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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.

- 1. PET basics II How to get numbers? Modeling for PET Turku PET Centre 2008-04-15 vesa.oikonen@utu.fi
- 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. Modeling for PET Tracer selection Comprehensive model Workable model Model validation Model application Huang & Phelps 1986
- 4. Dynamic processes in vivo • Translocation • Transformation • Binding Enzyme
- 5. 1. Translocation • Delivery and removal by the circulatory system • Active and passive transport over membranes • Vesicular transport inside cells
- 6. 2. Transformation • Enzyme-catalyzed reactions: (de)phosphorylation, (de)carboxylation, (de)hydroxylation, (de)hydrogenation, (de)amination, oxidation/reduction, isomerisation • Spontaneous reactions Enzyme
- 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. Dynamic processes • Dynamic process is of ”first-order”, when its speed depends on one concentration only • Standard mathematical methods assume first-order kinetics
- 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. 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. 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. 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. Compartmental model • By convention, in the nuclear medicine literature, the first compartment is the blood or plasma pool
- 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. 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. Three-tissue compartment model C0 C1 C2 C3 K1 k2 k3 k4 k5 k6
- 17. Three-tissue compartment model ( ) )()( )( )()( )( )()()()( )( 3615 3 2413 2 3624153201 1 tCktCk dt tdC tCktCk dt tdC tCktCktCkkktCK dt tdC −= −= ++++−=
- 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. Customized compartmental models • Glucose transport and phosphorylation in skeletal muscle with [18 F]FDG CA CEC CIC CM K1 k3 k4 k2 k5
- 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. 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. ... 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. 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. Applying differential equations • Simulation: calculate regional tissue curve based on – arterial plasma curve – model – physiological model parameters
- 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. 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. 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. 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. 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. 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. 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. PETO http://petintra/Instructions/PETO_manual.pdf
- 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. Additional information
- 35. Additional information
- 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. 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. 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. 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. 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. 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. 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. 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. ... continued • Convolution ∫ −=⊗ t dttbatbta 0 )()()()( ττ
- 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. ... continued • Solution for SRTM using Laplace transformation: t BP k REF I REFIT etC BP kR ktCRtC + − ⊗ + −+= 12 2 2 )( 1 )()(
- 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. ... 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. … 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. … 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

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