Economic Design of a Limiting Dilution Assay
Ian C. McKay, 23 February 1998
A limiting dilution assay is one in which serial dilutions of the substance to be assayed are each inoculated into several wells (or tubes or animals), and the number of wells (or tubes or animals) that show a particular effect (e.g. death, or infection) are counted. This paper discusses how to achieve an optimal balance between the number of different dilutions and the number of wells (or tubes or animals) inoculated per dilution. The aim is to achieve the smallest possible standard errors while keeping the total number of inocula within strict limits.
The main conclusion is that the best design of an assay is one in which the number of replicate wells per dilution is about equal to the number of different dilutions. It is also shown that in order to cut the standard errors by half it would be necessary to increase the total number of culture wells almost four-fold.
Let us assume that you know from experience what range of dilution factors need to be covered in order to obtain all infection rates from zero to 100 per cent. Let us also assume that economic constraints prevent you from using more than N culture wells in total for each assay. Then there will still be room for doubt about whether it is best to use closely spaced dilutions (e.g. 3-fold serial dilutions) with relatively few replicates per well, or more widely spaced dilutions (e.g. 10-fold serial dilutions) with a larger number of replicates per well.
The chosen criterion
The rather simplified criterion that I will use here is to aim for the smallest possible estimated standard error, as calculated by the formula
· d is the logarithm of the ratio of consecutive dilutions (e.g. d=0.699 if we are using 5-fold consecutive dilutions);
· n = the number of wells inoculated at each dilution (often 8).
· r = the total number of wells that become infected, counting all the dilutions used.
· r1 is the number of wells infected at the first dilution, r2 is the number infected at the next dilution and so on.
The mathematical model
Let us consider how changes to d or n will affect the value of SE. Suppose we were to number our tubes used in the serial dilution 1, 2, 3 . . . and plot a graph showing how the number of infected wells varies with tube number.
The area under this graph is approximately equal to the sum of the component areas created by joining up the points as shown above. Simple geometry tells us that this area is r - ½r1 , which will almost always be the same as r - ½n. It is also clear that the area will expand in direct proportion to n and in inverse proportion to d, so we can say that
where k1 is a constant of proportionality.
By plotting a similar graph showing how the square of the number of infected wells varies with tube number and using the same kind of proportionality argument we can show that
where k2 is another constant of proportionality.
If we now use equations 1 and 2 to substitute for r and for in equation 1, we get
which shows us how SE will be related to d and n, but not quite how it will relate to total cost.
The Cost of Precision
The cost of preparing the cultures and reading the results will be roughly proportional to the total number N of wells used, which will be n ´ m, where m is the number of different dilutions used. Now, to cover the necessary range of dilutions, the number of dilution intervals required will be inversely proportional to d, and the number of dilutions used will be the number of intervals plus 1. Therefore we can write another proportionality relation, namely . If we use this relationship to substitute for d in the above formula for SE, we get
It can be seen by simple calculation from the above formula (or by differential calculus) that if we fix the value of n ´ m by economic constraints, the smallest possible value of SE will be obtained by making n = m. In other words, the number of replicates per dilution should be about equal to the number of different dilutions.
It is easy to show by calculation that moderate departures from this ideal will have only a small detrimental effect on SE: it is only when there is a gross departure from the n = m rule that the SE begins to expand substantially. For example, consider an assay in which we can afford to use a total of 64 wells. We could use 8 dilutions, each with 8 replicates and get a standard error SE1, or we could use 16 dilutions, each with 4 replicates and get SE2, or we could use 32 dilutions, each with 2 replicates and get SE3. The above formula tells us that the values of SE1, SE2, SE3, will be in the ratio 1 : 1.043 : 1.257. So the second experimental design is only a little poorer than the first. It is not until we get to extremes that the SE becomes large enough to be noticeably wasteful.
What if we are just not satisfied with the SE that we get, even when we have made n = m ? Then the above formula tells us that in order to cut the SE by half we would need to increase (n - 1)(m - 1) four-fold. Since (n - 1)(m - 1) = nm - n - m +1, and nm is usually by far the largest term in this expression, we can see that the only way to have much impact on SE, apart from making n = m, is to increase the total number of wells very substantially. In effect, to reduce SE by a factor f we need to multiply the total number of wells N by about f2.
This relationship enables us to attach an economic cost to any departures from the n = m rule. In the case of 16 dilutions, each with 4 replicates, we could compensate for the poor design by increasing N by a mere 9 per cent, but in the case of 32 dilutions, each with 2 replicates, we could only compensate for the poor design by increasing N by about 58 per cent. Obviously, it would be much cheaper just to use a near-optimal design in the first place.