Multi-scale modeling of drug binding kinetics to predict drug efficacy

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

Mechanisms of antibiotic action shape the fitness landscapes of resistance mutations

Antibiotic-resistant pathogens are a major public health threat. A deeper understanding of how an antibiotic’s mechanism of action influences the emergence of resistance would aid in the design of new drugs and help to preserve the effectiveness of existing ones. To this end, we developed a model that links bacterial population dynamics with antibiotic-target binding kinetics. Our approach allows us to derive mechanistic insights on drug activity from population-scale experimental data and to quantify the interplay between drug mechanism and resistance selection. We find that both bacteriostatic and bactericidal agents can be equally effective at suppressing the selection of resistant mutants, but that key determinants of resistance selection are the relationships between the number of drug-inactivated targets within a cell and the rates of cellular growth and death. We also show that heterogeneous drug-target binding within a population enables resistant bacteria to evolve fitness-improving secondary mutations even when drug doses remain above the resistant strain’s minimum inhibitory concentration. Our work suggests that antibiotic doses beyond this ‘‘secondary mutation selection window” could safeguard against the emergence of high-fitness resistant strains during treatment

A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions

Mycobacterium tuberculosis (Mtb) is a widely diffused infection. However, in general, the human immune system is able to contain it. In this work, we propose a mathematical model which describes the early immune response to the Mtb infection in the lungs, also including the possible evolution of the infection in the formation of a granuloma. The model is based on coupled reaction-diffusion-transport equations with chemotaxis, which take into account the interactions among bacteria, macrophages and chemoattractant. The novelty of this approach is in the modeling of the velocity field, proportional to the gradient of the pressure developed between the cells, which makes possible to deal with a full multidimensional description and efficient numerical simulations. We perform a linear stability analysis of the model and propose a robust implicit-explicit scheme to deal with long time simulations. Both in one and two-dimensions, we find that there are threshold values in the parameters space, between a contained infection and the uncontrolled bacteria growth, and the generation of granuloma-like patterns can be observed numerically

Reconstruction of a nonlinear heat transfer law from uncomplete boundary data by means of infrared thermography

Heat exchange between a conducting plate and the environment is described here by means of an unknown nonlinear function F of the temperature u. In this paper we construct a method for recovering F by means of polynomial expansion, perturbation theory and the toolbox of thermal inverse problems. We test our method on two examples: In the first one, we heat the plate (initially at 20 °C) from one side, read the temperature on the same side and identify the heat exchange law on the opposite side (active thermography); in the second example we measure the temperature of one side of the plate (initially at 1500 °C) and study the heat exchange while cooling (passive thermography

A fluid dynamics model of the growth of phototrophic biofilms.

19 pagesInternational audienceA 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

A Mathematical Model for the Evolution of Nitric Oxide Concentration in Human Breathing

In physiology, the appearance of some inflammatory lung diseases is accompanied by an over production of nitric oxide which is detectable and measurable in the human breath. We propose a mathematical model describing the evolution of the concentration of such chemical compound based on a hyperbolic first order partial differential equation which takes into account the dominating effect of the advection due to the high velocity of the air with respect to diffusive effect. We take into account the geometry of the bronchial tree and we will study an entire breathing cycle. We will be able to give an explicit solution to the proposed equation, at least for some simple choice of the physiological functions. In turn, this makes it possible to describe accurately the concentration of nitric oxide in the bronchial tree as time varies, and to detect by its measure in the human breathing, the localization of the infections.

A fluid dynamics multidimensional model of biofilm growth: stability, influence of environment and sensitivity

International audienceIn this article, we study in details the fluid dynamics system proposed in Clarelli et al (2013) to model the formation of cyanobacteria biofilms. After analyzing the linear stability of the unique non trivial equilibrium of the system, we introduce in the model the influence of light and temperature, which are two important factors for the development of cyanobacteria biofilm. Since the values of the coefficients we use for our simulations are estimated through information found in the literature, some sensitivity and robustness analyses on these parameters are performed. All these elements enable us to control and to validate the model we have already derived and to present some numerical simulations in the 2D and the 3D cases

Reaction Kinetic Models of Antibiotic Heteroresistance

Bacterial heteroresistance (i.e., the co-existence of several subpopulations with different antibiotic susceptibilities) can delay the clearance of bacteria even with long antibiotic exposure. Some proposed mechanisms have been successfully described with mathematical models of drug-target binding where the mechanism’s downstream of drug-target binding are not explicitly modeled and subsumed in an empirical function, connecting target occupancy to antibiotic action. However, with current approaches it is difficult to model mechanisms that involve multi-step reactions that lead to bacterial killing. Here, we have a dual aim: first, to establish pharmacodynamic models that include multi-step reaction pathways, and second, to model heteroresistance and investigate which molecular heterogeneities can lead to delayed bacterial killing. We show that simulations based on Gillespie algorithms, which have been employed to model reaction kinetics for decades, can be useful tools to model antibiotic action via multi-step reactions. We highlight the strengths and weaknesses of current models and Gillespie simulations. Finally, we show that in our models, slight normally distributed variances in the rates of any event leading to bacterial death can (depending on parameter choices) lead to delayed bacterial killing (i.e., heteroresistance). This means that a slowly declining residual bacterial population due to heteroresistance is most likely the default scenario and should be taken into account when planning treatment length