Computational Modelling of Biosensors with Perforated and Selective Membranes

This paper presents a two-dimensional-in-space mathematical model of amperometric biosensors with perforated and selective membranes. The model is based on the diffusion equations containing a non-linear term of the Michaelis–Menten enzymatic reaction. Using numerical simulation of the biosensors action, the influence of the geometry of the perforated membrane on the biosensor response was investigated. The numerical simulation was carried out using finite-difference technique. The calculations demonstrated non-linear and non-monotonous change of the biosensor steady-state current at various degree of the surface of the perforated membrane covering. The non-monotonous behaviour of the biosensor response was also observed when changing the thickness of the perforated membrane.     

Computational Modelling of the Behaviour of Potentiometric Membrane Biosensors

This paper presents a mathematical model of a potentiometric biosensor based on a potentiometric electrode covered with an enzyme membrane. The model is based on the reaction–diffusion equations containing a non-linear term related to theMichaelis–Menten kinetics of the enzymatic reaction. Using computer simulation the influence of the thickness of the enzyme membrane on the biosensor response was investigated. The digital simulation was performed using the finite difference technique. Results of the numerical simulation were compared with known analytical solutions.      

An Analysis of Mixtures Using Amperometric Biosensors and Artificial Neural Networks

This paper presents a sensor system based on a combination of an amperometric biosensor acting in batch as well as flow injection analysis with the chemometric data analysis of biosensor outputs. The multivariate calibration of the biosensor signal was performed using artificial neural networks. Large amounts of biosensor calibration as well as test data were synthesized using computer simulation. Mathematical and corresponding numerical models of amperometric biosensors have been built to simulate the biosensor response to mixtures of compounds. The mathematical model is based on diffusion equations containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. The principal component analysis was applied for an optimization of calibration data. Artificial neural networks were used to discriminate compounds of mixtures and to estimate the concentration of each compound. The proposed approach showed prediction of each component with recoveries greater that 99% in flow injection as well as in batch analysis when the biosensor response is under diffusion control.     

Computational modelling of amperometric biosensors in the case of substrate and product inhibition

In this paper the response of an amperometric biosensor at mixed enzyme kinetics and diffusion limitations is modelled in the case of the substrate and the product inhibition. The model is based on non-stationary reaction–diffusion equations containing a non-linear term related to non-Michaelis–Menten kinetics of an enzymatic reaction. A numerical simulation was carried out using a finite difference technique. The complex enzyme kinetics produced different calibration curves for the response at the transition and the steady-state. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor response change shows a maximal value. The dependence of the biosensor sensitivity on the biosensor configuration is also investigated. Results of the simulation are compared with known analytical results and with previously conducted researches on the biosensors.     

Amperometric Determination of Hydrogen Peroxide and its Mathematical Simulation for Horseradish Peroxidase Immobilized on a Sonogel Carbon Electrode

A mathematical model of a horseradish peroxidase biosensor was applied to simulate the amperometric response for the detection of hydrogen peroxide. The development of the mathematical model was based on the Michaelis–Menten equation and Fick’s Second Law. The theoretical study is based on the determination of physico-chemical and geometric parameters of a horseradish peroxidase biosensor as well as the kinetic parameters of reaction mechanism such as diffusion coefficients of hydrogen peroxide, the thickness of enzymatic layer, and the Michaelis–Menten kinetic constant. The theoretical analysis provides an accurate estimate of parameters affecting the biosensor performance such as the diffusion coefficient of hydrogen peroxide in the biomembrane that was estimated to be 56?×?10^?12 m²/s. The thickness of diffusion layer was estimated to be 80–100?µm and the biomembrane 7.5?µm. The experimental and numerical values of kinetic parameters were 0.92 and 0.98?µM for the Michaelis–Menten constants and 0.010 and 0.012?µM/s for the catalytic activity rates. The model was validated for hydrogen peroxide detection and exhibited a good agreement with the experimental measurements.     

Modelling synergistic action of laccase-based biosensor utilizing simultaneous substrates conversion

The response of a laccase-based amperometric biosensor that acts in a synergistic manner was modelled digitally. A mathematical model of the biosensor is based on a system of non-linear reaction diffusion equations. The modelling biosensor comprises three compartments, an enzyme layer, a dialysis membrane and an outer diffusion layer. By changing input parameters the biosensor action was analysed with a special emphasis to the influence of the species concentrations on the synergy of the simultaneous substrates conversion. The digital simulation was carried out using the finite difference technique.     

Analytical expression of non-steady-state concentrations and current pertaining to compounds present in the enzyme membrane of biosensor

A mathematical model of trienzyme biosensor at an internal diffusion limitation for a non-steady-state condition has been developed. The model is based on diffusion equations containing a linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Analytical expressions of concentrations and current of compounds in trienzyme membrane are derived. An excellent agreement with simulation data is noted. When time tends to infinity, the analytical expression of non-steady-state concentration and current approaches the steady-state value, thereby confirming the validity of the mathematical analysis. Furthermore, in this work we employ the complex inversion formula to solve the boundary value problem.     

New theoretical expressions for the five adsorption type isotherms classified by BET based on statistical physics treatment

New theoretical expressions to model the five adsorption isotherm types have been established. Using the grand canonical ensemble in statistical physics, we give an analytical expression to each of five physical adsorption isotherm types classified by Brunauer, Emett, and Teller, often called BET isotherms. The establishment of these expressions is based on statistical physics and theoretical considerations. This method allowed estimation of all the mathematical parameters in the models. The physicochemical parameters intervening in the adsorption process that the models present could be deduced directly from the experimental adsorption isotherms by numerical simulation. We determine the adequate model for each type of isotherm, which fixes by direct numerical simulation the monolayer, multilayer, or condensation character. New equations are discussed and results obtained are verified for experimental data from the literature. The new theoretical expressions that we have proposed, based on statistical physics treatment, are rather powerful to better understand and interpret the various five physical adsorption type isotherms at a microscopic level.     

Mechanisms controlling the sensitivity of amperometric biosensors in flow injection analysis systems

This paper numerically investigates the sensitivity of an amperometric biosensor acting in the flow injection mode when the biosensor contacts an analyte for a short time. The analytical system is modelled by non-stationary reaction-diffusion equations containing a non-linear term related to the Michaelis-Menten kinetics of an enzymatic reaction. The mathematical model involves three regions: the enzyme layer where enzymatic reaction as well as the mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place, and a convective region. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor sensitivity can be increased and the calibration curve can be prolonged by changing the injection duration, the permeability of the external diffusion layer, the thickness of the enzyme layer and the catalytic activity of the enzyme. The apparent Michaelis constant is used as a main characteristic of the sensitivity and the calibration curve of the biosensor. The numerical simulation was carried out using the finite difference technique.     

Modelling Dynamics of Amperometric Biosensors in Batch and Flow Injection Analysis

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equations containing a non-linear term related to Michaelis–Menten kinetic of the enzymatic reaction. Using digital simulation, the influence of the substrate concentration as well as maximal enzymatic rate on the biosensor response was investigated. The digital simulation was carried out using the finite difference technique. The model describes the biosensor action in batch and flow injection regimes.     

Mathematical Model and Numerical Simulation of Conductometric Biosensor of Urea

In this article, a mathematical model was developed to describe and optimize the configuration of the urea biosensor. The biosensor is based on interdigitated gold microelectrodes modified with a urease enzyme membrane. The model presented here focuses on the enzymatic reaction and/or diffusion phenomena that occur in the enzyme membrane and in the diffusion layer. Numerical resolution of differential equations was performed using the finite difference technique. The mathematical model was validated using experimental biosensor data. The responses of the biosensor to various conditions were simulated to guide experiments, improve analytical performance, and reduce development costs.     

Modelling of Amperometric Biosensors with Rough Surface of the Enzyme Membrane

A two-dimensional-in-space mathematical model of amperometric biosensors has been developed. The model is based on the diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetic of the enzymatic reaction. The model takes into consideration two types of roughness of the upper surface (bulk solution/membrane interface) of the enzyme membrane, immobilised onto an electrode. Using digital simulation, the influence of the geometry of the roughness on the biosensor response was investigated. Digital simulation was carried out using the finite-difference technique.     

Use of emanation thermal analysis to characterize microstructure development during Portland cement hydration

Portland cement hydration has been investigated by emanation thermal analysis (ETA), based on the application of radon atoms as radioactive indicators. This method enabled us to characterize continuously changes in the microstructure of the cement paste at selected temperatures. The numerical simulation of time dependences of the emanating rate during cement hydration was carried out. An agreement between the mathematical model and experimental results of the ETA was obtained.     

Modeling and analysis of uranium isotope enrichment by chemical exchange

Starting with an accurate mathematical model a theoretical study for the analysis of the separation of uranium isotopes by chemical exchange has been presented. The experimental data used in this study were obtained by reverse breakthrough technique and the numerical algorithm developed for simulation in previous studies was adapted and found to be suitable for this kind of processes. The model parameters were identified from experimental data and simulations were carried out for different experimental conditions. This revised version was published online in July 2006 with corrections to the Cover Date.     

EMULSION POLYMERIZATION IN A SEED-FED CONTINUOUS STIRRED-TANK REACTOR

A numerical method has been developed to predict the particle size distribution (PSD) of the product latex from a steady-state polydisperse-seeded continuous reactor. Simulations have been carried out for the emulsion polymerization of vinyl chloride based on the experimental conditions reported by Berens(l). The simulation results can be reasonably well fitted to the PSD data published by Berens. The radical desorption constant, k_d, for Berens’ vinyl chloride emulsion polymerization can be estimated by fitting the simulated PSD to experimental measurements. The simulation work presented in this article demonstrates that the combination of mathematical modeling and PSD measurements can be a useful tool in studying radical transport rates and aqueous phase termination phenomena. The simulation results also indicate that the continuum diffusion theory for free radical absorption by the particles leads to a better PSD fit than a model based on equal diffusion rates per unit area.     

Analysis of droplet behavior in a de-oiling hydrocyclone

A mechanical separation process in a de-oiling hydrocyclone is described in which disperse oil droplets are separated from a continuous water phase. This separation process is influenced by droplet breakage and coalescence. Based on experimental data and simulation results in a stirred tank, a modified breakage model, which can be applied to droplet breakage in the de-oiling hydrocyclone, is developed. Then, a simulation model is developed coupling the numerical solution of the flow field in the hydrocyclone based on computational fluid dynamics (CFD) with population balances. The homogenous discrete method and the inhomogeneous discrete method are applied for solving the population balance model (PBM). The investigations show that the numerical results obtained by the simulation model coupled with the modified PBM using the inhomogeneous discrete method are in good accordance with experimental data under a high flow rate. According to this simulation model, the effect of three different inlet designs on the separation efficiency of the de-oiling hydrocyclone has been discussed. The results indicate that the separation efficiency of the de-oiling hydrocyclone can be improved with an appropriate inlet design.     

Modelling and simulation of hydro-gel growth mechanism by analysis of experimental data

We have experimentally studied the growth process of a water-absorbing polymer commonly known as hydro-gel. We took a number of spherical beads, immersed them in pure water, measured their diameters at regular intervals of time and plotted graphs showing the change in volume with the time of immersion, to analyze their swelling properties. We have formulated a dynamical model on the basis of curve-fitting to our experimental data. Our mathematical modelling is devoted solely towards the explanation of the swelling of spherical hydro-gels and we have interpreted our observations on these basis. And finally evaluating the constants of the process, we have shown that the results obtained from our modelling and numerical simulation exactly match the experimental data. The justification of this new approach lies in its successful application for the complete explanation of our observations through numerical and analytical ways.     

Mathematical modeling of a hydrophilic cylinder floating on water

In this paper, a hydrostatic model of the surface profile anchored to the upper edge of a vertical cylinder is proposed to explain why coins can float on water surface. The sharp edge of a cylinder is thus modeled as a round smooth surface on which the contact line may be anchored at a position according to the weight of the cylinder. The mathematical model of the surface profile is established based on the hydrostatics and a third order ordinary differential equation is resulted. Numerical solution of the model demonstrates under practical conditions the existence of the surface profiles that provide reasonable uplifting force at the contact line so that the force is available for floating coins on water surface. The proposed model explains the obviously enlarged apparent contact angle and the edge effect in the literature. The numerical simulation is found in very good agreement with the experimental data in the literature.     

Degradation of substrate and/or product: mathematical modeling of biosensor action

Mathematical model for evaluation of the multilayer heterogeneous biocatalytic system has been elaborated. The model consists of nonlinear system of partial differential equations with initial values and boundary conditions. An algorithm for computing the numerical solution of the mathematical model has been applied. Two cases: when product diffuses out of the biosensor and when the outer membrane is impermeable for product (product is trapped inside the biosensor) have been dealt with by adjusting boundary conditions in the mathematical model. Profiles of the impact of the substrate and product degradation rates on the biosensor response have been constructed in both cases. Value of the degradation impact has been analyzed as a function of the outer membrane thickness. The initial substrate concentration also affects influence of the degradation rates on the biosensor response. Analytical formulae, defining approximate values of relationships between the degradation rates and the outer membrane thickness or the initial substrate concentration, have been obtained. These formulae can be employed for monitoring of the biosensor response.     

Mathematical Modelling of a Non‐enzymatic Amperometric Electrochemical Biosensor for Cholesterol

Thickness of the electro‐polymerized layer grown on a substrate and used as the recognition element for the analyte is critical to measuring the response of a biosensor, with high sensitivity and accuracy. However, it is difficult to control the thickness during synthesis. A mathematical model is developed in this study that considers thickness of the electro‐polymerized layer in simulating the electrochemical response of a non‐enzymatic biosensor for cholesterol in blood. The model includes transient kinetics and one‐dimensional diffusion of the analyte in the poly‐methyl orange (PMO) recognition layer electrochemically grown on the electrode. The governing partial differential equations resulting from the species conservation balances in the PMO layer are numerically solved. Time and spatial concentration profiles of the analyte in the PMO layer are determined. Model predictions are calibrated with the experimental data for different PMO thicknesses. Interestingly, model predictions show a linear response over the calibrated concentration range of cholesterol for all PMO layer thicknesses. Based on the chronoamperometry measurements, the model predictions for the cholesterol concentrations measured in the laboratory samples were also found to be remarkably accurate. This is the first mathematical model developed to understand the transport and kinetics of an analyte in the electro‐polymerized layer used as the recognition element of a non‐enzymatic biosensor.