1. The Halvorson & Ziegler's relationship is widely used in thermobacteriology.
When a lot of units containing No living organisms is subjected to a lethal treatment, the lethality of the treatment is measured by comparing No and surviving organisms Ns levels.
According to Halvorson e Ziegler (1933) the expected average number of microorganisms per experimental unit [whether subjected to lethal treatments or not] can be calculated only if a fraction of the experimental units [tubes, cans, jars, vials, etc.] is sterile, i.e. does not contain living organisms.
The Halvorson & Ziegler's relationship is the following:µ = - Ln Po [HZ]
where ‘µ’ is the expected average number of microorganisms / sample unit, when the fraction of sterile samples [i.e. those receiving zero microorganisms] equals Po.
From [HZ] it follows that if Po = 80 sterile units / 400 units = 0.2 = 20% :
µ = - Ln (0.2)
µ = 1.6
and then the average number of organisms/sample is expected to be 1.6.
If Po = 5% , then:
µ = - Ln (0.05)
µ = 2.99
If Po = 37%, then:
µ = - Ln (0.37)
µ = 1
And so on.The relationship of Halvorson e Ziegler [1933] is largely employed in thermobacteriology to compute the value of D [D = decimal reduction time]:
DT = t / [Log No - Log Ns] [D1]
Where No and Ns are the concentration of organisms before and after the applied lethal treatment of t minutes at the temperature T.
By applying the [HZ], the eq. [D1] becomes:DT = t / [Log (No/unit) - Log µ] [D2]
Equation [D2] has a very wide application in thermobacteriology, where usually many single units [tubes, capillary tubes, vials, jars, cans] are subjected to single treatments and survival is measured from the spoilage frequency of unopened units after storage a suitable time at suitable temperature. This technique is a very valuable one because it allows to evaluate the probability of survival to a given treatment in the same substrate where the lethal condition is applied. That is it reflects what occurs in practice.
The technique of diluting and transfer to culture media for finding survivors, is more suitable for a different kind of acquisition.As an example, if 200 tubes of milk each containing No = 10,000 bacteria are treated at 100°C for 5 minutes and after suitable incubation only 120 tubes are sterile, then from the [HZ] we have the expected number of survivor/tube = µ = - Ln [120/200] = -Ln (0.6) = 0.51. According to [D2] the decimal reduction time D100°C of the tested organisms [if tubes were deliberately inoculated with a known microorganism] or anyway of the most resistant one, it is expected to be :
D = 5 / [Log 10000 - Log 0.51]
= 5 / [ 4 - (-0.29)]
= 5 / [ 4 + 0.29]
= 1.16 minutes
2. According to the Poisson Distribution, the probability Px that the rare event E
[the rare character E] will occur x times equals:
Px = (e -µ * µx ) / x! [0]Where ‘e’ is the base of natural logarithms and µ is the mean number of events.
From [0], the probability that event E will occur zero times is expected to be:
Po = (e -µ * µ0 ) /0! [1]
The probability for event E occurring 1 time will be:
P1 = (e -µ * µ1 ) /1! [2]
The probability for event E occurring 2 times will be:
P2 = (e -µ * µ2 ) /2! [3]
And so on.From [1] it follows that the probability for the event E to occur zero times, that is the probability that event E does not occur at all, will be obtained from:
Po = e -µ [4]
Since:
µ0 = 1
and
0! = 1
by definition.
The [4] is called the first term of the Poisson Distribution.
From [4] it comes:
Ln Po = - µ [5]
And then:µ = - Ln Po [6]
where µ is the expected average number of organisms/single unit in the population of all possible samples.
Equation [6] is exactly equivalent to the above reported relationship of Halvorson and Ziegler [1933].