Model effect of erythrocyte membrane on exchange of solutes between erythrocytes (red blood cells) and plasma water.

The erythrocyte membrane potentially limits the exchange of solutes between erythrocytes (red blodd cells) and plasma water. The multiple indicator dilution technique can be used to investigate the effect of this barrier on the exchange of tracer substances.

Theory

Single capillary

The concentrations in plasma, C_p and in erythrocytes, C_RBC are discribed by the following system of partial differential equations in time t and space x:

where C_p and C_RBC are concentrations in plasma and in erythrocytes, W is the linear velocity of erythrocytes within the sinusoidal lumen, k₁ and k₂ are transfer coefficients or rate constants, defining the transfer of tracer between plasma and the cytoplasm of perythrocytes, β is the ratio of the red cell water space to the plasma water space in the blood, t₀ is the large vessel transit time, and γ is the ratio of the extravascular water space to the sinusoidal plasma water space.

Preequilibrated injection

Tracer to be injected are added to a blood sample or the same hematocrit as in systemic blood some time before injection. In this case, the tracer will equilibrate between plasma and erythrocytes, and the boundary conditions are

• at t = 0: C_p = 0 and C_RBC = 0;
• at x = 0: C_p = (1 + β)/(1 + k₁/k₂ β) C_in and C_RBC = k₁/k₂C_p

where C_in is the tracer concentration at the inflow of the organ. 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 concentration in the blood leaving the organ through the portal vein is calculated as the average weighted according to the flow in all sinusoidal paths (analogous to the case of "barrier-limited transport") and the relative volume of plasma and erythrocytes:

Experimental design

The following tracers were injected into the portal vein of an anesthetized dog:

• Erythrocytes (Ery) labeled with ⁵¹Cr
• Labeled thiourea (NH₂-¹⁴CS-NH₂), a substance that penetrates easily into hepatocytes, and less easily into erythrocytes
• Tritiated water(³HOH)

Samples from a hepatic vein were analyzed for beta radioactivity, and the various isotopes where distinguished by scintillation spectrometry. In the following applet o click on "Run" to calculate the curves o click on plotpage_1 to see the data and the calculated curves

Calculations