Quote of the Day

Doing statistics is like doing crosswords except that one cannot know for sure whether one has found the solution.

— John Tukey, statistician and data analyst extraordinaire. If you get the chance, read his book Exploratory Data Analysis. It is a gem.

## Introduction

I was reading the Washington Post this weekend when I stumbled upon a 22-July-2017 article about concerns that ISIS in Mosul had access to an old medical radiation source. This source, which contains the radioactive isotope cobalt-60, is used in the treatment of cancer (Figure 1). However, cobalt-60 is extremely radioactive and could be used to build a dirty bomb. Fortunately, ISIS did not touch the source, but the concerns about a terrorist being able to use one of these radiation sources for a dirty bomb are real.

There have been encounters between radiation sources and unwary people – the encounters did not end well. For example, in one case, people tried to breakdown a cesium-137 radiation source for scrap. The incident ended with four people dead, twenty hospitalized, and 249 contaminated.

The Washington Post article mentioned three facts that we can easily verify using some simple math.

- The source contains 9 grams of cobalt-60, which generates a radiation level of 10,000 curies (Ci) when new. (Quote)
- Person standing three feet (~1 meter) from the unshielded source would receive a fatal does in less than 3 minutes. (Quote)
- The source is 30 years old, so its radiation level is significantly diminished with respect to a new source. (Quote)

## Background

### Definitions

- becquerel (Bq)
- One becquerel is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. (Source)
- curie (Ci)
- The curie (symbol Ci) is a non-SI unit of radioactivity, one Ci = 3.7 × 10
^{10}nucleus decays per second or one Ci = 3.73.7 × 10^{10}Bq. (Source) **sievert (Sv)**- The Wikipedia defines the Sievert (symbol: Sv) as the SI derived unit of equivalent radiation dose. The Sievert represents a measure of the biological effect, and should not be used to express the unmodified absorbed dose of radiation energy, which is a physical quantity measured in Grays.
- Dose Equivalent (H)
- Equivalent dose is a dose quantity H representing the stochastic health effects of low levels of ionizing radiation on the human body. It is derived from the physical quantity absorbed dose, but also takes into account the biological effectiveness of the radiation, which is dependent on the radiation type and energy. In the SI system of units, the unit of measure is the sievert (Sv). (Source)
- Half-Life (
*t*)_{HL} - Half-life is the time required for a quantity to reduce to half its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive, radioactive decay. (Source)

### Radiation Level Given Half-Life

Equation 1 tells us the amount of a radioactive species we have remaining after time *t* assuming that we had a 100% pure sample at time ø.

Eq. 1 |

where

*T*is the half-life of the radioactive species._{HL}*t*is elapsed time since having a pure sample.*N*is the initial amount of the substance. You can use mass or moles or even numbers of atoms._{0}*N*is the amount of the radioactive species left after time*t*.

To obtain the level of radioactivity (i.e. decays per second), we need to take the derivative of Equation 1 (see Figure 3).

## Analysis

### Setup

Figure 2 shows how I set up the calculations in Mathcad 15.

### Calculations

Figure 3 shows my calculations that duplicate the results in the Washington Post article. The purple check marks indicate the specific results. Note that the Post article computes that you would get a lethal dose of radiation from a new Co_{60} source ~2.5 minutes when it is new, a calculation which I duplicate below. The 30-year old source in question is shown to have its lethality reduced by a factor of over 50. This means you would get a lethal dose of radiation from this source with ~2 hours of exposure (thanks to Ronan in the comments for pointing this out).

## Conclusion

I had never thought about the potential security issues associated with medical radiation sources. I was surprised to the see how intense the radiation levels were from a cobalt source. While the cobalt source mentioned in the article is 30 years old and only 2% of its initial radiation level, it is still a very dangerous item.

## Appendix A: Quotes from the Article

### Radiation Level Quote

In a draft report written in November 2015, research fellow Sarah Burkhard calculated that the radioactive cores, when new, contained about nine grams of pure cobalt-60 with a potency of more than 10,000 curies — a standard measure of radioactivity.

### Fatal Dose Quote

A person standing three feet from the unshielded core would receive a fatal dose of radiation in less than three minutes.

### Reduction in Radiation Level Quote

Because cobalt-60 decays over time, the potency of the Mosul machines’ 30-year-old cobalt cores would have been far less than when the equipment was new, but still easily enough to deliver a lethal dose at close range, the report said.

Reference Link: Surgical technology

Your term Sv is not defined (it's late at night so I might have missed it). I believe you mean that it's a Sievert.

Hi Ronan,

You are correct. I have added some additional explanatory text and changed a variable name to the standard used in the Wikipedia (H for equivalent dose). Thanks for the review – it really help!

mark

Marc, thanks for sharing your formulas as I was not sure how to determine lethality. It seems that two sets of calculations are required. One for the initial 9 grams of Co_60 which you did and another lethality calculation for the 2% remaining Co_60 that might have been grabbed by ISIS.

You are correct. The level of radioactivity is always dropping and the lethality (i.e. exposure time for a lethal dose) increases over time. You have to perform the calculation for the times you are interested in.

I have read a number of stories about people killed by inadvertent exposure to medical radiation sources. One particularly sad one involved some bums who came across a radiation source that had been thrown into a dump. It was warm and the night was cold, so they slept next to it. That night did not end well.

mark

FYI, I redid your calculations allowing for the decreased amount of cobalt_60 after 30 years, and came up with about 2 hours before lethality is achieved.