Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers

Abstract

We propose a model of antibiotic diffusion through a bacterial biofilm when diffusion and/or absorption barriers develop in the biofilm. The idea of this model is: We deduce details of the diffusion process in a medium in which direct experimental study is difficult, based on probing diffusion in external regions. Since a biofilm has a gel-like consistency, we suppose that subdiffusion of particles in the biofilm may occur. To describe this process we use a fractional subdiffusion-absorption equation with an adjustable anomalous diffusion exponent. The boundary conditions at the boundaries of the biofilm are derived by means of a particle random walk model on a discrete lattice leading to an expression involving a fractional time derivative. We show that the temporal evolution of the total amount of substance that has diffused through the biofilm explicitly depends on whether there is antibiotic absorption in the biofilm. This fact is used to experimentally check for antibiotic absorption in the biofilm and if the biofilm parameters change over time. We propose a four-stage model of antibiotic diffusion in biofilm based on the mentioned above physical characteristics. The biological interpretation of the stages, in particular their relation with the bacterial defence mechanisms, is discussed. Theoretical results are compared with empirical results of ciprofloxacin diffusion through Pseudomonas aeruginosa biofilm, and ciprofloxacin and gentamicin diffusion through Proteus mirabilis biofilm.

11 pages, 6 figures

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• Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion
• First passage time for $g$--subdiffusion process of vanishing particles
• Generalized diffusion equation with nonlocality of space-time: analytical and numerical analysis
• Subdiffusion with particle immobilization process described by differential equation with Riemann--Liouville type fractional time derivative
• Subdiffusion in a system with a partially permeable partially absorbing wall