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Linearized models in PET

Linearized models in PET. Vesa Oikonen. http://pet.utu.fi/staff/vesoik/modelling/workshop2/linmod.ppt. 2003-06-05 Turku PET Centre – Modelling workshop Modelling workshop. 3-compartment model. K 1. k 3. C PLASMA. C FREE. C BOUND. k 2. k 4.

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Linearized models in PET

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  1. Linearized models in PET Vesa Oikonen http://pet.utu.fi/staff/vesoik/modelling/workshop2/linmod.ppt 2003-06-05 Turku PET Centre – Modelling workshop Modelling workshop

  2. 3-compartment model K1 k3 CPLASMA CFREE CBOUND k2 k4

  3. Differential equations for compartment concentrations K1 k3 CPLASMA CFREE CBOUND k2 k4

  4. Nonlinear estimation of model parameters Iterative minimization of weighted residual sum of squares

  5. Linearization #1 Sums, substitutions, rearrangements, integration http://pet.utu.fi/staff/vesoik/reports/tpcmod0007.pdf

  6. Linearization #1 ”Logan” plot Slope = DV Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol 2000;27:661-670.

  7. Linearization #2 Sums, substitutions, rearrangements, two integrations Blomqvist G. On the construction of functional maps in positron emission tomography. J Cereb Blood Flow Metab 1984;4:629-632

  8. Linearization #2 or after rearrangement Parameters can be solved by multilinear regression (Y=p1x1+p2x2+...)

  9. Same method applied to simplified reference tissue model (SRTM) Parameters can be solved by multilinear regression (Y=p1x1+p2x2+...) Zhou Y et al. Linear regression with spatial constraint to generate parametric images of ligand-receptor dynamic PET studies with a simplified reference tissue model. NeuroImage 2003;18:975-989.

  10. 3-compartment modelirreversible binding or trapped metabolite K1 k3 CPLASMA CFREE CBOUND k2 k4=0

  11. Differential equations for compartment concentrations K1 k3 CPLASMA CFREE CBOUND k2

  12. Linearization #1 (k4=0) Sums, substitutions, rearrangements, integration http://pet.utu.fi/staff/vesoik/reports/tpcmod0006.pdf

  13. Linearization #1 (k4=0) ”Patlak” plot Slope = Ki Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol 2000;27:661-670.

  14. Linearization #2 (k4=0) Sums, substitutions, rearrangements, two integrations Blomqvist G. On the construction of functional maps in positron emission tomography. J Cereb Blood Flow Metab 1984;4:629-632

  15. Linearization #2 (k4=0) or after rearrangement Parameters can be solved by multilinear regression (Y=p1x1+p2x2+...)

  16. Nonlinear models • Easy to set constraints for parameters • Applicable to all compartmental settings • Straightforward weighting • Predictable noise properties • Non-linearly affected by PVE and heterogeneity • Slow estimation of parameters • Commonly local minima • Cannot be applied to sinogram data

  17. Linear models • Fast calculation • Applicable to sinogram data • Linearly affected by PVE and heterogeneity • All models can not be linearized • Constraining parameters may be difficult • Noise may lead to bias • Weights are difficult to determine

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