2. The Kinetics of Thermal Death.  (TOC)

Putting aside for the moment consideration of heat transfer, assume that we know how to bring the temperature of a collection of microorganisms to any arbitrary level. For simplicity, assume the organisms are in an aqueous suspension, and that the temperature is above the threshold lethal level. Empirical observations have shown that at constant temperature, many microorganisms will die at a rate that is first order with respect to time. The time rate of change of viable organisms, that is organisms that have survived the heat treatment, can be described mathematically as:

where N is the number of a particular organism at time t, and K is a quantity characteristic of the organism. The number K is known as the thermal death constant. It is not really a constant, but is a function of temperature, as we shall see later. Actually, the number is also a function of numerous other factors as well. For very complex, and only partly understood reasons, the factor is also a function of the physico-chemical environment. Most important of these for good results is that the environment be moist, particularly a situation of condensing, saturated steam. Further considerations of the effects of the sterilizing environment will be treated later. It will suffice for now to state simply that with steam quality less than 100 percent, the thermal death constant will be smaller than with saturated condensing steam. Therefore the rate of thermal death will be less under such conditions.

Since we are considering isothermal conditions just now, the assumption that K is a constant is valid. Equation (1) is thus meaningful only at constant temperature. The equation holds true for a wide range of organisms even when more than one species of organism is present. However, it does not hold true for the collection itself. For example, in the simple case when two kinds of organisms are present:

for organism 1 and

for organism 2, where K₁ <> K₂. The total number of organisms is N = N₁ + N₂. It follows that

and in general, for n organism species, assuming that each species follows first order death kinetics:

Let’s look at the kinetics for a collection of a pure culture. Equation (1) is a nonhomogeneous ordinary differential equation of first order and first degree. Integrating Equation (1)

where C is a constant of integration. Differentiating Equation (6) and substituting in Equation (1) we find that C is identically zero for K <> 0. If we impose an initial condition, viz. the initial number of organisms, then

So, at time t

where N(t₀) is the number of organisms initially present at time t₀. It is sometimes conceptually simpler to express Equation (8) as a ratio, so that

Now, the number of remaining viable organisms will approach zero only as the treatment time approaches infinity. In other words, the solution will never be sterile according to this model. Yet the model is in nearly universal use. How can this be?

The answer lies in the nature of the model itself. Calculated results based on the model are based on probabilities, and therefore when we calculate a reduction in numbers, we express only a probability that the reduction will be as calculated. This is due to the fact that we are using a mathematically continuous model to describe the behavior of a collection of discrete organisms, each of which will react to the application of heat in a slightly different way. There will be a probability distribution of how organisms react. Of course, as the initial number of organisms becomes very large, a continuous model becomes a very good approximation to the discrete reality, but it is still a distribution of probabilities.