Barrier-limited model of Goresky, using Finite difference method

## 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}

where *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

### 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.

### 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.

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

- 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)

Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below:

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

**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.