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# Barrier-limited model

## Theory

### Single capillary

The concentrations in plasma, C1 and in tissue, C2, 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, k1 and k2 are transfer coefficients or rate constants, defining the transfer of tracer between the plasma and tissue, and k3 is the transfer coefficient of sequestration of tracer due to metabolism and/or biliary excretion.

The boundary conditions are

• at t = 0: C1 = 0 and C2 = 0;
• at x = 0: C1 = Cin

where Cin 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: Cin(t) = δ(t).

In order to make calculations easier, concentrations are normalized by dividing through the injected amount, and distances are normalized by dividung through the linear velocity W, thus assuiming W = 1.

### Whole organ

The tracer concentration leaving the whole organ, Cdiff, is obtained as the flow-weighted average of the cocentration in the individual sinusoids. Let fc) be the fraction of total organ blood flow emerging from paths with sinusoidal transit times between τc and τc + dτc. 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

Cref = fc + t0)

Thus,

### Closed solution

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

The solution cosists of two components:

1. the troughput component, e-k1(t - t0) Cref(t), representing tracer that remained in the extracellular space
2. 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 applet:

• 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:

2. Choose the desired parameter set. This will automatically change the paremters as well as the data for the reference curve.
3. Click on “Run” to use the new parameter set.
4. 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 “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

1. Click on the Optimizer tab (at the bottom of the left pane)
2. 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.
3. 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.

Key Terms: indicator dilution, barrier-limited, liver, transport, vascular volume, organ, disse space

Andreas J. Schwab andreas.schwab@mcgill.ca