Introduction
Electrochemical impedance spectroscopy (EIS) is commonly employed in the measurement of time-evolving biological tissue state[1]. As a small-signal linearization of the net electrochemical system, the measured impedance is a superposition of both the biological system under test as well as the current-voltage transfer characteristics of the counter and working electrodes. Consequently, a limitation of the two electrode (2E) configuration is the occlusion of the impedance of the biological system by the addition of the impedances of the electrodes. A common remedy is to employ a reference electrode (RE) in a three electrode (3E) configuration, effectively excluding the impedance contribution of the counter electrode[2]. However, in many cases, the 3E configuration is not considered practicable, as true REs are costly and challenging to integrate at scale into in vitro systems and in vivo implantations; while pseudo-reference electrodes (pREs) are considered inappropriate, due to their current-dependent interfacial potential, which introduces measurement uncertainty. We propose that for small-signal measurements of biological systems, pREs may, in fact, be utilized to improve measurement sensitivity, relative to the 2E configuration and present a framework for rationally designing pREs.

Results & Discussion
The two modes of error which arise from the use of pREs are distortion of the electric field applied to the biological system under test and instabilities in pRE potential which results in uncertainty in the measurement of the electrolyte potential (to which the working electrode potential is referred). Using finite element modelling we demonstrate an approach to the design of electrode geometry so as to minimize the distortion of the applied electric field, and consequently the distribution of ionic flux through the biological system. Using small-signal modelling of the net system, we derive an expression for the error introduced by fluctuations in pRE potential, which instructs the choice of material properties and key electrode geometries, and their impact on measurement certainty. A common use case of EIS for biological systems is the measurement of tissue barrier function, namely that of epithelial tissues, such as the intestinal mucosa and the skin[3]. Equivalent circuit fitting of the EIS spectra is utilized to extract metrics which are correlated to biological state, such as the paracellular resistance and transcellular capacitance. Through application of the proposed design framework, we construct a system for hosting and impedimetrically monitoring 3D cell cultures of barrier tissues, incorporating a stainless steel pRE. Using tissue phantoms, we extract relevant circuit parameters using the conventional 2E configuration and the pRE 3E configuration. Furthermore, we culture a 3D organotypic model of a commonly used epithelial tissue system and demonstrate a significant improvement in the accuracy of both the impedance spectra as well as the extracted paracellular resistance parameter, which is correlated to the well known transepithelial electrical resistance (TEER). 

Conclusion
We present a framework for rationally designing pREs such that the error introduced into the measurement is both quantified and mitigated.  We demonstrate that simple, cost effective, polarizable materials such as stainless steel can be used to improve the sensitivity of the measurement of tissue barrier function, which we validate using both tissue phantoms as well as an epithelial tissue model in a bioelectronic organ-on-chip platform.