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

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 subdiffusion and absorption parameters of the biofilm change over time. We propose a four-stage model of antibiotic diffusion in biofilm based on the following physical characteristics: whether there is absorption of the antibiotic in the biofilm and whether all biofilm parameters remain unchanged over time. The biological interpretation of the stages, in particular their relation with the bacterial defense 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.

Figure 1

Schematic of the system. The biofilm separates two regions in which normal diffusion occurs, $C$ is the antibiotic concentration, $D$ is the normal diffusion coefficient in regions $A$ and $B$, $D_M$ is the subdiffusion coefficient, $\alpha$ is the subdiffusion parameter and $\kappa$ is the absorption coefficient in the biofilm, $q_A$ and $q_B$ are probabilities of stopping a diffusing particle by the biofilm boundaries.

Figure 2

Random walk of a particle in a discrete system with one-sided fully permeable wall represented by the vertical line, more detailed description in the text.

Figure 3

Experimental results (squares) and theoretical function $W_{0B}$ Eq. (25) (dashed line) for diffusion of ciprofloxacin through P. mirabilis O18 biofilm, fitting parameters are $a_0=0.90\times {10}^{-5}\mathrm{m}/\sqrt{\mathrm{s}}$ and $b_0=0.95\times {10}^{-6}\mathrm{m}/{\mathrm{s}}^{0.05}$, and $\alpha =0.95$; here $C_0=1.5\mathrm{mol}/{\mathrm{m}}^3$ and $\mathrm{\Pi }=7.0\times {10}^{-5}{\mathrm{m}}^2$.

Figure 4

Experimental results (squares) and theoretical functions $W_{0B}$ Eq. (25) (dashed line) and $W_{\kappa B}$ Eq. (28) (solid line) for diffusion of ciprofloxacin through Psudomonas aeruginosa biofilm, the parameters are $a_0=0.86\times {10}^{-5}\mathrm{m}/\sqrt{\mathrm{s}}$, $b_0=1.90\times {10}^{-5}\mathrm{m}/{\mathrm{s}}^{0.05}$, $a_{\kappa }=0.44\times {10}^{-3}\mathrm{m}$, $b_{\kappa }=2.10\times {10}^{-3}\mathrm{m}/\sqrt{\mathrm{s}}$, $c_{\kappa }=8.57\times {10}^{-2}\mathrm{mol}/{\mathrm{s}}^{0.95}$, and $\alpha =0.95$; here $t_1=1560\mathrm{s}$, $C_0=3.0\mathrm{mol}/{\mathrm{m}}^3$, and $\mathrm{\Pi }=7.0\times {10}^{-5}{\mathrm{m}}^2$.

Figure 5

Experimental results (squares) and plots of the functions $W_{\kappa B}$ Eq. (28) (solid line) and $W_{\tilde{\kappa }\left(t\right)B}$ Eq. (35) (dot-dashed line) for diffusion of gentamicin in the system with P. mirabilis O18 biofilm, the parameters are $a_{\kappa }=0.30{10}^{-3}\mathrm{m}$, $b_{\kappa }=0.43{10}^{-3}\mathrm{m}/\sqrt{\mathrm{s}}$, $c_{\kappa }=17.1{10}^{-3}\mathrm{m}/{\mathrm{s}}^{0.95}$, $\alpha =0.95$, and $a=1.35$, $b=3501/\mathrm{s}$, $t_2=1000\mathrm{s}$; the experiment was performed for $C_0=1.5\mathrm{mol}/{\mathrm{m}}^3$ and $\mathrm{\Pi }=7.0\times {10}^{-5}{\mathrm{m}}^2$.

Figure 6

Different stages for the situation of Fig. 4. Dashed lines No. 1 and 2 represent $W_{0B}$ Eq. (25), solid lines No. 3 and 4 represent $W_{\kappa B}$ Eq. (28). Lines No. 1 and 3 are for $\alpha =0.95$, red lines No. 2 and 4 are for $\alpha =1.0$. The other parameters are $a_0=0.86\times {10}^{-5}\mathrm{m}/\sqrt{\mathrm{s}}$ and $b_0=1.90\times {10}^{-5}\mathrm{m}/{\mathrm{s}}^{0.05}$ for the functions No. 1 and 2, $a_{\kappa }=0.44\times {10}^{-3}\mathrm{m}$, $b_{\kappa }=2.10\times {10}^{-3}\mathrm{m}/\sqrt{\mathrm{s}}$, and $c_{\kappa }=8.57\times {10}^{-2}\mathrm{mol}/{\mathrm{s}}^{0.95}$ for the function No. 3, and $a_{\kappa }=0.42\times {10}^{-3}\mathrm{m}$ with the same $b_{\kappa }$ and $c_{\kappa }$ as in the previous case for the function No. 4.