## Kinetic Profiles and Appropriate Equations

This section summarizes the various shapes of kinetic profiles that have observed in the literature, the empirical equations that have been used to describe them, and the manner in which the parameters of the equations are estimated.

Four differently shaped kinetic profiles have been reported in various SSF systems: "linear", "exponential", "logistic", and "deceleration". The general shapes of these kinetic profiles are shown in Fig. 14.5 (Viccini et al. 2001).

The equations that describe these curves are shown in Table 14.1. The task is to select the curve that is best able to fit the particular experimental results for biomass, or some indicator of biomass. Note that other shapes of growth curves are possible, in which case it is necessary to propose a new equation that describes the shape of the new curve. Curve selection and fitting will typically be done by regression. In regression analysis the model parameters are adjusted until the sum of squares of deviations between the experimental results and the corresponding values on the fitted curve are at a minimum (Fig. 14.6). There are many software packages that can be used to do regression. After doing the regression for each of the different equations, the curve chosen will typically be the one for which the sum of squares of deviations is the smallest. However, there may also be reasons for preferring a specific equation, even if it does not give the best fit to the data. For example, the logistic equation is usually preferred because often it is possible to use it to describe the whole growth curve adequately, whereas with the other kinetic equations the growth cycle needs to be broken up into intervals, each with a different equation. The regression analysis also gives the values for the o a. E o o o </) </) CO