Monthly Archives: Июль 2014

TISSUE IMPEDANCE

Spectral information is of particular interest in medical appli­cations, where it can improve tissue characterization. A vari­ety of electrical models of tissue have been proposed to ex­plain the variation of impedance with frequency, the most widely used being the Cole plot. The tissue model in Fig. 3 would give rise to the Cole plot in Fig. 4. The difference be­tween the model and experimental findings is explained by assuming that the capacitive element has a complex re­actance given by K( ja>)—’a, where a = І would be a standard capacitor, but in tissue a = 0.8 typically. A different interpre-

Figure 3. Simplest tissue impedance model where Z(m) is the cell membrane capacitance, S is the intracellular impedance, and Z(dc) is extracellular impedance.

Figure 4. Locus of impedance versus frequency for the simple tis­sue model.

tation of tissue impedance is that the capacitance is distrib­uted, and this may give a similarly depressed Cole plot. A comprehensive treatment is given in a review by Rigaud (І4).

PHYSICAL THEORY

Take a body П in three-dimensional space with spatial vari­able x = (x, y, z) outward unit normal n. Suppose the body has possibly inhomogeneous isotropic conductivity a(x), per­mittivity e(x), and permeability p(x). A time-harmonic current density J(x, t) = J(x)e—jwt with angular frequency w is applied to the surface Ш, and this results after some settling time in an electric field E(x, t) = E(x)e—wt and magnetic field H(x, t) = H(x)e—wt in the body. Maxwell’s equations then give us

Vx E = irnaH

. (1)

VxH = (a — joe) E

Assuming sufficiently small permeability and frequency, we make the approximation V X E = 0 and therefore E = Чф, where ф is the electric potential. We now define the complex conductivity or admittivity у = a — j<we and we have the par­tial differential equation

(which is the continuum equivalent of Ohm’s law and Kirch — hoff’s law combined) subject to the Neumann boundary condi­tion

n ■ YVфЗQ = J (3)

For a known admittivity, solving the boundary value problem given by Eqs. (2) and (3) will be called the forward problem. It can be solved numerically using, for example, the finite ele­ment method (FEM). In EIT one applies a number of indepen­dent current patterns J at the surface and makes measure­ments of the potential ф|т also at the surface in an attempt to determine у in the interior. This is called the inverse problem.

Once the current density on the boundary and the admit — tivity is specified, the potential is determined up to an addi­tive constant, which we eliminate by setting

f ф dS = 0

Jsa

As there are no sources of current in the interior, Gauss’s law implies that the surface integral of the current density vanishes

f JdS = 0

Jsa

With these conditions surface current density and surface po­tential are related by a linear operator, the transfer imped­ance operator R(y)J = ф|м (referred to in the mathematical literature as the Neumann-to-Dirichlet mapping). The opera­tor R(y) represents a complete knowledge of boundary electri­cal data. In EIT we sample this operator using a system of electrodes to apply current and measure potential.

The first problem that needs to be addressed is the theoret­ical possibility of determining у from R(y). Specifically, the question ‘‘Does R(y1) = R(y2) imply у1 = у2?’’ has been an­swered in the affirmative under a variety of smoothness as­sumptions for the yt. For details of these results, including the case where the у; are complex, see Isakov (4). The closely related problem of recovering the resistance values of a pla­nar resistor network by boundary current and voltage mea­surement has been investigated by Curtis and Morrow (5) and Colin de Verdiere (6).

For the case where шр, is not negligible, Ola and Somersalo (7) show that the electrical parameters у and р are uniquely determined by a complete knowledge of boundary data n X E|sn and n X HU, provided w is not the resonant frequency.

The problem of actually recovering the admittivity from a noisy, sampled boundary data is difficult for two main rea­sons: The problem is nonlinear and ill posed. Notice that the potential ф depends on у, so that Eq. (2) is a nonlinear equa

Figure 2. The singular values of the sensitivity matrix give a clear illustration of ill conditioning of the linearized inverse problem. The singular values of S are the square roots of the eigenvalues of STS arranged as a decreasing sequence Ai > A2 > Л > 0. For a signal-to — noise ratio of S one would expect to be able to identify K conductivity parameters, where K is the largest integer with Ai/AK < S. These singular values were calculated using a two-dimensional finite ele­ment mesh, 16-point electrodes equally spaced, and trigonometric current patterns. [After Breckon (ii).]

tion for ф as a function of y. The nonlinearity is illustrated for a simple but typical example in Fig. 1.

The current is applied to the surface and voltage measure­ments made using a system of conducting electrodes. A typi­cal EIT system with l drive electrodes Db. . ., Dl and m mea­surement electrodes Mi, . . ., Mm will have single-ended digitally controlled current sources connected to all but one drive electrode (multiple-drive system), or a single double­ended current source connected to the drive electrodes by a system of multiplexers (pair drive system). The in phase and quadrature components of the voltage are measured each of the measurement electrodes.

A current pattern takes the form J, = I, x1 + • • • + Ix, where Xi is the normalized current density on the jth drive electrode (which for the moment we will assume to be one on the electrode and zero elsewhere), assuming the area of each drive electrode is the same Ii + • • • + Il = 0. Let us assume for simplicity that measurement electrodes are points and that voltage is measured relative to Mm. Let ф] be the poten­tial when Jj is applied. Then the measurements made are

Vjj = Фj (Mi) — ф} (Mm) = f Ф} (x)(S(x — Mi) — S(x — Mm )) dS

Jda

(4)

trodes, удфі/дп = S(x — Mi) — S(x — Mm). We can express the voltage measurement in an integral form as

We define the lead field ф to be the potential that would arise if a unit current were passed through the measurement elec­

RECONSTRUCTION ALGORITHMS

Now suppose that the admittivity is changed to у + Sy and the potential and lead fields change to фj + Sфj and ф, + Sфi, respectively, while the boundary current densities remain fixed. We then have the expansion

І фjудф’і/дn dS = I yV^ • ^Фj dV (5)

Jsa Ja

Z(0.6,s)

log 10(s)

Figure 1. The simple example illustrates the typical sigmoid re­sponse of boundary impedance measurement to interior conductivity change. Let П be a unit height unit radius cylinder. Suppose that the current density on the surface (using cylindrical coordinates (p, в, z) is J(i, в, z) = cos в and J(p, в, ±1/2) = 0. Let us assume a cylindrical anomaly with radius r and conductivity

(6)

j

Ja

Sijk =

(7)

Tk Vfi ■ Чф j dV

Y(p,9, z) =

The linear system V = Sg is highly ill conditioned (see Fig. 2), which means that it can only be solved with some regular­ization or smoothing of the admittivity. A simple example (Ti — chonov regularization) is to solve instead the system STV = (STS + e2I)g for some small parameter e. The resulting con­ductivity update Sy can be added to an assumed background admittivity to produce an approximate image. This simple lin­ear reconstruction algorithm is similar to the NOSER algo­rithm used by Rensselaer Polytechnic Institute (RPI) (8). As the inverse of STS + e2I can be precomputed assuming a suit­able background conductivity, the algorithm is quite fast (quadratic in the number of measurements used). However, as it is a linear approximation the admittivity contrast will be underestimated and some detail lost (see Fig. i). A fully nonlinear algorithm can be implemented by recalculating the Jacobian using the updated admittivity and solving the regu­larized linear system to produce successive updates to the admittivity until the numerical model fits the measured data to within measurement precision. This requires an accurate forward model, including the shape of the domain (8,9) and modeling of the electrode boundary conditions (i0). The non­linear algorithm is more computationally expensive as at each iteration the voltages have to be recalculated and the linear system solved.

There is still debate about the ideal current patterns Ji to drive. For a given constraint on the allowable current levels, an optimal set of current drives can be calculated. In the case where the total dissipated power is the active constraint, the optimal currents are as described by Cheney (І2). In the case of a two-dimensional disk where the unknown conductivity is rotationally symmetric, these are the trigonometric current patterns Iik = cos i ви, where вії is an angular coordinate of the kth drive electrode and І < i < l/2 (similar for sine). If the active constraint is the total injected current, only pairs of electrodes should be driven (І3); on the other hand, if the maximum current on each electrode is the only constraint, then all electrodes should be driven with positive or negative currents (Walsh functions). In medical applications the belief that limiting the dissipated power is the most important safety criterion has led to the design of systems with multiple current drives.

ELECTRIC IMPEDANCE IMAGING

The imaging of electrical conductivity and permittivity of the interior of a body from fixed-frequency electrical measure­ments at the boundary has come to be called electrical imped­ance tomography (EIT) although it is quite different from the true tomographic imaging methods in that slices cannot be imaged independently. The earliest specific references are Langer (1) and Slichter (2) in 1933, but little work was pub­lished until Henderson and Webster’s paper entitled ‘‘An Im­pedance Camera for Spatially Specific Measurements of the Thorax’’ (3) in 1979, which proposed EIT as a safe, noninva­sive imaging method and stimulated interest in the subject. By this time it was practical (economic) to implement elec­tronic systems capable of measuring with sufficient accuracy and of computing with sufficient speed for use in medical im­aging applications. The developments in this context led oth­ers to pursue the method, notably in process monitoring and geophysical prospecting, where there are different timescales and boundary constraints.

The method has potential for imaging because of the im­pedance differences of naturally occurring substances. For ex­ample, in the medical context

Resistivity, in П

• m

Human blood

1.5

Lung tissue

7.3-24.0

Bone

170

Muscle (longitudinal)

1.3-1.5

Muscle (transverse)

18-23

Brain (gray matter)

2.8

Brain (white matter)

6.8

Fat

21-28

Liver

3.5-5.5

Permittivity is measured when investigating low conduc­tivity substances like air/oil/water mixtures in pipelines by using capacitively coupled electrode systems. Resistive values are typically found when low-frequency excitation is used with directly coupled electrodes on conductive objects. Many of the resistive substances also have a small reactance that becomes measurable when high-frequency excitation is used.

PLANAR MICRO ELECTRODE ARRAYS FOR CULTURED NEURONS

Planar microelectrode arrays, consisting of transparent leads (indium tin oxide, or gold) to between 10 and 100 electrode sites (diameter typically 10 um), spaced at 100 um interdis­tance on glass plates, were used by Gross et al. (44,45), Novak and Wheeler (46), and others to study the activity and plastic­ity of developing cultured neuronal networks or brain slices. In this way, an attractive alternative was sought for the al­most impossible job of probing many neurons in a growing network by micropipettes.

An essential prerequisite for high-quality recordings is to lower the high impedance of the tiny electrode sites to below about 1 MH by additional electroplating of Pt-black (47) and to increase the sealing resistance between cell and substrate by promoting adhesion. The latter can be achieved by coating of the glass substrate with laminin-, polylysine-, or silane — based (mono)layers (48-50).

Yet a number of neurons will adhere too far away from the electrode sites to produce measurable action potentials. This led Tatic-Lucic et al. (51) to the design of arrays consisting of electrode wells, in which single embryonic neural somata were locked up. Only their neurites could protrude from the well to form neural networks. In this way, unique contacts are established, to be used as bidirectional probes into the network. Alternatively, one can improve the contact efficiency by patterning the adhesive layer; it is even possible to guide neural growth (52); for example, around and over electrodes. On the electrode side, improvements are sought by incorpo­rating an insulated gate field effect transistor (ISFET) in each electrode (53).

There is a considerable difference regarding whether stim­ulation or recording concerns an axon in a peripheral nerve trunk or a nerve cell body (called soma) lying over an elec­trode site on a multielectrode substrate. This is studied by modeling and measurement of electrode impedance as a func­tion of cell coverage and adhesion (54-56).

Except for neural network studies, cultured arrays may once be used as cultured neuron probes. They may be im­planted in living nerve tissue to serve as a hybrid interface between electronics and nerve. The advantage would be that the electrode-cell interface may be established and optimized in the lab, while the nerve network after implantation may be a realistic target for ingrowth of nerve (collaterals). Studies of the feasibility of this approach are currently underway.

CHRONIC IMPLANTATION AND BIOCOMPATIBILITY

For future use in humans, chronic implantation behavior and biocompatibility studies of microelectrode arrays will become of crucial importance.

McCreery et al. (57) implanted single Ir microwire elec­trodes in cat cochlear nucleus and found tissue damage after long stimulation, highly correlated to the amount of charge per phase. The safe threshold was 3 nC/phase (while the stimulus threshold was about 1 nC/phase). Lefurge et al. (32) implanted intrafascicularly Teflon-coated Pt-Ir wires, diame­ter 25 um. They appeared to be tolerated well by cat nerve tissue for six months, causing little damage. The influence of silicon materials silicon microshaft array rabbit and cat corti­cal tissue was investigated by Edell et al. (58) and Schmidt et al. (59). While neuron density around the 40 um shafts de­creased, tissue response along the shafts was minimal over six months (58), except at the sharp tips.

OTHER TYPES OF INTERFACES BETWEEN ELECTRODES AND NERVE TISSUE

Thus far, insertion of multielectrodes into peripheral nerve has been considered. As stated, one problem in this approach is that electrodes may have no target (fiber) close enough to be exclusive to one electrode (overlap problem). This lowers the efficiency of a multielectrode. Other ways to interface elec­trodes and nerve tissue are the regeneration of nerve through so-called sieves and the culturing of nerve cells on patterned multielectrode substrates. Both involve growth of nerve fibers or neurites. If successful, the principal advantage of such de­vices would be that each electrode has close contact to specific nerve fibers, reducing the overlap problem and increasing electrode efficiency.

Especially in neural culturing on planar substrates, a good understanding of the neuron-electrode interface is of primary concern and can directly be studied.

Both types of interfaces will be dealt with in subsequent sections.

REGENERATION SIEVE MICRO ELECTRODE ARRAYS

Another way of interfacing nerves to electrodes is the use of a 2-D (planar) sieve put in between the two cut end of a nerve. The silicon sieve permits nerve fibers to regenerate through metallized hole (or slit) electrodes in the sieve (39-43). The main advantage of this method is that microfabrication of flat devices is easier than that of 3-D devices. Another advantage

is that, once the nerve has been regenerated, the device is fixed firmly to the nerve. However, since the flats are typically only 10 um thick, there is a limited chance that nodes of Ran- vier will be close to an electrode (typical internode spacing of a 10 um fiber is 1 mm), thereby limiting the selectivity of stimulation/recording. Also, nerve fibers tend to grow through holes not as single fibers, but as a group (fasciculation), thereby reducing the possibility of selective stimulation. Zhao et al. (42) report that only when nerves are regenerated through 100 um hole diameters do they recover anatomically more or less normal, after 4 to 16 weeks of regeneration, but with about 40% loss of force in the corresponding muscle. Smaller holes yielded morphological and functional failures.