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

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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