PET basics II

How to get numbers?Modeling for PET

Turku PET Centre2008-04-15

vesa.oikonen@utu.fi

PET is quantitative

Concentrationsas a function of time

Bq/mL nCi/ccnmol/L

Model

Analysis reportRegion Receptor occupancy

striatum 45%putamen 43%caudatus 49%frontal 34%occipital 28%

Regional biochemical, physiological and pharmacological parameters per tissue volume

PerfusionGlucose consumptionEnzyme activityVolume of distributionBinding potentialReceptor occupancy...

Modeling for PET

Tracer selection

Comprehensive model

Workable model

Model validation

Model application

Huang & Phelps 1986

Dynamic processes in vivo

• Translocation

• Transformation

• Binding

Enzyme

1. Translocation

• Delivery and removal by the circulatory system

• Active and passive transport over membranes

• Vesicular transport inside cells

2. Transformation

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

• Spontaneous reactions

Enzyme

3. 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

Dynamic processes

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

• Standard mathematical methods assume first-order kinetics

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;k is independent of concentration of A and time;its unit is sec-1 or min-1.

Pseudo-first-order kinetics

• Dynamic processes in PET involve 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 kinetics

• This is one reason why we use tracer doses in PET(see Appendix 1)

Compartmental model

• Compartmental model assumes that:– injected isotope exists in the body in a fixed

number of physical or chemical states (compartments, see appendix 3),with specified interconnections among them; the arrows indicate the possible pathways the tracer can follow (dynamic processes)

– Compartmental models can be described in terms of a set of linear, first-order, constant-coefficient, ordinary differential equations (ODE)

Compartmental model

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

),(),(),()(

210 tCtCtCfdt

tdCi

i

Compartmental model

• By convention, in the nuclear medicine literature, the first compartment is the blood or plasma pool

One-tissue compartment model

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

)()()(

1"201

1 tCktCKdt

tdC

C0 C1

K1

k2”

Two-tissue compartment model

C0 C1 C2

K1

k2’

k3’

k4

)()(

)(

)()()()(

241'3

2

241'3

'201

1

tCktCkdt

tdC

tCktCkktCKdt

tdC

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

Customized compartmental models

• Perfusion (blood flow) with [15O]H2O

CA CT

f

f/p

)()()(

tCp

ftCf

dt

tdCTA

T

Customized compartmental models

• Glucose transport and phosphorylation in skeletal muscle with [18F]FDG

CA CEC CIC CM

K1 k3

k4

k2

k5

CAH2O

Customized compartmental models

• Oxygen consumption in skeletal muscle with [15O]O2

CAO2 CSM

O2 + CMbO2

K1

k3

k2O2

CSMH2O

K1

k2O2

Customized compartmental models

• Simplified reference tissue model for [11C]raclopride brain studies

• See appendix 4

CA

CF + CNS + CB

K1

k2

CF + CNS

K1’

k2’

Cerebellum

ROI

... continued

)(1

)(

)('

)(

2

2

11

tCBP

ktCk

dt

tdCKK

dt

tdC

ROI

CER

CERROI

• Simplified reference tissue model for [11C]raclopride brain studies

Solving differential equations

• Linear first-order ordinary differential equation (ODE) can be solved using– Laplace transformation;

see appendix 5– alternative method;

see appendix 6

Applying differential equations

• Simulation: calculate regional tissue curve based on– arterial plasma curve– model– physiological model parameters

Model fitting

• 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)()(

Model fitting

MinptCtCwN

iiSiPETi

1

22 ˆ,

Minimization of weighted residualsum-of-squares:

Otherwise

If measurement variance is known

2

1

iiw

1iw

Model fitting

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

Model comparison

• 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

Models that are independent on any specific compartment model structure• Spectral analysis• Multiple-time graphical analysis

(MTGA):– Gjedde-Patlak– Logan(see PET basics 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).

How to get numbers in practice?

• Follow the instructions in quality system: SOP, MET, DAN– Check that documentation is not

outdated

• PETO– Retrieve data for analysis– Record study documentation– Store final analysis results

PETO

http://petintra/Instructions/PETO_manual.pdf

Using analysis software

• Can be used on any PCwith Windows XP inhospital network and/orPET intranet

• Downloadable in WWW• Analysis instructions in

WWW• http://www.pet.fi/

or http://www.turkupetcentre.net/

• P:\bin\windows

Additional information

Additional information

Requesting software

• New software• Feature requests• Bug reports• Project follow-up• Software documents

• http://petintra/softaryhma/

• or ask IT or modelling group members

More reading

• Budinger TF, Huesman RH, Knittel B, Friedland RP, Derenzo SE (1985): Physiological modeling of dynamic measurements of metabolism using positron emission tomography. In: The Metabolism of the Human Brain Studied with Positron Emission Tomography. (Eds: Greitz T et al.) Raven Press, New York, 165-183.

• Cunningham VJ, Rabiner EA, Matthews JC, Gunn RN, Zamuner S, Gee AD. Kinetic analysis of neuroreceptor binding using PET. Int Congress Series 2004; 1265: 12-24.

• van den Hoff J. Principles of quantitative positron emission tomography. Amino Acids 2005; 29(4): 341-353.

• Huang SC, Phelps ME (1986): Principles of tracer kinetic modeling in positron emission tomography and autoradiography. In: Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart. (Eds: Phelps,M; Mazziotta,J; Schelbert,H) Raven Press, New York, 287-346.

• Ichise M, Meyer JH, Yonekura Y. An introduction to PET and SPECT neuroreceptor quantification models. J. Nucl. Med. 2001; 42:755-763.

• Lammertsma AA, Hume SP. Simplified reference tissue model for PET receptor studies. Neuroimage 1996; 4: 153-158.

• Lammertsma AA. Radioligand studies: imaging and quantitative analysis. Eur. Neuropsychopharmacol. 2002; 12: 513-516.

• Laruelle M. Modelling: when and why? Eur. J. Nucl. Med. 1999; 26, 571-572. • Laruelle M. Imaging synaptic neurotransmission with in vivo binding competition

techniques: a critical review. J. Cereb. Blood Flow Metab. 2000; 20: 423-451.

Even more reading

• Laruelle M, Slifstein M, Huang Y. Positron emission tomography: imaging and quantification of neurotransporter availability. Methods 2002; 27:287-299.

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

• Meikle SR, Eberl S, Iida H. Instrumentation and methodology for quantitative pre-clinical imaging studies. Curr. Pharm. Des. 2001; 7(18): 1945-1966.

• Passchier J, Gee A, Willemsen A, Vaalburg W, van Waarde A. Measured drug-related receptor occupancy with positron emission tomography. Methods 2002; 27:278-286.

• Schmidt KC, Turkheimer FE. Kinetic modeling in positron emission tomography. Q. J. Nucl. Med. 2002; 46:70-85.

• Slifstein M, Laruelle M. Models and methods for derivation of in vivo neuroreceptor parameters with PET and SPECT reversible radiotracers. Nucl. Med. Biol. 2001; 28:595-608.

• Turkheimer F, Sokoloff L, Bertoldo A, Lucignani G, Reivich M, Jaggi JL, Schmidt K. Estimation of component and parameter distributions in spectral analysis. J. Cereb. Blood Flow Metabol. 1998; 18: 1211-1222.

• Turkheimer FE, Hinz R, Cunningham VJ. On the undecidability among kinetic models: from model selection to model averaging. J. Cereb. Blood Flow Metab. 2003; 23: 490-498.

• Watabe H, Ikoma Y, Kimura Y, Naganawa M, Shidahara M. PET kinetic analysis - compartmental model. Ann Nucl Med. 2006; 20(9): 583-588.

Appendix 1: Tracer• PET tracer is a molecule labelled with positron

emitting isotope• 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.

Appendix 2: 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 for example 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 (Bq/mL —> nmol/L)

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

Appendix 3: Compartment

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

• Compartment is a chemical species in a physical place; for example, neither glucose or interstitial space is a compartment, but glucose in interstitial space is one

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

Appendix 4: Simplified reference tissue model (SRTM)

• Assumptions– K1/k2 is the same in all brain regions; specifically, in

regions of interest, and in reference region devoid of receptors (R1=K1/K1

REF)– One-tissue compartment model would fit all regional

curves fairly well• Differential equation for SRTM:

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

)(1

)()()( 2

21 tCBP

ktCk

dt

tdCR

dt

tdCTREF

REFT

Appendix 5: Laplace transformation

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

• Solution for one-tissue compartment model: tketCKtC

"2)()( 011

C0 C1

K1

k2”

... continued

• Convolution

t

dttbatbta0

)()()()(

… continued

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

... continued

• Solution for SRTM using Laplace transformation:

tBP

k

REFI

REFIT etCBP

kRktCRtC

12

2

2

)(1

)()(

Appendix 6: Alternative solution for ODEs

TT

dttCkdttCKTC0

1"2

0

011 )()()(

Example: solution for one-tissue compartment model

First step: ODE is integrated, assuming that at t=0 all concentrations are zero:

... continued

)(2

)(2

)()(00

TCt

tTCt

dttCdttC nn

tT

n

T

n

Second step: Integral of nth compartment is implicitly estimated for example with 2nd order Adams-Moulton method:

Integrals are calculated using trapezoidalmethod

… continued

• Finally, after substitution and rearrangement:

"2

1

0

1"2

0

01

1

21

)(2

)()(

)(k

t

tTCt

dttCkdttCK

TC

tTT

… continued

• Solution of SRTM:

BPkt

tTCt

dttCBP

kdttCkTCR

TCT

tT

T

T

REFREF

T

121

)(2

)(1

)()(

)(2

0

2

0

21