The Lambert Function Should Be in the Engineering Mathematical Toolbox

Analysis of dead-core formation in catalytic reaction and diffusion processes with generalized diffusion flux

Dead-core and non-dead-core solutions to the nonlinear diffusion-reaction equation based on the generalized diffusion flux with gradient-dependent diffusivity and the power-law reaction kinetics in catalyst slabs are established. The formation of dead zones where the reactant concentration vanishes is characterized by the critical Thiele modulus that is derived as a function of reaction order and diffusion exponent in the generalized diffusion flux. The effects of reaction order and diffusion exponent on the reactant concentration distribution in the slab and dead-zone length are analyzed. It is particularly demonstrated that by contrast to the model based on the standard Fick's diffusion, dead-core solutions exist in the case of first-order reactions. Also, the relationship between critical Thiele moduli for models based on the generalized and standard Fick's diffusion fluxes is established.

Deciphering longitudinal optical-density measurements to guide clinical dosing regimen design: A model-based approach

BACKGROUND

Model-based analysis of longitudinal optical density measurements from a bacterial suspension exposed to antibiotics has been proposed as a potentially efficient and effective method for extracting useful information to improve the individualized design of treatments for bacterial infections. To that end, the authors developed in previous work a mathematical modeling framework that can use such measurements for design of effective dosing regimens.

OBJECTIVES

Here we further explore ways to extract information from longitudinal optical density measurements to predict bactericidal efficacy of clinically relevant antibiotic exposures.

METHODS

Longitudinal optical density measurements were collected in an automated instrument where Acinetobacter baumannii, ATCC BAA747, was exposed to ceftazidime concentrations of 1, 4, 16, 64, and 256 mg/L and to ceftazidime/amikacin concentrations of 1/0.5, 4/2, 16/8, 64/32, and 256/128 (mg/L)/(mg/L) over 20 h. Calibrated conversion of measurements produced total (both live and dead) bacterial cell concentration data (CFU/mL equivalent) over time. Model-based data analysis predicted the bactericidal efficacy of ceftazidime and of ceftazidime/amikacin (at ratio 2:1) for periodic injection every 8 h and subsequent exponential decline with half-life of 2.5 h. Predictions were experimentally tested in an in vitro hollow-fiber infection model, using peak concentrations of 60 and 150 mg/L for injected ceftazidime and of 40/20 (mg/L)/(mg/L) for injected ceftazidime/amikacin.

RESULTS

Model-based analysis predicted low (<62%) confidence in clinically relevant suppression of the bacterial population by periodic injections of ceftazidime alone, even at high peak concentrations. Conversely, analysis predicted high (>95%) confidence in bacterial suppression by periodic injections of ceftazidime/amikacin combinations at a wide range of peak concentrations ratioed at 2:1. Both predictions were experimentally confirmed in an in vitro hollow fiber infection model, where ceftazidime was periodically injected at peak concentrations 60 and 150 mg/L (with predicted suppression confidence 38% and 59%, respectively) and a combination of ceftazidime/amikacin was periodically injected at peak concentrations 40/20 (mg/L)/(mg/L) (with predicted suppression confidence 98%).

CONCLUSIONS

The paper highlights the potential of clinicians using the proposed mathematical framework to determine the utility of different antibiotics to suppress a patient-specific isolate. Additional studies will be needed to consolidate and expand the utility of the proposed method.

A Kinetic Model for Predicting Trace Gas Uptake and Reaction

A model is developed to describe trace gas uptake and reaction with applications to aerosols and microdroplets. Gas uptake by the liquid is formulated as a coupled equilibria that links gas, surface, and bulk regions of the droplet or solution. Previously, this framework was used in explicit stochastic reaction-diffusion simulations to predict the reactive uptake kinetics of ozone with droplets containing aqueous aconitic acid, maleic acid, and sodium nitrite. With the use of prior data and simulation results, a new equation for the uptake coefficient is derived, which accounts for both surface and bulk reactions. Lambert W functions are used to obtain closed form solutions to the integrated rate laws for the multiphase kinetics; similar to previous expressions that describe Michaelis-Menten enzyme kinetics. Together these equations couple interface and bulk processes over a wide range of conditions and do not require many of the limiting assumptions needed to apply resistor model formulations to explain trace gas uptake and reaction.

Revisiting the standard for modeling the spread of infectious diseases

The COVID-19 epidemic brought to the forefront the value of mathematical modelling for infectious diseases as a guide to help manage a formidable challenge for human health. A standard dynamic model widely used for a spreading epidemic separates a population into compartments-each comprising individuals at a similar stage before, during, or after infection-and keeps track of the population fraction in each compartment over time, by balancing compartment loading, discharge, and accumulation rates. The standard model provides valuable insight into when an epidemic spreads or what fraction of a population will have been infected by the epidemic's end. A subtle issue, however, with that model, is that it may misrepresent the peak of the infectious fraction of a population, the time to reach that peak, or the rate at which an epidemic spreads. This may compromise the model's usability for tasks such as "Flattening the Curve" or other interventions for epidemic management. Here we develop an extension of the standard model's structure, which retains the simplicity and insights of the standard model while avoiding the misrepresentation issues mentioned above. The proposed model relies on replacing a module of the standard model by a module resulting from Padé approximation in the Laplace domain. The Padé-approximation module would also be suitable for incorporation in the wide array of standard model variants used in epidemiology. This warrants a re-examination of the subject and could potentially impact model-based management of epidemics, development of software tools for practicing epidemiologists, and related educational resources.

Ziegler and Nichols meet Kermack and McKendrick: Parsimony in dynamic models for epidemiology

The COVID-19 crisis popularized the importance of mathematical modeling for managing epidemics. A celebrated pertinent model was developed by Kermack and McKendrick about a century ago. A simplified version of that model has long been used and became widely popular recently, even though it has limitations that its originators had clearly articulated and warned against. A basic limitation is that it unrealistically assumes zero time to recovery for most infected individuals, thus underpredicting the peak of infectious individuals in an epidemic by a factor of as much as about 2. One could avoid this limitation by returning to the original comprehensive model, at the cost of higher complexity. To remedy that, we blend Ziegler-Nichols modeling ideas, developed for automatic controller tuning, with Kermack-McKendrick ideas to develop novel model structures that predict infectious peaks accurately yet retain simplicity. We illustrate these model structures with computer simulations on real epidemiological data.

SIMULTANEOUS IN VITRO SIMULATION OF MULTIPLE ANTIMICROBIAL AGENTS WITH DIFFERENT ELIMINATION HALF-LIVES IN A PRE-CLINICAL INFECTION MODEL

Combination therapy for treatment of multi-drug resistant bacterial infections is becoming common. In vitro testing of drug combinations under realistic pharmacokinetic conditions is needed before a corresponding combination is eventually put into clinical use. The current standard for design of such in vitro simulations for drugs with different half-lives is heuristic and limited to two drugs. To address that void, we develop a rigorous design method suitable for an arbitrary number of N drugs with different half-lives. The method developed offers substantial flexibility and produces novel designs even for two drugs. Explicit design equations are rigorously developed and are suitable for immediate use by experimenters. These equations were used in experimental verification using a combination of three antibiotics with distinctly different half-lives. In addition to antibiotics, the method is applicable to any anti-infective or anti-cancer drugs with distinct elimination pharmacokinetics.