ABSTRACT. A system of nonlinear hyperbolic partial differential equations is derived using mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme will be described in detail and several simulations will be presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing. Fluid dynamics model and Hyperbolic equations and Phototrophic biofilms and Front propagation AMS : 92C17, 35L50, 65M06.

The diffusion coefficient (also known as diffusivity) of an active pharmaceutical ingredient (API) is a fundamental physicochemical parameter that affects passive diffusion through biological barriers and, as a consequence, bioavailability and biodistribution. However, this parameter is often neglected, and it is quite difficult to find diffusion coefficients of small molecules of pharmaceutical relevance in the literature. The available methods to measure diffusion coefficients of drugs all suffer from limitations that range from poor sensitivity to high selectivity of the measurements or the need for dedicated instrumentation. In this work, a simple but reliable method based on time-resolved concentration measurements by UV-visible spectroscopy in an unstirred aqueous environment was developed. This method is based on spectroscopic measurement of the variation of the local concentration of a substance during spontaneous migration of molecules, followed by standard mathematical treatment of the data in order to solve Fick's law of diffusion. This method is extremely sensitive and results in highly reproducible data. The technique was also employed to verify the influence of the environmental characteristics (i.e., ionic strength and presence of complexing agents) on the diffusivity. The method can be employed in any research laboratory equipped with a standard UV-visible spectrophotometer and could become a useful and straightforward tool in order to characterize diffusion coefficients in physiological conditions and help to better understand the drug permeability process.

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Identifying optimal dosing of antibiotics has proven challenging—some antibiotics are most effective when they are administered periodically at high doses, while others work best when minimizing concentration fluctuations. Mechanistic explanations for why antibiotics differ in their optimal dosing are lacking, limiting our ability to predict optimal therapy and leading to long and costly experiments. We use mathematical models that describe both bacterial growth and intracellular antibiotic-target binding to investigate the effects of fluctuating antibiotic concentrations on individual bacterial cells and bacterial populations. We show that physicochemical parameters, e.g. the rate of drug transmembrane diffusion and the antibiotic-target complex half-life are sufficient to explain which treatment strategy is most effective. If the drug-target complex dissociates rapidly, the antibiotic must be kept constantly at a concentration that prevents bacterial replication. If antibiotics cross bacterial cell envelopes slowly to reach their target, there is a delay in the onset of action that may be reduced by increasing initial antibiotic concentration. Finally, slow drug-target dissociation and slow diffusion out of cells act to prolong antibiotic effects, thereby allowing for less frequent dosing. Our model can be used as a tool in the rational design of treatment for bacterial infections. It is easily adaptable to other biological systems, e.g. HIV, malaria and cancer, where the effects of physiological fluctuations of drug concentration are also poorly understood.

Optimizing drug therapies for any disease requires a solid understanding of pharmacokinetics (the drug concentration at a given time point in different body compartments) and pharmacodynamics (the effect a drug has at a given concentration). Mathematical models are frequently used to infer drug concentrations over time based on infrequent sampling and/or in inaccessible body compartments. Models are also used to translate drug action from in vitro to in vivo conditions or from animal models to human patients. Recently, mathematical models that incorporate drug-target binding and subsequent downstream responses have been shown to advance our understanding and increase predictive power of drug efficacy predictions. We here discuss current approaches of modeling drug binding kinetics that aim at improving model-based drug development in the future. This in turn might aid in reducing the large number of failed clinical trials.