Comprehensive model

All available qualitative information of physiology and biochemistry of tracer is collected to provide a basis for interpretation of tracer kinetic measurements.

Basic steps to describe the transport of tracer from blood to tissue are:

1. Tracer is transported to capillaries by blood flow (perfusion)
2. Tracer is extracted across the capillary wall into the interstitial space
3. Tracer crosses the cellular membrane
4. In cells, the tracer participates in various biochemical reactions, or binds to receptors in synaptic clefts.

Tracer delivery

In PET studies the tracers are administered into the vascular system by intravenous injection or inhalation (intraperitoneal injection may be preferred in mice studies). Tracer is well mixed in the blood in heart chambers, so all arterial blood has the same concentration. Therefore, the concentration of the tracer (model input function) delivered to the tissue capillaries can be obtained from any peripheral artery, and the amount of tracer delivered to tissue is proportional to the blood flow (perfusion).

Blood consists of plasma and red blood cells (RBCs). The volume fraction of red blood cells (haematocrit) is smaller in arterioles and capillaries because of the higher velocity of RBCs, but the ratio of RBC flow to plasma flow is the same as in large vessels. Radiotracer may be present in plasma or RBC, or both, and the rate of exchange affects the amount of radiotracer that is available for tissue uptake. Plasma contains proteins, particularly albumin, which bind radiotracers with different affinities, affecting the clearance of the radiotracers.

Tracer extraction into tissue

Fraction of the radiotracer is extracted into tissue across the capillary wall, while the unextracted fraction is washed away by the venous blood flow. According to compartmental model for tracer delivery and extraction to tissue

, the extraction fraction E is related to blood flow (perfusion, f) and the capillary permeability-surface product of the tracer (PS) with equation

$E = \frac{PS}{PS+f} \notag$

Transport mechanisms

Model developer must know the mechanism of radiotracer transport across the capillary wall. Active transport (requires energy) can have a net flux of substance moved against a concentration gradient, while in passive transport the direction of net flux is determined solely by the direction of the concentration gradient across the membrane. Passive transport can be subdivided into facilitated transport and passive diffusion. When the concentration gradient of the molecule is kept steady by constantly supplying and depleting molecules at opposite sides of membrane, the net flux in passive diffusion is related only to the diffusion coefficient and the gradient. In some tissues substances can also cross the capillary wall by bulk flow through openings between the capillary endothelial cells. Facilitated transport, by special carriers in the membrane, allows a selective and regulated uptake of many important substrates, such as glucose and amino acids.

Facilitated transport can be considered as consisting of three steps: the combination of the substrate molecule with the carrier molecule, the assisted movement of the substrate-carrier complex, and the release of the substrate on the other side of the membrane. This process is very similar to the enzyme catalysed chemical reaction, being saturable and affected by competitive inhibition, and can usually be characterized with similar terms by its maximal transport rate (Vmax) and half-saturation concentration (KM). For natural substrate S, the facilitated transport is not a linear process, but for labelled tracer, which has a very low concentration, the linearity holds, and transport process is describable by compartmental models. Permeability-surface product (PS) can be expressed in terms of Michaelis-Menten constants and the concentration of natural substrate [S] as

$PS = \frac{V_{max}}{K_M+[S]} \notag$

Enzyme catalysed reactions

Most chemical reactions in living tissues are catalysed by enzymes, and can be described with Michaelis-Menten kinetics. For the tracer, the equation is linear with respect to its concentration; linearity holds even when the reactions are not following strictly Michaelis-Menten kinetics, as far as the concentration of tracer is low as compared to the natural substance.

Receptor binding

As an example, below is a representation of a comprehensive model for labelled neuroreceptor-binding ligand:

In this model, after the ligand is extracted from the vascular space to the free space in tissue (dissolved to water), it binds to non-specific binding sites in the interstitial and the cellular spaces. Some ligand molecules diffuse to the synapses of neural cells and bind to the specific neuroreceptors. The bindings to both specific and non-specific binding sites are reversible.

This model is far too complicated to be used to analyse PET studies, but based on that the modeller can start to construct a workable model.

Literature

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Liu D, Chalkidou A, Landau DB, Marsden PK, Fenwick JD. Interstitial diffusion and the relationship between compartment modelling and multi-scale spatial-temporal modelling of 18F-FLT tumour uptake dynamics. Phys Med Biol. 2014; 59(17): 5175-5202. doi: 10.1088/0031-9155/59/17/5175.

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Updated at: 2019-01-19
Created at: 2011-11-22
Written by: Vesa Oikonen