Medical professionals often are interested not only in the value obtained by a blood test, but in how those values are changing over time. By analogy with the description of motion in physics, the change in the measured parameter divided by the time in the measurements is referred to as the velocity of the parameter.

A measurement of one particular antigen (PSA) is commonly used as a signal of the possible presence of a slow-growing cancer. If the PSA begins to climb rapidly, that is a possible signal that the cancer has mutated into a more dangerous metastatic form. The velocity of the PSA is taken as a serious warning sign.

As we know from our study of velocities derived from position, the numerical construction of velocity is difficult since when you subtract two quantities that have some uncertainty, the magnitude of the quantity tends to cancel and give a smaller result, but the uncertainties add and get larger. The numerical construction of a velocity is therefore "noisy".

To get a sense of this for the velocity of an antigen measured in a blood test, consider a PSA measured as a value of 4.3 ng/ml (nanograms of the antigen per deciliter of blood) at one time and then as 4.6 ng/ml six months later. 

1. Calculate the velocity of the antigen measurement in ng/ml/month.

In order to get an idea of the uncertainty in the velocity, we need the uncertainty of the individual measurements. Unfortunately, medical measurements from laboratories are rarely reported with uncertainties. For the purposes of this problem, let us assume that the standard deviation is represented by a change of one unit in the last significant figure reported: ± 0.1 ng/ml. This means that is the laboratory took the same blood sample and did the lab test again, that the probability is ~67% that the result would fall in that range. So we would say that a value of "4.3 ± 0.1 ng/ml" means that there is a 67% probability that the true result lies between 4.2 and 4.4.  

2. Estimate the range of uncertainty in the velocity of the antigen that you calculated for part 1. Use the maximum and minimum of the "67% probability of a true result" ranges for the two values to get a maximum and minimum value and an estimate of the uncertainty for the antigen's velocity.

3. The way you calculated in part 2 isn't really the way that uncertainties add. Since it is not very likely that one measurement is at its extreme positive range while the other is at the extreme negative range, to get a "67% probability of a true result" for the velocity, the uncertainty in the change in the antigen needs to be calculated "in quadrature" -- that is, if we measured the initial value of the antigen as A₁ with a range σ₁, and the final value of the antigen as A₂ with a range σ₂, then the uncertainty in A₂-A₁ is

Use this principle to calculate the uncertainty in the antigen's velocity.

4. There are a number additional factors (not counting that the lab might have made a mistake!) that could result in the obvious number calculated for the antigen's velocity in part 1 not being a good description of how the antigen is changing over time. Discuss one such factor.