Linear Flow Coupling. Model I. Const Capillary Transit Time (TT) and Varying Large Vessel TT. Model II. Constant Large Vessel TT and Varying Capillary TT. Model III: Linear relation between capillary and large-vessel TT.

Model number: 0159

Linear Flow Coupling

To model the movement of substances through an organ, the movement along a single path of blood flow through the microcirculation is considered first.

Each flow path consists of two portions: a capillary-bed portion, where exchange with extravascular space occurs, and a noncapillary-bed portion consisting of arteries, arterioles, venules, and veins where all indicators are confined to the vasculature. In linear systems, the order of serial connection is irrelevant, such that arteries, arterioles, venules, and veins (the large vessels) can be lumped together and described by a single distribution of transit times.

The tracer concentration at the outflow of the organ is the flow-weighted average of the outflow concentrations from all the single flow paths. Flow paths of given total transit times, τ, vary in capillary transit time, τ_c, according to the conditional probability distribution, P(τ_c|τ).

The following special cases can be considered:

1. uniform capillary transit time independent of τ, with variable large-vessel transit time
2. variable capillary transit time τ_c = τ – t₀ with uniform large-vessel transit time, t₀.
3. variable capillary and variable large-vessel transit times
   1. random coupling: capillary and non-capillary transit times are stochastically independent from each other
   2. flow coupling: for each total flow-path transit time τ, there is a single capillary transit time τ_c
      1. linear relation between capillary and large-vessel transit times
      2. non-linear relation between capillary and large-vessel transit times

 

Theory

Model I: Constant Capillary Transit Time and Varying Large Vessel Transit Times

For the intravascular tracer (albumin), the output concentration is

C_ref(t) = (q/Fr(tc) = (q/F) r(τ_l)

where r(τ_l) is the distribution of large vessel transit times, τ_c is the common capillary transit time, q is the total amount of tracer injected, and F is total coronary flow.

For the exchanging diffusible tracer:


Model parameters to be fitted are:

• k_s, the permeability-surface product for efflux from and influx into the capillary per volume of accessible extravascular space
• Φ = 1/(γ τ_c), the flow per volume of accessible extravascular space

t_app is the time when the label first appears at the exit of the organ, and γ is the ratio of the volume of accessible extravascular space to that of the vascular space.

 

Model II. Constant Large Vessel Transit Time and Varying Capillary Transit Times

This case is the same as the barrier-limited case for the liver.

For the reference tracer C_ref(t) = (q/F) n(t – τ_l) = (q/F) n(τ_c) where n(τ_c) is the distribution of capillary transit times, and τ_l is the common large vessel transit time.

For the diffusible tracer

Model parameters to be fitted are:

• k_s, the permeability-surface product for efflux from and influx into the capillary per volume of accessible extravascular space
• γ, the ratio of the volume of accessible extravascular space to that of the vascular space
• τ_l, the large vessel transit time

τ_cm = t_app – τ_l is the minimal capillary transit time, where t_app is the appearance time of the tracer.

 

Model III: Linear relation between capillary and large-vessel transit time

The capillary transit time of a flow path with transit time t is

τ_c(t) = τ_cm + b(t – τ_cm – τ_lm )

and the large-vessel transit time of a flow path with transit time t is

τ_l(t) = τ_lm + (1 – b)(t – τ_cm – τ_lm)

where τ_cm and τ_lm are the minimal capillaray and large vessel transit times, respectively, with τ_cm + τ_lm = t_app.

For the reference tracer C_ref(t) = (q/F) w(t – τ_l – τ_lm) where w(τ_c) is the distribution of capillary transit times.

For the diffusible tracer
 

Substitution of τ_c and τ_l yields

where t₀ = (1-1/b)τ_cm + τ_lm = t_app – a´/b´ and b´ = k_s γ b.

Model parameters to be fitted are:

• k_s, the permeability-surface product for efflux from and influx into the capillary per volume of accessible extravascular space
• a´ = k_s γ τ_cm
• b´ = k_s γ b.

 

Relation between the models

Model III can be formulated as a reparametrization of Model II, such that b´/k_s in Model III is equivalent to γ in Model II, and t₀ in Model III is equivalent to τ_l. Model II is a realization of the data only if the fitted value for τ_l> 0. If τ_l < 0, only Model III can be applied to the data. In the latter case, the heterogeneity of the capillariy transit times is bounded by

b < (t_appb´)/a´

and the extravascular/vascular volume ratio is bounded by

γ > a´/(t_app k_s)

Experimental design

The following tracers were injected into the coronary artery of an anesthetized dog:

• ¹²⁵I-albumin (as an indicator for the vascular space)
• [U-¹⁴C]sucrose (as an indicator for the combined vascular and interstitial spaces)

Samples obtained at the coronary sinus were analyzed for beta and gamma radioactivity.

 

With each dog, a control injection (run 1) was performed, followed by a second injection (run 2) performed in a vasodilated state. In Experiment 1, vasodilation was achieved by intracoronary infusion of flavaspidic acid. In Experiment 6, vasodilatation was achieved by intracoronary infusion of 2-bromopalmitate.

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References

CP Rose and CA Goresky. Vasomotor control of capillary transit time heterogeneity in the canine coronary circulation. Circ. Res. 39:541-554, 1976

Author: Andreas J. Schwab (andreas.schwab@mcgill.ca)

Related Models

Back to Goresky Modeling of transport and metabolism tutorial

• Flow limited model
• Simple elimination with flow-limited distribution
• Barrier limited model
• Linear flow coupling
• Two-barrier model
• Red cell model

Key Terms

indicator dilution, flow-limited, liver, transport, vascular volume, organ, disse space, Goresky transport tutorial

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Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.