## Friday, November 3, 2017

### Countercurrent Transport of Oxygen in the Gills of a Fish

In Section 5.8 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss countercurrent transport.
The countercurrent principle is found in the renal tubules (Hall 2011, p. 309; Patton et al. 1989, p. 1081), in the villi of the small intestine (Patton et al., 1989, p. 915), and in the lamellae of fish gills (Schmidt-Nielsen 1971, p. 45). The principle is also used to conserve heat in the extremities—such as people’s arms and legs, whale flippers, or the leg of a duck. If a vein returning from an extremity runs closely parallel to the artery feeding the extremity, the blood in the artery will be cooled and the blood in the vein warmed. As a result, the temperature of the extremity will be lower and the heat loss to the surroundings will be reduced.
In a homework problem, Russ and I ask the student to analyze a countercurrent heat exchanger. In this blog post, I present a new exercise studying countercurrent oxygen exchange in fish gills.
Problem 19 ½. Fish use countercurrent transport to increase uptake of oxygen in their gills. Consider a capillary extending from x = 0 to x = L, with blood flowing in the positive x direction. Blood entering the capillary has a low oxygen concentration that we take as zero. Seawater flowing outside the capillary has an oxygen concentration of [O2] where it enters the gill. Consider the case when |ain|=|aout|=a in Eq. 5.24. The goal is to calculate the oxygen concentration in the blood, Cin, and the oxygen concentration in the seawater, Cout, as functions of x, and in particular to determine the blood oxygen concentration at the end of the gill where it reenters the fish's body, Cin(L).

a) Consider the case when the seawater flows in the same direction as the blood. Draw a picture illustrating this case. Derive expressions for Cin(L) and Cout(x) in terms of [O2], a, and L. Plot qualitatively Cin(x) and Cout(x) versus x when aL is greater than 1.

b) Now consider the case of countercurrent transport, when the seawater flows in the opposite direction as the blood. Draw a picture, derive expressions, and make plots.

c) Which case results in the highest oxygen concentration in the blood when it leaves the gill and enters the fish’s body?

d) Explain why countercurrent transport is so effective using words instead of equations.
 How Animals Work, by Knut Schmidt=Nielsen.
The solution is given at the bottom of this blog post. You can probably guess that countercurrent transport is more efficient for absorbing oxygen. In one of my favorite books, How Animals Work, Knut Schmidt-Nielsen describes countercurrent transport in the fish gill. His description would be an excellent answer to part d) of the new homework problem.
In the lamellae of the fish gill, water and blood flow in opposite directions (figure 27). As a consequence, the blood, just as it is about to leave the gill, encounters the incoming water which still has all its oxygen; that is, the oxygen tension of the blood will approach that of the water before any oxygen has been removed. At the other end of the lamellae, the water that is about to exit encounters venous blood, so that, even though much oxygen has already been removed from the water, more can still be taken from it by the blood. As a result of this arrangement, fish may extract as much as 80 to 90% of the oxygen in the water, an efficiency which could not easily be achieved without a countercurrent flow.