Behind Modeling Electrochemical Impedance Response of a Continuous Glucose Monitor
Authors: Gao Ming , Kong Rui , Orazem Mark E 
Published: February 2020

The continuous glucose monitor is used to measure the glucose concentration in interstitial fluids by means of a glucose oxidase enzyme that converts glucose into hydrogen peroxide, which can be detected electrochemically. The development of a model for the electrochemical impedance response of the sensor may be envisioned as a sequential series of steps.
The electrochemical oxidation of hydrogen peroxide, in the absence of the enzyme, can be treated as a charge-transfer resistance coupled with a diffusion impedance. In the present case, however, the enzyme provides a homogeneous production of the hydrogen peroxide. The coupling of a homogeneous reaction with a heterogeneous electrochemical reaction was addressed, in the 1950s, by Gerischer, who assumed that homogeneous kinetics can be described by a linear equation, that all diffusion coefficients were equal, and that diffusion took place within a film. The resulting diffusion impedance had two loops, the high-frequency loop corresponding to the homogeneous reaction and the low-frequency loop corresponding to diffusion.
While the simplified model represented by the electrochemical oxidation of hydrogen peroxide coupled with a Gerischer impedance yields insight into the impedance response of a glucose sensor, the chemistry of the sensor is much more complex. Three electrochemical reactions may take place in the potential range of interest. The enzymatic production of hydrogen peroxide involves more than one homogeneous reaction, and these reactions may not be modeled by linear equations. In addition, the diffusion coefficients of species associated with the coupled heterogeneous and homogeneous reactions are not equal.
The present work describes the development of a mathematical model for the impedance response of glucose-oxidase based electrochemical biosensors [3]. The homogeneous reactions included anomerization between -D-glucose and -D-glucose, four reversible enzymatic catalytic reactions transforming -D-glucose and oxygen into gluconic acid and hydrogen peroxide, weak acid dissociation equilibrium, two reactions accounting of the pH-dependence of enzymatic activity, and a series of reactions associated with the buffer system. The electroactive hydrogen peroxide was considered to be oxidized or reduced at the electrode. In addition, reduction of oxygen was considered as a potential cathodic reaction. The heterogeneous reactions were coupled by the concentration of hydrogen ions, which appear as reactants for the cathodic reactions and as a product of the anodic reaction. Thus, the faradaic impedances associated with the three heterogeneous reactions could not be considered independent.
The mathematical model was solved numerically by using the finite-difference method and Newman’s BAND algorithm [4]. The model demonstrates how the coupled non-linear homogeneous reactions affect the diffusion impedance, which has broadened the scope of the Gerischer impedance. The model can be used to explore the influence of various system parameters on limiting current, reaction profiles, and diffusion impedance. The system parameters, including interstitial glucose concentration, oxygen concentration, active enzyme concentration, diffusion coefficients, reaction rate constants and layer thickness, are related to various sensor working conditions such as body sugar level, inflammation, sensor degradation and sensor design.
References

H. Gerischer, “Wechselstrompolarisation Von Elektroden Mit Einem Potentialbes- timmenden Schritt Beim Gleichgewichtspotential,” Zeitschrift fur Physikalische Chemie, 198 (1951) 286-313.

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

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, 275 (2018), 119-132

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

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