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
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.
The tracer concentration leaving the whole organ, Cdiff, is obtained as the flow-weighted average of the cocentration in the individual sinusoids. Let f(τc) 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
The closed (“analytical”) solution for the whole-organ response is
The solution cosists of two components:
Download the JSim project file.
There are five parameter sets in this applet:
To change the parameter set:
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.
If you want to optimize the parameters of the newly selected experiments
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.
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