Calculating sample size?

[Andy Resnick]

There has to be a simple way to calculate this, I just don't have a clue how.

The setup:

I'm teaching cell culture methods to some students, and part of the process is checking for contamination- after making culture media, a small volume is put into the incubator overnight and checked for bacterial growth in the morning.

The question (using 'normal' units for things)
"When you made 100 milliliters of solution, there could also be a small number of bacteria present, perhaps as much as 0.1 nanomolar. How much volume do you need to sample in order to be at least 99% confident that you captured at least 1 bacterium?"

Thanks in advance....

 

[jedishrfu]

It seems 1ml would work. I guess you could imagine from 0.1 nanomolar = numerically how many bacteria (6.9E12). Then assume they are evenly distributed.

Take your volume of 100ml and divide it into 100 1ml. So each 1ml would have 1/100th the number of bacteria (6.0E10).

 

[Ibix]

I'd approach it by noting that if there were one bacterium in a volume $V$ ($V=100\mathrm{ml}$ in your case) and I drew a volume $v$, the probability of not getting the bacterium would be $1-v/V$. The probability of not getting any of $n$ bacteria whose locations are independent random variables is therefore $(1-v/V)^n$. Requiring this to be less than 1% gives us $$
\begin{eqnarray*}
0.01&>&(1-v/V)^n\\
0.01^{1/n}&>&1-v/V\\
v&>&V(1-0.01^{1/n})
\end{eqnarray*}$$For your case where the number of bacteria is $n\approx 6×10^{12}$, $v$ is pretty much indistinguishable from zero. You would have to try really really hard not to get a bacterium unless they all clump in one corner or something.

 

[berkeman]

checking for contamination- after making culture media,

Then assume they are evenly distributed.

Is that a valid assumption, though? @Andy Resnick -- what are the typical causes of contamination in this setup? Does it usually come from contaminated dishes (so likely evenly distributed), or contaminated dispensing gear (so likely the contamination will be washed out with the first solution dispensed), or something else?

 

[jedishrfu]

Sometimes you need to put a stick in the ground and make a guess.

 

[Andy Resnick]

Thanks to everyone for the replies, let me address them:

Take your volume of 100ml and divide it into 100 1ml. So each 1ml would have 1/100th the number of bacteria (6.0E10).

You would have to try really really hard not to get a bacterium unless they all clump in one corner or something

Apparently, I grossly overstated the (maximum) contamination level- obviously, as the number of contaminants decrease, I need to sample larger and larger volumes, but that would seem to quickly approach a limit when I need to sample the entire volume (or a substantial fraction of it. This is clearly not what is done in the biomedical or food industries:

"[xxx] brand, [YYY] cell culture liquid products are prepared by an aseptic process for which each step has been validated to ensure that all products meet the industry standard sterility assurance level of 10^-3; i.e. product that demonstrates a contamination level of no more than 1 of 1,000 units during the manufacturing process. The highest level of sterility assurance (equal to or greater than 10^-6) cannot be achieved without terminal sterilization which is harmful to the performance of cell culture products."

If I understand this correctly, the "industry standard sterility assurance level of 10^-3" means less than 0.01 bacteria/milliliter. If I make up a 1 liter batch, there can be up to 1 bacterium present. How much volume do I need to sample to meet the assurance level?

Is that a valid assumption, though? @Andy Resnick -- what are the typical causes of contamination in this setup?

Other than grossly overestimating the contamination level, it a valid assumption. Ultimately, the typical causes of contamination are 'user error': touching any surface with a sterile pipette tip; improper sterilization of gloves or work area; rapid movement that brings outside air into the culture hood; stuff like that.

Sometimes you need to put a stick in the ground and make a guess.

Which is what I have been doing all along :) So far it's worked out well (using the secret weapon of 'geometrical mean', I typically withdraw 2ml out of a 100 ml batch), I am/was hoping for some rational calculation to support my guess.