modelling and parameter estimation in pet vesa oikonen turku pet centre 2004-06-03
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PET provides
• Quantitation of biochemical and physiological processes
• ... per organ volume• Noninvasive measurement• In vivo
PET provides
• Perfusion: ml blood / (min * 100 g tissue)• Glucose consumption: μmol glucose /
(min * 100 g tissue)• Oxygen consumption: ml O2 / (min * 100
g tissue)• Amino acid uptake• Fatty acid uptake• Concentration and affinity of receptor
(Bmax , KD)
PET also provides
• Change in perfusion (brain activation)
• Change in binding potential (receptor occupancy, endogenous ligand)
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
Input curve
t t
Mixing in the heartExchange with interstitial volume
Exchange with intracellular volumeTime delay
Intravenous bolus infusion Measured arterial plasma TAC
Correction for labeled metabolite in plasma
0 15 30 450.0
0.2
0.4
0.6
0.8
1.0
Fra
ctio
n o
f au
then
tic
trac
er
Time (min)
0 15 30 45 600
10
20
30
40
50
60
70
80
90
100
Rad
ioac
tivi
ty c
on
c. (
kBq
/mL
)
Time (min)
Plasma TAC Authentic tracer Metabolized tracer
PET data
0 15 30 45 60 75 900
10
20
30
40
50
60
70
80
90
Co
nce
ntr
atio
n o
f au
then
tic
trac
er (
kBq
/mL
)
Time (min)
0 15 30 45 60 75 900
10
20
30
40
50
60
70
80
90
Co
nce
ntr
atio
n in
tis
sue
(kB
q/m
L)
Time (min)
”input” ”output”Authentic tracerconcentrationavailable in
arterial blood
Concentrationin tissue
measured byPET scannerPerfusion
Endothelial permeabilityVascular volume fraction
Transport across cell membranes
Specific binding to receptorsNon-specific binding
Enzyme activity
Translocation
• Delivery and removal by circulatory system
• Active and passive transport over membranes
• Vesicular transport inside cells
Transformation
• Enzyme-catalyzed reactions: (de)phosphorylation, (de)carboxylation, (de)hydroxylation, (de)hydrogenation, (de)amination, oxidation/reduction, isomerization
• Spontaneous reactions
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
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
First-order kinetics
A Pk
For a first-order process A->P, the velocity vcan be expressed as
)()()(
tCkdt
tdC
dt
tdCv A
AP
, where k is a first-order rate constant;it is independent of concentration and time;its unit is sec-1 or min-1.
First-order kinetics -radioactive decay
eOF k 1818
dtktC
tdCdtk
tC
tdCtkC
dt
tdC
F
F
F
FF
F
)(
)(
)(
)()(
)(
18
18
18
1818
18
Integrate:)0(ln)(ln 1818 FF CdtktC
Subtract ln CF-18(0) from both sides, and take exponentials:
tk
F
F eC
tC
)0(
)(
18
18
We have linear first-order ordinary differential equation (ODE):
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
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.
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/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
• High SA is required to reach sufficient count level without injecting too high mass
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
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
Compartment models
• Change of tracer concentration in one of the compartments is a linear function of the concentrations in all other compartments:
),(),(),()(
210 tCtCtCfdt
tdCi
i
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).
One-tissue compartment model
• Change over time of the tracer concentration in tissue, C1(t) :
)()()(
1"201
1 tCktCKdt
tdC
C0 C1
K1
k2”
One-tissue compartment model
• Linear first-order ordinary differential equations (ODEs) can be solved using Laplace transformation:
tketCKtC"2)()( 011
C0 C1
K1
k2”
Alternative solution of ODEs
TT
dttCkdttCKTC0
1"2
0
011 )()()(
1. ODE is integrated, assuming that at t=0all concentrations are zero:
Alternative solution of ODEs
)(2
)(2
)()(00
TCt
tTCt
dttCdttC nn
tT
n
T
n
2. Integral of nth compartment is implicitely estimated for example with 2nd order Adams-Moulton method:
Integrals are calculated using trapezoidalmethod.
Alternative solution of ODEs
"2
1
0
1"2
0
01
1
21
)(2
)()(
)(kt
tTCt
dttCkdttCK
TC
tTT
3. After substitution and rearrangement:
Two-tissue compartment model
C0 C1 C2
K1
k2’
k3’
k4
)()(
)(
)()()()(
241'3
2
241'3
'201
1
tCktCkdt
tdC
tCktCkktCKdt
tdC
Two-tissue compartment model
C0 C1 C2
K1
k2’
k3’
k4
)()(
)()(
012
'31
2
0421412
11
21
21
tCeekK
tC
tCekekK
tC
tt
tt
, where 24
24
4'2
2
4'3
'24
'3
'22
4'2
2
4'3
'24
'3
'21
kkkkkkkk
kkkkkkkk
Phelps ME et al. Ann Neurol 1979;6:371-388
Three-tissue compartment model
)()()(
)()()(
)()()()()(
36153
24132
36241532011
tCktCkdt
tdC
tCktCkdt
tdC
tCktCktCkkktCKdt
tdC
Three-tissue compartment model
• Specific binding (k3,k4) and nonspecific binding (k5,k6) cannot be distinguished unless (k5,k6) >> (k3,k4)
• If (k5,k6) >> (k3,k4), then the system reduces to two-tissue compartment model
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:
i
iBBBS tCVtCVtC )(1)()(
Fitting of compartment models to measured data
MinptCtCwN
iiSiPETi
1
22 ˆ,
Minimization of weighted residualsum-of-squares:
Otherwise
If measurement variance is known
2
1
iiw
1iw
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
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
Major steps in modelling
Tracer selection
Comprehensive model
Workable model
Model validation
Model applicationHuang & Phelps 1986
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
Macroparameters
• Combination of model parameters can be computed with better reproducibility
• Reversible models:Distribution volume (DV)
• Irreversible models:Net influx rate (Ki)
Distribution volumeOne-tissuecompartmentmodel
Two-tissuecompartmentmodel
Three-tissuecompartmentmodel
6
5
4
3
2
1 1k
k
k
k
k
KDV
4
'3
'2
1 1k
k
k
KDV
"2
1
k
KDV
Distribution volume ratio
• Ratio between DV in region of interest and reference region (region without specific binding)
REF
ROI
DV
DVDVR
Binding potential (BP)
• Binding potential equals the concentration of free receptors, multiplied by affinity (1/KD) and fraction of free tracer in C1’ (combined C1 and C3)
• BP=DVR-1
DK
Bf
k
kBP
'max
24
'3
Net influx rateOne-tissuecompartmentmodel
Two-tissuecompartmentmodel
Three-tissuecompartmentmodel
32
31'3
'2
'31
kk
kK
kk
kKK i
32
132
123
:
:
kk
KKkk
KKkk
i
i
32
31
kk
kKK i
Simplified reference tissue model (SRTM)
• Assumption #1: K1/k2 is the same in all regions (RI=K1/K1REF)
• Assumption #2: 1-tissue compartment model would fit all TACs fairly well
)(1
)()()( 2
2 tCBP
ktCk
dt
tdCR
dt
tdCTREF
REFI
T
Lammertsma AA, Hume SP. Neuroimage 1996;4:153-158
Simplified reference tissue model (SRTM)
• Solution using Laplace transformation:
tBP
k
REFI
REFIT etCBP
kRktCRtC
12
2
2
)(1
)()(
• Solution using 2nd order Adams-Moulton:
BPkt
tTCt
dttCBPk
dttCkTCR
TCT
tT
T
T
REFREFI
T
121
)(2
)(1
)()(
)(2
0
2
0
2
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 (Ki)
Logan analysiswith plasma input
0 5 10 15 20 25 300
20
40
60
80
100
120
140
160
CR
OI i
nte
gra
l / C
RO
I
CPLASMA
integral / CROI
Distribution volume=
Slope of the Logan plot
Distribution volumeratio =Ratio of slopes of theROI and referenceregionLogan J. Graphical analysis of PET data applied to reversible
and irreversible tracers. Nucl Med Biol 2000;27:661-670
Logan analysiswith reference region input
Logan J. Graphical analysis of PET data applied to reversibleand irreversible tracers. Nucl Med Biol 2000;27:661-670
Distribution volume ratio=
Slope of the Logan plotcalculated using
reference region input
0 10 20 30 40 500
20
40
60
80
100
120
140
160
CR
OI i
nte
gra
l / C
RO
I
CREFERENCE
integral / CROI
BP = DVR - 1
Gjedde-Patlak analysiswith plasma input
0 20 40 60 80 1000.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
CR
OI/C
PL
AS
MA (
mL
/mL
)
CPLASMA
integral/CPLASMA
(min)
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.
Gjedde-Patlak analysiswith reference region input
0 10 20 30 40 500
1
2
3
4
5
6
CR
OI/C
RE
F
CREF
integral/CREF
(min)
Net influx rate Ki=
Slope of the Patlak plot=
k2*k3/(k2+k3)
Unit of referenceinput 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.
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)
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