Fig. 8.2  Geometry of the 5 × 5 electrode array used in the model (not to scale). The electrode array is positioned inside the ocular globe, in close contact with the retina. In the model simulated the electrodes are 10 mm away from the retinal surface, and the gap in between is filled with vitreous humor. The electrodes are encased in a block of a dielectric. The current return is positioned on the back of the assembly, and exposed to the vitreous humor, which is a relatively good electrical conductor

As arrays become denser, electrodes are packed closer and have smaller size. There is a compromise between electrode size, stimulation rate, and charge injected. A smaller electrode must use higher current densities to inject the same amount of charge in the same time compared to a larger electrode. Current densities must be limited so non-reversible electrochemical effects in the electrode-tissue interface are minimized, and there is no permanent damage to the living tissue [2]. Because of inherent difficulties in performing in vivo experiments, modeling and simulation are valuable aids in designing retinal implants [6, 13, 15, 17].

The admittance method is attractive for bioelectromagnetic problems because it can solve complex heterogeneous models using a wide variety of electromagnetic stimulation types in a computationally efficient way.

8.2  Quasistatic Numerical Methods: The Admittance Method

Quasistatic electromagnetic methods have been successfully used over the last 25 years to model bioelectromagnetic interactions; in particular, variations of the finite difference method – the admittance method – and its complement, the impedance method have proven useful for diverse bioelectromagnetic problems, including calculation of specific absorption rate (SAR) and novel ways to induce hyperthermia in patients [1, 5]. Armitage and Ghandi were the first to study realistic models with generic lumped circuit element quasistatic electromagnetic numerical methods. Before them, other authors have used analytic formulations of gross geometric body tissues approximations such as cylinders and multilayered spheres (see references of [1]). Our research group extended the admittance method

formulation introducing a general two-dimensional multiresolution meshing algorithm, in which cells close to tissue boundaries are discretized with fine resolution while cells surrounded by large areas of homogeneous space are progressively larger [3]; more recently, a three-dimensional method based on the same principle has also been developed [18].

In the impedance method, the electrical properties of the model are described using an impedance network; similarly, an admittance network is used in the admittance method; otherwise the methods are equivalent. For both methods there are several possible formulations suited to different electromagnetic stimulation mechanisms, be it an external electromagnetic field, a capacitive electrode, a metallic electrode, an implanted coil, etc. For the rest of this chapter we will focus on the admittance method.

The quasistatic constraint assumes that the highest significant frequency component of the electromagnetic fields involved in the simulation has a wavelength much larger than the size of the model simulated. Another way of looking at the validity of the quasistatic assumption is to consider if at the single highest frequency component used for excitation it is reasonable to assume that the phase differences across the model are negligible. For the model sizes used in this application, the quasistatic condition is met up to frequency components in the range of tens of megahertz.

The general idea of the admittance method is to obtain an equivalent circuit model starting from the geometry of the implant and surrounding tissue and the bulk electrical properties of the substances involved, and then apply electrical or magnetic stimulation using ideal current and voltage sources. The resulting equivalent circuit is solved for the node voltages using circuit theory and a numerical linear solver. Branch currents can then be calculated from the node voltages and circuit components values. Noting that each node in the equivalent circuit will correspond to the location of a spatial point in the model, the numerical solution of the simulation using the admittance method is the current vector field and the matching scalar potential field at each significant point in the model. Electric field, current densities, equivalent impedances, etc., can then be derived from these results.

8.2.1  Layered Retinal Model

For retinal implants, detailed three-dimensional models of retinal tissues and surrounding areas can be constructed using a layered retinal model [17]. In the multi-layer retinal models each layer represents a different tissue type. The values for conductivity (s) used for the simulation results presented have been obtained from experimental measurements performed on a frog’s retina [11]. The thickness values for each layer have been measured from electron microscopy images of a transverse cut of a mammalian retina [16]. The retinal layers considered and their thickness and conductivity are shown in Table 8.1.

In addition to the retina, models for retinal stimulation often must include the implant, electrodes, current return, and surrounding tissue, including choroid

The conductivity values used for this set of simulations correspond to frog retinal tissue [11]

Fig. 8.3  (a) Simulation model is obtained from a geometric description of the retinal layered substrate, electrode array and current return. The 2.5 mm × 2.5 mm × 0.48 mm model was discretized to a resolution of 10 mm, resulting in a mesh having 250 × 250 × 48 voxels. In this view, current sources are noted as triangles under the electrodes, and the position of the current return marked by the central triangle. (b) Transverse slice of the model at 45Y = 125, showing detail of the retinal layers and the implanted electrode array, dielectric backing and current return (topmost layer of assembly)

(s = 0.92 s m−1), sclera (s = 0.50 s m−1) and vitreous humor (s = 1.5 s m−1) [4]. The model used is depicted in Fig. 8.3. For the example presented in this chapter, a 25 electrode array having cylindrical electrodes of 200 mm diameter spaced 500 mm between centers, and with the current return exposed to the vitreous humor on the back of the electrode array dielectric substrate has been included. Each electrode is constantly injecting – 50 mA for cathodic stimulation, and the simulation considers the model to be static, i.e. purely resistive.

A discrete model is then obtained by spatially sampling a three-dimensional geometric description of the tissue and implant at regular intervals on the three Cartesian coordinate axes. This sampled model can then be interpreted as a collection of voxels, each surrounding a sampled point and considered made of a single material. Since the thickness of individual retinal layers are in the order of tens of micrometers, to calculate electrical activity inside a layer the model must be resolved with voxels of size 15 mm or less.

In order to reduce the voxel count and thus the size of the resulting linear system, the spatial sampling can optionally be made to create an expanding grid in one of more spatial dimensions and additional multiresolution techniques can be applied

An equivalent electrical circuit is created by considering all the resistors for every voxel, and knowing that contiguous voxels share two nodes at each shared edge. The ground node is assigned, and the stimulation is added using current sources. In the example case considered in this chapter we have used one current source per electrode; since we considered cathodic stimulation, for each current source the negative terminal is connected to a node belonging to an electrode and the positive terminal to ground. Additional circuit elements can also be added to the equivalent circuit to model other electrical behaviors of the tissue or implant.

8.2.3  Electric Potentials and Current Density Magnitude

Calculation

This equivalent electric circuit can be solved using standard circuit theory tools. Using Kirchoff current law, the model’s equations can then expressed as a linear combination, in terms of an admittance matrix G, an unknown voltage vector V, and a current vector I, as shown in (8.2).

On a circuit having n nodes including ground, G is a symmetric, sparse, (n−1) × (n−1) admittance matrix, V is the unknown voltage vector, and I is the vector of independent current sources modeling the stimulating electrodes. While any method can be used for small systems, the size of the matrix G grows with the square of the number of nodes. Because of this, direct methods such as Gaussian elimination may not be the most suitable for this application; instead, iterative approaches including Krylov sub-space methods present advantages. In particular, the biconjugate gradient method with appropriate preconditioning has proven to be efficient in the considered cases (Figs. 8.5 and 8.6).

The solution of the system provides the value for the potentials at each node of the circuit, which are equivalent to the potentials at the vertexes of each voxel. The components of the current density vector field at each point in the model can then be determined from the cross-sectional areas of each voxel, the equivalent resistors, and the branch currents flowing through each of them:

For each voxel, the branch currents parallel to one coordinate axis are calculated by considering the voltage at the nodes and the four resistors along that direction. For the X axis, for example, the current density component is determined by taking

Fig. 8.5  Top: Model slice (Y = 125) and contour plot of the electric potential resulting from admittance method simulation of the retinal prosthesis applying (center) cathodic and (bottom) anodic stimulation through the central electrode of the array. Units on the X and Y axes are voxels

Fig. 8.6  Contour plot of the electric potential resulting from admittance method simulation of the retinal prosthesis applying (a) cathodic and (b) anodic stimulation through all 25 electrodes. Values correspond to Y = 125 slice. Units on the X and Y axes are voxels

into account the contribution of each sub-voxel in the X direction (8.3), where JX is the current density component for direction X, In is the branch current going through each of the resistors parallel to the X axis, and Yn · Zn is the transverse area for each sub-voxel in the X direction. The current densities in the remaining axes are calculated in a similar way.

Figures 8.7–8.9 show the resulting current density magnitudes for the model slice considered in Fig. 8.5, and the values on a line at the ganglion cell layer, under the middle row of electrodes, for both the case of a single electrode and all 25 electrodes excited.