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). Tracers are 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).

Tracer extraction into tissue

Fraction of the tracer is extracted into tissue across the capillary wall, while the unextracted fraction is washed away by the venous blood flow. Compartmental model for tracer delivery and extraction to tissue (Graham, 1985):

Tissue extraction

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

Because of the back transport from tissue to blood, the net extraction fraction (arteriovenous concentration difference divided by the arterial concentration) is usually lower than unidirectional extraction fraction.

Transport mechanisms

Model developer must know the mechanism of tracer 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 catalyzed 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 labeled tracer, which has a 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

Enzyme catalyzed reactions

Most chemical reactions in living tissues are catalyzed 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 labeled neuroreceptor-binding ligand:

Comprehensive model for receptor tracer

In this model, after the ligand is extracted from the vascular space to the free space in tissue (dissolved to water), it binds to nonspecific 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 nonspecific binding sites are reversible.

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


See also:


References:

Budinger TF, Huesman RH, Knittel B, Friedland RP, Derenzo SE. Physiological modeling of dynamic measurements of metabolism using positron emission tomography. In: Greitz T et al. The metabolism of the human brain studied with positron emission tomography. Raven Press, New York, 1985.

Carson RE. The development and application of mathematical models in nuclear medicine. J Nucl Med. 1991; 32:2206-2208.

Garfinkel D. Computer modeling, complex biological systems, and their simplifications. Am J Physiol. 1980; 239: R1-R6.

Graham MM. Model simplification: complexity versus reduction. Circulation 1985; 72:IV63-IV68.

Khoshmanesh K, Kouzani AZ, Nahavandi S, Baratchi S, Kanwar JR. At a glance: cellular biology for engineers. Comput Biol Chem. 2008; 32: 315-331.

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.

Phair RD. Development of kinetic models in the nonlinear world of molecular cell biology. Metabolism 1997; 46:1489-1495.

Stillwell W: An Introduction to Biological Membranes - From Bilayers to Rafts. Academic Press, 2013.



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Created at: 2011-11-22
Updated at: 2016-03-08
Written by: Vesa Oikonen