Decay Plus Input at a Constant Rate

Section 2.7 of the 4th edition of Intermediate Physics for Medicine and Biology is titled Decay Plus Input at a Constant Rate. When I taught Biological Physics last fall (using for my textbook—you guessed it—Intermediate Physics for Medicine and Biology), I found that we kept coming back to this section over and over. Russ Hobbie and I write

Suppose that in addition to the removal of y from the system at a rate –by, y enters the system at a constant rate a, independent of y and t. The net rate of change of y is given by

dy/dt = a – by…  (2.24)

The solution is

y = a/b (1 –  e^−bt).

One of the first applications of this equation is to the speed of an animal falling under the force of gravity and air friction (Chapter 2, Problem 28). One can show that the terminal speed of the animal is a/b. If further one proves that the gravitational force (a) is proportional to volume, and the frictional force (−by) is proportional to surface area, then the implication is that larger animals fall faster than smaller ones. This led to Haldane’s famous quote “You can drop a mouse down a thousand-yard mine shaft: and arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes.”

We see this equation again in Chapter 3 when analyzing Newton’s law of cooling (Problem 45). The surrounding temperature plays the role of a, and the exponential cooling from convection is represented by a term like −by. The solution to the resulting differential equation is just the exponential solution presented in Sec. 2.7.

In Section 5.7 about the artificial kidney, the equation arises again in governing the concentration of solute in the blood when the concentration of the solute in the dialysis fluid is a constant. As with the animal falling, the ratio of blood volume V to membrane area S is a key parameter.

The equation appears twice in Chapter 6 (Impulses in Nerve and Muscle Cells). First, the gate variables m, h, and n in the Hodgkin and Huxley model obey this same differential equation. In the voltage-clamp case, the gates approach their steady-state values exponentially. Then, Problem 35 analyzes electrical stimulation of a space-clamped passive axon using a constant current, and finds that the transmembrane potential approaches its steady-state value exponentially also. This result is used to derive two quantities with colorful names—rheobase and chronaxie—that are important in neural stimulation.

By Chapter 10, some of the students have almost forgotten the equation when it appears again in the study of feedback loops (particularly Section 10.4). I am sure that the equation would appear even more times, except my one-semester class ended with Chapter 10.

Some might wonder why Intermediate Physics for Medicine and Biology contains an entire chapter (Chapter 2) about exponential growth and decay. I believe that the way we are constantly returning to the concepts introduced in Chapter 2 justifies why we organize the material the way we do. In fact, Chapter 2 has always been one of my favorite chapters in Intermediate Physics for Medicine and Biology.