Mathematical Modeling of Biofilm Structures Using COMSTAT Data

Mathematical modeling holds great potential for quantitatively describing biofilm growth in presence or absence of chemical agents used to limit or promote biofilm growth. In this paper, we describe a general mathematical/statistical framework that allows for the characterization of complex data in terms of few parameters and the capability to (i) compare different experiments and exposures to different agents, (ii) test different hypotheses regarding biofilm growth and interaction with different agents, and (iii) simulate arbitrary administrations of agents. The mathematical framework is divided to submodels characterizing biofilm, including new models characterizing live biofilm growth and dead cell accumulation; the interaction with agents inhibiting or stimulating growth; the kinetics of the agents. The statistical framework can take into account measurement and interexperiment variation. We demonstrate the application of (some of) the models using confocal microscopy data obtained using the computer program COMSTAT

Determination of Tobramycin in M₉ Medium by LC-MS/MS: Signal Enhancement by Trichloroacetic Acid

It is well known that ion-pairing reagents cause ion suppression in LC-MS/MS methods. Here, we report that trichloroacetic acid increases the MS signal of tobramycin. To support studies of an in vitro pharmacokinetic/pharmacodynamic simulator for bacterial biofilms, an LC-MS/MS method for determination of tobramycin in M9 media was developed. Aliquots of 25 μL M9 media samples were mixed with the internal standard (IS) tobramycin-d5 (5 µg/mL, 25 µL) and 200 µL 2.5% trichloroacetic acid. The mixture (5 µL) was directly injected onto a PFP column (2.0 × 50 mm, 3 µm) eluted with water containing 20 mM ammonium formate and 0.14% trifluoroacetic acid and acetonitrile containing 0.1% trifluoroacetic acid in a gradient mode. ESI+ and MRM with ion m/z 468 → 324 for tobramycin and m/z 473 → 327 for the IS were used for quantification. The calibration curve concentration range was 50–25000 ng/mL. Matrix effect from M9 media was not significant when compared with injection solvents, but signal enhancement by trichloroacetic acid was significant (∼3 fold). The method is simple, fast, and reliable. Using the method, the in vitro PK/PD model was tested with one bolus dose of tobramycin

Fractional compartmental models and multi-term Mittag–Leffler response functions

Systems of fractional differential equations (SFDE) have been increasingly used to represent physical and control system, and have been recently proposed for use in pharmacokinetics (PK) by (J Pharmacokinet Pharmacodyn 36:165–178, 2009) and (J Phamacokinet Pharmacodyn, 2010). We contribute to the development of a theory for the use of SFDE in PK by, first, further clarifying the nature of systems of FDE, and in particular point out the distinction and properties of commensurate versus non-commensurate ones. The second purpose is to show that for both types of systems, relatively simple response functions can be derived which satisfy the requirements to represent single-input/single-output PK experiments. The response functions are composed of sums of single- (for commensurate) or two-parameters (for non-commensurate) Mittag–Leffler functions, and establish a direct correspondence with the familiar sums of exponentials used in PK

Fractional dynamics pharmacokinetics–pharmacodynamic models

While an increasing number of fractional order integrals and differential equations applications have been reported in the physics, signal processing, engineering and bioengineering literatures, little attention has been paid to this class of models in the pharmacokinetics–pharmacodynamic (PKPD) literature. One of the reasons is computational: while the analytical solution of fractional differential equations is available in special cases, it this turns out that even the simplest PKPD models that can be constructed using fractional calculus do not allow an analytical solution. In this paper, we first introduce new families of PKPD models incorporating fractional order integrals and differential equations, and, second, exemplify and investigate their qualitative behavior. The families represent extensions of frequently used PK link and PD direct and indirect action models, using the tools of fractional calculus. In addition the PD models can be a function of a variable, the active drug, which can smoothly transition from concentration to exposure, to hyper-exposure, according to a fractional integral transformation. To investigate the behavior of the models we propose, we implement numerical algorithms for fractional integration and for the numerical solution of a system of fractional differential equations. For simplicity, in our investigation we concentrate on the pharmacodynamic side of the models, assuming standard (integer order) pharmacokinetics