate:)0(ln)(ln 1818 FF CdtktC

Subtract ln CF-18(0) from both sides, and take exponentials:

tk

F

F eC

tC

)0(

)(

18

18

We have linear first-order ordinary differential equation (ODE):

Pseudo-first-order

• Usually process involves two or more reactants

• If the concentration of one reactant is very small compared to the others, equations simplify to the same form as for first-order process

• In PET: trace-dose

Definition: TRACER

• Tracer is a positron emitting isotope labeled molecule• Tracer is either structurally related to the natural

substance (tracee) or involved in the dynamic process• Tracer is introduced to system in a trace amount i.e.

with a high specific activity; process being measured is not perturbed by it. In general, the amount of tracer is at least a couple of orders of magnitude smaller than the tracee.

• Dynamic process is evaluated in a steady state: rate of process is not changing with time, and amount of tracee is constant during the evaluation period. Steady state of the tracer is not required

• When these requirements are satisfied, the processes can be described with pseudo-first-order rate constants.

Definition: Specific activity

• Only few of tracer molecules contain radioactive isotope; others contain ”cold” isotope

• Specific activity (SA) is the ratio between “hot” and “cold” tracer molecules

• SA is always measured; its unit is MBq/μmol or mCi/μmol

• All radioactivity measurements, also SA, are corrected for physical decay to the time of injection

• SA can be used to convert measured radioactivity concentrations in tissue and blood to mass

• High SA is required to reach sufficient count level without injecting too high mass

Compartment models

• Tracer is injected intravenously as a bolus• Tracer is well mixed with blood at the heart• Tracer is distributed by arterial circulation to

the capillary bed, where exchange with tissue takes place

• Tracer concentration in tissue increases by extraction of tracer from plasma

• Concentration in tissue is reduced by backward transfer

Compartment models

• Physiological system is decomposed into a number of interacting subsystems, called compartments

• Compartment is a chemical species in a physical place

• Inside a compartment the tracer is considered to be distributed uniformly

Compartment models

• Change of tracer concentration in one of the compartments is a linear function of the concentrations in all other compartments:

),(),(),()(

210 tCtCtCfdt

tdCi

i

Distributed models

• Distributed models are generally accepted to correspond more closely to physiological reality than simpler compartment models

• In PET imaging, compartment models have been shown to provide estimates of receptor concentration that are as good as those of a distributed model, and are assumed to be adequate for analysis of PET imaging data in general (Muzic & Saidel, 2003).

One-tissue compartment model

• Change over time of the tracer concentration in tissue, C1(t) :

)()()(

1"201

1 tCktCKdt

tdC

C0 C1

K1

k2”

One-tissue compartment model

• Linear first-order ordinary differential equations (ODEs) can be solved using Laplace transformation:

tketCKtC"2)()( 011

C0 C1

K1

k2”

Convolution

t

dttbatbta0

)()()()(

Alternative solution of ODEs

TT

dttCkdttCKTC0

1"2

0

011 )()()(

1. ODE is integrated, assuming that at t=0all concentrations are zero:

Alternative solution of ODEs

)(2

)(2

)()(00

TCt

tTCt

dttCdttC nn

tT

n

T

n

2. Integral of nth compartment is implicitely estimated for example with 2nd order Adams-Moulton method:

Integrals are calculated using trapezoidalmethod.

Alternative solution of ODEs

"2

1

0

1"2

0

01

1

21

)(2

)()(

)(kt

tTCt

dttCkdttCK

TC

tTT

3. After substitution and rearrangement:

Two-tissue compartment model

C0 C1 C2

K1

k2’

k3’

k4

)()(

)(

)()()()(

241'3

2

241'3

'201

1

tCktCkdt

tdC

tCktCkktCKdt

tdC

Two-tissue compartment model

C0 C1 C2

K1

k2’

k3’

k4

)()(

)()(

012

'31

2

0421412

11

21

21

tCeekK

tC

tCekekK

tC

tt

tt

, where 24

24

4'2

2

4'3

'24

'3

'22

4'2

2

4'3

'24

'3

'21

kkkkkkkk

kkkkkkkk

Phelps ME et al. Ann Neurol 1979;6:371-388

Three-tissue compartment model

C0 C1 C2

C3

K1

k2

k3

k4

k5 k6

Three-tissue compartment model

)()()(

)()()(

)()()()()(

36153

24132

36241532011

tCktCkdt

tdC

tCktCkdt

tdC

tCktCktCkkktCKdt

tdC

Three-tissue compartment model

• Specific binding (k3,k4) and nonspecific binding (k5,k6) cannot be distinguished unless (k5,k6) >> (k3,k4)

• If (k5,k6) >> (k3,k4), then the system reduces to two-tissue compartment model

Fitting of compartment models to measured data

• Tissue TAC measured using PET is the sum of TACs of tissue compartments and blood in tissue vasculature

• Simulated PET TAC:

i

iBBBS tCVtCVtC )(1)()(

Fitting of compartment models to measured data

MinptCtCwN

iiSiPETi

1

22 ˆ,

Minimization of weighted residualsum-of-squares:

Otherwise

If measurement variance is known

2

1

iiw

1iw

Fitting of compartment models to measured data

Initial guess of parameters

Simulated PET TACMeasured PET TAC

Measured plasma TAC

Weighted sum-of-squares

Final model parameters

New guess of parametersModel

if too large

if small enough

Fitting of compartment models to measured data

• Optimization algorithm is used for iteratively moving from one set of parameters to a better set until progress is stalled or until a fixed maximum number of iterations has passed

• If the criterion function has multiple local minima, the iterative search may end up at any one of these

• If no constraints are imposed on the parameters, the minimum could correspond to a physically unrealizable set of parameters

Major steps in modelling

Tracer selection

Comprehensive model

Workable model

Model validation

Model applicationHuang & Phelps 1986

Comparing models

• More complex model allows always better fit to noisy data

• Parameter confidence intervals with bootstrapping

• Significance of the information gain by additional parameters: F test, AIC, SC

• Alternative to model selection: Model averaging with Akaike weights

Albert Einstein :

”Everything should be madeas simple as possible,but not simpler”

Macroparameters

• Combination of model parameters can be computed with better reproducibility

• Reversible models:Distribution volume (DV)

• Irreversible models:Net influx rate (Ki)

Distribution volumeOne-tissuecompartmentmodel

Two-tissuecompartmentmodel

Three-tissuecompartmentmodel

6

5

4

3

2

1 1k

k

k

k

k

KDV

4

'3

'2

1 1k

k

k

KDV

"2

1

k

KDV

Distribution volume ratio

• Ratio between DV in region of interest and reference region (region without specific binding)

REF

ROI

DV

DVDVR

Binding potential (BP)

• Binding potential equals the concentration of free receptors, multiplied by affinity (1/KD) and fraction of free tracer in C1’ (combined C1 and C3)

• BP=DVR-1

DK

Bf

k

kBP

'max

24

'3

Net influx rateOne-tissuecompartmentmodel

Two-tissuecompartmentmodel

Three-tissuecompartmentmodel

32

31'3

'2

'31

kk

kK

kk

kKK i

32

132

123

:

:

kk

KKkk

KKkk

i

i

32

31

kk

kKK i

Simplified reference tissue model (SRTM)

• Assumption #1: K1/k2 is the same in all regions (RI=K1/K1REF)

• Assumption #2: 1-tissue compartment model would fit all TACs fairly well

)(1

)()()( 2

2 tCBP

ktCk

dt

tdCR

dt

tdCTREF

REFI

T

Lammertsma AA, Hume SP. Neuroimage 1996;4:153-158

Simplified reference tissue model (SRTM)

• Solution using Laplace transformation:

tBP

k

REFI

REFIT etCBP

kRktCRtC

12

2

2

)(1

)()(

• Solution using 2nd order Adams-Moulton:

BPkt

tTCt

dttCBPk

dttCkTCR

TCT

tT

T

T

REFREFI

T

121

)(2

)(1

)()(

)(2

0

2

0

2

Multiple-time graphical analysis (MTGA)

• Data is transformed to a linear plot• Macroparameter estimated directly

as the slope of linear phase of plot• Independent of compartments• Reversible models:

Logan analysis (DV, DVR)• Irreversible models:

Gjedde-Patlak analysis (Ki)

Logan analysiswith plasma input

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

CR

OI i

nte

gra

l / C

RO

I

CPLASMA

integral / CROI

Distribution volume=

Slope of the Logan plot

Distribution volumeratio =Ratio of slopes of theROI and referenceregionLogan J. Graphical analysis of PET data applied to reversible

and irreversible tracers. Nucl Med Biol 2000;27:661-670

Logan analysiswith reference region input

Logan J. Graphical analysis of PET data applied to reversibleand irreversible tracers. Nucl Med Biol 2000;27:661-670

Distribution volume ratio=

Slope of the Logan plotcalculated using

reference region input

0 10 20 30 40 500

20

40

60

80

100

120

140

160

CR

OI i

nte

gra

l / C

RO

I

CREFERENCE

integral / CROI

BP = DVR - 1

Gjedde-Patlak analysiswith plasma input

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

CR

OI/C

PL

AS

MA (

mL

/mL

)

CPLASMA

integral/CPLASMA

(min)

Net influx rate Ki=

Slope of the Patlak plot

Unit of Ki =ml plasma * min-1 * ml tissue-1

Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. J Cereb Blood Flow Metab 1985;5:584-590.

Gjedde-Patlak analysiswith reference region input

0 10 20 30 40 500

1

2

3

4

5

6

CR

OI/C

RE

F

CREF

integral/CREF

(min)

Net influx rate Ki=

Slope of the Patlak plot=

k2*k3/(k2+k3)

Unit of referenceinput Ki = min-1

Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. J Cereb Blood Flow Metab 1985;5:584-590.

Mathematical model validation

• Residual curve must not show any time-dependent pattern (underparameterization)

• Considering the noise, standard errors of the fitted parameters should be small (overparameterization)

• Variable parameters must not be correlated (overparameterization)

Biochemical model validation

• Absolute accuracy of model parameters must be tested with a "gold standard", if one is available for the measurement of interest

• Intervention studies must be performed to estimate the sensitivity of the estimated parameters to the physiologic parameter of interest.

• Parameters of interest must not change in response to a perturbation in a different factor

Clinical model validation

• Repeatability coefficient (RC) and intraclass correlation coefficient (ICC) must be high (test-retest setting)

• Effect size and discriminating power must high (patient-control or treatment-placebo study)