- Email: [email protected]
Impedance spectroscopy of the electrode-tissue interface of living heart with isoösmotic conductivity perturbation
Chemical Physics Letters 390 (2004) 445–453 www.elsevier.com/locate/cplett
Impedance spectroscopy of the electrode-tissue interface of €smotic conductivity perturbation q living heart with isoo Marc Ovadia
a,b,c,*
, Daniel H. Zavitz
a,b,d
a
Laboratory of Materials Characterization for Cardiac Pacing, College of Medicine, University of Illinois, 840 South Wood Street, Mail Code 856, Chicago, IL 60612, USA b Department of Pediatrics, College of Medicine, University of Illinois, 840 South Wood Street, Mail Code 856, Chicago, IL 60612, USA c Department of Medicine, College of Medicine, University of Illinois, 840 South Wood Street, Mail Code 715, Chicago, IL 60612, USA d Department of Chemistry, University of Illinois, 845 West Taylor Street, Mail Code 111, Chicago, IL 60607, USA Received 2 January 2004 Available online 6 May 2004
Abstract Impedance spectroscopy was used to solve the Pt electrode interface with metabolically active perfused living heart. Three impedance spectra were observed: the Warburg impedance (ZW1 ), a single high angle constant-phase-element, and a thin-film impedance (ZD ). When characterized again after cyclic change of ionic strength (and hence conductivity j) each interface had one of only two spectra, with exclusion of ZW1 . The in vivo interfacial impedance spectrum is thus neither single-valued nor stable in time. Because metaljliving tissue interfaces are obligatory circuit elements in biosensors and electrodes in heart and brain, the multiplevalued and thin-film character of its impedance are significant. 2004 Published by Elsevier B.V.
1. Introduction The use of electrodes in living tissue is fundamental in the area of sensors. For biophysical recording and stimulation, electrodes in living tissue are fundamental in biology and medicine. In sensors, a signal is generated at the electrode-living tissue interface; biophysical recording involves signal transmission and cardiac pacemakers involve current passage through the interface. The equivalent circuit of this interface has never been solved [1,2]. Extrapolations from non-living systems are inadequate [3] and it has not been possible to study this interface at sufficient bandwidth. In a perfused living beating rat heart system [4–7], we performed impedance spectroscopy at the platinum (Pt)electrodejtissue interface using iso€ osmolar substitution q
Presented in part at the American Chemical Society, the Gordon Research Conferences, Frontiers in Electrochemistry and an Invited NIST Colloquium. * Corresponding author. Fax: 708-346-4788. E-mail addresses: [email protected], [email protected] (M. Ovadia). 0009-2614/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.cplett.2004.04.046
of perfusate to change ionic strength and thereby vary conductance of perfused elements. The data disclosed an in situ generated thin-film diffusional impedance existing alongside conventional circuit elements predicted for noble metaljelectrolyte interfaces. 2. Experimental 2.1. Rat preparation Long-Evans rats (wt. 250 g) were sacrificed and the aorta cannulated via median sternotomy,then perfused immediately to avoid no flow ischemia. 2.2. Electrodes The platinum (Pt) working electrode was placed in the ventricular wall tangentially with no exposure to the ventricular cavity (Levy Type 3 insertion [4]); the AgjAgCl reference was placed in the right atrium in contact with the blood pool; and a large surface area auxiliary electrode was placed in the right heart. Plati-
446
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
num (Pt) wire was obtained from Aldrich Chemical (0.25 mm diameter, 99.9% Pt, No. 34,940-2). 2.3. Instrumentation Sine waves of 2–4 mV amplitude (RMS) and frequency range 30 Hz–1.24 MHz were applied to the preparation, where the excitation signal was summed with a DC bias of )500 mV vs. AgjAgCl; current density was 3–11 mA/cm2 . Impedance spectra were obtained for a smooth Pt working electrode, in three-electrode potentiostatic configuration. Digitization at up to 100 MS/ s, signal-averaging and signal extraction were performed according to published techniques after inspection of current traces to exclude diodic behavior [5]. Important aspects of the preparation include (1) the broadband AD744 BiFET current-to-voltage converter, (2) DC bias potential based on cyclic voltammetry at a flat region of the i–V plot to avoid Faradaic peaks due to redox-active endogenous substances which might introduce diodic or active elements (resulting from their N-type negative impedance i–V relation), and (3) short time period between sacrifice and data gathering. Spectral purity and analog validation of the digital signal extraction algorithms have been published [5]. 2.4. j-Perturbation (iso-PTGO) The preparation was perfused with oxygenated buffered modified Tyrode’s solution from the time of aortic cannulation. To solve the equivalent circuit, systematic perturbation of all impedance elements parallelly bridged with a perfused ionic conductance, was performed by iso€ osmolar perfusate substitutions to vary conductivity j. In this technique, serial perfusate solution substitutions are made by replacing the initial Tyrode’s solution 1 with iso€ osmolar sucrose x : (1 x), (typical series x ¼ 0, 0.2, 0.25, 0.35, 0.75, 0.9, 0.99) to a nominal 99:1 Sucrose: Tyrode’s stock mixture with a final return to Tyrode’s (x ¼ 0) solution. The solution was kept saturated with 95% O2 :5% CO2 mixture and pH adjusted to 7.35 0.03 with a NaHCO3 /CO2 buffer. Conductivity j of the perfusate was varied over two orders of magnitude with pressure, temperature, geometry, and osmolality held constant (iso-PTGO); we refer to this as a j-perturbation (iso-PTGO). 2.5. Analysis Impedance spectra were determined as the complex impedance plane plots of impedance Z; Randles’ plots 1 The Tyrode’s solution had the following concentrations (in mM): NaCl 136.9, KCl 2.68, MgCl2 0.5, NaHCO3 12.0, NaH2 PO4 0.4, glucose 6.0 in deionized water (Sybron Barnstead NANOpure IITM ; Boston, MA) of measured resistivity 17.7 MX cm.
pffiffiffiffi1=2 ) were employed to aid analysis. (Re{Z} vs x LEVM (Solartron, Farnborough, ENGLAND) was used to simulate model equivalent circuits. MATLAB-5 (MathWorks, Natick, MA) was used for plotting interpolated three-dimensional data sequences (termed H–Nyquist plots) and to perform Monte Carlo calculations. Other data manipulations are described in the text.
3. Results and discussion Impedance spectra were determined in the living beating heart with a Pt electrode placed tangentially in ventricular wall. Fig. 1a shows a typical impedance spectrum derived for a 12.4 mm2 Pt electrode in the living heart at normal ionic strength (153 mM), conductivity j ¼ 15:9 mS/cm and osmolality 315 mOs/L. pffiffi Z is a complex quantity Z ¼ RefZg þ iImfZg, i ¼ ð 1Þ, whose complex conjugate Z is here plotted. With increasing frequency the impedance approaches the Re{Z} axis, indicating a pure resistance of magnitude equal to the Re{Z} (x-axis) intercept. This impedance locus follows in the lower frequency range the pattern defined as a constant phase angle element (CPE), described in non-living systems [8]. We termed this CPE1, with phase angle j/j > p=4 radians and no low frequency bound on the CPE line segment [5]. The CPE is a heuristic concept; genesis of such CPE’s has been attributed to dispersion of permittivity in some systems. Similar results in nonbiologic systems are attributed to dispersion of conductivity [8–10]. Fig. 1b shows the same interface in the same experiment, after perfusion with iso€ osmolar perfusate of reduced conductivity (j ¼ 11 mS/cm). Despite an electrodejtissue interface identical in all elements to that of Fig. 1a, the impedance plane plot has undergone a quite remarkable transition. In the high frequency region a second CPE has emerged, of lower phase angle, (slightly less than p=4 radians), bounded by intersections with CPE1 and the Re{Z}-axis. We term this CPE2. The presence of two CPEs as the impedance locus is characteristic of thin film behavior. This thin film impedance locus 2 can be derived analytically from the diffusion equation by application of appropriate boundary conditions [11,12]. Thus this experiment at the present level of resolution offers insight into generation of CPEs at biologic interfaces. The CPE at j/j > p=4 is the first of two CPE’s comprising the thin film diffusional impedance. At usual ionic strength, only CPE1 appears; j-perturbation (iso-PTGO) proves its origin in the thin film
2
Familiar in solid state ionics.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
447
Fig. 1. (a–c) Impedance spectra of a single living beating heartjPt electrode interface (12.4 mm2 ) under j-perturbation (iso-PTGO). j ¼ 15:9 mS/cm (a), j ¼ 11 mS/cm (b), and j ¼ 0:2 mS/cm (c). (d–e) Impedance spectra of a single living beating heart-Pt electrode interface (11.8 mm2 ) under j-perturbation (iso-PTGO). j ¼ 15:9 mS/cm (d), j ¼ 0:2 mS/cm (e). (f) Plot of high frequency capacitance versus ionic strength showing square root dependency and a zero intercept.
impedance locus. This is the first proof of the genesis of a high-angle CPE in any biological system of the present type. Fig. 1c shows the impedance plane plot for the same interface after perfusion with iso€ osmotic buffer of lower conductivity (j ¼ 0:2 mS/cm). The impedance locus has undergone further transition and now displays a depressed semicircular element at high frequency [13]. Fig. 1d shows an experiment at normal ionic strength (153 mM) for an 11.8 mm2 electrode. The impedance plot demonstrates two CPEs. When the heart is perfused through its normal vasculature with iso€ osmotic perfusate of conductivity j ¼ 0:2 mS/cm (Fig. 1e) the same transition occurs to a semicircular locus. Again the center is depressed below the Re{Z} axis, which usually indicates a non-Debye capacitor parallel to a resistor. The resistive element, existing in liquid phase electrolyte, has been created by now relatively resistive perfusate. The reduction of conductivity by pure j-perturbation (iso-PTGO) in perfused regions of current passage thus
brings about reproducible changes in the impedance locus by perturbing equivalent circuit elements. The question arises, What part of the interfacial equivalent circuit is perfused, and therefore affected? Fig. 1f shows a plot of capacitance 3 [5] versus ionic strength. Because capacitance arises from charge separation at the interface, it should behave as a type of double layer capacitance if the interface is fully perfused. The figure shows this behavior, with square root dependency of capacitance on ionic strength and zero intercept. The relation of capacitance and ionic strength was consistent with a square root relationship (p < 0:0001) with no higher order terms [14]. This is compatible with a Go€ uy–Chapman–Stern diffuse layer [15,16], and predicts that the interface is perfused to 7.9– from the electrode. Because muscle is highly vas90 A cular tissue the entire interfacial equivalent circuit is
3
Calculated at the high frequency limit.
448
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
RB ¼ RB ðjÞ; H ¼ HðUÞ
Fig. 2. Equivalent circuit of the living beating heart electrode interface (see text).
perfused, and j-perturbation (iso-PTGO) is a probe of the living electrodejtissue interface 4. These data permit deduction of the equivalent circuit (Fig. 2). This is RB Ra1 ½ðCÞðRa2 HZD Þ where RB is a bulk resistance (the Re{Z} intercept at high j), Ra1 is an access resistance (the increment in the Re{Z} intercept at low j) 5, C is the high frequency interfacial capacitance 6, Ra2 is an access resistance (the base of the semicircle at low j), H the Faradaic impedance (H 0 by experimental design), and ZD is the previously discussed thin-film diffusional impedance that confers double constant phase element behavior whose presence is proven in this communication. At high j, the access resistance Ra2 is low: there is no manifestation of the parallel capacitance C, but only ZD and RB are observed (since Ra1 is also low). By contrast at low j, the augmented magnitude of Ra2 brings out the parallel capacitance. The equivalent circuit bears similarities to that determined by reducing surface area of the physical electrode, where the experimental variable (perturbation) is increased access resistance, with the unfortunate concomitant variable of altered geometry [5]. Since an identical macroscopic interface is present before and after j-perturbation, the new data definitively exclude the alternative hypothesis that semicircular behavior in these systems implies transition to ultramicroelectrode character due to a non-affine changed vector field symmetry (for P, D, E and P [or Z], Hertz’s polarization potential [5,18–20]) reflected in the transition to the semicircle. Functional dependencies under j-perturbation are: 4
In contrast to such perturbations as reductions of surface area or electrode conductance, this perturbation affects only the living tissue side of the living tissuejelectrode interface. 5 Ra1 is in fact an approximation to the tensor impedance quantity Zij whose off-diagonal elements’ functionality reflects critical phenomena for excitable systems that therefore may be modeled by percolation theory (excitable systems is a term that includes cardiac muscle, nerve, and skeletal muscle inter alia). For non-excitable living systems we have studied to date (cartilage, a poroelastic medium and bone, both of which geometrically resemble lattices where percolation processes [17] are known to occur) Zij has zero off-diagonal elements and the phenomena may be modeled as diffusional. 6 Whether this does or does not include a Debye capacitance within this non-Debye capacitance is not specified here; in fact, for all systems we have studied a detectable Debye capacitance can be identified (vide infra, Footnote 5 and legend to Figs. 4 and 3) that we attribute to organic adsorbate on metal faces.
Ra1 ¼ Ra1 ðx; jÞ; and
Ra2 ¼ Ra2 ðx; jÞ;
C ¼ Cðe; N ðjÞÞ;
where U is the offset potential and N ðjÞ the distribution of conductivities. Electrode material is subsumed in U. For the character of the variation of the equivalent circuit under jperturbation (iso-PTGO) is identical to that observed under surface area perturbation [5] (where in addition to surface area of the ionically-conducting electrolyte element of the interface, the electronically conducting metallic/admetallic elements are concomitantly varied). Since surface area perturbation simultaneously varies both the ionic and the electronic conducting parts of the equivalent circuit, whereas j-perturbation varies only the ionic conductivity, the identical character of the equivalent circuit under either perturbation implies that the variation of electronically conducting impedance elements (metal-side) has no measurable effect on the equivalent circuit 7. As a corollary, there is no electronically conducting true physical thin film 8 underlying thin-film behavior 9. Additional observations bear comment regarding evolution in time of the impedance spectrum that occurs for the same experimental conditions. It is axiomatic in impedance spectroscopy that a single impedance spectrum corresponds to each particular physical circuit. For macroscopic systems this follows directly from the first law of thermodynamics 10. As mentioned, we observed a single high angle CPE locus at highest conductivity (similar to Fig. 1a), as well as of a dual CPE locus in similar preparations (similar to Fig. 2a). We made the observation also of the appearance of the alternative locus after j was perturbed in cyclic fashion (changed and then returned to its original value). Occasionally a third pattern was observed, that of a single CPE resembling CPE2 and the classic Warburg impedance ZW1 which also appeared to interconvert. Experimental protocol dictated that perfusate used in the final stage of each experiment was identical to the initial, returning the preparation to its initial state. We tallied the occasions where a transition was observed
7 And in U variation is the unimportant work function variation subsumed. We have shown recently in living tissue that for metals as disparate as Pt and Au, whose work functions differ by 7%, the open circuit potential (the standard electrode potential for the virtual electrode) differs by less than 1% and is undetectable even by a null current measurement carried out in 3-electrode potentiostatic configuration [21]. 8 E.g., Corrosion product. 9 This does not exclude the possibility of an ionically conducting true physical thin film or, as will be discussed in the figure caption for Figs. 4 and 3, a kinetically electronic but non-conducting lamina. 10 Alternatively stated, the impedance locus is a single-valued function of the physical circuit; there is no degeneracy for particular circuit elements in a particular geometry.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
from one impedance locus to another during the same experiment. Using ZD [5] to designate the double constant phase angle defined visually above 11, CPE1 for the higher phase angle CPE, and identifying the CPE2 plot by the symbol [5,8,22,23] ZW1 , all possible transitions are ZD ! ZD ZD ! CPE1 ZD ! ZW1
CPE1 ! ZD CPE1 ! CPE1 CPE1 ! ZW1
ZW1 ! ZD ZW1 ! CPE1 ZW1 ! ZW1 :
Diagonal elements have identical initial and final states; * transitions were not observed 12. Each other element represents an observed transition with initial and final states as noted. The fact that any of the non-diagonal transitions occur within this set of impedance spectra, implies that the different impedance spectra are different manifestations of the same living biophysical interface. The experimental observation of transition between different members of this equivalence class may be written as an evolution matrix of transition probabilities. Writing kaij kði ¼ 1; 2; 3; j ¼ 1; 2; 3; 1 ¼ ZD ; 2 ¼ CPE1; 3 ¼ ZW1 Þ for the transition probability to impedance spectrum i and from impedance spectrum j gives a11 a12 a13 0:64 0:56 1 Piso ¼ a21 a22 a23 ¼ 0:35 0:44 0 a31 a32 a33 0 0 0 applying to any experimental preparation, substituting 0’s for unobserved processes 13. Sample standard deviations for the first two terms may be included: a11 ¼ 0:64 0:05 and a21 ¼ 0:35 0:08. From the theory of reducible matrices with nonnegative elements [25] a final state under repeated transitions may exist and in the limit as t ! 1 N
1 ¼ lim ðPiso Þ ¼ lim ðPiso Piso Piso Piso Þ Piso N !1 N !1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} N terms 1 1 1 0:61 0:61 0:61 a a a 12 13 11 1 a1 a1 0:39 0:39 ¼ 22 23 ¼ 0:39 ; a21 a1 a1 a1 0 0 0 31 32 33
449
noting that in the time period of the experimental protocol the transition matrix is constant. This result may be obtained numerically because of rapid convergence under Cayley multiplication 14.
14
The question may be asked, is convergence of this infinite product highly dependent on the exact values of the transition probabilities or is convergence assured for a large range or for the entire range of likely values? This is a question somewhat akin to the question of whether this is stable as a dynamical system, but it differs from that question as posed ordinarily for dynamic systems that enter a chaotic regime based on slight alteration of the initial conditions, i.e. their canonical Hamiltonian variables characterizing the state or equivalently the region of phase space in which the system is allowed to operate. Here, the transition probabilities are physicochemical constants determined for the particular system, and change of material or ambient concentrations of mineral or biological species alone would be necessary to change them. We performed a Monte Carlo calculation [26] assuming sampling from a Poisson distribution, and found convergence and were able to derive error bars for the final transition matrix, with convergence of all tested matrices. This proves that there is a neighborhood of convergence around the determined values for the transition probabilities, that is, that in the six-dimensional space represented by the 3 · 3 transition matrix, the point is a H€aufungspunkt [27] for the domain of convergence. This is quite remarkable actually. For consider the obvious comparison matrix for such a determination, the identity matrix for transition probabilities between the different states which would characterize monostable interfaces 1 0 0 0 1 0 : 0 0 1 An infinite product of the identity matrix with itself yields the identity matrix, so clearly there is convergence of the infinite product at the point represented by the identity matrix. But does this matrix lie embedded in a region where nearby points converge similarly, as does the matrix we determined experimentally? The answer is no. For if one deviates any of the coefficients ever so slightly from the stated values, e.g. 1 0 0
0 1d d
0 0 ; 1
where d is some vanishingly small number, the matrix does not converge to the identity matrix. Even if the probability is, say 11
Analytically in [5]. An example of an alternative notation is j1 jCPE1 ¼ ZD (in place of CPE1 ! ZD used in the text), where j represents the j perturbation, j1 the return to original ionic strength and conductivity, and the symbol j1 j CPE1 signifies a system characterized by CPE1 that is subjected to a j perturbation followed by the return to original conductivity j. The equation j1 jCPE1 ¼ ZD states that after perturbation and then return to the original conductivity j, the system is found experimentally to be characterized by a ZD impedance locus. This notation calls attention to some important pecularities of the system, as the system and the operation as defined are not a group but at most a semigroup [24], since the equation j1 j ¼ 1 (the identity element, implying no change in time) does not hold for the system when a single impedance locus is taken as the argument (thus there is no inverse element). 13 Transition processes whose initial state has not been observed, as well as those (where the transition probability may be estimated more precisely) where the state is uncommon. 12
d¼
1 101012
a number so small as to be meaningless chemically, the system will eventually diverge away from the identity matrix. Thus the identity matrix is not a H€aufungspunkt, it has no neighborhood where other points exist where the infinite product converges. This discussion is placed in a footnote because of its mathematical character. But what it implies chemically is that monostable interfaces may be less stable in biological systems than bistable interfaces, because the monostable interface does not exist in a domain of convergence or stability for the interfacial system. Parenthetically, extreme care must be taken in defining the meaning of neighborhood since at this point in the exposition, no metricization has been performed on the space, though it is well defined experimentally, in the clause ‘We tallied the occasions where a transition was observed from one impedance locus to another during the same experiment, etc’. By way of anticipation, by the end of Footnote 9 (Vide Infra), these difficulties will have been resolved.
450
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
In the limit, the proportions observed will be 0.61:0.39:0 for impedance spectra ZD :CPE1:ZW1 . With uncertainties, these are 0.61 0.05:0.39 0.08:0. The multiple patterns of impedance spectra imply that the surface structure of the electrode is dynamic in living tissue with conversion between different spectra occurring spontaneously. One introduces hereby the concept of an equivalence class of impedance spectra (necessary because the impedance of the interface can be defined only to such precision as allowed by the multiple-valued impedance spectra of the equivalence class). Name the equivalence class EDf , Df standing for D€ unnfilm, the biologic thin film. EDf is a set of 3 impedance spectra 15 [6] ZD , CPE1 and ZW1 . Each signifies for the living biological preparation the presence of the same biophysical interface. Specification of the members of the class defines the interface electrically. This concept resolves confusion in the electrochemical literature, where a single impedance spectrum is assumed sufficient to characterize an interface. The absence of ZW1 in the limit implies that the apparent ZW1 is a transient (metastable) state. The definition of the true impedance element of semi-infinite linear diffusion ZW1 (a constant phase element whose phase angle is p=4 and whose Randles’ plot is linear [8,22,23,28,29]) is inadequate, as it fails to distinguish the true ZW1 from the ZW1 identified in this communication that is part of the EDf equivalence class. To specify the classic Warburg impedance and exclude the latter element, one must add the criterion that the ZW1 impedance spectrum be stable in time and under cyclic j perturbation 16. By this criterion, ZW1 has never been observed in electrodejliving tissue interfaces. This conclusion is equally applicable to the CPE1. On the basis of the present data, this CPE is the lowest frequency part of ZD . CPE1 is merely a manifestation of the EDf equivalence class. It implies the existence of a thin film entirely covering the electrode, and is caused not by dispersion of dielectric properties but by transport characteristic of a laminate phase. The significance of this latter remark cannot be overemphasized, providing an important corrective to the temptation to assume, based on the literature of impedance spectroscopy in solid state physics, that for a distributed element the dielectric permittivity is the important dispersed quantity. In fact conductivity may be the relevant quantity
15
Or, from an alternative but equivalent viewpoint, of transfer functions corresponding to the spectra. Whether the cardinal number of this set may be the transfinite @0 or 2@0 in the limit as period increases without bound, is a question that may be deferred to another communication. 16 By the alternative notation, this criterion may be expressed as j1 jZW1 ¼ ZW1 . Such a hypothetical system and the operations as defined would constitute a group, albeit one that prior to the present communication has never been conceptualized theoretically.
whose distribution is reflected in the appearance of the distributed element. Hook et al. [30] have identified two states of adsorbed hemoglobin on octadecathiol self-assembled monolayers existing under a form of j-perturbation. j-perturbation promotes ordering of the adsorbed protein layer, the physicochemical basis being ionic strength dependency of protein solubility. The thin film is not amorphous but in fact has 2 or more ordered states each corresponding to a member of the equivalence class. In order to answer an additional question, What solid state circuit elements are most similar to these observed in living tissue?, we performed a simulation not of a single impedance plot but one taking into account the range of the perturbation. Figs. 3 and 4 show data in a 3-space (x; y; z;) H–Nyquist plot, where )Im{Z} is plotted on the z-axis, the perturbation fractional ionic strength (Fig. 3) or surface area (Fig. 4) the x-axis and Re{Z} the y-axis. A yz plane is drawn at the location of the unperturbed system’s impedance plot, and each projection of the locus onto this plane represents a impedance plot for the perturbed system. Color maps either the magnitude of )Im{Z} or the frequency. Note the similarities between the plots for the two perturbations. The rich fine structure (nested dispersed elements) is apparent in the limits of low ionic strength and high frequency. In Fig. 4 a simulation is depicted whose salient feature is inclusion of a reactance in the Hawriliak-Negami element. The simulation of the actual H–Nyquist locus is evident. In summary, the interface is an in situ generated ordered thin film determinate only to within a degeneracy of n ¼ 3, the members of the equivalence class. (Different impedance spectra may be considered congruent to within EDf 17 [35].) This is important when interpreting signals transmitted through an electrode tissue interface, and is relevant to analytical chemistry (for sensors), biology, neuroscience, and cardiology where sensing electrodes are essential to pacemakers and defibrillators. Degeneracy of the impedance spectrum implies that the transfer function is multiple-valued as well, important in the frequency band 0.5–50 Hz relevant to electrophysiologic signals. This multiple-valued character makes it impossible to recover source signal from signal transmitted through the interface without simultaneously interrogating the interface to determine the impedance spectrum extant at the relevant frequencies. This is caused by the fundamental indeterminacy of the applicable impedance spectrum 17 If the equivalence class as a whole is taken to be the argument to which the perturbation and the inverse perturbation are applied, then the following expression is true: j1 jEDf ¼ EDf . This system constitutes a group.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
451
Fig. 3. 3-Space instantaneous representations using data from the present experiments. The z-axis is impedance in Ohms for )Im{Z} (e.g. for Panel (a) )400–800 X), the x-axis is fractional ionic strength x (e.g. for Panels (a), (c), (d) and (f) ranging from 0 to 1; Panel (b) uses a logarithmic scale) and the y-axis is Re{Z} (e.g. for Panel (a) 0–2000 X). Panel (a) shows the rich fine structure of the high frequency limit. At fractional ionic strength of 1 (x ¼ 1), the aspect of the spatially confined Warburg impedance (ZD ) can be discerned with a two constant phase element intersection with the yzplane depicted. At lower ionic strengths semicircular elements arise and break into more finely structured elements. At lowest ionic strength a depressed semicircle is appreciated, a distributed element with admittance and reactance in parallel. Omitted are low frequency points whose loci is a high xy-plane intersection angle CPE surface similar to that in Panels (a)–(c) of Fig. 1. The color spectrum represents )Im{Z} magnitude Panel (b) is a redrawing of the data represented in Panel (a), with a logarithmic scale applied to the fractional ionic strength. This allows for appearance of fine structure at the low ionic strength limit, where 1–2 smooth depressed semicircles are clearly visible. The color spectrum is as in Panel (a). Panel (c) is a 3-space instantaneous representation of a different data set. In contradistinction to Panel (a), the limit at fractional ionic strength of x ¼ 1 is an apparent single high-angle CPE. The entire graph, again from actual data, shows similarities and differences with Panel (a) but a rich fine structure of nested dispersed elements is again present. The color spectrum is as in Panels (a) and (b). Panel (d) shows a third experiment where the appearance of a depressed semicircle is again manifest at low ionic strength. The color spectrum corresponds to the log jxj with red depicting high frequency. We refer to these plots as H–Nyquist plots. Attempts to model these loci with the full array of known impedance element models meet with failure; only the Panel (e) model produced low residuals. Panel (e) is a diagram of the equivalent circuit optimal for simulating the data. RBulk is the bulk resistance [4,5]. Cp or ZCp is evaluated in the limit of computation as ZCp increases without bound (i.e. capacitance goes to zero). CDebye is a Debye capacitor required to achieve excellent fit and is responsible for reducing residuals to 10%. We attribute it to an average difference of electronegativities of the protein (or other biological species adsorbed) and the metal for the species in closest proximity to the metal. CPE indicates a non-Warburg CPE whose frequency dependency is jZj ¼ kx0:4 distinct from the Warburg impedance of semi-infinite linear diffusion and the finite (spatially-confined) Warburg impedance. HN is the distributed element of Havriliak–Negami [31], with parameters so constrained as toreduce to an asymmetrical Cole– Davidson impedance [10,32,33]. Panel (f) shows the finest example of an H–Nyquist plot with dual CPE behavior on the x ¼ 1 plane (a yz-plane) with a rich fine structure of nested parallel dispersed elements at lower fractional ionic strength. It should be noted that the successful modeling of the time evolution of the true loci (as presented) applies only to the traces of the loci on the fractional ionic strength x ¼ 1 plane.
452
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
Fig. 4. Panels (a)–(c) show, from different vantage points, a 3-space representation of a fitted model for perturbation of surface area. This perturbation emulates to some degree the ionic strength perturbation of the present experiments and the axes correspond to those of the preceding figure except that fractional ionic strength is replaced by a generalized resistance perturbation. This therefore applies both to j-perturbation and to perturbation of surface area (Data available in [5]) Each series of points is at constant surface area. In (a) it is seen clearly that for the high impedance tail (which corresponds to the low x limit) jZj increases without bound. Panel (b) superimposes the impedance plots, allowing appreciation of increased dispersed element resistance (depressed semicircular element) as surface area is reduced. Panel (c) shows the absence of this element at higher surface areas, and its presencein the lower surface area limit. Panel (d) shows the high surface area low frequency locus in 3-space from actual experimental data from surface area perturbation experiments. The color spectrum reflects magnitude of )Im{Z}. Note the close correspondence of (c) and (d) with actual data. Critique of the model might include comment that a true diffusional element jZj ¼ kx1=2 exists in all biologic interfaces studied [34].
(and transfer function [6]), an inherent indeterminacy reminiscent of quantum mechanical phenomena.
reported here. The authors acknowledge the helpfulness of Professor Ahmed Zewail and of Reviewer 1.
Acknowledgements
References
The authors thank Professor J. Ross Macdonald for his hospitality and kindness at the time of the author’s visit to the Department of Physics and Astronomy at The University of North Carolina, Chapel Hill for the purpose of LEVM simulation of the equivalent circuit
[1] F.W. Lindemans, A.N.E. Zimmerman, in: C. Meere (Ed.), Proceedings of the VI World Symposium on Cardiac Pacing, Montreal, III-2, 1979. [2] K.B. Stokes, G.N. Kay, in: K.A. Ellenbogen, G.N. Kay, B.L. Wilkoff (Eds.), Clinical Cardiac Pacing, Saunders, Philadelphia, 1995, p. 3.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453 [3] E.T. Mc Adams, J. Jossinet, Innov. Technol. Biol. Med. 12 (1991) 330. [4] M.H. Schoenfisch, J.E. Pemberton, M. Ovadia, M. Levy, Electroanalysis 9 (1997) 135. [5] M. Ovadia, D.H. Zavitz, Electroanalysis 10 (1998) 262. [6] H.A. Chou, M. Ovadia, M. Moskowitz, D.H. Zavitz, PACE 23 (2000) 386. [7] H.A. Chou, D.H. Zavitz, M. Ovadia, Biosens. Bioelectron. 18 (2003) 11. [8] J.R. Macdonald, Impedance Spectroscopy. Emphasizing Solid Materials and Systems, Wiley, New York, 1987. [9] K.S. Cole, R.H. Cole, J. Chem. Phys. 9 (1941) 341. [10] J.R. Macdonald, J. Appl. Phys. 82 (1997) 3962. [11] C. Ho, I.D. Raistrick, R.A. Huggins, J. Electrochem. Soc. 127 (1980) 343. [12] D.R. Franceschetti, J.R. Macdonald, J. Electrochem. Soc. 129 (1982) 1754. [13] A.K. Jonscher, Phys. Thin Films 11 (1980) 206. [14] A.W. Adamson, Physical Chemistry of Surfaces, Wiley, New York, 1990, pp. 203–257. [15] M.G. Go€ uy, Veme partie, Annales de Physique (Paris) 9 (ser 7) (1917) 129. [16] D.L. Chapman, Lond., Edinburgh Dublin Philos. Mag. J. Sci. 6(25) (1913) 475. [17] S.R. Broadbent, J.M. Hammersley, Proc. Camb. Phil. Soc. 53 (1957) 629. [18] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, pp. 28 and 185. [19] G. Eder, Elektrodynamik, Bibliographisches Institut AG, Mannheim FRG, 1967, p. 184.
453
[20] M. Fleischmann, S. Pons, D.R. Rolison, P.P. Schmidt, Ultramicroelectrodes, Datatech Sys. Inc, Morgantown, North Carolina, 1987. [21] H.A. Chou, D. Frenkel, M. Ovadia, PACE 25 (2002) 1053. [22] E. Warburg, Annalen der Physik (Wiedemann-Drudes Annalen der Physik [Dresden]) 3 (1899) 493. [23] J.R. Sandifer, R.P. Buck, Electroanal. Chem. Interf. Electrochem. 56 (1974) 385. [24] E. Hille, Functional Analysis and Semigroups, American Mathematical Society, New York, 1948, p. 147. [25] F.R. Gantmacher, The Theory of Matrices, 2, Chelsea, New York, 1959. [26] J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, Chapman and Hall, London, 1979. [27] N.I. Achieser, I.M. Glasmann, Theorie der linearen Operatoren im Hilbert-Raum, Akademie-Verlag, Berlin, 1954, p. 5. [28] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. [29] M. Sluyters-Rehbach, J.H. Sluyters, in: E. Yeager, J.O’M. Bockris, B.E. Conway, S. Sarangapani (Eds.), Electrodics: Experimental Techniques, Comprehensive Treatise of Electrochemistry, vol.9, Plenum Press, New York, 1984, p. 177. [30] F. Hook, M. Rodahl, B. Kasemo, P. Brzezinski, Proc. Nat. Acad. Sci (USA) 95 (1998) 12271. [31] S. Havriliak, S.J. Negami, Polym. Sci. (Part C Polymer Symposium) 14 (1966) 99. [32] D.W. Davidson, R.H. Cole, J. Chem. Phys. 19 (1951) 1484. [33] C. Svanberg, J. Appl. Phys. 94 (2003) 4191. [34] M. Ovadia, H.A. Chou, D.H. Zavitz, Comp. Cardiol. 27 (2000) 579. [35] H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel, Leipzig, 1928, p. 6.
Impedance spectroscopy of the electrode-tissue interface of €smotic conductivity perturbation q living heart with isoo Marc Ovadia
a,b,c,*
, Daniel H. Zavitz
a,b,d
a
Laboratory of Materials Characterization for Cardiac Pacing, College of Medicine, University of Illinois, 840 South Wood Street, Mail Code 856, Chicago, IL 60612, USA b Department of Pediatrics, College of Medicine, University of Illinois, 840 South Wood Street, Mail Code 856, Chicago, IL 60612, USA c Department of Medicine, College of Medicine, University of Illinois, 840 South Wood Street, Mail Code 715, Chicago, IL 60612, USA d Department of Chemistry, University of Illinois, 845 West Taylor Street, Mail Code 111, Chicago, IL 60607, USA Received 2 January 2004 Available online 6 May 2004
Abstract Impedance spectroscopy was used to solve the Pt electrode interface with metabolically active perfused living heart. Three impedance spectra were observed: the Warburg impedance (ZW1 ), a single high angle constant-phase-element, and a thin-film impedance (ZD ). When characterized again after cyclic change of ionic strength (and hence conductivity j) each interface had one of only two spectra, with exclusion of ZW1 . The in vivo interfacial impedance spectrum is thus neither single-valued nor stable in time. Because metaljliving tissue interfaces are obligatory circuit elements in biosensors and electrodes in heart and brain, the multiplevalued and thin-film character of its impedance are significant. 2004 Published by Elsevier B.V.
1. Introduction The use of electrodes in living tissue is fundamental in the area of sensors. For biophysical recording and stimulation, electrodes in living tissue are fundamental in biology and medicine. In sensors, a signal is generated at the electrode-living tissue interface; biophysical recording involves signal transmission and cardiac pacemakers involve current passage through the interface. The equivalent circuit of this interface has never been solved [1,2]. Extrapolations from non-living systems are inadequate [3] and it has not been possible to study this interface at sufficient bandwidth. In a perfused living beating rat heart system [4–7], we performed impedance spectroscopy at the platinum (Pt)electrodejtissue interface using iso€ osmolar substitution q
Presented in part at the American Chemical Society, the Gordon Research Conferences, Frontiers in Electrochemistry and an Invited NIST Colloquium. * Corresponding author. Fax: 708-346-4788. E-mail addresses: [email protected], [email protected] (M. Ovadia). 0009-2614/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.cplett.2004.04.046
of perfusate to change ionic strength and thereby vary conductance of perfused elements. The data disclosed an in situ generated thin-film diffusional impedance existing alongside conventional circuit elements predicted for noble metaljelectrolyte interfaces. 2. Experimental 2.1. Rat preparation Long-Evans rats (wt. 250 g) were sacrificed and the aorta cannulated via median sternotomy,then perfused immediately to avoid no flow ischemia. 2.2. Electrodes The platinum (Pt) working electrode was placed in the ventricular wall tangentially with no exposure to the ventricular cavity (Levy Type 3 insertion [4]); the AgjAgCl reference was placed in the right atrium in contact with the blood pool; and a large surface area auxiliary electrode was placed in the right heart. Plati-
446
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
num (Pt) wire was obtained from Aldrich Chemical (0.25 mm diameter, 99.9% Pt, No. 34,940-2). 2.3. Instrumentation Sine waves of 2–4 mV amplitude (RMS) and frequency range 30 Hz–1.24 MHz were applied to the preparation, where the excitation signal was summed with a DC bias of )500 mV vs. AgjAgCl; current density was 3–11 mA/cm2 . Impedance spectra were obtained for a smooth Pt working electrode, in three-electrode potentiostatic configuration. Digitization at up to 100 MS/ s, signal-averaging and signal extraction were performed according to published techniques after inspection of current traces to exclude diodic behavior [5]. Important aspects of the preparation include (1) the broadband AD744 BiFET current-to-voltage converter, (2) DC bias potential based on cyclic voltammetry at a flat region of the i–V plot to avoid Faradaic peaks due to redox-active endogenous substances which might introduce diodic or active elements (resulting from their N-type negative impedance i–V relation), and (3) short time period between sacrifice and data gathering. Spectral purity and analog validation of the digital signal extraction algorithms have been published [5]. 2.4. j-Perturbation (iso-PTGO) The preparation was perfused with oxygenated buffered modified Tyrode’s solution from the time of aortic cannulation. To solve the equivalent circuit, systematic perturbation of all impedance elements parallelly bridged with a perfused ionic conductance, was performed by iso€ osmolar perfusate substitutions to vary conductivity j. In this technique, serial perfusate solution substitutions are made by replacing the initial Tyrode’s solution 1 with iso€ osmolar sucrose x : (1 x), (typical series x ¼ 0, 0.2, 0.25, 0.35, 0.75, 0.9, 0.99) to a nominal 99:1 Sucrose: Tyrode’s stock mixture with a final return to Tyrode’s (x ¼ 0) solution. The solution was kept saturated with 95% O2 :5% CO2 mixture and pH adjusted to 7.35 0.03 with a NaHCO3 /CO2 buffer. Conductivity j of the perfusate was varied over two orders of magnitude with pressure, temperature, geometry, and osmolality held constant (iso-PTGO); we refer to this as a j-perturbation (iso-PTGO). 2.5. Analysis Impedance spectra were determined as the complex impedance plane plots of impedance Z; Randles’ plots 1 The Tyrode’s solution had the following concentrations (in mM): NaCl 136.9, KCl 2.68, MgCl2 0.5, NaHCO3 12.0, NaH2 PO4 0.4, glucose 6.0 in deionized water (Sybron Barnstead NANOpure IITM ; Boston, MA) of measured resistivity 17.7 MX cm.
pffiffiffiffi1=2 ) were employed to aid analysis. (Re{Z} vs x LEVM (Solartron, Farnborough, ENGLAND) was used to simulate model equivalent circuits. MATLAB-5 (MathWorks, Natick, MA) was used for plotting interpolated three-dimensional data sequences (termed H–Nyquist plots) and to perform Monte Carlo calculations. Other data manipulations are described in the text.
3. Results and discussion Impedance spectra were determined in the living beating heart with a Pt electrode placed tangentially in ventricular wall. Fig. 1a shows a typical impedance spectrum derived for a 12.4 mm2 Pt electrode in the living heart at normal ionic strength (153 mM), conductivity j ¼ 15:9 mS/cm and osmolality 315 mOs/L. pffiffi Z is a complex quantity Z ¼ RefZg þ iImfZg, i ¼ ð 1Þ, whose complex conjugate Z is here plotted. With increasing frequency the impedance approaches the Re{Z} axis, indicating a pure resistance of magnitude equal to the Re{Z} (x-axis) intercept. This impedance locus follows in the lower frequency range the pattern defined as a constant phase angle element (CPE), described in non-living systems [8]. We termed this CPE1, with phase angle j/j > p=4 radians and no low frequency bound on the CPE line segment [5]. The CPE is a heuristic concept; genesis of such CPE’s has been attributed to dispersion of permittivity in some systems. Similar results in nonbiologic systems are attributed to dispersion of conductivity [8–10]. Fig. 1b shows the same interface in the same experiment, after perfusion with iso€ osmolar perfusate of reduced conductivity (j ¼ 11 mS/cm). Despite an electrodejtissue interface identical in all elements to that of Fig. 1a, the impedance plane plot has undergone a quite remarkable transition. In the high frequency region a second CPE has emerged, of lower phase angle, (slightly less than p=4 radians), bounded by intersections with CPE1 and the Re{Z}-axis. We term this CPE2. The presence of two CPEs as the impedance locus is characteristic of thin film behavior. This thin film impedance locus 2 can be derived analytically from the diffusion equation by application of appropriate boundary conditions [11,12]. Thus this experiment at the present level of resolution offers insight into generation of CPEs at biologic interfaces. The CPE at j/j > p=4 is the first of two CPE’s comprising the thin film diffusional impedance. At usual ionic strength, only CPE1 appears; j-perturbation (iso-PTGO) proves its origin in the thin film
2
Familiar in solid state ionics.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
447
Fig. 1. (a–c) Impedance spectra of a single living beating heartjPt electrode interface (12.4 mm2 ) under j-perturbation (iso-PTGO). j ¼ 15:9 mS/cm (a), j ¼ 11 mS/cm (b), and j ¼ 0:2 mS/cm (c). (d–e) Impedance spectra of a single living beating heart-Pt electrode interface (11.8 mm2 ) under j-perturbation (iso-PTGO). j ¼ 15:9 mS/cm (d), j ¼ 0:2 mS/cm (e). (f) Plot of high frequency capacitance versus ionic strength showing square root dependency and a zero intercept.
impedance locus. This is the first proof of the genesis of a high-angle CPE in any biological system of the present type. Fig. 1c shows the impedance plane plot for the same interface after perfusion with iso€ osmotic buffer of lower conductivity (j ¼ 0:2 mS/cm). The impedance locus has undergone further transition and now displays a depressed semicircular element at high frequency [13]. Fig. 1d shows an experiment at normal ionic strength (153 mM) for an 11.8 mm2 electrode. The impedance plot demonstrates two CPEs. When the heart is perfused through its normal vasculature with iso€ osmotic perfusate of conductivity j ¼ 0:2 mS/cm (Fig. 1e) the same transition occurs to a semicircular locus. Again the center is depressed below the Re{Z} axis, which usually indicates a non-Debye capacitor parallel to a resistor. The resistive element, existing in liquid phase electrolyte, has been created by now relatively resistive perfusate. The reduction of conductivity by pure j-perturbation (iso-PTGO) in perfused regions of current passage thus
brings about reproducible changes in the impedance locus by perturbing equivalent circuit elements. The question arises, What part of the interfacial equivalent circuit is perfused, and therefore affected? Fig. 1f shows a plot of capacitance 3 [5] versus ionic strength. Because capacitance arises from charge separation at the interface, it should behave as a type of double layer capacitance if the interface is fully perfused. The figure shows this behavior, with square root dependency of capacitance on ionic strength and zero intercept. The relation of capacitance and ionic strength was consistent with a square root relationship (p < 0:0001) with no higher order terms [14]. This is compatible with a Go€ uy–Chapman–Stern diffuse layer [15,16], and predicts that the interface is perfused to 7.9– from the electrode. Because muscle is highly vas90 A cular tissue the entire interfacial equivalent circuit is
3
Calculated at the high frequency limit.
448
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
RB ¼ RB ðjÞ; H ¼ HðUÞ
Fig. 2. Equivalent circuit of the living beating heart electrode interface (see text).
perfused, and j-perturbation (iso-PTGO) is a probe of the living electrodejtissue interface 4. These data permit deduction of the equivalent circuit (Fig. 2). This is RB Ra1 ½ðCÞðRa2 HZD Þ where RB is a bulk resistance (the Re{Z} intercept at high j), Ra1 is an access resistance (the increment in the Re{Z} intercept at low j) 5, C is the high frequency interfacial capacitance 6, Ra2 is an access resistance (the base of the semicircle at low j), H the Faradaic impedance (H 0 by experimental design), and ZD is the previously discussed thin-film diffusional impedance that confers double constant phase element behavior whose presence is proven in this communication. At high j, the access resistance Ra2 is low: there is no manifestation of the parallel capacitance C, but only ZD and RB are observed (since Ra1 is also low). By contrast at low j, the augmented magnitude of Ra2 brings out the parallel capacitance. The equivalent circuit bears similarities to that determined by reducing surface area of the physical electrode, where the experimental variable (perturbation) is increased access resistance, with the unfortunate concomitant variable of altered geometry [5]. Since an identical macroscopic interface is present before and after j-perturbation, the new data definitively exclude the alternative hypothesis that semicircular behavior in these systems implies transition to ultramicroelectrode character due to a non-affine changed vector field symmetry (for P, D, E and P [or Z], Hertz’s polarization potential [5,18–20]) reflected in the transition to the semicircle. Functional dependencies under j-perturbation are: 4
In contrast to such perturbations as reductions of surface area or electrode conductance, this perturbation affects only the living tissue side of the living tissuejelectrode interface. 5 Ra1 is in fact an approximation to the tensor impedance quantity Zij whose off-diagonal elements’ functionality reflects critical phenomena for excitable systems that therefore may be modeled by percolation theory (excitable systems is a term that includes cardiac muscle, nerve, and skeletal muscle inter alia). For non-excitable living systems we have studied to date (cartilage, a poroelastic medium and bone, both of which geometrically resemble lattices where percolation processes [17] are known to occur) Zij has zero off-diagonal elements and the phenomena may be modeled as diffusional. 6 Whether this does or does not include a Debye capacitance within this non-Debye capacitance is not specified here; in fact, for all systems we have studied a detectable Debye capacitance can be identified (vide infra, Footnote 5 and legend to Figs. 4 and 3) that we attribute to organic adsorbate on metal faces.
Ra1 ¼ Ra1 ðx; jÞ; and
Ra2 ¼ Ra2 ðx; jÞ;
C ¼ Cðe; N ðjÞÞ;
where U is the offset potential and N ðjÞ the distribution of conductivities. Electrode material is subsumed in U. For the character of the variation of the equivalent circuit under jperturbation (iso-PTGO) is identical to that observed under surface area perturbation [5] (where in addition to surface area of the ionically-conducting electrolyte element of the interface, the electronically conducting metallic/admetallic elements are concomitantly varied). Since surface area perturbation simultaneously varies both the ionic and the electronic conducting parts of the equivalent circuit, whereas j-perturbation varies only the ionic conductivity, the identical character of the equivalent circuit under either perturbation implies that the variation of electronically conducting impedance elements (metal-side) has no measurable effect on the equivalent circuit 7. As a corollary, there is no electronically conducting true physical thin film 8 underlying thin-film behavior 9. Additional observations bear comment regarding evolution in time of the impedance spectrum that occurs for the same experimental conditions. It is axiomatic in impedance spectroscopy that a single impedance spectrum corresponds to each particular physical circuit. For macroscopic systems this follows directly from the first law of thermodynamics 10. As mentioned, we observed a single high angle CPE locus at highest conductivity (similar to Fig. 1a), as well as of a dual CPE locus in similar preparations (similar to Fig. 2a). We made the observation also of the appearance of the alternative locus after j was perturbed in cyclic fashion (changed and then returned to its original value). Occasionally a third pattern was observed, that of a single CPE resembling CPE2 and the classic Warburg impedance ZW1 which also appeared to interconvert. Experimental protocol dictated that perfusate used in the final stage of each experiment was identical to the initial, returning the preparation to its initial state. We tallied the occasions where a transition was observed
7 And in U variation is the unimportant work function variation subsumed. We have shown recently in living tissue that for metals as disparate as Pt and Au, whose work functions differ by 7%, the open circuit potential (the standard electrode potential for the virtual electrode) differs by less than 1% and is undetectable even by a null current measurement carried out in 3-electrode potentiostatic configuration [21]. 8 E.g., Corrosion product. 9 This does not exclude the possibility of an ionically conducting true physical thin film or, as will be discussed in the figure caption for Figs. 4 and 3, a kinetically electronic but non-conducting lamina. 10 Alternatively stated, the impedance locus is a single-valued function of the physical circuit; there is no degeneracy for particular circuit elements in a particular geometry.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
from one impedance locus to another during the same experiment. Using ZD [5] to designate the double constant phase angle defined visually above 11, CPE1 for the higher phase angle CPE, and identifying the CPE2 plot by the symbol [5,8,22,23] ZW1 , all possible transitions are ZD ! ZD ZD ! CPE1 ZD ! ZW1
CPE1 ! ZD CPE1 ! CPE1 CPE1 ! ZW1
ZW1 ! ZD ZW1 ! CPE1 ZW1 ! ZW1 :
Diagonal elements have identical initial and final states; * transitions were not observed 12. Each other element represents an observed transition with initial and final states as noted. The fact that any of the non-diagonal transitions occur within this set of impedance spectra, implies that the different impedance spectra are different manifestations of the same living biophysical interface. The experimental observation of transition between different members of this equivalence class may be written as an evolution matrix of transition probabilities. Writing kaij kði ¼ 1; 2; 3; j ¼ 1; 2; 3; 1 ¼ ZD ; 2 ¼ CPE1; 3 ¼ ZW1 Þ for the transition probability to impedance spectrum i and from impedance spectrum j gives a11 a12 a13 0:64 0:56 1 Piso ¼ a21 a22 a23 ¼ 0:35 0:44 0 a31 a32 a33 0 0 0 applying to any experimental preparation, substituting 0’s for unobserved processes 13. Sample standard deviations for the first two terms may be included: a11 ¼ 0:64 0:05 and a21 ¼ 0:35 0:08. From the theory of reducible matrices with nonnegative elements [25] a final state under repeated transitions may exist and in the limit as t ! 1 N
1 ¼ lim ðPiso Þ ¼ lim ðPiso Piso Piso Piso Þ Piso N !1 N !1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} N terms 1 1 1 0:61 0:61 0:61 a a a 12 13 11 1 a1 a1 0:39 0:39 ¼ 22 23 ¼ 0:39 ; a21 a1 a1 a1 0 0 0 31 32 33
449
noting that in the time period of the experimental protocol the transition matrix is constant. This result may be obtained numerically because of rapid convergence under Cayley multiplication 14.
14
The question may be asked, is convergence of this infinite product highly dependent on the exact values of the transition probabilities or is convergence assured for a large range or for the entire range of likely values? This is a question somewhat akin to the question of whether this is stable as a dynamical system, but it differs from that question as posed ordinarily for dynamic systems that enter a chaotic regime based on slight alteration of the initial conditions, i.e. their canonical Hamiltonian variables characterizing the state or equivalently the region of phase space in which the system is allowed to operate. Here, the transition probabilities are physicochemical constants determined for the particular system, and change of material or ambient concentrations of mineral or biological species alone would be necessary to change them. We performed a Monte Carlo calculation [26] assuming sampling from a Poisson distribution, and found convergence and were able to derive error bars for the final transition matrix, with convergence of all tested matrices. This proves that there is a neighborhood of convergence around the determined values for the transition probabilities, that is, that in the six-dimensional space represented by the 3 · 3 transition matrix, the point is a H€aufungspunkt [27] for the domain of convergence. This is quite remarkable actually. For consider the obvious comparison matrix for such a determination, the identity matrix for transition probabilities between the different states which would characterize monostable interfaces 1 0 0 0 1 0 : 0 0 1 An infinite product of the identity matrix with itself yields the identity matrix, so clearly there is convergence of the infinite product at the point represented by the identity matrix. But does this matrix lie embedded in a region where nearby points converge similarly, as does the matrix we determined experimentally? The answer is no. For if one deviates any of the coefficients ever so slightly from the stated values, e.g. 1 0 0
0 1d d
0 0 ; 1
where d is some vanishingly small number, the matrix does not converge to the identity matrix. Even if the probability is, say 11
Analytically in [5]. An example of an alternative notation is j1 jCPE1 ¼ ZD (in place of CPE1 ! ZD used in the text), where j represents the j perturbation, j1 the return to original ionic strength and conductivity, and the symbol j1 j CPE1 signifies a system characterized by CPE1 that is subjected to a j perturbation followed by the return to original conductivity j. The equation j1 jCPE1 ¼ ZD states that after perturbation and then return to the original conductivity j, the system is found experimentally to be characterized by a ZD impedance locus. This notation calls attention to some important pecularities of the system, as the system and the operation as defined are not a group but at most a semigroup [24], since the equation j1 j ¼ 1 (the identity element, implying no change in time) does not hold for the system when a single impedance locus is taken as the argument (thus there is no inverse element). 13 Transition processes whose initial state has not been observed, as well as those (where the transition probability may be estimated more precisely) where the state is uncommon. 12
d¼
1 101012
a number so small as to be meaningless chemically, the system will eventually diverge away from the identity matrix. Thus the identity matrix is not a H€aufungspunkt, it has no neighborhood where other points exist where the infinite product converges. This discussion is placed in a footnote because of its mathematical character. But what it implies chemically is that monostable interfaces may be less stable in biological systems than bistable interfaces, because the monostable interface does not exist in a domain of convergence or stability for the interfacial system. Parenthetically, extreme care must be taken in defining the meaning of neighborhood since at this point in the exposition, no metricization has been performed on the space, though it is well defined experimentally, in the clause ‘We tallied the occasions where a transition was observed from one impedance locus to another during the same experiment, etc’. By way of anticipation, by the end of Footnote 9 (Vide Infra), these difficulties will have been resolved.
450
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
In the limit, the proportions observed will be 0.61:0.39:0 for impedance spectra ZD :CPE1:ZW1 . With uncertainties, these are 0.61 0.05:0.39 0.08:0. The multiple patterns of impedance spectra imply that the surface structure of the electrode is dynamic in living tissue with conversion between different spectra occurring spontaneously. One introduces hereby the concept of an equivalence class of impedance spectra (necessary because the impedance of the interface can be defined only to such precision as allowed by the multiple-valued impedance spectra of the equivalence class). Name the equivalence class EDf , Df standing for D€ unnfilm, the biologic thin film. EDf is a set of 3 impedance spectra 15 [6] ZD , CPE1 and ZW1 . Each signifies for the living biological preparation the presence of the same biophysical interface. Specification of the members of the class defines the interface electrically. This concept resolves confusion in the electrochemical literature, where a single impedance spectrum is assumed sufficient to characterize an interface. The absence of ZW1 in the limit implies that the apparent ZW1 is a transient (metastable) state. The definition of the true impedance element of semi-infinite linear diffusion ZW1 (a constant phase element whose phase angle is p=4 and whose Randles’ plot is linear [8,22,23,28,29]) is inadequate, as it fails to distinguish the true ZW1 from the ZW1 identified in this communication that is part of the EDf equivalence class. To specify the classic Warburg impedance and exclude the latter element, one must add the criterion that the ZW1 impedance spectrum be stable in time and under cyclic j perturbation 16. By this criterion, ZW1 has never been observed in electrodejliving tissue interfaces. This conclusion is equally applicable to the CPE1. On the basis of the present data, this CPE is the lowest frequency part of ZD . CPE1 is merely a manifestation of the EDf equivalence class. It implies the existence of a thin film entirely covering the electrode, and is caused not by dispersion of dielectric properties but by transport characteristic of a laminate phase. The significance of this latter remark cannot be overemphasized, providing an important corrective to the temptation to assume, based on the literature of impedance spectroscopy in solid state physics, that for a distributed element the dielectric permittivity is the important dispersed quantity. In fact conductivity may be the relevant quantity
15
Or, from an alternative but equivalent viewpoint, of transfer functions corresponding to the spectra. Whether the cardinal number of this set may be the transfinite @0 or 2@0 in the limit as period increases without bound, is a question that may be deferred to another communication. 16 By the alternative notation, this criterion may be expressed as j1 jZW1 ¼ ZW1 . Such a hypothetical system and the operations as defined would constitute a group, albeit one that prior to the present communication has never been conceptualized theoretically.
whose distribution is reflected in the appearance of the distributed element. Hook et al. [30] have identified two states of adsorbed hemoglobin on octadecathiol self-assembled monolayers existing under a form of j-perturbation. j-perturbation promotes ordering of the adsorbed protein layer, the physicochemical basis being ionic strength dependency of protein solubility. The thin film is not amorphous but in fact has 2 or more ordered states each corresponding to a member of the equivalence class. In order to answer an additional question, What solid state circuit elements are most similar to these observed in living tissue?, we performed a simulation not of a single impedance plot but one taking into account the range of the perturbation. Figs. 3 and 4 show data in a 3-space (x; y; z;) H–Nyquist plot, where )Im{Z} is plotted on the z-axis, the perturbation fractional ionic strength (Fig. 3) or surface area (Fig. 4) the x-axis and Re{Z} the y-axis. A yz plane is drawn at the location of the unperturbed system’s impedance plot, and each projection of the locus onto this plane represents a impedance plot for the perturbed system. Color maps either the magnitude of )Im{Z} or the frequency. Note the similarities between the plots for the two perturbations. The rich fine structure (nested dispersed elements) is apparent in the limits of low ionic strength and high frequency. In Fig. 4 a simulation is depicted whose salient feature is inclusion of a reactance in the Hawriliak-Negami element. The simulation of the actual H–Nyquist locus is evident. In summary, the interface is an in situ generated ordered thin film determinate only to within a degeneracy of n ¼ 3, the members of the equivalence class. (Different impedance spectra may be considered congruent to within EDf 17 [35].) This is important when interpreting signals transmitted through an electrode tissue interface, and is relevant to analytical chemistry (for sensors), biology, neuroscience, and cardiology where sensing electrodes are essential to pacemakers and defibrillators. Degeneracy of the impedance spectrum implies that the transfer function is multiple-valued as well, important in the frequency band 0.5–50 Hz relevant to electrophysiologic signals. This multiple-valued character makes it impossible to recover source signal from signal transmitted through the interface without simultaneously interrogating the interface to determine the impedance spectrum extant at the relevant frequencies. This is caused by the fundamental indeterminacy of the applicable impedance spectrum 17 If the equivalence class as a whole is taken to be the argument to which the perturbation and the inverse perturbation are applied, then the following expression is true: j1 jEDf ¼ EDf . This system constitutes a group.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
451
Fig. 3. 3-Space instantaneous representations using data from the present experiments. The z-axis is impedance in Ohms for )Im{Z} (e.g. for Panel (a) )400–800 X), the x-axis is fractional ionic strength x (e.g. for Panels (a), (c), (d) and (f) ranging from 0 to 1; Panel (b) uses a logarithmic scale) and the y-axis is Re{Z} (e.g. for Panel (a) 0–2000 X). Panel (a) shows the rich fine structure of the high frequency limit. At fractional ionic strength of 1 (x ¼ 1), the aspect of the spatially confined Warburg impedance (ZD ) can be discerned with a two constant phase element intersection with the yzplane depicted. At lower ionic strengths semicircular elements arise and break into more finely structured elements. At lowest ionic strength a depressed semicircle is appreciated, a distributed element with admittance and reactance in parallel. Omitted are low frequency points whose loci is a high xy-plane intersection angle CPE surface similar to that in Panels (a)–(c) of Fig. 1. The color spectrum represents )Im{Z} magnitude Panel (b) is a redrawing of the data represented in Panel (a), with a logarithmic scale applied to the fractional ionic strength. This allows for appearance of fine structure at the low ionic strength limit, where 1–2 smooth depressed semicircles are clearly visible. The color spectrum is as in Panel (a). Panel (c) is a 3-space instantaneous representation of a different data set. In contradistinction to Panel (a), the limit at fractional ionic strength of x ¼ 1 is an apparent single high-angle CPE. The entire graph, again from actual data, shows similarities and differences with Panel (a) but a rich fine structure of nested dispersed elements is again present. The color spectrum is as in Panels (a) and (b). Panel (d) shows a third experiment where the appearance of a depressed semicircle is again manifest at low ionic strength. The color spectrum corresponds to the log jxj with red depicting high frequency. We refer to these plots as H–Nyquist plots. Attempts to model these loci with the full array of known impedance element models meet with failure; only the Panel (e) model produced low residuals. Panel (e) is a diagram of the equivalent circuit optimal for simulating the data. RBulk is the bulk resistance [4,5]. Cp or ZCp is evaluated in the limit of computation as ZCp increases without bound (i.e. capacitance goes to zero). CDebye is a Debye capacitor required to achieve excellent fit and is responsible for reducing residuals to 10%. We attribute it to an average difference of electronegativities of the protein (or other biological species adsorbed) and the metal for the species in closest proximity to the metal. CPE indicates a non-Warburg CPE whose frequency dependency is jZj ¼ kx0:4 distinct from the Warburg impedance of semi-infinite linear diffusion and the finite (spatially-confined) Warburg impedance. HN is the distributed element of Havriliak–Negami [31], with parameters so constrained as toreduce to an asymmetrical Cole– Davidson impedance [10,32,33]. Panel (f) shows the finest example of an H–Nyquist plot with dual CPE behavior on the x ¼ 1 plane (a yz-plane) with a rich fine structure of nested parallel dispersed elements at lower fractional ionic strength. It should be noted that the successful modeling of the time evolution of the true loci (as presented) applies only to the traces of the loci on the fractional ionic strength x ¼ 1 plane.
452
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453
Fig. 4. Panels (a)–(c) show, from different vantage points, a 3-space representation of a fitted model for perturbation of surface area. This perturbation emulates to some degree the ionic strength perturbation of the present experiments and the axes correspond to those of the preceding figure except that fractional ionic strength is replaced by a generalized resistance perturbation. This therefore applies both to j-perturbation and to perturbation of surface area (Data available in [5]) Each series of points is at constant surface area. In (a) it is seen clearly that for the high impedance tail (which corresponds to the low x limit) jZj increases without bound. Panel (b) superimposes the impedance plots, allowing appreciation of increased dispersed element resistance (depressed semicircular element) as surface area is reduced. Panel (c) shows the absence of this element at higher surface areas, and its presencein the lower surface area limit. Panel (d) shows the high surface area low frequency locus in 3-space from actual experimental data from surface area perturbation experiments. The color spectrum reflects magnitude of )Im{Z}. Note the close correspondence of (c) and (d) with actual data. Critique of the model might include comment that a true diffusional element jZj ¼ kx1=2 exists in all biologic interfaces studied [34].
(and transfer function [6]), an inherent indeterminacy reminiscent of quantum mechanical phenomena.
reported here. The authors acknowledge the helpfulness of Professor Ahmed Zewail and of Reviewer 1.
Acknowledgements
References
The authors thank Professor J. Ross Macdonald for his hospitality and kindness at the time of the author’s visit to the Department of Physics and Astronomy at The University of North Carolina, Chapel Hill for the purpose of LEVM simulation of the equivalent circuit
[1] F.W. Lindemans, A.N.E. Zimmerman, in: C. Meere (Ed.), Proceedings of the VI World Symposium on Cardiac Pacing, Montreal, III-2, 1979. [2] K.B. Stokes, G.N. Kay, in: K.A. Ellenbogen, G.N. Kay, B.L. Wilkoff (Eds.), Clinical Cardiac Pacing, Saunders, Philadelphia, 1995, p. 3.
M. Ovadia, D.H. Zavitz / Chemical Physics Letters 390 (2004) 445–453 [3] E.T. Mc Adams, J. Jossinet, Innov. Technol. Biol. Med. 12 (1991) 330. [4] M.H. Schoenfisch, J.E. Pemberton, M. Ovadia, M. Levy, Electroanalysis 9 (1997) 135. [5] M. Ovadia, D.H. Zavitz, Electroanalysis 10 (1998) 262. [6] H.A. Chou, M. Ovadia, M. Moskowitz, D.H. Zavitz, PACE 23 (2000) 386. [7] H.A. Chou, D.H. Zavitz, M. Ovadia, Biosens. Bioelectron. 18 (2003) 11. [8] J.R. Macdonald, Impedance Spectroscopy. Emphasizing Solid Materials and Systems, Wiley, New York, 1987. [9] K.S. Cole, R.H. Cole, J. Chem. Phys. 9 (1941) 341. [10] J.R. Macdonald, J. Appl. Phys. 82 (1997) 3962. [11] C. Ho, I.D. Raistrick, R.A. Huggins, J. Electrochem. Soc. 127 (1980) 343. [12] D.R. Franceschetti, J.R. Macdonald, J. Electrochem. Soc. 129 (1982) 1754. [13] A.K. Jonscher, Phys. Thin Films 11 (1980) 206. [14] A.W. Adamson, Physical Chemistry of Surfaces, Wiley, New York, 1990, pp. 203–257. [15] M.G. Go€ uy, Veme partie, Annales de Physique (Paris) 9 (ser 7) (1917) 129. [16] D.L. Chapman, Lond., Edinburgh Dublin Philos. Mag. J. Sci. 6(25) (1913) 475. [17] S.R. Broadbent, J.M. Hammersley, Proc. Camb. Phil. Soc. 53 (1957) 629. [18] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, pp. 28 and 185. [19] G. Eder, Elektrodynamik, Bibliographisches Institut AG, Mannheim FRG, 1967, p. 184.
453
[20] M. Fleischmann, S. Pons, D.R. Rolison, P.P. Schmidt, Ultramicroelectrodes, Datatech Sys. Inc, Morgantown, North Carolina, 1987. [21] H.A. Chou, D. Frenkel, M. Ovadia, PACE 25 (2002) 1053. [22] E. Warburg, Annalen der Physik (Wiedemann-Drudes Annalen der Physik [Dresden]) 3 (1899) 493. [23] J.R. Sandifer, R.P. Buck, Electroanal. Chem. Interf. Electrochem. 56 (1974) 385. [24] E. Hille, Functional Analysis and Semigroups, American Mathematical Society, New York, 1948, p. 147. [25] F.R. Gantmacher, The Theory of Matrices, 2, Chelsea, New York, 1959. [26] J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, Chapman and Hall, London, 1979. [27] N.I. Achieser, I.M. Glasmann, Theorie der linearen Operatoren im Hilbert-Raum, Akademie-Verlag, Berlin, 1954, p. 5. [28] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. [29] M. Sluyters-Rehbach, J.H. Sluyters, in: E. Yeager, J.O’M. Bockris, B.E. Conway, S. Sarangapani (Eds.), Electrodics: Experimental Techniques, Comprehensive Treatise of Electrochemistry, vol.9, Plenum Press, New York, 1984, p. 177. [30] F. Hook, M. Rodahl, B. Kasemo, P. Brzezinski, Proc. Nat. Acad. Sci (USA) 95 (1998) 12271. [31] S. Havriliak, S.J. Negami, Polym. Sci. (Part C Polymer Symposium) 14 (1966) 99. [32] D.W. Davidson, R.H. Cole, J. Chem. Phys. 19 (1951) 1484. [33] C. Svanberg, J. Appl. Phys. 94 (2003) 4191. [34] M. Ovadia, H.A. Chou, D.H. Zavitz, Comp. Cardiol. 27 (2000) 579. [35] H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel, Leipzig, 1928, p. 6.