# A Fermi Problem About Pharmaceuticals

Rebecca Skloot tweets the following question, which she asks on behalf of Melinda Wenner (@lindy2350):

Anyone know how I can find the # of people worldwide taking a particular class of drug?

This is an interesting problem, and as well it’s a nice example of what has come to be known as a “Fermi problem.” The physicist Enrico Fermi famously posed the question “How many piano tuners are there in Chicago?” to his students, to be answered without consulting any reference sources: the answer to the question must come entirely from “back of the envelope” calculations. A similar situation faces a scientist approaching a new problem. How much voltage will be required for a new apparatus, made for an experiment never performed before? What is the most critical quantity in a new, never-before-tested theory? How large an effect should be expected if our theory is correct? Even an estimate of the order of magnitude of the quantity of interest would be useful, since nothing is known about it. Fermi problems are intended to train a student to think about quantities whose values can be discovered and what their relationships are, so that the answer to the Fermi problem can be deduced.

I’d like to consider Melinda’s question, but first, I’d like to work through the piano tuners problem, direct readers to some more sites with Fermi problems, and describe an experiment conducted by Fermi with nothing more than scraps of paper and “back of the envelope” calculations—to measure the power of the first atomic bomb.

## How many piano tuners are there in Chicago?

A solution to the piano tuner question can be generated as follows. First, estimate the percentage of people in a large city like Chicago who have pianos. From this, deduce an estimate of how many pianos there are in Chicago. Next, estimate how often a piano needs to be tuned, and how long it takes to tune a single piano. From this, deduce the number of hours in a week are spent tuning pianos. Now, together with an estimate of how many hours per week a piano tuner works, the number of tuners can be deduced. Even though there’s no obvious way to measure the number of piano tuners in Chicago, it’s easy to come up with an estimate. Perhaps this information has never been used for any significant purpose; but it’s easy to imagine one.  The piano tuners want to start a credit union: what’s the maximum number of potential members? If each buys a single share for \$10, how much capital will the Credit Union have to work with when it starts? What’s reasonable to charge for a loan, given the level of income of most piano tuners? The answers to these questions can be refined as information is obtained about the number of pianos, how long it takes to tune one, how many hours the tuners work each week, and the other quantities mentioned above. Similarly, worst- and best-case scenarios can be generated.

## Fermi measures the power of the atomic bomb

Richard Rhodes, in The Making of the Atomic Bomb, reports that Enrico Fermi was one of the scientists at Los Alamos in the 1940’s who worked on the atomic bomb. He was present when the first completed bomb was tested. Although precision instruments were used to measure the bomb’s power, Fermi didn’t want to wait until the data from the test were analyzed. So, he conducted a simple test. When the bomb was detonated, he released, from his hand, a number of small scraps of paper. The scraps of paper were blown away from the bomb blast by the shock wave it created. (He was far away from the test site, and so was not at risk himself of being harmed by the shock wave.) By measuring how far the shock wave had carried the scraps of paper, together with his knowledge of the laws of physics and how far away he was from the blast, he was able to estimate the bomb’s power right away.

About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during, and after the passage of the blast wave. Since, at the time, there was no wind, I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2-1/2 meters which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of the TNT.  (Rhodes, p. 674)

He was wrong by less than a factor of two. More sensitive instruments measured it at 18.6 Kilotons of TNT.

## Where to find more Fermi Problems

The University of Maryland Physics Education Research Group has created the University of Maryland Fermi Problems Site. Some, like “Estimate the number of square inches of pizza consumed by all the students at the University of Maryland during one semester,” are fun, but probably not particularly significant; the solutions to others are of the greatest significance, and probably were considered by scientists at one time or another, in pursuit of cutting-edge results.

The effect of air pressure was demonstrated in 1654 in Magdeburg, Germany with an impressive experiment. Two hollow metal hemispheres with a flat metal rim (flange) were placed together and the air removed from the interior with a primitive vacuum pump. The external pressure of the air pushed them together. When they tried to pull the hemispheres apart using two rings welded to the hemispheres, they could not be pulled apart even by a number of people pulling together. The sphere produced by putting the two hemispheres together is about the size of a basketball. Estimate the force needed to pull them apart.

More can be found at the Hampton College Garden of Fermi Problems, which links to the Old Dominion University Fermi Problems Site; and Old Dominion faculty members Lawrence Weinstein (“I smash atoms for a living”) and John A. Adam have recently published Guesstimation: Solving the World’s Problems on the Back of a Cocktail Napkin. Ben Lorica, blogging at O’Reilly Radar, approaches the question, “Of the 300 million users [of facebook], how many are actively using [it] right now?” He concludes that “there are 1.6 to 6 million people actively using Facebook right now”—a wide range, but, as Lorica points out, “in classic Fermi problems, being within a factor of 10 is considered acceptable.” I found some of these links by way of Steve Hsu’s blogHsu, a professor in the Physics Department at the University of Oregon, is no stranger to Fermi problems. He reports that “When I took my oral exam as a first year graduate student at Berkeley, theoretician Geoff Chew (a former student of Fermi’s) asked me, ‘How many blades of grass are on your front lawn?’ and ‘What is the ratio of paved to unpaved surface area in Iowa?’ (He had earlier asked where I grew up.)”

Before moving on to the drugs problem, I would like to formulate a Fermi Problem which I first thought of in response to one of Rebecca’s tweets. Rebecca has recently completed a book about Henrietta Lacks. Lacks provided (unknowingly and probably in violation of her rights) cells from her body, now called HeLa cells, which have been used by researchers for many decades—the cell line was maintained and continues to be because HeLa cells have important biological properties, including a propensity for rapid growth. Reading Rebecca’s tweets recalled to my mind homework problem from my introductory chemistry class. The assignment was to estimate the quantity of oxygen molecules in any given breath we take which were also inhaled by Julius Caesar. A similar question can be asked about the Lacks cell line. I conjecture that, especially since cells deriving from the initial sample taken from Lacks are so frequently used in medical research, each of us probably has at least one stretch of DNA which has been transferred horizontally by a virus or some other means. The Fermi Problem here is:

What is the probability that the lineage of any given gene in the body of any given person alive today passes through the initial sample of cells taken from Lacks? Use any definition of “gene” to indicate whatever stretch of DNA you like—an arbitrary set of three adjacent DNA codons;  a longer stretch coding for some biologically important molecule; the entire open reading frame for a gene having an important effect on an important trait; or any other way.

## The drugs problem

There are three ways that come to mind right away for arriving at an estimate of how many people worldwide are taking a given class of drug. I interpret the problem to be about prescription drugs used to treat illness, and not, for instance, different classes of recreational drugs such as stimulants, hallucinogens, and the like. Of course, methods similar to those I propose here about the former might be used for the latter as well.

Method #1 First, estimate the number of people suffering from all the diseases treated with drugs of that class.  Data collected by the World Health Organization or United Nations might help here. Then, estimate the proportion of people being treated for those diseases worldwide. Again, the WHO or UN data would probably be useful here. Now an estimate of the total number of people taking drugs of that class can be deduced simply by multiplying the two figures.

This probably requires fine-tuning. Some of the diseases treated by drugs of class A might also sometimes be treated by drugs of another class, B, so an estimate of the proportion of those suffering from diseases treated by B-class drugs rather than A-class drugs is needed so as not to overestimate the number of people taking drugs in class A. How could this information be obtained? Price information might be useful. Suppose that drugs in class A cost \$1 per dose, while drugs in class B cost \$10 per dose. (Suppose these numbers are averages of the price of the different drugs in each class.) Suppose that this means that there are 10 people taking drug A for every one person taking drug B. Information about drug prices could probably be obtained from US government sources, or by sampling formularies of a range of health plans. Usually there are “tiers” of drugs: brand-name drugs in tier 1 cost \$30 per month, generics \$20 per month; drugs in tier 2 cost \$40 per month and \$30 per month, respectively, for brand name and generics; and so on.

Another way to fine-tune the estimate would be to generate figure for different countries, or different parts of the world, and then sum them to obtain the worldwide number. There are bound to be wide discrepancies in the number of people being treated in a given locale rather than another, due to differences in economic, social, and political situations across locales. Maybe dividing countries up by GDP would create meaningful groupings; I suppose the quality of health care, and accordingly the number of people being treated for the illnesses in question, is correlated with GDP.

Method #2 A second approach to the problem uses financial information from the start. How much did each drug company make from selling drugs of the class we’re interested in? Estimate the cost of a daily dose, and the number of doses are usually required for a complete course of treatment, and it’s possible to estimate how many people took the drug in a given year.

Method #3 A third approach is to try to estimate the number of prescriptions directly, rather than estimating other quantities on which the total depends. At http://findarticles.com/p/articles/mi_m0NKV/is_9_3/ai_91832686/, BNET magazine, an online newsletter about management, reports the number of prescriptions filled in the United States by different sources, e.g., how many are filled at neighborhood pharmacies rather than national chains. Information about the number of prescriptions written for antidepressants in England was obtained by the “Liberal Democrats’ health spokesman, Norman Lamb” (http://www.guardian.co.uk/society/2009/jun/21/mental-health-antidepressants-recession-prescriptions); Lamb found that the number has increased dramatically during the last year, an increase he attributed to mental health stresses brought on by the recent recession. Neither of these sources are part of any systematic or widely-distributed information source about prescriptions, which suggests that finding that kind of information probably requires an expert, or at least, an ingenious Internet search strategy. These two were identified by searching Google with the key “prescriptions written.”

Review articles about diseases or drugs of a given class might provide information needed for any one or another of the three approaches sketched above. Surveillance information from the Centers for Disease Control or the National Institutes of Health would be likely sources for information about the distribution and number of different diseases worldwide.