Finite element analysis of a floating microstimulator

Abstract

Analytical solutions for voltage fields in a volume conductor are available only for ideal electrodes with radially symmetric contacts and infinitely extending substrates. Practical electrodes for neural stimulation may have asymmetric contacts and finite substrate dimensions and hence deviate from the ideal geometries. For instance, it needs to be determined if the analytical solutions are adequate for simulations of narrow shank electrodes where the substrate width is comparable to the size of the contacts. As an extension to this problem, a "floating" stimulator can be envisioned where the substrate would be finite in all directions. The question then becomes how small this floating stimulator can be made before its stimulation strength is compromised by the decrease in the medium impedance between the contacts as the contacts are approaching each other. We used finite element modeling to solve the voltage and current profiles generated by these radially asymmetric electrode geometries in a volume conductor. The simulation results suggest that both the substrate size and the bipolar contact separation influence the voltage field when these parameters are as small as a few times the contact size. Both of these effects are larger for increasing elevations from the contact surface, and even stronger for floating electrodes (finite substrate in all directions) than the shank-type electrodes. Location of the contacts on the floating electrode also plays a role in determining the voltage field. The voltage field for any device size and current, and any specific resistance of the volume conductor can be predicted from these results so long as the aspect ratios are preserved.

Fig. 1

Electrode geometries modeled in this study. Contacts are drawn relatively larger than they are to show details. A: the cubic shaped volume conductor is shown with the electrode geometry in C placed half way from the bottom. Figs. B, C, and D show various electrode geometries from the top view. B: a monopolar electrode on a substrate that extends to the borders of the volume conductor in the horizontal plane to simulate infinity. The electrode current (1μA) current is applied to the bottom surface of the metal contact embedded on the top surface of the substrate. The current returns through an electrode (thickness 15 μm) that covers the entire top surface of the volume conductor. C: finite width substrate (simulating shank-type electrodes) with a monopolar contact is shown. The return current is split in half between the return electrodes placed on the top and the bottom of the volume conductor for symmetry. For the bipolar case two contacts are placed symmetrically on the substrate. D: a floating stimulator is modeled. Two square contacts (each 20 × 20 μm) are placed 10 μm from each end symmetrically on the top surface. The positive current (1μA) is applied to one of the contacts and returned through the other. The device size is 40 × 40 × 100 μm. E: A sample 3-D voltage plot taken from the top surface of a floating microstimulator shows the details of the meshing structure.

Fig. 2

The effect of contact shape and substrate width on the voltage field for monopolar electrodes. A: Potential profiles for circular and square monopolar electrodes in the x direction (see Fig. 1(b) for the geometry). All voltage profiles start at the contact center. The FE solution for a square (20 × 20 μm) and circular (r=11.28 μm) contacts with the same area is plotted to demonstrate the effect of contact shape on the voltage profile. Analytical solution for a circular contact (radius=11.28 μm) on an infinite substrate is also plotted for comparison. B and C: Voltage profiles in the y and z directions for a circular contact on 60 μm and 30 μm substrates demonstrate the effect of substrate width. D: Potential fields for a single square contact at increasing elevations from its center (z = 0 to 30 μm) as a function of the substrate width (see Fig. 1(c) for the geometry). Each trace is normalized with respect to its own maximum for an infinite substrate.

Fig. 3

The differential voltage measured in the volume conductor due to bipolar contacts placed on A: an infinite substrate, and B: a floating electrode, as the distance between the contacts is varied. The voltage difference between the contact centers is plotted at elevations of 0, 10, 20, 30, and 40 μm from the surface.

Fig. 4

Potential profiles along the x(A) and y(B) axes for a floating electrode. Inset shows the line positions at which the potentials are extracted, i.e., at incremental steps of 10 μm starting from z = 0 at the contact surface. Black and gray squares are the cathodic and anodic contacts, respectively. The size of the microstimulator is 40 × 40 × 100 μm and the contacts are 20 × 20 μm.

Fig. 5

Potential profiles along the x(A) and y(B) axes for a floating electrode where the contacts are placed on the end walls and centered in the middle of the wall. Insets show the line positions at which the potentials are extracted in each case, which are at 10 μm incremental elevations from z = −10 μm to z = 40 μm in A (z = 0 is the contact surface), and at horizontally increasing distances from the contact surface in B(x = 0 to x = −50 μm). The contact and device sizes are same as in Fig. 4.

Fig. 6

Potential profiles along the x(A) and y(B) axes for a floating electrode where the contacts cover the entire walls at the ends. The stimulator size is reduced from that of Fig. 5 to 20 × 20 × 100 μm. The potentials are extracted at 10 μm incremental elevations from z = 0 to z = 50 μm in A, and at horizontally increasing distances from the contact surface in B(x = 0 to x = −50 μm).

Fig. 7

Potential profiles along the x axis for a floating electrode with one contact placed on the side wall and the other on the top. The device size is 40 × 40 × 100 μm. The potentials are extracted at 10 μm incremental elevations from z = −10 to z = 40 μm where z = 0 is the contact surface.