A Mathematical Model for Electrochemical Impedance Response of a Continuous Glucose Monitor
Authors: Gao Ming , Hazelbaker Morgan S. , Kong Rui , Orazem Mark E 
Published: February 2020

Enzymatic biosensors commonly used for management of diabetes must be maintained in good working condition to provide reliable real-time data. Electrochemical impedance spectroscopy has potential applications in in-vivo sensor diagnostics and sensor calibration.
In the present work, a mathematical model for the impedance response of glucose oxidase electrochemical biosensors has been developed [1,2]. The coupling between the homogeneous reactions and heterogeneous reactions considered in the model included anomerization between α-D-glucose and β-D-glucose and four reversible enzymatic catalytic reactions transforming β-D-glucose and oxygen into gluconic acid and hydrogen peroxide. The electroactive hydrogen peroxide was considered to be reversibly oxidized or reduced at the electrode. In the model, the transport of the species were accounted for and the homogeneous enzymatic reaction followed ping-pong kinetics and the laws of mass action.
The electrochemical system was modeled mathematically as a one-dimensional boundary value problem and solved using Newman’s BAND algorithm [3]. This model accounts for two layers, a layer adjacent to the electrode where the glucose oxidase is immobilized and the glucose-limiting membrane (GLM), which controls the amount of glucose participating in the enzymatic reaction. The steady-state concentrations of reacting species were calculated, including the concentrations of α- and β-D-glucose, oxygen, gluconic acid, hydrogen peroxide, oxidized and reduced enzyme, and enzymatic-complex intermediates. The solution of the nonlinear coupled set of differential equations for the steady-state was used as input for the set of coupled linear differential equations for the frequency-domain calculations. The corresponding impedance was calculated for each specified frequency [4].
The model was used to explore the influence of system parameters on limiting current, reaction profiles, and diffusion impedance. The system parameters, including interstitial glucose concentration, oxygen concentration, active enzyme concentration, reaction rate constants and layer thickness, are related to various sensor working conditions such as body sugar level, inflammation, sensor degradation and sensor design. A condition was found in which a reduced activity sensor exposed to a high glucose concentration yielded the same current as would be generated by an intact sensor exposed to a lower glucose concentration. The simulation showed measurement of impedance to be a potential tool for differentiating these cases. The value of diffusion resistance and capacitance can be extracted from the impedance at the potential on the mass-transfer limited plateau, which may have some discriminatory value. At a potential below the plateau, the impedance can contain more information. The present mathematical model can guide sensor design and can be used for diagnosing potential sensor failure mechanisms.
References

M. S. Harding, Mathematical Models for Impedance Spectroscopy, PhD dissertation, University of Florida, 2017.

M. Gao, M. S. Hazelbaker, R. Kong, and M. E. Orazem, “Mathematical Model for the Electrochemical Impedance Response of a Continuous Glucose Monitor,” Electrochimica Acta, submitted, 2018.

J. S. Newman and K. E. Thomas-Alyea, Electrochemical Systems (John Wiley & Sons, Hoboken, NJ, 2004), 3rd edition.

M. E. Orazem and B. Tribollet, Electrochemical Impedance Spectroscopy (John Wiley & Sons, Hoboken, NJ, 2017), 2nd edition, p. 279-293.

Acknowledgement
The support of Medtronic Diabetes (Northridge, CA) and Andrea Varsavsky, program monitor, is gratefully acknowledged.