PET basics II
How to get numbers?
Modeling for PET
Turku PET Centre
2008-04-15
vesa.oikonen@utu.fi
PET is quantitative
Concentrations
as a function of time
Bq/mL nCi/cc
nmol/L
Model
Analysis report
Region Receptor occupan...
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
• Vesicula...
2. Transformation
• Enzyme-catalyzed reactions:
(de)phosphorylation,
(de)carboxylation,
(de)hydroxylation,
(de)hydrogenati...
3. Binding
• Binding to plasma proteins
• Specific binding to receptors and
activation sites
• Specific binding to DNA and...
Dynamic processes
• Dynamic process is of ”first-order”,
when its speed depends on one
concentration only
• Standard mathe...
First-order kinetics
A P
k
For a first-order process A->P, the
velocity v
can be expressed as
)(
)()(
tCk
dt
tdC
dt
tdC
v ...
Pseudo-first-order kinetics
• Dynamic processes in PET involve
two or more reactants
• If the concentration of one reactan...
Compartmental model
• Compartmental model assumes that:
– injected isotope exists in the body in a fixed
number of physica...
Compartmental model
• Change of tracer concentration in one
of the compartments is a linear
function of the concentrations...
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
tCktCK
dt...
Two-tissue compartment model
C0 C1 C2
K1
k2’
k3’
k4
( )
)()(
)(
)()()(
)(
241
'
3
2
241
'
3
'
201
1
tCktCk
dt
tdC
tCktCkkt...
Three-tissue compartment model
C0 C1 C2
C3
K1
k2
k3
k4
k5 k6
Three-tissue compartment model
( )
)()(
)(
)()(
)(
)()()()(
)(
3615
3
2413
2
3624153201
1
tCktCk
dt
tdC
tCktCk
dt
tdC
tCkt...
Customized compartmental models
• Perfusion (blood flow) with [15
O]H2O
CA CT
f
f/p
)()(
)(
tC
p
f
tCf
dt
tdC
TA
T
×−×=
Customized compartmental models
• Glucose transport and
phosphorylation in skeletal muscle
with [18
F]FDG
CA CEC CIC CM
K1...
CA
H2O
Customized compartmental models
• Oxygen consumption in skeletal
muscle with [15
O]O2
CA
O2
CSM
O2
+ CMb
O2
K1
k3
k...
Customized compartmental models
• Simplified reference tissue model for
[11
C]raclopride brain studies
• See appendix 4
CA...
... continued
( )
)(
1
)(
)(
'
)(
2
2
11
tC
BP
k
tCk
dt
tdC
KK
dt
tdC
ROI
CER
CERROI
×
+
−
×+
×=
• Simplified reference ti...
Solving differential equations
• Linear first-order ordinary
differential equation (ODE) can be
solved using
– Laplace tra...
Applying differential equations
• Simulation: calculate regional tissue
curve based on
– arterial plasma curve
– model
– p...
Model fitting
• Tissue TAC measured using PET is
the sum of TACs of tissue
compartments and blood in tissue
vasculature
• ...
Model fitting
( ) ( )[ ] MinptCtCw
N
i
iSiPETi =−=Χ ∑=1
22
ˆ,
Minimization of weighted residual
sum-of-squares:
Otherwise
...
Model fitting
Initial guess of parameters
Simulated PET TACMeasured PET TAC
Measured plasma TAC
Weighted sum-of-squares
Fi...
Model comparison
• More complex model allows always
better fit to noisy data
• Parameter confidence intervals with
bootstr...
Models that are independent on any
specific compartment model structure
• Spectral analysis
• Multiple-time graphical anal...
Distributed models
• Distributed models are generally
accepted to correspond more closely to
physiological reality than si...
How to get numbers in practice?
• Follow the instructions in quality
system: SOP, MET, DAN
– Check that documentation is n...
PETO
http://petintra/Instructions/PETO_manual.pdf
Using analysis software
• Can be used on any PC
with Windows XP in
hospital network and/or
PET intranet
• Downloadable in ...
Additional information
Additional information
Requesting software
• New software
• Feature requests
• Bug reports
• Project follow-up
• Software documents
• http://peti...
More reading
• Budinger TF, Huesman RH, Knittel B, Friedland RP, Derenzo SE (1985):
Physiological modeling of dynamic meas...
Even more reading
• Laruelle M, Slifstein M, Huang Y. Positron emission tomography: imaging and
quantification of neurotra...
Appendix 1: Tracer
• PET tracer is a molecule labelled with positron
emitting isotope
• Tracer is either structurally rela...
Appendix 2: Specific activity
• Only few of tracer molecules contain radioactive
isotope; others contain ”cold” isotope
• ...
Appendix 3: Compartment
• Physiological system is decomposed into
a number of interacting subsystems,
called compartments
...
Appendix 4: Simplified reference
tissue model (SRTM)
• Assumptions
– K1/k2 is the same in all brain regions; specifically,...
Appendix 5: Laplace transformation
• Linear first-order ordinary differential
equations (ODEs) can be solved using
Laplace...
... continued
• Convolution
∫ −=⊗
t
dttbatbta
0
)()()()( ττ
… continued
C0 C1 C2
K1
k2’
k3’
k4
( ) ( )[ ]
[ ] )()(
)()(
0
12
'
31
2
04214
12
1
1
21
21
tCee
kK
tC
tCekek
K
tC
tt
tt
⊗−...
... continued
• Solution for SRTM using Laplace
transformation:
t
BP
k
REF
I
REFIT etC
BP
kR
ktCRtC +
−
⊗





+
−+=...
Appendix 6: Alternative solution for
ODEs
∫∫ −=
TT
dttCkdttCKTC
0
1
"
2
0
011 )()()(
Example: solution for one-tissue
comp...
... continued
)(
2
)(
2
)()(
00
TC
t
tTC
t
dttCdttC nn
tT
n
T
n
∆
+∆−
∆
+= ∫∫
∆−
Second step: Integral of nth compartment
...
… continued
• Finally, after substitution and
rearrangement:
"
2
1
0
1
"
2
0
01
1
2
1
)(
2
)()(
)(
k
t
tTC
t
dttCkdttCK
TC...
… continued
• Solution of SRTM:
BP
kt
tTC
t
dttC
BP
k
dttCkTCR
TC
T
tT
T
T
REFREF
T
+





 ∆
+








∆−
∆...
Upcoming SlideShare
Loading in …5
×

users.utu.fi/vesoik/presentations/PET_Basics_II_20...

345 views

Published on

0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
345
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
14
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide
  • 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.
  • This is also the best place to store the final analysis results.
  • When reporting a bug, ALWAYS provide us with the same data files with which the software or IT system failed, and tell us the exact procedures and commands and computer platform that you used. The software have been verified with standard procedures, and the bug probably can not be found with our current test data sets.
  • Analytic solutions for linear first-order differential equations can be derived using Laplace transformation.
  • users.utu.fi/vesoik/presentations/PET_Basics_II_20...

    1. 1. PET basics II How to get numbers? Modeling for PET Turku PET Centre 2008-04-15 vesa.oikonen@utu.fi
    2. 2. PET is quantitative Concentrations as a function of time Bq/mL nCi/cc nmol/L Model Analysis report Region Receptor occupancy striatum 45% putamen 43% caudatus 49% frontal 34% occipital 28% Regional biochemical, physiological and pharmacological parameters per tissue volume Perfusion Glucose consumption Enzyme activity Volume of distribution Binding potential Receptor occupancy ...
    3. 3. Modeling for PET Tracer selection Comprehensive model Workable model Model validation Model application Huang & Phelps 1986
    4. 4. Dynamic processes in vivo • Translocation • Transformation • Binding Enzyme
    5. 5. 1. Translocation • Delivery and removal by the circulatory system • Active and passive transport over membranes • Vesicular transport inside cells
    6. 6. 2. Transformation • Enzyme-catalyzed reactions: (de)phosphorylation, (de)carboxylation, (de)hydroxylation, (de)hydrogenation, (de)amination, oxidation/reduction, isomerisation • Spontaneous reactions Enzyme
    7. 7. 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
    8. 8. Dynamic processes • Dynamic process is of ”first-order”, when its speed depends on one concentration only • Standard mathematical methods assume first-order kinetics
    9. 9. First-order kinetics A P k For a first-order process A->P, the velocity v can be expressed as )( )()( tCk dt tdC dt tdC v 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 .
    10. 10. 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)
    11. 11. 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)
    12. 12. Compartmental model • Change of tracer concentration in one of the compartments is a linear function of the concentrations in all other compartments: ( )),(),(),( )( 210 tCtCtCf dt tdC i i =
    13. 13. Compartmental model • By convention, in the nuclear medicine literature, the first compartment is the blood or plasma pool
    14. 14. One-tissue compartment model • Change over time of the tracer concentration in tissue, C1(t) : )()( )( 1 " 201 1 tCktCK dt tdC −= C0 C1 K1 k2”
    15. 15. Two-tissue compartment model C0 C1 C2 K1 k2’ k3’ k4 ( ) )()( )( )()()( )( 241 ' 3 2 241 ' 3 ' 201 1 tCktCk dt tdC tCktCkktCK dt tdC −= ++−=
    16. 16. Three-tissue compartment model C0 C1 C2 C3 K1 k2 k3 k4 k5 k6
    17. 17. Three-tissue compartment model ( ) )()( )( )()( )( )()()()( )( 3615 3 2413 2 3624153201 1 tCktCk dt tdC tCktCk dt tdC tCktCktCkkktCK dt tdC −= −= ++++−=
    18. 18. Customized compartmental models • Perfusion (blood flow) with [15 O]H2O CA CT f f/p )()( )( tC p f tCf dt tdC TA T ×−×=
    19. 19. Customized compartmental models • Glucose transport and phosphorylation in skeletal muscle with [18 F]FDG CA CEC CIC CM K1 k3 k4 k2 k5
    20. 20. CA H2O Customized compartmental models • Oxygen consumption in skeletal muscle with [15 O]O2 CA O2 CSM O2 + CMb O2 K1 k3 k2 O2 CSM H2O K1 k2 O2
    21. 21. Customized compartmental models • Simplified reference tissue model for [11 C]raclopride brain studies • See appendix 4 CA CF + CNS + CB K1 k2 CF + CNS K1 ’ k2 ’ Cerebellum ROI
    22. 22. ... continued ( ) )( 1 )( )( ' )( 2 2 11 tC BP k tCk dt tdC KK dt tdC ROI CER CERROI × + − ×+ ×= • Simplified reference tissue model for [11 C]raclopride brain studies
    23. 23. Solving differential equations • Linear first-order ordinary differential equation (ODE) can be solved using – Laplace transformation; see appendix 5 – alternative method; see appendix 6
    24. 24. Applying differential equations • Simulation: calculate regional tissue curve based on – arterial plasma curve – model – physiological model parameters
    25. 25. 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)()(
    26. 26. Model fitting ( ) ( )[ ] MinptCtCw N i iSiPETi =−=Χ ∑=1 22 ˆ, Minimization of weighted residual sum-of-squares: Otherwise If measurement variance is known 2 1 i iw σ = 1=iw
    27. 27. 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
    28. 28. 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
    29. 29. Models that are independent on any specific compartment model structure • Spectral analysis • Multiple-time graphical analysis (MTGA): – Gjedde-Patlak – Logan (see PET basics I)
    30. 30. 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).
    31. 31. 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
    32. 32. PETO http://petintra/Instructions/PETO_manual.pdf
    33. 33. Using analysis software • Can be used on any PC with Windows XP in hospital network and/or PET intranet • Downloadable in WWW • Analysis instructions in WWW • http://www.pet.fi/ or http://www.turkupetcentre.net/ • P:binwindows
    34. 34. Additional information
    35. 35. Additional information
    36. 36. Requesting software • New software • Feature requests • Bug reports • Project follow-up • Software documents • http://petintra/softaryhma/ • or ask IT or modelling group members
    37. 37. 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.
    38. 38. 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.
    39. 39. 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.
    40. 40. 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
    41. 41. 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
    42. 42. 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 tC BP k tCk dt tdC R dt tdC TREF REFT + −+=
    43. 43. Appendix 5: Laplace transformation • Linear first-order ordinary differential equations (ODEs) can be solved using Laplace transformation • Solution for one-tissue compartment model: tk etCKtC " 2 )()( 011 − ⊗= C0 C1 K1 k2”
    44. 44. ... continued • Convolution ∫ −=⊗ t dttbatbta 0 )()()()( ττ
    45. 45. … continued C0 C1 C2 K1 k2’ k3’ k4 ( ) ( )[ ] [ ] )()( )()( 0 12 ' 31 2 04214 12 1 1 21 21 tCee kK tC tCekek K tC tt tt ⊗− − = ⊗−+− − = −− −− αα αα αα αα αα , where ( ) ( ) 24 24 4 ' 2 2 4 ' 3 ' 24 ' 3 ' 22 4 ' 2 2 4 ' 3 ' 24 ' 3 ' 21     −+++++=     −++−++= kkkkkkkk kkkkkkkk α α Phelps ME et al. Ann Neurol. 1979;6:371-388
    46. 46. ... continued • Solution for SRTM using Laplace transformation: t BP k REF I REFIT etC BP kR ktCRtC + − ⊗      + −+= 12 2 2 )( 1 )()(
    47. 47. Appendix 6: Alternative solution for ODEs ∫∫ −= TT dttCkdttCKTC 0 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:
    48. 48. ... continued )( 2 )( 2 )()( 00 TC t tTC t 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 trapezoidal method
    49. 49. … continued • Finally, after substitution and rearrangement: " 2 1 0 1 " 2 0 01 1 2 1 )( 2 )()( )( k t tTC t dttCkdttCK TC tTT ∆ +         ∆− ∆ +− = ∫∫ ∆−
    50. 50. … continued • Solution of SRTM: BP kt tTC t dttC BP k dttCkTCR TC T tT T T REFREF T +       ∆ +         ∆− ∆ + + −+ = ∫∫ ∆− 12 1 )( 2 )( 1 )()( )( 2 0 2 0 21

    ×