wave-guiding principles to construct a focused point of stimulation at any .... [3] Simon Ramo, John R. Whinnery, and Theodore Van Duzer,. Fields and Waves in ...
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High Spatial Resolution, Focused Electrical Stimulation of Electrically-Excitable Tissue∗ J.B. Laflen1,2 , T.M. Talavage1,2 , A.K. Sarychev2 1
2
Department of Biomedical Engineering, Purdue University, West Lafayette, IN School of Electrical & Computer Engineering, Purdue University, West Lafayette, IN
Abstract— A theoretical method is presented which uses wave-guiding principles to construct a focused point of stimulation at any location within a region of electricallyexcitable tissue. The excitation is formed through the constructive interference of different frequency wave energies, traveling at frequency-dependent velocities, which coalesce at the stimulation point. Further energy is brought to the stimulation point through reflection(s) at the boundaries of the wave-guiding structure. The resulting excitation signal depends upon several factors, including pulse envelope and duration, allowable frequency range, and stimulation time. A necessary trade-off exists amongst stimulation time, pulse duration, and the degree of focus.
fc ), the resulting “pulses” would travel at differing velocities. If the generation of faster velocity pulses is delayed relative to the slow pulses, they will eventually coalesce in space and produce an interference pattern. Using this principle, we have constructed a waveform in which the constituent frequency energies are appropriately phase-delayed to coalesce at the target stimulation location within the waveguide. In general, the spectral content of the stimulation waveform is: AX Z tM p F (2πf ) = A(τ ) · (2) IN τ =tM p
Keywords— electrical stimulation, implants
Z
B(ξ)dp(τ, ξ)S(τ, 2π(f − ξ))∂ξ∂τ
I. Introduction Traditional stimulation paradigms limit themselves almost exclusively to the use of electrodes. Although electrode solutions are successfully used for both single-point stimulation (e.g. cardiac stimulation [1]) and for multipoint stimulation (e.g. cochlear implants [2]), their efficacy is constrained by the physical limitations of their placement as well as their degree of resolution. Within electrically-excitable tissues, individual neighboring cells can be functionally distinct units. In some cases, this results in a high level of resolution of function within a region of cells. Sensory organs commonly express this resolution. Therefore stimulation resolution relates directly to perceptual fidelity in sensory systems. Our method of focusing constructive interference patterns of excitatory energies is a novel approach to creating stimulation with a high degree of resolution, in both the location of the stimulus as well as in the size of the affected area. II. Methodology Acoustic and electromagnetic waveguides exhibit frequency-dependent propagation velocity [3], [4] generalized by the frequency-dependent propagation “constant:” s 2 f (1) β ∝ 1− fc where fc is the intrinsic cutoff frequency of the waveguide. It follows from (1) that if a pulse envelope is amplitude modulated to varying center frequencies (above ∗ Patent
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fM AX (τ )
ξ=fM IN (τ )
with 2 dp(tp , f ) = −[(1+νM (1−(fc /f )2 ))(cot−1 νH −cot−1 νL )]−1 IN is a function of tp is the pulse envelope duration. tM p AX maximum allowed frequency, fM AX , and tM is specp ified for the application. νM is the speed of light within the waveguide (dependent upon fc ) while νH and νL are functions of fc and tp . S(tp , 2πf ) is the spectrum of the pulse envelope and dp(tp , f ) scales the envelope spectra to minimize interference away from the target. The waveguide excitation waveform is attained by applying phase distortion from (1) to the spectrum from (2). Additional interference is achieved by changing the far end of the waveguide to a reflecting boundary. We then double the relative gain in the coalescing waveform by first exciting a negative waveform, and delaying a second, positive waveform to interfere with the first as it approaches the stimulation location following reflection. We envision construction of the waveguide such that its field geometry intersects with the tissue.
III. Results Simulations of the waveforms described by (2) were computed using fast-Fourier-transform (FFT) techniques. The sampling rates were always chosen above the Nyquist criterion (at least three times the highest relevant frequency). A one-dimension waveguide was assumed for the purposes of this presentation, although the method can be extrapolated to two and three dimensional tissue geometries. The resulting data, therefore, was two-dimensional — time and distance along the waveguide.
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A
Fig. 1. Two intercepting waveforms, constructed with a pulse envelope duration of 10.0 ms. The target was the point 20.0 mm along the 30.0 mm waveguide. The vertical axis denotes distance along waveguide (top: 0 mm, bottom: 30 mm), and the horizontal axis denotes passage of time (left: 0 ms, right: 57.9 ms), thus the upper-left corner is the point (t = 0, z = 0). Red indicates maximal interference.
Fig. 1 shows a typical “bounce-diagram” of the simulation. The vertical axis corresponds to distance and the horizontal axis corresponds to time. Thus, the waveform begins in the upper-left corner (t = 0, z = 0), travels the length of the waveguide, reflects, and travels back to the waveguide entrance. Fig. 2 depicts the maximum absolute amplitude of the wave energy as a function of distance along the waveguide, across the entire simulation time. The pulse duration parameter, tp , influences the gain of the peak over the input signal, with longer envelopes producing higher gain at the cost of longer stimulation time. In Fig. 2A, tp was selected at 1 ms, while in Fig. 2B, tp = 100 ms. These results are typical of any selected stimulation target along the length of the waveguide. The target location influences the phase-delay in the waveform construction, and can be readily altered.
B
Fig. 2. Maximum absolute electric field within the waveguide for two intercepting waveforms, across the entire simulation. The stimulation target in both diagrams was 15.0 mm along the 30.0 mm waveguide. A) Pulse duration was set to 1 ms, and the stimulation time was 14.9 ms. B) Pulse duration was set to 100 ms, and the stimulation time was 241 ms.
ing an advantage to prosthetic devices where perceptual fidelity is an issue. Furthermore, the level of focus for the stimulation region is arbitrary, although there is a trade-off with stimulation time.
IV. Discussion
References
Energy used to stimulate electrically-excitable tissue must be restricted to a range of frequencies below thermal excitation, fM AX , which is typically between 10 and 100 kHz. Additionally, tissue geometries constrain the cross-section dimensions of the wavguide to 1–10 mm. These considerations place restrictions upon the waveguide, namely the intrinsic parameters fc and νM . Specifically, the propagation velocities must be slow enough that resolution along the waveguide distance is not compromised. These considerations can be addressed through acoustic waveguides with piezo-electric materials [4], or electromagnetic waveguides arranged into “slow-wave” structures [3], [5] and utilizing materials with high permeability and permittivity [6], [7]
[1] L. A. Geddes and L. E. Baker, Principles of Applied Biomedical Instrumentation, John Wiley and Sons, New York, third edition, 1989. [2] John K. Niparko, Karen Iler Kirk, Nancy K. Mellon, Amy McConkey Robbins, Debara L. Tucci, and Blake S. Wilson, Eds., Cochlear Implants, Principles & Practices, Lippincott Williams & Wilkins, Philadelphia, 2000. [3] Simon Ramo, John R. Whinnery, and Theodore Van Duzer, Fields and Waves in Communication Electronics, John Wiley and Sons, Inc., New York, third edition, 1994. [4] B. A. Auld, Acoustic Fields and Waves in Solids, vol. One and Two, Robert E. Krieger Publishing Company, Malabar, Florida, second edition, 1990. [5] R. E. Collin, Field Theory of Guided Waves, McGraw-Hill Book Company, Inc., New York, 1960. [6] Nora E. Hill, Worth E. Vaughan, A. H. Price, and Mansel Davies, Dielectric Properties and Molecular Behaviour, The Van Nostrand Series in Physical Chemistry. Van Nostrand Reinhold Company, New York, 1969. [7] S. S. Bellad, S. C. Watawe, and B. K. Chougle, “Some AC electrical properties of Li-Mg ferrites,” Materials Research Bulletin, vol. 34, no. 7, pp. 1099–1106, 1999.
V. Conclusion The presented method offers an alternative to electrode and electrode-array stimulation. The stimulation point can be constructed at any desired location, offer-
