# Fund BioImag 2012 7-1 7: Two compartment modeling 1.What is compartmental modeling ? 2.How can tracer kinetics be mathematically described ? 3.How do 2-deoxyglucose.

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Fund BioImag 2012 7-1 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 you 1.Understand how mass conservation can be used to model tracer kinetics and estimate metabolic rates 2.Understand the mathematical principle underlying metabolic modeling of imaging data 3.Can apply the principle of modeling tracer uptake to simple kinetic situations 4.Understand the basics of modeling deoxyglucose uptake into tissue to extract metabolic rates

Fund BioImag 2012 7-2 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 II Situation 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)

Fund BioImag 2012 7-3 Imaging intracellular glucose metabolism [ 18 F]FDG (2-[ 18 F]Fluoro-2-Deoxy-Glucose) Hexokinase GLUT-3 GLUT-5 O HOCH 2 OH 18 F GLUT-2,4,7 G6-Phosphate- isomerase X X O 18 F CH 2 OH OH CH 2 OPO 3 2- GLUT-1 FDG uptake depends on: 1. GLUT-Expression 2. Hexokinase-Activity Cell Autoradiography: Glucose metabolism using deoxyglucose

Fund BioImag 2012 7-4 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 measured output time Inaccessible portion Accessible Portion ? A B C Key elements of compartmental modeling 1.Predict inaccessible features of system 2.Measurement in the accessible portion 3.Estimation of specific parameters of interest. Steady-state assumption: 1. metabolic rate of process is not changing withtime 2. concentrations are constant during the evaluationperiod. k1k1 k2k2 FDG (intracellular)FDG-6-Phosphate k3k3 k4k4 Hexokinase Glucose-6-Phosphatase ATP ADP Glucose transporter (GLUT-1) O H OH H OH O H H H 18 F H2CH2C OH O H OH H OH O H H H 18 F H2CH2C OH O H OH H OH O H H H 18 F H2CH2C OH P FDG (plasma) processes can be described with pseudo-first- order rate constants.

Fund BioImag 2012 7-5 Fick Principle (steady state conditions) Metabolic rate (MR) of X consumption, MR X Flow, f f xMR X = {[X] in – [X] out } X concentration of blood entering tissue, X in Fick’s principle Conservation of mass X = O 2, glucose, ammonia, water tissue blood Brain physiology: O 2 consumption increases less than Flow Q: What is the consequence? Flow rate of O 2 consumption [O 2 ] leaving – [O 2 ] 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 O 2 (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, X out introduced in a trace amount (=orders ofmagnitude below tracee); process beingmeasured is not perturbed by it.

Fund BioImag 2012 7-6 7-2. First-order tracer kinetics One-tissue compartment model First-order process S->T Reaction velocity V [µmol/g/min] : K 1, k 3 - (pseudo) first-order rate constants;  independent of concentration and time; unit: [ sec -1 or min -1 ] Unidirectional chemical reaction S → T: CSCS CTCT K1K1 k3k3 V= The rate of labeled molecules entering C T dC T */dt = Metabolic flux V x probability of precursor C S labeled Need to add efflux from C T : k 3 : Metabolic efflux V x probability of molecule C T being labeled Infuse tracer with concentration C s * Measure tracer enrichment/specific activity C T * How many labeled (red) molecules/per min ? (Assume the rate is V=10/min) k  V/C

Fund BioImag 2012 7-7 One-tissue compartment model Linear first-order ordinary differential equations (ODEs): → Laplace transformation Example: C s * increased from 0 to  at t=0 CSCS CTCT K1K1 k3k3 Plateau enrichment Tracer enrichmenttime C T *(0)=0 time  C s *(t)

Fund BioImag 2012 7-8 Input curve t Mixing (heart) Exchange (interstitial, intracellular volume) Intravenous bolus infusion Measured arterial plasma (arterial input function) C s * tracer concentration in arterial blood ”output”, C T * Concentration in 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 permeability Vascular volume fraction Transport across cell membranes Specific binding to receptors Non-specific binding Enzyme activity

Fund BioImag 2012 7-9 7-3. Deoxyglucose (DG) measurement of glucose metabolism (autoradiography, FDG PET) The problem: wantedunwanted Rapid glucose transport : C S *(t)  C free *(t) Metabolic rate of glucose MR Glc Measured when C free *~0 (why?) Lumped constant (LC): differences between glucose and DG (affinities for transporters and hexokinase) C S : blood glucose concentration Unit of MR glc : Voxel Measurement Blood DG (C S *) Free (tissue)DG Tissue DG6P K1K1 k2k2 k3k3 Parameters: K 1, k 2, k 3 0 0.5 1 1.5 2 2.5 3 3.5 020406080100120 Time Concentration Plasma measurement (arterial input function) T MR Glc  k 3

Fund BioImag 2012 7-10 The typical FDG PET scan 45 min uptake phase (minimal tissue FDG) then scan FDG-6P Rodent FDG PET

Fund BioImag 2012 7-11 PET Reporter Gene Imaging Vector Reporter Gene Enzymes HSV1-TK HSV1-sr39TK Substrates FIAU FHBG Receptors D2R SSTR Ligands FESP SST-Analogues Transporters Na-I-S Iodine Paradigm: Reporter Gene  Gene Product  Reporter Probe Therapeutic effector gene product Effector Gene mRNA Translation

Fund BioImag 2012 7-12 Tracking tumor metabolism and cell proliferation [ 11 C]Methionine / [ 18 F]F-Ethyl-Tyrosine Protein Synthesis Amino acid transporter Tumor cell Proteins NH 3 + COO - 11 CH 3 S 18 O F NH 3 + COO - X Transferase Protein catalysis

Mathematical Models Parameters: Slope, Intercept Model: Concentration = Slope x Time + Intercept

Mathematical Models Parameters: Slope, Intercept Model: Concentration = Slope x Time + Intercept Arterial Input Function Image Derived Samples Parameters: K 1, k 2, k 3, k 4

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