# Limits on hair growth

By making some simple assumptions we can establish an equation governing hair growth. The equations shows that generally the rate of hair growth decreases exponentially and hair length is bounded by a constant which depends on the initial assumptions.

Before beginning I need to clarify what hair length means since not all hairs have the same length. I have treated hair length as some sort of average over all hairs. The nature of the average is a little vague but hopefully won’t affect the result in any significant way.

Here are some simple assumptions which I have used to derive the equations. Let A(t) denote hair length at time t.

Assumption 1. The rate of increase in the length of each individual hair is a constant, say g. g is the same for each hair. This says that input into the ‘hair system’ is constant over time.

Assumption 2. The number of active hair follicles, say N, is fixed. This means that there is no hair loss.

Assumption 3. The number of hairs shed per unit of time , say L, is constant. The shed hairs will be replaced over time in accordance with assumption 2 above.

We derive the following first order differential equation from the above assumptions:

dA/dt = g – L*A/N

If L = 0 (no shedding of hair) then the general solution is

A = g*t + c

for some constant c. Thus the average hair length increases linearly over time without bound. If L is not zero then the general solution is

A = gN/L + cexp(-L*t/N)

for some constant c which is determined from the initial conditions. If c > 0 then average hair length decreases over time.  If c < 0 then average hair length increases over time. In both cases A approaches g*N/L as t approaches infinity and the rate at which A approaches this bound decreases exponentially.  For a fixed initial hair length and fixed N, c would be > 0 when L is large compared to g and c would be < 0 when L is small compared to g. You would expect this since if L >> g the natural hair growth rate cannot on average replace the hair that is being shed so the average length should decrease over time. Conversely if L << g then growth is outstripping shedding and the average hair length should increase over time.