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:

- uniform capillary transit time independent of τ, with variable large-vessel transit time
- variable capillary transit time τ
_{c}= τ –*t*_{0}with uniform large-vessel transit time,*t*_{0}. - variable capillary and variable large-vessel transit times
- random coupling: capillary and non-capillary transit times are stochastically independent from each other
- flow coupling: for each total flow-path transit time τ, there is a single capillary transit time τ
_{c}- linear relation between capillary and large-vessel transit times
- 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*(*t*c) = (*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 *t*_{0} = (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 *t*_{0} 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*_{app}*b*´)/*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:

^{125}I-albumin (as an indicator for the vascular space)- [U-
^{14}C]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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