fund bioimag 2012 7-1 7: two compartment modeling 1.what is compartmental modeling ? 2.how can...
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Fund BioImag 20127-1
7: Two compartment modeling
1. What is compartmental modeling ?
2. How can tracer kinetics be mathematically described ?
3. How do 2-deoxyglucose methods trace glucose metabolism ?
After this course you1. Understand how mass conservation can be used to model tracer kinetics and
estimate metabolic rates2. Understand the mathematical principle underlying metabolic modeling of
imaging data3. Can apply the principle of modeling tracer uptake to simple kinetic situations4. Understand the basics of modeling deoxyglucose uptake into tissue to extract
metabolic rates
Fund BioImag 20127-2
Importance of understanding basic modeling of linear systems
Situation I: Team A measures a high expression (mRNA) of a gene. Team B finds a low protein level at the same time point
Situation II: A year later, Team C reports low mRNA of the same system. Team D finds a high protein level
mRNA
protein
Controversy or is there a common underlying explanation?
Situation IISituation I
NB. Underlying mathematical principles also applicable to• Contrast agent dynamics
• Enzyme kinetics (increase in reaction velocity vs. product buildup)
• Diabetes (insulin vs. glucose uptake)
• Sailing (rudder/sail position vs. direction/speed change)
• Economics (financial incentives vs. production)
It’s all about inertia (resistance to change)It’s all about inertia (resistance to change)
Fund BioImag 20127-3
Imaging intracellular glucose metabolism
[18F]FDG (2-[18F]Fluoro-2-Deoxy-Glucose)
Hexokinase
GLUT-3GLUT-5
O
HOCH2
OH
OH
OH
18F
GLUT-2,4,7
G6-Phosphate-isomerase
X
X
O
18F
CH2OH
OH
OHOH
CH2OPO32-
GLUT-1
FDG uptake depends on:1. GLUT-Expression 2. Hexokinase-Activity
Cell
Autoradiography: Glucose metabolism using deoxyglucose
Fund BioImag 20127-4
7-1. Compartment models using tracers
Definition: Compartment
Concept: Physiological system - decomposed into N interacting subsystems
Subsystem = chemical species in a physical place (compartment)
NB. Tracer is considered to be distributed uniformly in compartment
Blood, pixels
(known)input
measuredoutputtime
Inaccessible portion
Accessible Portion
? AB
C
Key elements of compartmental modeling
1. Predict inaccessible features of system 2. Measurement in the accessible portion3. Estimation of specific parameters of
interest.
Steady-state assumption:1. metabolic rate of process is not changing with
time2. concentrations are constant during the
evaluation period.
k1
k2
FDG (intracellular) FDG-6-Phosphate
k3
k4
Hexokinase
Glucose-6-Phosphatase
ATP ADP
Glucose transporter
(GLUT-1)
OH
OH
H
OH
O
H
H H
18F
H2C OH
OH
OH
H
OH
O
H
H H
18F
H2C OH
OH
OH
H
OH
O
H
H H
18F
H2C OHP
FDG (plasma)
processes can be described with pseudo-first-order rate constants.
Fund BioImag 20127-5
Fick Principle (steady state conditions)
Metabolic rate (MR) of X consumption, MRX
Flow, f
f x
MRX = {[X]in – [X]out}
X concentration of blood entering tissue , Xin
Fick’s principleConservation of mass
X = O2, glucose, ammonia, water
tissue
blood
Brain physiology: O2 consumption increases less than FlowQ: What is the consequence?
Flow
rate of O2 consumption [O2] leaving – [O2] entering =
Definition Tracer• radio-activity emitting, labelled molecule• structurally related to the natural
substance (tracee) or involved in the dynamic process
– See earlier examples, but also O2 (left)
few tracer molecules contain radioactive isotope; others contain ”cold” isotope
Specific activity (SA) = “hot” / “cold” tracer molecules
SA is always measured; [MBq/μmol or mCi/μmol]
→ convert measured radioactivity concentrations in tissue and blood to mass (correct for physical decay)
X concentration in blood leaving tissue, Xout
introduced in a trace amount (=orders of magnitude below tracee); process being measured is not perturbed by it.
Fund BioImag 20127-6
7-2. First-order tracer kineticsOne-tissue compartment model
First-order process S->TReaction velocity V [µmol/g/min] :
K1, k3 - (pseudo) first-order rate constants;
independent of concentration and time;unit: [ sec-1 or min-1]
Unidirectional chemical reaction S → T:
)()()(
31 tCktCKdt
tdCTS
T
CS CT
K1
k3
V=
)()( *
1
**
tCKC
CV
dt
tdCS
S
ST
)()()( *
3*
1
*
tCktCKdt
tdCTS
T
The rate of labeled molecules entering CT dCT*/dt = Metabolic flux V x probability of
precursor CS labeled
Need to add efflux from CT: k3: Metabolic efflux V x probability of molecule CT being labeled
Infuse tracer with concentration Cs*
Measure tracer enrichment/specific activity CT*
How many labeled (red) molecules/per min ? (Assume the rate is V=10/min)
?)(*
dt
tdCT
kV/C
Fund BioImag 20127-7
One-tissue compartment model
Linear first-order ordinary differential equations (ODEs):
→ Laplace transformation
tkST etCKtC 3)(*)(* 1
t
dtbatbta0
)()()()(
)()( *
31
*
tCkkdt
tdCT
T Example: Cs* increased from 0 to a at t=0
tkT e
k
ktC 31)(
3
1*
CS CT
K1
k3
Plateau enrichmentT
race
r en
richm
ent
timeCT*(0)=0
)()()( *
3*
1
*
tCktCKdt
tdCTS
T
time
a
Cs*(t)
Fund BioImag 20127-8
Input curve
t
Mixing (heart)Exchange (interstitial, intracellular volume)
Intravenous bolus infusion
Measured arterial plasma (arterial input function) Cs*
0 15 30 45 60 75 900
10
20
30
40
50
60
70
80
90
Co
nc
en
tra
tio
n o
f a
uth
en
tic
tra
cer
(k
Bq
/mL
)
Time (min)
tracerconcentration
inarterial blood
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)
”output”, CT*
Concentrationin tissue
measured
Tracer: injected intravenously (as a bolus, i.e. Short time period)
1. well mixed with blood ( heart)
2. distributed to capillary bed→ exchange with tissue
3. Tracer concentration in tissue increases by extraction of tracer from plasma
4. Concentration in tissue is reduced by backward transfer
Uptake into tissue, e.g.Perfusion
Endothelial permeabilityVascular volume fraction
Transport across cell membranes
Specific binding to receptorsNon-specific binding
Enzyme activity
Fund BioImag 20127-9
7-3. Deoxyglucose (DG) measurement of glucose metabolism(autoradiography, FDG PET)
The problem:
wantedunwanted
)()( *
3
*
tCkdt
tdCfree
T
)()()()( *
32*
1
*
tCkktCKdt
tdCfreeS
free
Rapid glucose transport : CS*(t)Cfree*(t)
Metabolic rate of glucose MRGlc
Measured when Cfree*~0 (why?)
dttC
TC
LC
CMR T
S
TSGlc
0
*
*
)(
)(
dttCkTCT
freeT 0
*3
* )()(
Lumped constant (LC): differences between glucose and DG (affinities for transporters and hexokinase) CS: blood glucose concentration
Unit of MRglc: min
tissuemL
GlcmolMRglu
Voxel Measurement
BloodDG (CS*)
Free (tissue)DG
TissueDG6P
K1
k2
k3
Parameters:K1, k2, k3
0
0.5
1
1.5
2
2.5
3
3.5
0 20 40 60 80 100 120Time
Con
cent
ratio
n
Plasma measurement
(arterial input function)
T
MRGlc k3dttC
TCk T
S
T
0
*
*
3
)(
)(
Fund BioImag 20127-10
The typical FDG PET scan
45 min uptake phase (minimal tissue FDG)
then scan FDG-6P
Rodent FDG PET
Fund BioImag 20127-11
PET Reporter Gene Imaging
VectorReporter Gene
EnzymesHSV1-TK
HSV1-sr39TK
SubstratesFIAUFHBG
ReceptorsD2R
SSTR
LigandsFESP
SST-Analogues
TransportersNa-I-S
Iodine
Paradigm: Reporter Gene Gene Product Reporter Probe
Therapeutic effector gene productEffector Gene
mRNA
Translation
Fund BioImag 20127-12
Tracking tumor metabolism and cell proliferation
[11C]Methionine / [18F]F-Ethyl-Tyrosine
Protein Synthesis
Amino acid transporter
Tumor cell
Proteins
NH 3+
COO -11 CH
3
S
18OF NH
3+
COO -
X
TransferaseProtein catalysis