Inundative Phage Density Calculator

∞ generated and posted on 2022.05.19 ∞

That phage titer required, at a minimum, to reduce a given bacterial number to some new especially phage-sensitive bacterial number over some predetermined interval of time.

Please cite as:

Stephen T. Abedon
Inundative Phage Density Calculator.

Click here for calculator or see immediately below for further explanation and discussion.

With this calculator you can get a feel for how many phages may be required to sufficiently eliminate phage-sensitive bacteria from an environment, as well as how terribly wrong it can be to assume that 100% of phages are adsorbing, i.e., that so-called MOIactual might approximate MOIinput particularly given phage treatment of lower bacterial concentrations.


Enter either the starting bacterial number (N0) or starting bacterial concentration (also N0, but in this case per ml), the latter in combination with the volume (ml) those bacteria occupy (V).

Then enter the desired number (not concentration) of phage-sensitive bacteria found at the end of treatments (NF), that is, after some specific interval of time (t, also entered).

Then either indicate the adsorption rate constant of the phage-bacterium-environment combination you are working with (k) or instead use the default value supplied (2.5 × 10-9 ml-1 min-1, but which is often written instead as 2.5 × 10-9 ml/min).

Returned will be (a) the minimum phage titer required to achieve that goal, assuming that phages adsorb with replacement (IPDmin w/), (b) the minimum phage titer required assuming that free phages are lost upon their adsorption (IPDmin w/o), and (c) the minimum phage titer required assuming instead 100% phages adsorption.

(The latter is called an inundative phage number or IPNmin and for which both the time and volume components are ignored.)

Overall, it is (b), the minimum phage titer required assuming that free phages are lost upon their adsorption (IPDmin w/o), which is the most useful output. This is unless calculations do not take volumes into account, in which case it is (c), the minimum phage titer required assuming instead 100% phage adsorption (IPNmin), which is most useful. IPNmin in many cases, and particularly when concentrations of target bacteria are low, can be grossly unrealistic, however, which you can tell by comparing (b) (realistic) with (c) (unrealistic).


The following is a detailed explanation of why and when (b), IPDmin w/o, is the preferred output of this calculator:

Unless bacterial concentrations are very high, e.g., ~108/ml or higher, then it is highly unlikely that even approaching 100% of supplied phages will adsorb. Also, at lower concentrations of target bacteria the following is expected:

(b) IPDmin w/o (× volume)  >  (a) IPDmin w/ (× volume)  >  (c) IPNmin.

This is because IPDmin w/o (b) at lower bacterial concentrations will result in somewhat less than 100% adsorption

IPDmin w/ (a) by definition also results in less than 100% adsorption but since phage titers do not decline over time in effect more phages are present even if the indicated inundative number is lower.

As noted, IPNmin (c) assumes 100% adsorption, so more phages that are actually impacting bacteria are present from the start, i.e., the calculated inundative number of initially free phages therefore is lower.

When phage adsorptions come to approach 100% for (b), i.e., when concentrations of target bacteria are higher (e.g., 107/ml), then the inequality fails, and indeed well prior to that the IPDmin w/ calculation (a) will have become unrealistic. Alternatively, when concentrations of target bacteria are lower, then (a) IPDmin w/ will come to approximate (b) IPDmin w/o.

IPDmin w/ (a) thus may be used as an approximation if bacterial concentrations are relatively low, e.g., such as approaching 105 bacteria/ml and lower. The advantage of using (a) however is that it is a simpler calculation, but this is an advantage that no longer holds if you are using this calculator as here the calculation is done for you. Also, (a) works even if volume is not defined, which can be preferable to assuming 100% phage adsorption (c).

Note that you can test whether the above inequality holds using this calculator, rejecting using (a) especially if the inequality does not robustly hold, particularly if (a) is not much larger than (c) such as ten-fold larger, though be sure to set the volume to greater than 0 as this is a requirement for calculating IPDmin w/o (b). Look for the '???' tag in the output.


Always assumed in the calculations are that phages do not replicate in the course of bacterial-inundation process, that bacteria also do not replicate, and that all targeted bacteria are equally susceptible to phage adsorption.

See instead the Active Phage Therapy Calculator for calculations which inlude of both phage and bacterial replication.


For further discussion, see Abedon, S.T. (2022). Further Considerations on How to Improve Phage Therapy Experimentation, Practice, and Reporting: pharmacodynamic perspectives. Phage 3:98-111.

See also the Phage Half-Life Calculator, the Bacterial Half-Life Calculator, and the Multiplicity of Infection Calculator.

= m
above is set to 1 if left blank or not a number or ≤ 0
= n
above is set to 0 if left blank or not a number or < 0

= V

= m
above is set to 1 if left blank or not a number or ≤ 0
= n
above is set to 0 if left blank or not a number or < 0

= t

= m
above is set to 2.5 if left blank or not a number or ≤ 0
= n
above is set to -9 if left blank or not a number or > 0

k value is defaulted at 2.5 × 10-9 ml-1 min-1 from Stent (1963)
may be greater or lower but must be defined to calculate MOIactual
the larger the number, the faster the phage adsorption