Exponential growth model

A practical introduction to using the exponential growth model for modelling bacterial growth in batch processes. Use the exponential growth model to predict and control bacterial growth.

Introduction to growth modelling

There are many ways to model bacterial growth, from detailed mechanistic models to complex dynamic simulations. For most batch processes, however, where the goal is to understand how input parameters affect the trajectory of growth and process performance, a simple exponential growth model often provides the most practical insight.

It is not a perfect model, but it is a powerful first-order approximation. It enables intuitive testing of how changes in inoculum size, specific growth rate, or target concentration affect process timing and output.

Exponential growth model

In the exponential growth model, the concentration of cells, N, at time, t, is given by

$$N(t) = N_0 \, e^{\mu (t-\lambda)}$$

where:

• N₀ is the initial concentration of cells,

• µ is the specific growth rate

• λ is the apparent lag phase.

This model describes bacterial growth during the exponential phase and remains valid until the culture enters the deceleration phase, as it approaches stationary conditions.

Obtaining input parameters for the model

Only a few input parameters are needed to model and predict bacterial growth under a given set of process conditions. These are N₀, µ_max, and λ - all of which can be readily derived using BactoBox.

Additionally, knowledge of the maximum carrying capacity (Κ) in the process is useful for identifying when the exponential model ceases to apply.

Note: A detailed protocol for deriving growth parameters using BactoBox will be available soon.

Visualising the simple exponential growth model

Consider a step in your seed train for which you have determined the following parameters:

• N₀ = 5×10⁶ cells/mL

• λ = 2.4 h

• µ_max = 1.04 h^-1

• Κ = 1×10¹⁰ cells/mL

Using these parameters, the bacterial growth profile can be visualized as shown below.

Figure legend: A visual representation of the exponential growth model. The model assumes constant growth rate and therefore does not account for the acceleration and deceleration that occur in real bacterial growth. It is important to note that this model is not accurate for predicting cell concentrations immediately after the culture exits the lag phase or just before it enters the stationary phase. Nevertheless, this simplification is often acceptable when the goal is to establish an intuitive understanding of the process.