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We analyze a mathematical model for drug delivery monitoring subject to flow-through bioreactor. We use a convection–diffusion–reaction model combined with PID control. Enzyme-containing bioreactor converts a prodrug into an active drug. This approach connects two aspects of drug delivery, mechanical pumps and prodrugs. Drug delivery monitoring is performed by adjusting the prodrug inflow pressure. We model several dynamic treatment regimes. A combined algorithm of treatment regimes can be used for personalized treatment. A control-aided system allows us to reduce the impact of pumping fluctuations, diffusion slowdown, and drug production rate reduction.

We describe the action of electrochemical enzyme-based biosensor by applying mathematical modeling. We consider two types of biosensors: a biosensor containing a single heterogeneous enzyme layer and biosensor containing an additional protecting polymer-based layer. The initial parameters of the biosensor were selected on the basis of typical immobilized glucose oxidase-based electrochemical biosensor. A phenomenon of the accumulation of the electrochemically active product inside the biocatalytic layer was evaluated. It was shown that accumulation of the product can increase sensitivity of the biosensor no more than 2.6 times. Due to the asymmetric distribution of the electrochemically active product inside the enzyme-containing membrane and asymmetric diffusion of the substrate, it was shown that the thickness of the membrane possesses an optimal value. For the selected set of initial parameters, the optimal thickness of the enzyme-containing layer was 2.9–4.5 $$\upmu $$ m. Real profiles of the impact of the thickness of the membranes were evaluated. A method for the evaluation of acceptable fluctuations of the membrane diffusion parameters on biosensor response was created, and the profiles of the dependence were calculated. These dependencies can be used for development of the software for biosensor monitoring.

Electrochemical biosensor containing flat semi-permeable membrane covering enzyme-containing layer has been investigated. Mathematical modeling of the action modes of electrochemical biosensors with outer diffusion membrane was performed. Operation of the biosensor under the conditions when the permeability of the membrane and the activity of the biocatalytic layer depend on the parameters of the probe has been examined. The pH and temperature were selected as the main parameters which often affect the action of biosensors. A set of parameters was selected when the biosensor operates in kinetic and diffusion modes of action. The response time of the biosensor was shown to be sensitive to the mode of the biosensor action especially in the boundary region of the biosensor action. The linearity of the biosensor (the linear dependence of the biosensor response on the substrate concentration) in the deep diffusion mode can be increased by several magnitudes, whereas the response time increases only several times.

Plate-gap model of enzyme doped porous electrode was utilized in order to calculate apparent Michaelis constants ( $$K_{\rm M}^{\rm app}$$ ) and apparent maximal currents ( $$I_{\rm max}^{\rm app}$$ ) of modeled amperometric biosensor for the wide range of given reaction/diffusion parameters. It was found that $$K_{\rm M}^{\rm app}$$ of plate-gap biosensor linearly depends on $$I_{\rm max}^{\rm app}$$ when rates of enzymatic reaction are lower than critical. Theoretically predicted linear correlation between apparent parameters was observed experimentally for the case of carbon paste electrodes, which were modified by PQQ-dependent alcohol dehydrogenases. At overcritical rates (or apparent maximal currents), $$K_{\rm M}^{\rm app }$$ is practically independent on Michaelis constant of soluble enzyme. Therefore, apparent Michaelis constant can be regarded as biosensor’s topology representing parameter which, in fact, is not related to the specificity of enzyme kinetics. High and rate-independent values of $$K_{\rm M}^{\rm app}$$ indicate that reaction proceeds at substrate-exposed top layer of the gap. In this case, reaction–diffusion system formally is stratified into separate reaction (top) and diffusion (bottom) zones. Topology of such reaction–diffusion system reminds “inverted” planar electrode, which contains diffusion layer below reaction layer. The net effect of plate-gap topology of working electrode on apparent Michaelis constant is similar to the effect of diffusion layer covering enzymatic planar electrode.

The plate-gap model of porous enzyme doped electrode has been proposed and analyzed. It was suggested that reaction diffusion conditions in pores of bulk electrode resemble particular conditions in thin gap between parallel conducting plates. The model is based on the diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetic of the enzymatic reaction inside gap. Steady state current was calculated for the wide range of given parameters and substrate concentrations. All dependences of current on substrate concentration were approximated by hyperbolas in order to obtain “apparent” parameters (maximal currents and apparent Michaelis constants) of modelled biosensors. Simple approximate relationships between given and apparent parameters were derived. The applicability of theoretical plate-gap model was tested for the case of carbon paste electrodes which were doped with PQQ – dependent glucose dehydrogenase. It was found, that soluble glucose dehydrogenase based biosensors exhibit characteristic features of the theoretical plate-gap biosensors.