Team:Leiden/javascript/katex

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Theoretical Model

For interpretation of disk diffusion assays, a model-dependent analysis is used. Both the MIC value and the diffusion constant can be estimated from a series of experiments using a simple diffusion model. Traditionally, the diffusion of the compound is modelled as free, linear diffusion using Fick's second law: $$ \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2} $$ From this equation, the following relation between diffusion constant ($ D $), MIC value and concentration can be derived: $$ \ln(\text{MIC}) = \ln(c) - \frac{x^2}{4Dt} $$ This approximation has been applied successfully to diffusion studies of a number of antibiotics, such as penicillin. However, certain hydrophobic or amphiphilic compounds cause zones of inhibitions that scale linearly rather than quadratic with concentration. Therefore Bonev et al. proposed a modification on Fick's law to account for this deviation. By adding a dissipative term describing the loss of antibiotic as it moves through the agar, the concentration over time becomes: $$ \frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2} + V\frac{\partial c}{\partial x}$$ This equation yields the following linear relationship between the diameter of the zone of inhibition and ln(c): $$ \ln(\text{MIC}) = \ln(c) - \frac{x^2}{4Dt} $$