Bio-Sensing and Biological/Electrical Interfaces

The shape and size of electrode's head may have a strong influence on the sensing performance, both of intra-cavitary electrodes and intra-tissue electrodes, whenever the sensed region is close enough to electrode's surface. The purpose of this paper is to discuss the performance of concave electrodes when used as sensing devices. The shape and size of a critical region of higher sensing sensitivity to the electric activity of biological tissues is studied using a mathematical model of a homogeneous and isotropic volume conductor. A formula is derived for the measured potential difference between a proximal concave electrode and a distal indifferent one, due to tissues' electrical activity.

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Introduction: The geometry of the electrical current field generated by a given electrode in the volume conductor formed by biological tissues and the properties and position of the target region in excitable tissues, both determine the performance of the electrode as a stimulating and as a sensing device. Both from experimental facts and from theoretical reasons, the geometry of the current field of certain concave electrodes deserves special attention.
Materials and Methods: For a field with a symmetry axis the distribution of current sources at electrode’s surface is substituted by multipolar coefficients assigned to the electric centre of the electrode. Using Levenberg-Marquardt algorithm, the multipolar coefficients can be determined from suitable experimental results. Besides this, the current densities at any two points over the same field line are related by a functional of the mean curvatures of the points of the equipotential surfaces crossed by said field line between these two points. Then, Prim’s theorem for solenoidal and almost gradient fields can be applied to the analysis of the critical region of the electrode, experimentally found as described in the first part of this work.
Results and Conclusions: At points not too far away from the symmetry axis, the field can be well represented by three or four terms of the multipolar expansion, even in the case of concave electrodes. The electric potential, considered as a function of distance on the symmetry axis, shows an inflection point for concave electrodes and none for convex electrodes. This inflection point is the centre of the critical region of the electrode. The critical region of the electrode can be defined as a certain neighbourhood of the surface formed by the points where the equipotential surfaces have zero mean curvature. Its position and extension is the related with the multipolar coefficients of the field.
Key words: concave electrodes, clothoidal electrode, multipolar expansion of a potential field, solenoidal fields, almost gradient fields, critical regions, mean curvature,

As shown in the first part of this work, the shape and size of electrode's head may have a strong influence on the sensing performance, both of intra-cavitary electrodes and intra-tissue electrodes, whenever the sensed region is close enough to electrode's surface. The purpose of this second paper is threefold: (a) Review some experimental results obtained for the lead fields and the corresponding sensitivities of a set of metal electrodes of different shapes and suitably enlarged size to use an electrolytic tank, and comparing these results with the mathematical model for concave electrodes immersed in homogeneous and isotropic volume conductors developed in the previous paper. (b) Derive in the time domain a formula for the measured potential difference between a polarisable proximal concave electrode and a distal indifferent one, due to tissues' electrical activity. (c) Give some guidelines to be applied in the conceptual design of concave electrode heads intended to sense bioelectric signals generated in electromotive sources imbedded in heterogeneous and anisotropic tissues. A virtual space (alibi), homogeneous and isotropic, is introduced in correspondence with physical space. The design problem is easier to pose and analyze in this auxiliary space. Then the results may be transformed back to the physical space.

Introduction: The design of concave electrodes able to generate a critical region (C.R.) where current density is higher and field lines are relatively parallel, located at a certain distance from electrode's surface may be of interest both for electrical simulation of living tissues and for sensing tissue's own electrical activity (1). We present here a draft of a new design method that allows a sequential determination of size and shape of electrode's head, such that the electrode will be able to generate a C.R. of pre-established position and shape in given volume conductor, assuming an axis of symmetry for the desired field.
Materials and Methods: The connection between geometric parameter of electrode’s head and its C.R. is established trough the multipolar coefficients of the electric potential field (2). In a first stage we obtain an approximation to the monopole, quadrupole and octupole coefficients as functions of geometric parameter of C.R. (This relation is obtained from Whittaker’s classical solution of Laplace’s equation). In a second stage we simulate the potential field by means of a finite analytical approximation and make comparison of obtained C.R. with the desired one. If necessary we add some multipoles and proceed to a new digital simulation, and so on until we reach the desired degree of accuracy.
Results and Conclusions: Then, we identify the point on the axis of symmetry of the field, where the potential has its extreme value before going into monotone variation towards zero. The equipotential surface that crosses the axis in this point can be adopted as the best model for size and shape of electrode’s head. The proposed method doesn’t involve the use of ab-initio numerical algorithms for the solution of the exterior boundary value problem of potential theory.
Key words: concave electrodes, clothoidal electrode, size and shape of the critical region, Whittaker’s solution of Laplace equation, multipole expansions.