An Incomplete Medical Physics Review

Tracer Kinetic Modeling

Tracers and compartment modeling is a critical tool for more advanced imaging applications.

Kinetic modeling uses time varying radiotracer distribution to extract additional information from an imaging session.

Useful to assess receptor binding, diffusion, active transport, metabolism, washout, excretion, etc. 

Requires dynamic imaging - frames at specific times and durations during and after the introduction of a tracer.

Model applied to kinetics can extract physiologic quantities, e.g., perfusion in mL/min/g. 

Ideal tracer: identical/relatable behavior to natural substance, does not alter physiological process (tracer <1% of endogenous compound). High specific activity  →  quality imaging and blood samples without altering physiology. Change in mass of isotope should not affect biochemistry ("isotope effect").
If labeled with a different radionuclide than naturally occurs, it should behave the same or in a predictably different manor, e.g. FDG not going through full cycle that glucose does (actually an advantage).

Compartment models

Compartment: space where a tracer rapidly become uniformly distributed.  E.g., blood pool or a particular cell type.  
Closed (cannot escape) or open (can move to other compartments). Depends on compartment and tracer. 

Distribution volume: $V_d = A/C$ where $A$ is activity of the tracer (Bq) and $C$ is the concentration of the tracer (Bq/mL).  "Dilution principle", conservation of mass.

Partition Coefficient: $\lambda \text{(mL/g)} = C_t \text{(Bq/g)} / C_b \text{(Bq/mL)}$ where $C_t$ and $C_b$ are tissue and blood concentrations at equilibrium.

Flux: Amount of substance that crosses a boundary or surface (or between compartments) per unit time. 

Rate Constants (k): describe relationship between concentrations and fluxes of a substance between compartments. 
Flux = k × amount of substance.  $k$ in units of inverse time, flux in mass/time or mass/volume/time. 
Transport often described exponentially: $e^{-kt}$ for transport out of a compartment at a rate of $k$ at time $t$.

Steady State: process not changing with time. E.g., flux in steady state if concentrations of reactants and products are not changing over time. Never really exists biologically, but can get close enough.
Kinetic models can use a tracer steady state, but often incorporate periods of change.

Blood flow, extraction, clearance: inject, delivery to capillary, extract across capillary wall into tissue, incorporate into biochemical reaction. 

Blood flow through vessels in volume/time (e.g. mL/min) 
perfusion: blood flow per mass of tissue (mL/min/g)

Net extraction: difference in steady-state between input and output blood flow of an organ. $E_n = (C_A-C_V)/C_A$
where $C$s are arterial and venous concentrations. 
Unidirectional extraction: tracer extracted from blood to tissue only. $E_u \gt E_n$ (but equal for O2). 
Fick principle: conservation of mass, in steady-state, net uptake of tracer is difference between input to and output from tissue. Net uptake rate: $U = F \times (C_A - C_V)$ where $F$ is blood flow.
Extraction depends on permeability $P$, surface area $S$, and blood flow $F$: $E_u = 1 - e^{-(P \times S / F)}$ (simplified model).  $F$ = blood flow through capillary tube while $P \times S$ is flow across capillary wall.
Clearance: $F \times E$ amount of material entering tissue. 

Transport

  • Active - requires energy (e.g. ATP), possible to move across concentration gradients. E.g. sodium-potassium pump
  • Passive - no energy required, in direction of concentration gradient
    • Carrier mediated diffusion: e.g. glucose, amino acid from blood to brain
      • Carrier + substrate crosses, then dissociates.
      • Can be saturated if limited carriers (e.g. protein enzyme)
    • Passive diffusion: along concentration gradient, e.g., 99mTc-pertechnetate from blood to brain in disrupted BBB. 
      • Permeability $P \text{(cm/min)} = D \text{(cm^2/min)} / x \text{(cm)}$ where $D$ is a diffusion constant and $x$ is membrane thickness (diffusion path length).
      • Examples: water, oxygen, ammonia, CO2

Formulation

Typically use first-order rate constants to describe flux between compartments. 

Two-compartments A and B: $dC_b (t)/dt = k_1 C_A(t) - k_2 C_B(t)$ (first order ODE). 
$C_A(t)$ is input function, $k$s are model parameters into and out of B. Measure $C_A$ from blood samples, get $C_B(t)$ from dynamic images, and estimate $k$s through regression analysis. 

Examples

Renal function (glomerular filtration rate) with Tc-99m DTPA (planar scintigraphy). ROI on each kidney for many frames + blood samples to measure tracer concentration over time. 

Cardiac function and ejection fraction: label red blood cells (RBCs) with 99mTc. Inject as bolus, image dynamically with gamma camera to show blood flow through veins to right atrium, right ventricle, lungs, left atrium, left ventricle, aorta to body. Look for abnormal flow patterns. e.g. intracardiac shunt from right to left ventricle, then left ventricle get activity sooner than expected. 
Once RBCs are uniformly distributed, images reflect regional blood volume. Can see changes in ventricular volume over time, calculate ejection fraction from end-diastolic and end-systolic time points (counts proportional to blood volume): $EF = (EDV - ESV) / EDV$.

Blood flow: typically either trapping, clearance or equilibrium technique.
Trapping: distributed according to blood flow, then then stay in tissue (e.g. 99mTc-MAA, 13NH3).
Clearance: inert, washout also depends on blood flow, may remain in vasculature, or diffuse into/out of tissues.
Equilibrium: continuous supply of diffusible tracer, reach steady-state, image distribution (e.g. 15O, not common)

Receptor Ligands

Receptors on cell surfaces are useful molecular targets the expression of which can be used to assess the presence/state of a certain disease (AD, Parkinson's, cancers, etc). 

Radioligands can be used with PET and SPECT to bind to receptors and imaging the binding. 
Binding is usually reversible. At equilibrium, $k_{on} L\,\,R = k_{off} LR$ where $L$, $R$ and $LR$ are the concentrations of unbound ligand and receptor, and the bound complex. 
Equilibrium dissociation constant: $K_d = k_{off}/k{on}$
Affinity: $1/K_d$, strength of ligand-receptor binding.
Total concentration of receptors: $B_{max} = R + RL$
$RL = \frac{B_{max} L}{L + K_d}$ where $RL$ can be measured as a function of $L$ in vitro, and data fit to determine $K_d$ and $B_max$
For imaging, use trace amounts of $L$ such that $RL < 0.1\,R$ → $BP = \frac{B_{max}}{K_d} = \frac{RL}{L}$ where $BP$ is binding potential, i.e., ratio of bound to free ligand concentration at equilibrium. Can be used as the relative concentration of receptors, $B_{max}$.

Kinetic modeling of binding with dynamic images: 
Ligand can be in several states: in blood, free in tissue, non-specifically bound in tissue, bound to receptor in tissue.
Simplify by assuming rapid transfer between free and non-specific, combine in to "non-displaceable" compartment. Now have three compartment / two tissue model with four rate constants: K1 plasma to ND, k2 ND to plasma, k3 ND to bound, k4 bound to ND.

Differential equations:
$$dC_{ND} (t) / dt = K_1 C_p (t) - k_2 C_{ND} (t) - k_3 C_{ND}(t) + k_4 C_b (t)$$
$$dC_{b} (t) / dt = k_3 C_{ND} (t) - k_4 C_b (t)$$

At equilibrium, assume $k_3 C_{ND} = k_4 C_b \rightarrow C_b / C_{ND} = k_3/k_4 = BP_{ND}$

$K_1 C_p (t) + k_4 C_b (t) = k_2 k_4 C_b (t) / k_3 + k_4 C_b (t) \rightarrow C_b/C_p = \frac{K_1 k_3}{k_2 k_4} = BP_p$

Input function $C_p$ can be obtained from arterial sampling, and the combination of $C_{ND}$ and $C_b$ from imaging. 
Displacement studies: $BP$ can be altered by administration of certain drugs or actions, adding cold ligands, or activating endogenous ligands.

Note: all models are wrong, but some are useful!


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