A mathematical model for determining minimal inhibitory concentrations (MICs) via diffusion assays

A mathematical model is presented for the description of inhibition zones in a diffusion bioassay. In such an experiment the drug is placed at the center of a Petri dish containing a bacterial lawn in an agar gel and after a certain incubation period one observes a concentric ring around the center marking the toxic area. From the knowledge of the radius rtox of the toxic zone, the lower limit ctox at which the inhibitory response is observed can be readily calculated. This quantity is very important in evaluating the sensitivity of microorganisms to toxic substances. The mathematical model of the assay is given by a two-dimensional diffusion equation describing the changes in drug concentration due to diffusion, decay of the chemical and consumption by bacteria. The diffusion equation being mildly non-linear is solved numerically with the aid of a computer. For this purpose a numerical solver was developed as well as a "best-fit" simulation program that fits the parameters for which experimental values could not be obtained. The method was tested with N-methyl-N'-nitro-N-nitrosoguanidine and ethylmethanesulfonate and was seen to be fast, efficient, and inexpensive. In principle it could be used for routine quantitative screening for toxicity of chemicals.