# barrier

Barrier-limited model of Goresky, using Finite difference method

Model number: 0076

## Theory

### Single capillary

The concentrations in plasma, *C*_{1} and in tissue, *C*_{2}, follow the following system of two differential equations in time, *t*, and space, *x*:

where *W* is the linear velocity of the tracer the sinusoidal plasma, *k*_{1} and *k*_{2} are transfer coefficients or rate constants, defining the transfer of tracer between the plasma and tissue, and *k*_{3} is the transfer coefficient of sequestration of tracer due to metabolism and/or biliary excretion.

The boundary conditions are

- at
*t*= 0:*C*_{1}= 0 and*C*_{2}= 0; - at
*x*= 0:*C*_{1}=*C*_{in}

*C*

_{in}is the tracer concentration at the inflow of the organ. For quasi-instantaneuse injection, administration of tracer will be represented by the Dirac impulse function:

*C*

_{in}(

*t*) = δ(

*t*). In order to make calculations easier, concentrations are normalized by dividing through the injected amount, and distances are normalized by dividing through the linear velocity

*W*, thus assuiming

*W*= 1.

### Whole organ

The tracer concentration leaving the whole organ, *C*_{diff}, is obtained as the flow-weighted average of the cocentration in the individual sinusoids. Let *f*(τ_{c}) be the fraction of total organ blood flow emerging from paths with sinusoidal transit times between τ_{c} and τ_{c} + dτ_{c}. The mixed outflow concentration from the whole organ, *C*(t), will then be the flow-weighted average of the outflow concentrations, according to the integral

For the reference tracer (sucrose), the normalized concentration at the portal vein is

*C*_{ref} = *f*(τ_{c} + t_{0})

Thus,

### Closed solution

The closed ("analytical") solution for the whole-organ response is

The solution consists of two components:

- The throughput component,
*e*^{-k1(t - t0)}*C*_{ref}(*t*), representing tracer that remained in the extracellular space. - The returning component, repesented by the second line of the above equation, representing tracer that has entered the tissue at least one and has returned to the extracellular space.

## Calculations

### Download JSim model project file

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### Parameter sets

There are five parameter sets in this JSim model project file:

- Gal1: Galactose experiment with no galactose infused
- Gal2: Galactose experiment with high galactpse concentration
- Pal1: Palmitate acid experiment (normal control)
- Pal2: Palmitate acid experiment with infusion of α-bromopalmitate
- Rb: Rubidium experiment

To change the parameter set:

- Select Load project parameter set from the ParSet pulldown menu
- Choose the desired parameter set. This will automatically change the paremters as well as the data for the reference curve.
- Click on "Run" to use the new parameter set.
- In order to show the correct data in the plot, select the data set and the tracer from the pulldown menus labeled "data".

### Closed solution

Select model "ClosSol" from the pulldown menu activated by the "Models" tab. Notice that the calcuation using the closed solution is much faster than the finite-different caculation. The parameter set has to be changed separately for each calculation method.

## Model Feedback

We welcome comments and feedback for this model. Please use the button below to send comments:

### Optimization

If you want to optimize the parameters of the newly selected experiments

- Click on the Optimizer tab (at the bottom of the left pane)
- Change all the DataSet entries under the "Data to Match" heading.
- Click on the Dataset entries to get a selection of data sets to choose from.
- Click on the Curve entries to get a selection of data columns to choose from.

- Hit the "Run" button.

## References

- Goresky CA, Bach GC, Nadeau BE. On the uptake of materials by the intact liver. The concentrative transport of rubidium-86.
*J Clin Invest*52:975-990, 1973. - Goresky CA, Bach GC, Nadeau BE. On the uptake of materials by the intact liver. The transport and net removal of galactose.
*J Clin Invest*52:991-1009, 1973. - Goresky CA, Daly DS, Mishkin S, Arias IM. Uptake of labeled palmitate by the intact liver: role of intracellular binding sites.
*Am J Physiol*234:E542-53, 1978.

**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, barrier-limited, liver, transport, vascular volume, organ, disse space, transport physiology, whole organ model, Goresky, Data, Tutorial, PDE

## Model History

Get Model history in CVS.## Acknowledgements

Please cite **www.physiome.org** in any publication for which this software is used and send an email
with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.

Or send a copy to:

The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

[This page was last modified 30Jul13, 10:13 am.]

**Model development and archiving support at
physiome.org 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.