1. MODELLING AND PARAMETER ESTIMATION IN PET Vesa Oikonen Turku PET Centre 2004-06-03

2. PET provides Quantitation of biochemical and physiological processes... per organ volume Noninvasive measurement In vivo

3. PET provides Perfusion: ml blood / (min * 100 g tissue) Glucose consumption: μ mol glucose / (min * 100 g tissue) Oxygen consumption: ml O 2 / (min * 100 g tissue) Amino acid uptake Fatty acid uptake Concentration and affinity of receptor (B max, K D )

4. PET also provides Change in perfusion (brain activation) Change in binding potential (receptor occupancy, endogenous ligand)

5. What PET actually provides Time course of the radioactivity concentration (TAC) in each image voxel (in units Bq/ml) Radioactivity concentration in blood plasma (input curve) is measured separately, or is replaced by reference region TAC from the image

6. Input curve tt Mixing in the heart Exchange with interstitial volume Exchange with intracellular volume Time delay Intravenous bolus infusionMeasured arterial plasma TAC

7. Correction for labeled metabolite in plasma

8. PET data ”input””output” Authentic tracer concentration available in arterial blood Concentration in tissue measured by PET scanner Perfusion Endothelial permeability Vascular volume fraction Transport across cell membranes Specific binding to receptors Non-specific binding Enzyme activity

9. Dynamic processes 1.Translocation 2.Transformation 3.Binding

10. Translocation Delivery and removal by circulatory system Active and passive transport over membranes Vesicular transport inside cells

11. Transformation Enzyme-catalyzed reactions: (de)phosphorylation, (de)carboxylation, (de)hydroxylation, (de)hydrogenation, (de)amination, oxidation/reduction, isomerization Spontaneous reactions

12. 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

13. First-order kinetics Models that can be reasonably analyzed with standard mathematical methods assume first- order processes Process is of ”first-order”, when its speed depends on one concentration only

14. First-order kinetics AP k For a first-order process A->P, the velocity v can be expressed as, where k is a first-order rate constant; it is independent of concentration and time; its unit is sec -1 or min -1.

15. First-order kinetics - radioactive decay Integrate: Subtract ln C F-18 (0) from both sides, and take exponentials: We have linear first-order ordinary differential equation (ODE):

16. Pseudo-first-order Usually process involves 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 process In PET: trace-dose

17. Definition: TRACER Tracer is a positron emitting isotope labeled molecule 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.

18. Definition: 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 MBq/ μ mol or mCi/ μ mol 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 High SA is required to reach sufficient count level without injecting too high mass

19. Compartment models Tracer is injected intravenously as a bolus Tracer is well mixed with blood at the heart Tracer is distributed by arterial circulation to the capillary bed, where exchange with tissue takes place Tracer concentration in tissue increases by extraction of tracer from plasma Concentration in tissue is reduced by backward transfer

20. Compartment models Physiological system is decomposed into a number of interacting subsystems, called compartments Compartment is a chemical species in a physical place Inside a compartment the tracer is considered to be distributed uniformly

21. Compartment models Change of tracer concentration in one of the compartments is a linear function of the concentrations in all other compartments:

22. 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).

23. One-tissue compartment model Change over time of the tracer concentration in tissue, C 1 (t) : C0C0 C1C1 K1K1 k2”k2”

24. One-tissue compartment model Linear first-order ordinary differential equations (ODEs) can be solved using Laplace transformation: C0C0 C1C1 K1K1 k2”k2”

25. Convolution

26. Alternative solution of ODEs 1. ODE is integrated, assuming that at t=0 all concentrations are zero:

27. Alternative solution of ODEs 2. Integral of nth compartment is implicitely estimated for example with 2nd order Adams-Moulton method: Integrals are calculated using trapezoidal method.

28. Alternative solution of ODEs 3. After substitution and rearrangement:

29. Two-tissue compartment model C0C0 C1C1 C2C2 K1K1 k2’k2’ k3’k3’ k4k4

30. C0C0 C1C1 C2C2 K1K1 k2’k2’ k3’k3’ k4k4, where Phelps ME et al. Ann Neurol 1979;6:371-388

31. Three-tissue compartment model C0C0 C1C1 C2C2 C3C3 K1K1 k2k2 k3k3 k4k4 k5k5 k6k6

33. Specific binding (k 3,k 4 ) and nonspecific binding (k 5,k 6 ) cannot be distinguished unless (k 5,k 6 ) >> (k 3,k 4 ) If (k 5,k 6 ) >> (k 3,k 4 ), then the system reduces to two-tissue compartment model

34. Fitting of compartment models to measured data Tissue TAC measured using PET is the sum of TACs of tissue compartments and blood in tissue vasculature Simulated PET TAC:

35. Fitting of compartment models to measured data Minimization of weighted residual sum-of-squares: Otherwise If measurement variance is known

36. Fitting of compartment models to measured data 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

37. Fitting of compartment models to measured data 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

38. Major steps in modelling Tracer selection Comprehensive model Workable model Model validation Model application Huang & Phelps 1986

39. Comparing models 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

40. Albert Einstein : ”Everything should be made as simple as possible, but not simpler”

41. Macroparameters Combination of model parameters can be computed with better reproducibility Reversible models: Distribution volume (DV) Irreversible models: Net influx rate (K i )

42. Distribution volume One-tissue compartment model Two-tissue compartment model Three-tissue compartment model

43. Distribution volume ratio Ratio between DV in region of interest and reference region (region without specific binding)

44. Binding potential (BP) Binding potential equals the concentration of free receptors, multiplied by affinity (1/K D ) and fraction of free tracer in C 1 ’ (combined C 1 and C 3 ) BP=DVR-1

45. Net influx rate One-tissue compartment model Two-tissue compartment model Three-tissue compartment model

46. Simplified reference tissue model (SRTM) Assumption #1: K 1 /k 2 is the same in all regions (R I =K 1 /K 1REF ) Assumption #2: 1-tissue compartment model would fit all TACs fairly well Lammertsma AA, Hume SP. Neuroimage 1996;4:153-158

47. Simplified reference tissue model (SRTM) Solution using Laplace transformation: Solution using 2nd order Adams-Moulton:

48. Multiple-time graphical analysis (MTGA) Data is transformed to a linear plot Macroparameter estimated directly as the slope of linear phase of plot Independent of compartments Reversible models: Logan analysis (DV, DVR) Irreversible models: Gjedde-Patlak analysis (K i )

49. Logan analysis with plasma input Distribution volume = Slope of the Logan plot Distribution volume ratio = Ratio of slopes of the ROI and reference region Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol 2000;27:661-670

50. Logan analysis with reference region input Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol 2000;27:661-670 Distribution volume ratio = Slope of the Logan plot calculated using reference region input BP = DVR - 1

51. Gjedde-Patlak analysis with plasma input Net influx rate Ki = Slope of the Patlak plot Unit of Ki = ml plasma * min -1 * ml tissue -1 Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. J Cereb Blood Flow Metab 1985;5:584-590.

52. Gjedde-Patlak analysis with reference region input Net influx rate Ki = Slope of the Patlak plot = k 2 *k 3 /(k 2 +k 3 ) Unit of reference input Ki = min -1 Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. J Cereb Blood Flow Metab 1985;5:584-590.

53. Mathematical model validation Residual curve must not show any time- dependent pattern (underparameterization) Considering the noise, standard errors of the fitted parameters should be small (overparameterization) Variable parameters must not be correlated (overparameterization)

54. Biochemical model validation Absolute accuracy of model parameters must be tested with a "gold standard", if one is available for the measurement of interest Intervention studies must be performed to estimate the sensitivity of the estimated parameters to the physiologic parameter of interest. Parameters of interest must not change in response to a perturbation in a different factor

55. Clinical model validation Repeatability coefficient (RC) and intraclass correlation coefficient (ICC) must be high (test-retest setting) Effect size and discriminating power must high (patient-control or treatment-placebo study)