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

2. 3-compartment model C PLASM A C FREE C BOUND k3k3 k4k4 K1K1 k2k2

3. C PLASM A C FREE C BOUND k3k3 k4k4 K1K1 k2k2 Differential equations for compartment concentrations

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=p 1 x 1 +p 2 x 2 +...)

9. Same method applied to simplified reference tissue model (SRTM) Parameters can be solved by multilinear regression (Y=p 1 x 1 +p 2 x 2 +...) 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 model irreversible binding or trapped metabolite C PLASM A C FREE C BOUND k3k3 k 4 =0 K1K1 k2k2

11. C PLASM A C FREE C BOUND k3k3 K1K1 k2k2 Differential equations for compartment concentrations

12. Linearization #1 (k 4 =0) Sums, substitutions, rearrangements, integration http://pet.utu.fi/staff/vesoik/reports/tpcmod0006.pdf

13. Linearization #1 (k 4 =0) ”Patlak” plot Slope = K i Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol 2000;27:661-670.

14. Linearization #2 (k 4 =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 (k 4 =0) or after rearrangement Parameters can be solved by multilinear regression (Y=p 1 x 1 +p 2 x 2 +...)

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

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