Monitoring living tissues by electrical impedance spectroscopy 1994 monitoring living... · 2014. 1. 15. · pathology, Electrical models of living tissues. INTRODUCTION Electric

Annals of Biomedical Engineering, Vol. 22, pp. 328-337, 1994 0090-6964/94 $10.50 + .00 Printed in the USA. All rights reserved. Copyright �9 1994 Biomedical Engineering Society

Monitoring Living Tissues by Electrical Impedance Spectroscopy

PAUL HI~ROUX and MICHEL BOURDAGESt

Department of Occupational Health, Faculty of Medicine, McGill University; Electropathology Laboratory, Royal Victoria Hospital, Montreal, Qu6bec, Canada, and tInstitut de Recherche d'Hydro-Qu6bec, Varennes, Qu6bec, Canada

Abstract--Solving the experimental difficulties associated with measurement of the electrical impedance of living tissues gives access to valuable tissue compartment parameters which are sensed within seconds using minimally invasive, simple metallic electrodes. Extracellular conductivity and cell membrane capacitance can be followed over time under conditions of metabolic toxicity, perfusion loss and thermal stress in liver, brain cortex, and muscle, respectively. Application of this technique in burns therapy allows an accurate estimation of the severity of thermal injury to skeletal muscle, supporting predictions on tissue sur- vival.

Keywords--Edema, Wound assessment, Toxicity, Bums, Dis- sipation factor, Spectral analysis, 60-Hz, Microwaves, Electro- pathology, Electrical models of living tissues.

INTRODUCTION

Electric current is limited in living tissues by highly insulating cell membranes; however, with rising alternating current frequency, the membranes impede current less. Electrical impedance readings over a frequency range then allow separation of extracellular space and of the membranes themselves. Simple metallic electrodes in di- rect contact with tissue produce a phenomenon of electrical polarization, induced by the driving electrodes themselves, among other problems (9,31). Solution of these problems and interpretation the impedance spectrum signatures (2,8,12,18,21,25,30,33), yield the desired tissue- compartment variables, using: 1) a simple pair of thin metallic electrodes, minimally invasive; 2) an unorthodox variable, the dissipation factor, read at many frequencies to specify tissue characteristics (dissipation factor = D = R/X, the ratio of resistance to reactance), and 3) microprocessor-based mathematical algorithms for data interpretation. We call the technique Electrical Impedance Spectroscopy (EIS).

Acknowledgment--Jean Dumas, Yves Brissette, Frank Huang, and David Evans for collaboration. Carolyn Kerrigan and Robert W. Dykes for discussions. This research was funded by the Electrical Power Re- search Institute, Canadian Electrical Association, and Hydro-Qu6bec- Environnement.

Address correspondence to Paul H6roux, D e p ~ e n t of Occupa- tional Health, Faculty of Medicine, McGill University, Electropathology Lab, Royal Victoria Hospital, Montreal, Quebec, Canada.

(Received 200ct92, Revised 14Sep93, Revised 7Feb94, Accepted 25Feb94)

Continuous monitoring of tissue compartments was achieved for 1) intra-cellular edema produced in rat liver by ketamine-xylazine-induced hepatic toxicity (19,23,28); 2) cell membrane decay in the rabbit brain cortex by blood flow interruption (3-6,13,20,24,29,34); and 3) extracellular edema produced in rat gluteus muscle by thermal stress (10,11,14,27). Data analysis confirms known pathophysiology, and sometimes improves it. Fast diag- nostics based on this electrical technique have been developed to help surgeons gauge intraoperatively the viability of muscle in electrical burn victims (7,16). Such victims sometimes present scattered thermal damage because of the complexity of electric current patterns in the human body (17,15,35).

Integral proteins transit water and ions through cell membranes, homeostatically maintaining osmotic bal- ance. Under stress from some physical or pharmacological agents, electrolyte balances shift, producing micro- compartmental changes in cells and tissues.. Pathological manifestations of substantial shifts occur in some clinical conditions such as compartment syndrome (26).

Monitoring micro-compartment shifts using electrical measurements in bulk tissue is feasible because extracellular paths contribute a resistive component in parallel with an intracellular path made of resistive and reactive components in series. When cell membrane integrity is nearly completely eliminated, such as occurs at 135 hrs post-death, muscle has progressed towards tissue liquefac- tion and shows impedance characteristics close to those of a free-electrolyte resistor. Electrical determination allows gains in speed and simplicity over classical techniques for the measurement of extracellular space, such as the dilu- tion of radioactive ions.

IMPEDANCE PROBE

Our probe has two slender (0.17 mm diameter) electrodes held 5 mm from each other by an insulating plate and connected to two miniature coaxial cables (16). Elec- trode length is task-specific; we frequently use 3.5 ram. Implanting the probe parallel or at a right angle to tissue fibers affects the readings little since, from the small diameter-to-spacing of the electrodes, more than 50% of the impedance dwells within 0.8 mm of the electrodes. This

328

Monitoring Living Tissues by EIS 329

short reach indicates that contributing impedances are in close proximity to the electrodes in this probe geometry, which ensures a small influence of tissue boundaries on measurements. Proximity to an air interface affects the results very little, if at all (32), but histological transitions (for example, from muscle to tendon) must be guarded against when making assessments. Electrode-insulating layers, mostly oxidized metal, are countered by using gold-plated stainless steel electrodes, which are also resil- ient mechanically, although softer platinum was found to be most stable chemically. The electrodes induce at their surface and in the tissue a phenomenon of polarization, documented by Schwann (31) and others (9). Basically, the electrical discontinuities at the cell membranes and at the electrode-tissue interface are the site of substantial series impedances which are sensitive to the electrode surface state and prevent reliable readings of tissue values.

data to a narrow numerical range. Although we covered 200 Hz to 13 MHz, determinations in this paper use twenty D readings spread logarithmically between 1 and 500 kHz, acquired using a HP4192A Impedance Ana- lyzer.

It was shown early in our work that a given tissue (muscle) has a similar D signature across animal species (human, monkey, swine, rabbit, and rat) and that various organs (see Fig. 2) have specific D signatures. Because all tissues consist mostly of cells thinly surrounded by extracellular fluid, the spectral signatures differ only quantita- tively. Computation of the tissue extracellular resistance from the D curves yields an ordering (kidney cortex, liver, skeletal muscle, heart muscle, and spleen, which is the lowest (22), due to the venous sinuses and permeability of the splenic pulp) which is compatible with our knowledge of micro-anatomical variations between tissues.

DISSIPATION FACTOR

An effective way of attenuating electrode-tissue interface problems is to gather data, not as resistance or capacitance, but as the ratio of resistance to reactance, also known as the dissipation factor (D = R/X; X = 1/o~C). Since D is blind to any change in R accompanied by a proportional change in X, deepening of electrode penetration or change in effective electrode area are not sensed. In tissue implantations, standard deviations on D (or = 1.8%) are ten times smaller than on R or X. D also displays polarization characteristics as increasing linearly with frequency, allowing discrimination between authen- tic tissue contributions and those due to polarization. D also allows dealing with a single variable, and restricts

EQUIVALENT CIRCUIT AND MODELING

Electrode polarization and the mostly capacitive con- necting cable are compensated for in the impedance measurements. Polarization (Fig. 1) is modeled as R and C (below), frequency-dependent elements in series. The constants in Eqs. 1 and 2 are obtained from calibration of the electrodes in a saline solution of suitable resistance (130 l~-cm):

R = a • 10 [3.8+0.03295 (log3")-0.2367 (1ogj)2+0.02699 (log f) 3]

(1)

C(nF) = b x 10 I2"233+0"3929 (1ogf)-0.1632 (1ogj)2+0.01363 (log f) 3]

(2)

FIGURE 1. The Electrical Impedance Spectroscopy probe, with its small electrode diameter, is minimally invasive, and implantable chronically in living animals. Gold-plated stainless steel was used most often in our laboratory, although solid platinum is chemically the most stable. With proper electrical compensation, almost any conductor could be used for the electrodes. The tissue-equivalent circuit is in the center, with the twin lateral circuits representing the parasitic polarization of each electrode.

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FIGURE 2. Comparison of dissipation factor signatures for various organs of Wistar rats and rabbits. In these measurements, the impedance probe is implanted into the tissue through a skin incision after pentobarbital anesthesia. Each curve was averaged from five to nine measurements (each measurement takes four minutes) in various parts of the organ, using two rabbits and five to nine rats. A larger database on intact rat gluteus (n = 31) shows some variation within the normal physiological state, so that confirmation of exact species differences would require measurements on larger numbers of animals. The maximums in D between 2 kHz and 30 kHz are due to the presence of the cellular bodies. Scale is in 4~/~.

The tissue is modeled as a resistive extracellular path (Rext) in parallel with cell membrane capacitance (Cm), intracellular plus intercellular (gap between cells) resistance, and an inductive element, K, in series. K is be- lieved to account mostly for relaxation of voltage-sensitive and Ca-activated K + currents (8,18,25), and for some minor circuit parasites. It is generally inversely correlated with Cm in modeling results. Using baseline muscle data, a curve-fitting iteration was performed on the value of the exponent at the denominator of K. Our value of 0.63 is identical to that retained by Bao (2) in his constant-phase- angle element for the erythrocyte membrane (34~ A fractional exponent is an indication of fractal-like porosity in the underlying structure (12,21,30,33). The simulation of membranes we use is a simpler (25) one, which cannot account for effects of pore-controlling substances, for example. For the calculations, a knowledge of Rhf , resistance of the tissue at minimum reactance (usually in the 1-10 MHz range) is also required. Our equivalent circuit contains fewer elements than do classical models of tissues and cells, or than would be desirable from a physical point of view. For example, polarization in the tissue and at the electrodes is lumped together, and the conductivity of cell membranes is assumed infinite. These concessions are necessary in order to supply the curve-fitting algorithm with a minimum number of variables, thereby supporting simulation stability, and restricting computation time.

Eq. 3 for the dissipation factor, D, was developed based on the equivalent circuit of Fig. 1, assuming geometrically regular cylindrical cells (muscle). The expres-

sion was fitted to data using standard methods of numerical analysis, allowing the five independent variables (Rex t and Rin t .. . . llular q- Rint . . . . llular are geometrically coupled) to be optimized from any given dissipation factor spectrum. The error in the fit is generally low: the root-mean- square error averages 7% with a minimum of 0.9% for microwave [MW] burns data, but increases when cell membrane damage is present in the tissue.

D = 2w (D1 + D2 + D3 + D4)

(D5 + D6 + D7 + D8) (3)

D 1 =

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e e 2 f - 2Rhfe2e~xt )J 5"26

4"rr 2 (4)

= k2C 2 . n D2 C2(Rhf - Rext) 2 mem(/(ext + R ) f 4 (5)

f,2R2 l ~ 2 2R)f4.63 D3 = ,-, extt/~ext - Rhf) kCmem(Rext + Rh f +

(6)

t-~2R4 t--,2 / n e) f5 .26 0 4 = t.. extWmemt_,Xhf h_ (7)

C(eex t -- g h f ) 2 f 2.26

D5 = 4'rfl (8)

2 2 3 D6 = C(Rhf -- Rext) k2Cmemf (9)

2 2 3.63 D 7 = 2CRext(Rext -- R h f ) k C m e m f (10)


= 2 2 D8 CRextCmem(CRext - 2CRextRhf + CR~f 2 .26

h- e e x t f m e m ) f 4 ( l 1 )

The data were assessed for their repeatability by comparison of their D spectra (39 livers, 8 brains, and - 2 0 0 muscles). 25 were mathematically fitted to the equations above, from which the representative examples below were selected. Note that for each individual determination of tissue variables shown in Figs. 4, 5, and 6 there is no a priori assumption on the amplitude of the polarization

FIGURE 3. EIS Module. The apparatus allows chronic EIS monitoring of six rats under light anesthesia. The computer gathers impedance and other test data and manages anesthesia for each rat individually without human intervention. It includes, from top to bottom, a display screen for the computer, bags for anesthetic and maintenance fluids, custom interface electronics, manifolds, electric valves, metering pumps, a multiplexer unit, six cradles (with motion detectors, pad heating, and temperature monitoring), keyboard, disc drive, computer, control unit, and impedance analyzer. Data is selected by an intelligent information manager. The station is programmable using HP Basic.

elements, R and C (a and b in Eqs. 1 and 2). a and b are actually determined using the impedance data on the basis of assumed spectral characteristics of polarization, as the curve-fitting program simultaneously optimizes cell and polarization parameters. Procedures on animals and their care were in accordance with institutional guidelines.

CHRONIC HEPATIC TOXICITY

EIS was used to monitor the evolution of tissue compartments in the liver of Wistar rats under chronic anesthesia for up to 72 hr. In this test, all animals were first induced with pentobarbital. Thereafter, a ketamine- xylazine anesthetic was computer-delivered on demand, only upon detection of animal motion by the station pic- tured in Fig. 3. Because of this, control animals for these measurements were conveniently provided by three rare cases, in which subjects slept for 24 hr while receiving almost no drug. These three livers show only very small changes in their tissue parameters, compatible with a single dose of pentobarbital. In the normal tests, a rat receives 156 mg of ketamine and 2 mg of xylazine per day in small doses more or less uniformly delivered at 20 min intervals.

A typical case of the resulting evolution of tissue compartments as inferred from dissipation factor measurements and the equations above is shown in Fig. 4. Of interest in this experiment are the chronic falls in extracellular conductance, intercellular resistance and cell membrane capacitance. As the animal dies and blood circulation stops, there is a rapid loss of cell membrane capacitance, indicative of catastrophic events taking place in the tissue. Hematoxylin and eosin histological plates corresponding to normal liver and to liver intoxicated for 16 hours are shown in Figs. 4A and 4B. The dyes are more uniformly diffused in the intoxicated liver, which is compatible with the electrically measured increase in membrane permeability.

Anesthetics as a class of drugs are known to produce partial depolarization of cell membranes (19) by interfer- ing with cell membrane stability. The rapid change in intercellular resistance after the first two hours probably coincides with calcium entry into the hepatocytes (28). The increased membrane permeability manifests in the progressive loss of hepatocyte membrane capacitance. This leads to intracellular edema (23) and elimination of extracellular space sensed electrically as vanishing intercellular resistance (Rin t . . . . . llular is 10 ~t initially and 1 after death). When extracellular conductance approaches 30% of baseline, micro-diffusion in the liver is exces- sively limited, and the animal dies.

ACUTE LOSS OF BLOOD SUPPLY IN THE BRAIN CORTEX

EIS monitors rapid events following blood perfusion loss in the rabbit brain cortex. Lethal stress in delicate

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FIGURE 4. Progressive toxic changes in the Wistar rat liver followed by death of the animal. A thin sheet of mylar is fixed to the surface of the Wistar rat liver with tiny amounts of histo-acryl glue. The EIS probe is implanted to 3.5 mm in depth through the liver and mylar, the last serving to stabilize the probe mechanically in the breathing animal. The sutured animal is then positioned in one of the stalls of the apparatus shown in Fig. 3, the liver electrically monitored by computer for anesthetic-related impedance changes. (A) Permeability of membranes in tissues can be assessed by the diffusion of dyes, such as hematoxylin and eosin. In this normal liver, the pattern left behind by the washout procedure shows a mottled appearance. (B) In liver intoxicated by the ketamine-xylazine mixture for 16 hours, the dyes show more uniform penetration, compatible with increased membrane permeability.

tissues such as the cortex can occur over minutes (5,13,20,29,34), rather than hours as in liver and muscle. Anoxic damage is often accompanied by loss of integrity of the cell membranes (6,24). In this test, following anesthesia and parietal craniotomy, the dura is delicately removed and the EIS probe implanted 1 mm deep in the animal's cortex. After baseline is established, an intra- peritoneal barbiturate overdose is given to the animal. The

exact moment when breathing stops is monitored, and blood circulation is assumed to be interrupted shortly thereafter. Observed through EIS measurements following loss of blood supply are rapid cytotoxic edema (4) and loss of cell membrane integrity in the cortical tissue (6).

In Fig. 5, the significant alterations in EIS parameters 10 min after intra-peritoneal injection are interpreted as cell swelling causing a reduction of extracellular space to


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FIGURE 5. Changes in the rabbit brain cortex from pentobarbital-induced respiratory and cardiac arrest. 1 mm deep implantation of the EIS probe into the parietal aspect of brain cortex allows monitoring of rapid decays in extracellular conductance, intercellular resistance, and cell membrane capacitance. (A) Histology of the rat's pristine brain cortex shows the typical pockets (between the nuclei) that cannot fast or do not allow penetration of the dyes. (B) Histology of the brain cortex of the overdosed rat shows a softening of the previous mottled pattern, again compatible with loss of membrane integrity.

45% of baseline, together with a 30% reduction of cell membrane capacitance. It is likely that changes of this magnitude coincide with irreversible changes in the tissue, the loss of capacitance not larger because a subset of cells is initially affected. Both liver and brain EIS measurements show an elimination of micro-diffusion, although on a different time scale (4). Another important difference from the liver is the increase in apparent cell capacitance immediately before the final fall. Since decay of internal cell membranes can increase the measured tissue capaci-

tance, these results support fluorescent probe and electron microscopy evidence of early changes in the endoplasmic reticulum and Golgi apparatus in hypoxia (3,20). EIS may here display capacity to probe inside the cell, potentially revealing a site of action. Corresponding histology is shown in Figs. 5A and 5B.

THERMAL STRESS IN MUSCLE

Burns produce large amounts of edema and represent a choice case for EIS. Great care was taken in quantitative

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FIGURE 6. Extracellular conductance and intercellular resistance as a function of t ime after burn measured by an EIS probe implanted in the gluteus of a rat. The animal received a microwave burn rated at 52.7~ in a limited region of the muscle and was thereafter monitored using the EIS module. (A) Because of the axial symmetry of muscle, the amount of extracellular space can be readily appreciated from a transverse histological slide. Normal muscle shows barely any visible extracellular space. (B) In this 24-hr post-burn hematoxylin and eosin slide of muscle (microwaves, 52.7~ the extracellular space has grown substantially. This volume is free to conduct ions without interference from insulating membranes, which translates to the electrical characteristics shown at the right of Fig. 6.

EIS assessment of bums, with the aim of supporting operating room interventions.

After skin incision, anesthetized Wistar rats receive an electrode (9 mm diameter) bum to the gluteus. Rating the 40- to 80-sec bums, delivered using 60 Hz or MW currents involves computer control and four real-time temperature readings within the target. Software is used to estimate integral bum temperature, which we believe accurate to less than I Kelvin. After EIS probe implantation,

the skin is sutured over and computer-anesthetized animals are monitored by the station (Fig. 3).

As shown in Fig. 6, EIS modeling of the MW burn results shows only modest changes in intracellular resistance and cell membrane capacitance. However, extracellular edema, quantified by extracellular conductance (1/ Rext) displays large, smoothly progressing increases, in step with burn temperature. From 11,870 12 at burn baseline, the value for R e x t c a n shrink 50-fold (at 24 hr) in


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FIGURE 7. EIS(C/R) index vs. temperature in the gluteus muscle of 72 rats at 12 hr post-trauma. 60-Hz burns (squares) delivered at 10 V, mJmm for an average of 76 secs. 2.45 GHz microwaves (diamonds) at a specific absorption rate of - 1 W/g ( -1 V, ms/mm) for an average of 78 sec. The EIS(C/R) index is adjusted so that intact muscle will yield negative values. The circuit at upper left is the fast-EIS network fitted to measured D spectra, yielding C and R values for computation of the index. The lines represent linear regressions for lethal and non-lethal burns. The double arrow represents our best estimate (---1 Kelvin} of the inaccuracy attributable to temperature delivery. F_IS(C/R) can be tightly linked to 1/Rex t in the complete model, so that a slightly distorted Fig. 6 could have extracellular conductivity as a vertical axis. Another fast-EIS index, EIS(X), in the early stages of development, detects dielectric damage in tissues.

heavy (55~ bums, to as little as 200 ~ . The huge ex- pansion of extracellular space due to fluid passed from the circulation into the tissue through damaged intima (10), the innermost layer of blood vessels, is prominent on 24- hr transverse section histological slides, as shown in Fig. 6B.

60-Hz burns results were more irregular, generally showing more reduction in cell membrane capacitance, possibly revealing dielectric injury to membranes with di- minished capacity for recovery because of thermal death (11). This situation would tend to increase earlier systemic reaction in the case of 60-Hz bums, from both humoral and cell chemotaxis responses.

FAST EIS

In the surgical treatment of electrical bums, damage cannot be completely assessed in the limbs by gross appearance, color, bleeding from a scalpel cut, or contrac- tility under a 2 mV stimulator. Conventional histology (27), histochemistry, and vital microscopy have been used to determine damage limits, but each method presents problems either of practicality or interpretation.

The correct damage limits in the tissue should correlate tightly with temperature exposure integrated using the Ar- rhenius equation (14). When adjusted with experimental evidence, the equation's predictions are quite coherent in placing the lethal limit at 49~ (15,17), tissue death oc- curring over a very small temperature range (I~ This suggests that an EIS criterion tightly correlated with temperature would also be tightly related to tissue viability. To make an EIS viability index practical and faster, more specialized algorithms must be used than the general EIS curve-fitting procedure described above, which needs almost two hours per determination on a 386-7, 25 MHz microprocessor.

From modeling, variations in polarization elements, R and C, have little effect on the horizontal position of the maximum in the D signature, while changes in Rex t exert a large influence. A new electric circuit model with a curve similar to muscle was devised consisting of four elements, two of which are kept fixed, while two others are adjusted to match the position of the maximum in the EIS data (see insert in Fig. 7). In this new evaluation procedure, 15 dissipation factor measurements were performed over the frequency range and a fast curve-fitting algorithm used to yield a single index labeled EIS(C/R).

336 P. HI~ROUX and M. BOURDAGES

The C and R in this index do not relate to tissue parameters, but are tied numerically to 1/eext, and empirically to bum temperature (see below). Since the new measurement- analysis routine is typically complete in 8 seconds, it is ac- ceptable in demanding theaters, such as operating rooms.

Formal series of tests of this indicator were performed in the range of 45~ to 55~ with the results shown in Fig. 7. This range around the lethal temperature (49~ corresponds to burns that cannot be discriminated by conventional surgical criteria. The ability of EIS to do so constitutes a substantial gain in diagnostic capacity.

Both 60-Hz and MW curves show a discontinuity of 49~ and the 60-Hz tests also appear to segregate into two groups separated by a gap. From the detailed EIS analysis, it is known that the cell membrane capacitances in lethal (>5 l~ 60-Hz burns are more irregular than is the corresponding MW bums. This may indicate focused de- struction and a stronger early inflammatory reaction, ma- terialized as the gap. Similarly rated 60-Hz burns produce more edema than do MW burns at 12 hr, as seen in the figure, but the difference narrows during the next 12 hr. Statistical analysis of the EIS(C/R) 60-Hz results showed that the low (44.4~ to 47.5~ and high (50.5~ to 55.2~ temperature groups were reliably distinguished at P < 0.001 and that the group of rats formed around the lethal temperature showed a step in the index curve which satisfies conventional significance (1). If a lethal C/R index of 15 for 60 Hz was adopted, sensitivity (true positive) of EIS would rate 95% and specificity (true negative) 97%. A clinically oriented chronic study (ten days) using a different group of rats and measurement techniques closely simulating clinical conditions showed assessment based on color, bleeding, and contraction to be unreliable up to three days, while EIS values were significantly different between control, 46~ and 56~ at P < 0.001 and predictive of outcome at 4 hr (7). To support transition to human applications, specifically in the detection of damage hidden within tissues (35), an EIS probe with electrodes two cm long and with software allowing incremen- tal tissue impedance estimations in successively deeper penetrations was developed.

DISCUSSION

A simple push of a probe into tissue followed by acti- vation of micro-processor-based measurement sequences yields an EIS determination. Although the algorithms of the method may be complex, this complexity, resident in micro-electronics, is invisible to the user. The technique is fast and robust enough to allow in vivo and on-line measurement of tissue structures in the most demanding set- tings and average results over thousands of undisturbed cells at a time. Its qualities are related to the accuracy of electrical measurements, to the reliability of the dissipa-

tion factor as a descriptive variable, and to the inherent power of spectral analysis. Beyond expensive laboratory instruments, EIS can be implemented in dedicated, mostly analog, low-priced instruments, or in the time domain using a specialized digitizing card inserted in a personal computer.

As shown above, EIS can, at present, resolve compartments in a diversity of tissues. There are indications, bur- ied in the present data, that some cell compartments may be accessible. In the future, a coupling with cellular ionics may be possible.

Because EIS possesses the power of adaptable software to interpret impedance, it may, in time, profiting from progress in cell modeling and from improvements in electronics and numerical methods, contribute significantly to biological research.

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7. Evans, D.; H6roux, P. EIS as a diagnostic aid in the evaluation of high-voltage electrical injury. Canadian Electrical Association research report 165-D-286A: Appendix XI; 1992.

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9. Fricke, H. The theory of electrolytic polarization. Phil. Mag. 14:310-318; 1932.

10. Gabbiani, G.; Badonnel, M.C. Early changes of endothelial clefts after thermal injury. Microvascular Research 10:65- 75; 1975.

11. Hahn, G. Metabolic aspects of the role of hyperthermia in mammalian cell inactivation and their possible relevance to cancer treatment. Cancer Res. 34:3117-3123; 1974.

12. Halsey, T.C.; Leibig, M. Random walks and the double- layer impedance. Europhys. Lett. 14(8):815-820; 1991.

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14. Henle, K.J. Arrhenius analysis of thermal responses. In: Hyperthermia in Cancer Therapy. G.K. Hall Medical Pub- lishers; 1983: pp. 47-53.

15. H6roux, P. Thermal damage: mechanisms, patterns and detection in electrical bums. In: Lee, Cravalho, Burke, eds. Electrical Trauma. Cambridge University Press, in press.


16. H6mux, P.; Ballard, P.A.; Daniel, R.K.; Howard, C.R. Studies on electrical bums. San Francisco: Seventh annual meeting of the Bioelectromagnetics Society; June 1985:5 p.

17. H6roux, P.; Ballard, P.A.; Daniel, R.K.; Howard, C.R.; Zelt, R.G. Experimental investigation of electrical burns: analysis of recorded temperatures. In: Rudolf Hauf, ed. Beitr~ige zur Ersten Hilfe und Behandlung von Unfiillen durch elektrischen Strom. Freiburg, W. Germany; 1986: pp. 147-180.

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TABLE OF NOMENCLATURE

O R = X = o)

C =

Rex I =- C m Rhf =

Rintercellula r =

Rintracellula r = EIS =

f = a , b =

K + = Ca = K = EIS[C/R] =

p

tins EIS[X] =

Dissipation factor Resistance Reactance Angular frequency Capacitance Extracellular resistance Cell membrane capacitance Resistance at high frequency Resistance contributed by the lumen between two cells Resistance contributed by the cytosol Electrical Impedance Spectroscopy Frequency Polarization parameters Potassium ion Calcium Ion relaxation inductive variable Variable obtained from fitting an RC model to spectral dielectric data from tissues Probabili ty level specification Root mean square The frequency at which reactance becomes zero, related to dielectric damage in tissues

G R E E K S Y M B O L S

cr = Sigma (standard deviation) f~ = Ohm (resistance)

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