16 - Osmotic pressure

Summary

Consider a membrane of thickness 10 μm that has a number of tiny cylindrical pores (of radius 10 nm) passing through it. The density of pores in the membrane is such that the porosity (fractions of water-filled space) of the membrane is 0.1%.

1. (a) Find the hydraulic conductivity (L_p, flow rate per unit area per unit pressure drop) of this membrane.

2. (b) Consider a 4 mM solution of a large protein on one side of this membrane and physiologic saline on the other, with the same pressure on both sides of the membrane. Assume that the protein is sufficiently large that it cannot pass through the membrane and that van ’t Hoff’s law holds for this solute. Calculate the initial flow rate of saline through a membrane of area 5 cm² at a temperature of 300 K.

The graph shown in the figure overleaf is adapted from a 1927 paper [16] in which Landis proved the existence of Starling’s phenomenon by occluding capillaries. The ordinate is the volume of fluid leaking out of (or re-entering) the capillary per unit capillary wall area, j. Although it is not precisely true, for the purposes of this question you may assume that the reflection coefficient of this capillary wall to plasma proteins is unity.

1. (a) Assuming that p – Π for the interstitium is –5 cm H₂O, estimate the plasma osmotic pressure (Π) from the figure. Note that the plasma proteins are the main species influencing the osmotic pressure difference across the capillary wall.

2. (b) Estimate the filtration coefficient L_p for this capillary.

3. (c) Consider a capillary 0.05 cm long of diameter 8 µm, for which the arteriolar and venular luminal pressures are 25 and 5 cm H₂O, respectively. Assume that L_p and Π are constant and that the pressure drop varies linearly along the capillary. What is the net rate of fluid loss (gain) from the capillary?