Simulations of the Frequency Response of Implantable Glucose

The response of enzyme electrode glucose sensors implanted in tissues to physiologic blood glucose oscillations is simulated. Models describe both ...
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Anal. Chem. 2000, 72, 1853-1859

Simulations of the Frequency Response of Implantable Glucose Sensors Michael Jablecki and David A. Gough*

Department of Bioengineering, University of California, San Diego, La Jolla, California 92093

The response of enzyme electrode glucose sensors implanted in tissues to physiologic blood glucose oscillations is simulated. Models describe both oxygen-based and peroxide-based glucose sensors in spatially homogeneous medium simulating some mass transfer properties of tissue. Pass-through ratios and delays are reported as a function of frequency for the oxygen-based sensor, and the effects on continuous blood glucose monitoring are illustrated using data from the literature. Certain peroxidebased sensor designs may produce common signals for different glucose concentrations, a characteristic not found in oxygen-based sensors. The dynamic response depends on the frequency of glucose oscillation and is sensitive to sensor type, enzyme activity, and diffusional resistance. Biosensors must be capable of following the dynamics of the analyte concentration, regardless of mass transfer effects of the medium in which they are employed. An example is the implantable sensor under development by many groups for continuous monitoring of glucose concentration in diabetes.1-12 Most investigators intend for the sensor to be implanted at a tissue site for reasons of safety, but sensor signals will have to be interpreted in terms of blood or plasma glucose concentrations for clinical utility, rather than tissue fluid glucose concentrations. A variety of studies has shown that, at tissue implant sites, the sensor output often contains apparent offsets, lags to blood glucose changes, * Corresponding author: David A. Gough, Department of Bioengineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0412. Phone: 858-822-3446. FAX: 858-822-3467. E-mail: [email protected]. (1) Shichiri, M.; Asakawa, N.; Yamasaki, Y.; Kawamori, R.; Abe, H. Diabetes Care 1986, 9, 298. (2) Abel, P.; Mu ¨ ller, A.; Fischer, U. Biomed. Biochim. Acta 1984, 43, 577. (3) Moatti-Sirat, D.; Capron, F.; Poitout, V.; Reach, G.; Bindra, D. S.; Zhang, Y.; Wilson, G. S.; Thevenot, D. R. Diabetologia 1992, 35, 224. (4) Johnson, K. W.; Mastrototaro, J. J.; Howey, D. C.; Brunelle, R. L.; BurdenBrady, P. L. Biosens. Bioelectron. 1992, 7, 709. (5) Gilligan, B. J.; Shults, M. C.; Rhodes, R. K.; Updike, S. J. Diabetes Care 1994, 17, 882. (6) Sternberg, F.; Meyerhoff, C.; Mennel, F. J.; Mayer, H.; Bischof, F.; Pfeiffer, E. F. Diabetologia 1996, 39, 609. (7) Atanasov, P.; Wilkins, E. Biotechnol. Bioeng. 1994, 43, 262. (8) Moussy, F.; Harrison, D. J.; Rajotte, R. V. Int. J. Artif. Organs 1994, 17, 88. (9) Csoregi, E.; Schmidtke, D. W.; Heller, A. Anal. Chem. 1995, 67, 1240. (10) Claremont, D. J.; Sambrook, I. E.; Penton, C.; Pickup, J. C. Diabetologia 1986, 29, 817. (11) Ertefai, S.; Gough, D. A. J. Biomed. Engin. 1989, 11, 362. (12) Armour, J. C.; Lucisano, J. Y.; McKean, B. D.; Gough, D. A. Diabetes 1990, 39, 1519. 10.1021/ac991018z CCC: $19.00 Published on Web 03/17/2000

© 2000 American Chemical Society

damping of high-frequency signal components, and other undesirable effects that may be related to mass transfer. It may be possible to reduce or eliminate certain dynamic artifacts by appropriate design of the sensor. If the maximum possible physiologic frequency of analyte variation is known, the sensor could be designed to follow transients that occur at that frequency or a lesser one, but be unresponsive to more rapid concentration-independent effects. For blood glucose, the maximum rate of change has been estimated as 0.2 mM/min,13 and the bandwidth has been estimated to be e0.8 × 10-3 Hz. This information can be used to define the required dynamic response of the sensor. In this paper, we describe simulations of the response of two types of enzyme electrode glucose sensors, based, respectively, on differential oxygen detection and hydrogen peroxide detection. The models used in the simulations include lumped mass transfer resistances, to represent the effects of membrane layers in various sensor designs, and are cast in a format that accepts arbitrary glucose input functions. The simulations include reasonable values for mass transfer in homogeneous tissue, although actual values remain to be experimentally determined. Not included are the effects of spatially heterogeneous mass transfer and enzyme inactivation of both sensor types, the nonunique response to glucose of peroxide-based sensors, and the electrochemical reaction of interferants characteristic of peroxide-based sensors. The studies employ sinusoidally varying glucose concentrations over the physiologic frequency range as the forcing function, and the sensor pass-through ratio and delay are reported as a function of frequency for typical sensor designs. The response to a blood glucose recording from a diabetic subject in the literature is also simulated for various values of tissue permeability. The simulation approach is appropriate for this study, as certain models have been shown previously to quantitatively describe sensor response. Moreover, an in vitro experimental approach would require the construction and testing of many sensors with specified properties and the design of a physical testing device that can produce rapid sinusoid concentration changes without associated flow effects. In vivo studies can also benefit from the simulation approach as an aid to interpretation of sensor response. Once validated, the simulation approach can be an efficient tool for design and analysis. (13) Baker, D. A.; Gough, D. A. Anal. Chem. 1996, 68, 1292. (14) Gough, D. A.; Bremer, T.; Lucisano, J. Y., in review.

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BACKGROUND Several implantable glucose sensors based on the enzyme electrode principle are under development. Most glucose sensors employ immobilized glucose oxidase, which catalyzes the reaction

glucose + O2 + H2O f gluconic acid + H2O2

(1)

The hydrogen peroxide-based version of the sensor1-8 is based on a membrane containing immobilized glucose oxidase coupled to a noble-metal anode and produces a signal current derived from electrochemical oxidation of peroxide. Additional outer membranes in front of the enzyme layer are usually included to control glucose diffusion and enhance biocompatibility, and a porous inner membrane is included between the enzyme layer and the electrode surface to reduce transport and direct the electrochemical reactions of other molecules present in the medium.6,8 In the alternative oxygen-based version of the sensor,11-13 glucose oxidase can be employed alone or with co-immobilized catalase, which catalyses the reaction

H2O2 f 1/2O2 + H2O

(2)

When catalase is present in excess, the overall reaction becomes

glucose + 1/2O2 f gluconic acid

(3)

In this sensor, the enzyme is coupled to a membrane-covered cathode that produces a signal current by electrochemical reduction of unconsumed oxygen remaining from the enzymatic reaction. A similar oxygen reference sensor without the enzyme produces a signal that indicates the ambient background oxygen, which is subtracted. The sensor may include outer membranes as before, but the inner membrane is a nonporous layer of silicone rubber that allows oxygen transport and is completely impermeable to polar molecules. There is a controversy about the electrochemical processes of these sensors that is relevant to modeling. It has been welldocumented that enzyme electrodes based on two-electrode Clark type oxygen sensors and hydrogen peroxide sensors produce signals that decay under continuous use.6,8,15 In the Clark type oxygen sensor, where an oxygen-permeable hydrophobic membrane prevents electrochemical poisoning by polar molecules, signal decay can be a result of reference electrode polarization and subsequent deterioration. Electrode polarization also occurs in the two-electrode peroxide sensor, but the dominant effect here may be electrochemical poisoning due to anodic oxidation of small organics that pass the necessarily porous inner membrane of that sensor. There is little quantitative information about the process. For modeling purposes, this requires some form of time-dependent boundary condition. (15) Putnam, S. P.; Rolfe, P.; Albery, W. J. J. Biomed. Eng. 1984, 6.

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Various strategies have been used in practice to deal with signal decay. In many commercial applications, the problem is solved by operating sensors in a discrete mode in which the sensor is exposed briefly to a sample, followed by a rinse and automatic recalibration. In continuous applications, electrochemical interference in the peroxide-based sensor can be minimized, but not entirely eliminated, over the long term, by substantially restricting the pore size of the inner membrane. For implant applications, the peroxide-based sensor may be better suited for short-term applications or applications in which frequent recalibration by comparison with blood values is possible. The potentiostatic oxygen sensor was designed to overcome these shortcomings.16 This sensor consists of a potentiostatic, three-electrode system (working, high-impedance reference, lowimpedance counter) covered by a nonporous, hydrophobic polymer layer. This sensor has been documented to operate continuously in vitro16 and in vivo12 as the basis of an oxygen-based enzyme electrode glucose sensor implanted in the bloodstream, for periods of months in both cases without the need for recalibration. For modeling purposes, this stable performance justifies stationary boundary conditions. Models of peroxide- and oxygen-based glucose sensors have been proposed previously for both the steady state and transient responses to glucose, assuming stationary boundary conditions.17-22 Numerical and closed-form solutions to steady state and transient differential equations governing the reaction-diffusion processes have been obtained. The transient analyses have typically been cast as initial-value problems, in which individual time-dependent signal segments are computed in response to a single glucose concentration step or ramp input. That approach is useful to develop an initial understanding of sensor behavior and for sensor testing, but is less effective for simulating the sensor response to a prolonged or continuous series of glucose variations, as would be seen by actual functioning sensors. The numerical methods used here are computationally efficient and can accept a continuously varied input forcing function. Comparison of the oxygen- and peroxide-based sensors using a single modeling approach has not been previously attempted. It is arguable that such comparisons may not be fully effective prior to optimization of each sensor type, but a preliminary comparison can be useful to provide a relative understanding of the sensor types and suggest directions for further improvement. THEORETICAL The mathematical modeling method employed here is an extension of methods we have used previously,21,22 but employs a new recursive formulation and graphical representation of the numerical computation schemes. The governing equations describe the homogeneous reaction and diffusion of, respectively, glucose, oxygen, and hydrogen peroxide in the immobilized (16) Lucisano, J. Y.; Armour, J. C.; Gough, D. A. Anal. Chem. 1987, 59, 736. (17) Atkinson, B.; Lester, D. E. Biotechnol. Bioeng. 1974, 16, 1321. (18) Brady, J. E.; Carr, P. W. Anal. Chem. 1980, 52, 977. (19) Mell, L. D.; Maloy, J. T. Anal. Chem. 1976, 48, 1597. (20) Rhodes, R. K.; Shults, M. C.; Updike, S. J. Anal. Chem. 1994, 66, 1520. (21) Leypoldt, J. K.; Gough, D. A. Anal. Chem. 1984, 56, 2896. (22) Gough, D. A.; Lucisano, J. Y.; Tse, P. H. S. Anal. Chem. 1985, 57, 2351.

enzyme membrane as follows

boundary conditions for both sensors. For oxygen-based sensors

[

]

1/D1 0 0 ∂ 1/D2 0 B(b) c ) 0 b c ∆b c + φ1 R ∂t 1/D3 0 0 2

(4)

∂ c ) 0|x1)0 ∂x1 1 c2 ) 0|x1)0

(11a,b,c)

∂ c ) 0|x1)0 ∂x1 3

where the column vector

b c ) b( c b) x ) (c1(b), x c2(b), x c3(b)) x

(5) For peroxide-based sensors

with 1, 2, and 3 for glucose, oxygen, and peroxide, respectively.

b c B ) b( c b)| x x1)L

(6)

R B(b) c ) (-R(b),-γR( c b),R( c b)) c b x ) bx ı 1 + bx j 2+B k x3

(7)

∂ ∂ ∂ ∇ ) bı + bj +B k ∂x1 ∂x2 ∂x3

(8c)

The normalized reaction rate expression R B is the two-substrate Michaelis-Menten reaction expression given previously21 and the square of the Thiele modulus φ12, or ratio of reaction to internal diffusion, and its components are given by21

( )

c2* )

c2* + κ

σ12 )

R2D2b c2 νR1D1b c1

κ)

Vmax K1

K′2 )

V′max )

V′maxδ2m D1

c3 ) 0|x1)0

D2K′2 νD1

(9c,d)

K2 K1

(9e,f)

∂ b( c b)|x x 1)L ∂x1

B Bi ) (Bi1, Bi2, Bi3)

[

Bi1 0 0 diag(B Bi) ) 0 Bi2 0 0 0 Bi3

]

(13)

(14)

with terms defined previously21 as the respective ratios of external to internal mass transfer

Bii )

hiδm RiDm,i

i ) 1,2,3

(15)

(9a,b)

The parameters K1 and K2 are the respective Michaelis constants at infinite concentration of the other substrate, and Vmax is the maximal velocity. For the purposes of compactness, the concentrations of glucose, oxygen, and peroxide are kept in b. c The stoichiometry coefficient, ν, equals 1 without catalase and 1/2 with catalase. For both sensors, the boundary conditions at the membrane-medium interface are

diag(B Bi)‚(b c B - b( c b)) x )

(12a,b,c)

The parameters and variables are given by

(8b)

c 2*

∂ c ) 0|x1)0 ∂x1 2

(8a)

∂2 ∂2 ∂2 ∆ ) ∇‚∇ ) 2 + 2 + 2 ∂x1 ∂x2 ∂x3

φ12 ) σ12

∂ c ) 0|x1)0 ∂x1 1

(10)

The boundary conditions at the membrane-electrode interface are specific for sensor type. We consider the case of stationary

where hi is the external mass transfer coefficient given by the external substrate diffusivity divided by the effective length of the local mass transfer gradient, Dm,i is the solute diffusion coefficient within the membrane, Ri is the solute partition coefficient in the membrane, and δm is the membrane thickness. The flux at the membrane-electrode interface is equated to current density, taking into account the respective electron number, n (2 for peroxide, 4 for oxygen), the Faraday constant, and electrode area, A. METHODS Model Representation. The algorithms to solve model equations were represented in Simulink23 by information flow pathways, functional operators, and switches in a format that corresponds to the numerical computation sequence. The volume of the homogeneous computational elements used in this system was reduced as an inverse geometric series at each iteration to ensure maximal accuracy. The glucose input forcing function was independent of the model and given as an arbitrary function of time or as a digitized series. The respective membranes between the enzyme layer and electrode surface of both types of sensors were included as lumped elements in the models. (23) Simulink is a product of The MathWorks, Inc.,

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Table 1. Some Fundamental Parameter Values Used in Simulations16,21,22 K1 ) 125. mM δm ) 25.0 × 10-6 m Do ) 3.92 × 10-10 m2/s Ri ) 1.0

K2 ) 0.70 mM Dg ) 1.20 × 10-10 m2/s Dp ) 1.50 × 10-10 m2/s

Parameter and Input Values. Independently determined parameter values used in the simulations were taken from the literature. The values are listed in Table 1. The oxygen concentration was fixed at 0.1 mM in all cases to demonstrate the effects of other parameters and variables. Quasi-steady-state responses were created by challenging the models with a slowly varying linear glucose concentration ramp. The input glucose concentration forcing functions used in simulations to determine sensor bandwidth were sine waves of amplitude ( 2.5 mM around a mean of 5 mM with increasing frequency. The reported maximal rate of 0.2 mM/min blood glucose change in diabetes,13 in conjunction with the forcing function amplitude of 2.5 mM, suggests that a maximal required frequency, 2πωmax of 0 2 × 10-3 Hz is needed to adequately test sensors for diabetic applications. Other methods have suggested a maximum frequency of 0.8 × 10-3 Hz for glucose monitoring.14 As an example of the simulation of the sensor in physiologic conditions, a quasi-continuous blood glucose recording from a diabetic subject obtained using conventional glucose analysis methodology24 was digitized and used as an input function. The bulk hydrogen peroxide concentration was assumed to remain at zero due to the abundance of catalase in tissues. RESULTS AND DISCUSSION Examples of quasi-steady-state solutions are shown in Figure 1A-D. The simulated signal current for oxygen-based sensors in Figure 1 parts A and B and that for peroxide-based sensors in Figure 1 parts C and D is plotted as a function of bulk glucose concentration for indicated values of σ1, which, for practical purposes, can be taken as relative enzyme loading. Figure 1A and C are sensors with all values of Bii, or the ratio of external to internal mass transfer for each chemical species, set equal to unity for purposes of illustration of the role enzyme loading. Figures 1B and D are sensors with different and relatively high external mass transfer for each species, as might be expected under certain operational conditions. Quasi-Steady-State Results for the Oxygen-Based Sensor. The simulated responses of oxygen-based sensors are shown in parts A and B of Figure 1. The reference oxygen sensor was not used, and the signal current is plotted as increasing with concentration for purposes of comparison. At high enzyme loading, the signal rises with concentration to the point of oxygen limitation and thereafter remains independent of glucose concentration. At lower loading, the signals rise nonlinearly and monotonically with concentration, asymptotically approaching the oxygen-limiting plateau. The oxygen-based sensors with higher external mass transfer, shown in Figure 1B, behave qualitatively the same regardless of enzyme loading. These results have been (24) Service, F. J.; Molnar, G. D.; Rosevear, J. W.; Ackerman, E.; Taylor, W. F.; Cremer, G. M.; Moxness, K. E. Mayo Clin. Proc. 1969, 44, 466.

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validated quantitatively in previous experimental studies.21 Previous functional oxygen-based sensors have been designed with relatively high enzyme loading and substantial mass-transfer resistance, resulting in long-term stability of signal and a linear response to relatively high glucose concentrations.21 Quasi-Steady-State Results for the Peroxide-Based Sensor. The responses of peroxide-based sensors are shown in parts C and D of Figure 1. For uniform mass transfer in Figure 1C, at the higher values of enzyme loading the signal rises with concentration to the point of oxygen limitation and thereafter remains independent of glucose concentration. At lower enzyme loading, the signals rise nonlinearly with concentration over a greater glucose concentration range, asymptotically approaching the oxygen-limiting plateau. This is qualitatively similar to the response of the oxygen-based sensor in Figure 1A, although the range of glucose dependence is slightly lower because of the absence of catalase-driven oxygen recycling and the lower electron number. In Figure 1D, with a different representative value for mass transfer for each molecule, quite a different quasi-steady-state response is observed. Sensors with higher loading show a similar steep rise to a maximum, but the signal passes through a maximum, followed by a gradual decline with increasing concentration to the final, oxygen-limited value. In contrast, the same sensor at lower loading produces only a monotonic rise to the asymptotic limits. The signal inflection at high loading is observable when Bip > Bio, which is likely to be common because of the slight difference in diffusivity of the two molecules. In contrast, it was not possible to produce a quasi-steady-state inflection with oxygen-based sensors, regardless of parameter values. The fact that the same signal was obtained at two different glucose concentrations for certain peroxide-based sensors precluded the creation of a unique set of response curves. Nevertheless, with carefully chosen but restrictive parameter values, a set of frequency response curves can be constructed for a limited set of peroxide-based sensors that are similar to the oxygen-based sensor curves. This quasi-steady-state overshoot phenomenon in peroxidebased sensors can be explained as a result of a substantial fraction of peroxide not reaching the electrode surface because of its relatively high production a short distance within the membrane and diffusion back into the external medium. At lower enzyme loading, reaction is less, substrates diffuse farther within the membrane before peroxide is produced, and a greater fraction of peroxide traverses the membrane, although some loss to back diffusion may still occur. The fact that most investigators working with peroxide-based sensors have reported a quasi-linear calibration curve over a broad concentration range with no inflection suggests that practical sensors are made, intentionally or otherwise, with relatively low enzyme loading. Immobilized enzyme activity may be further reduced over time by inactivation mediated by hydrogen peroxide itself.25 Low enzyme loading may suffice for short-term applications, but low loading combined with enzyme inactivation may lead to reduction of sensor lifetime and the need for frequent recalibration. This phenomenon compromises the ability to use excess loading to counter enzyme inactivation as a strategy for extending sensor lifetime. (25) Tse, P. H. S.; Gough, D. A. Biotechnol. Bioeng. 1986, 29, 705.

Figure 1. Simulations of quasi-steady-state sensor responses. A and B: Oxygen-based sensors. C and D: Peroxide-based sensors. Other parameter values are given in Table 1.

Electrochemical Interference at the Peroxide-Based Sensor. Certain features of the peroxide-based sensor are not easily simulated. Many small polar molecules present in tissue fluids, including glucose, itself, may react in parallel with hydrogen peroxide on the metal electrode surface in slow, nonspecific processes, depending on the composition of the medium, membrane properties, electrode-surface characteristics, electrode potential, and time.26 Although a number of investigators have observed electrochemical interference in peroxide-based sensors, there have been few quantitative studies of the effect on the glucose signal. The effects of electrochemical interference are therefore not included in this model. The limited available information about direct glucose oxidation on a rotated disk platinum anode26 is used here to illustrate the (26) Lerner, H.; Giner, J.; Soeldner, J. S.; Colton, C. K. J. Electrochem. Soc. 1979, 126, 237.

issues. At constant potentials, the current of a bare, rotated disk electrode can decay in the presence of glucose in a slowly varying function of time without attaining a steady state, suggesting a slow electrode deactivation process. The signal current shows no simple functional dependence on time at most anodic potentials, but in the range of -200 to 50 mV vs a standard calomel electrode (SCE), the current decays logarithmically (not exponentially) with time according to the following equation26

i ) kec(1 - θ)

(16)

where ke is a constant at a particular electrode potential, c is concentration at the electrode surface, and θ is the fraction of the electrode surface covered by the deactivating species as a Analytical Chemistry, Vol. 72, No. 8, April 15, 2000

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Figure 2. Simulations of the pass-through ratio and the delay as functions of the glucose concentration sine wave frequency for oxygenbased sensors. The pass-through ratio (A and C) is normalized by the concentration-specific, quasi-steady-state values. Delays (B and D) correspond to parts A and C. σ1 ) 0.1. Other parameter values are given in Table 1.

function of time, given by

θ ) k2 ln(t/k3)

for

1 e t/k3 < e1/k2

(17)

The variable t is time after exposure to glucose, and k2 and k3 are constants having, respectively, no units and units of time. The values of the constants are a function of the electrode roughness factor and have been given elsewhere.27 It is unclear whether this process prevails when peroxide is the electrochemical reactant and glucose and other small organics are the interferants. However, if this or a similar decay process is validated in future studies, the inner membrane permeability and (27) Gough, D. A.; Leypoldt, J. K. J. Electrochem. Soc. 1980, 127, 1278.

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total time of sensor exposure to the interferant-containing medium may have to be included in the model. It is also possible, depending on the kinetic form and respective rate constants of the decay process, that there may be an initial relatively rapid signal decay followed by a much slower decay over a long period, which may allow less frequent recalibration after an initial adjustment period. There is a need for detailed studies on this question. Frequency Response of Oxygen-Based Sensors. Simulations of the frequency response of the oxygen-based sensor are summarized in Figure 2A-D. The mean pass-through ratio, or output signal, normalized by the quasi-steady-state input signal for the same concentration is plotted as a function of sinusoidal input frequency, ω, in parts A and C of Figure 2. The correspond-

properties. The sensor signals are normalized by the maximum and minimum values to allow visualization of delays. As the relative mass transfer resistance of the tissue increases, potentially due in practice to implant encapsulation, microvessel exclusion, or other processes, the value of Big becomes smaller. The results indicate that delays can become substantial when the value of Big is low, that is, when external mass transfer resistance is relatively large. In Figure 3B, the same signals are shown on a log scale without normalization, as they would be directly recorded. The blood glucose recording (dark line) from the literature is indicated on an arbitrary scale. The recorded signals (light lines) show very little deflection to blood glucose variations, indicating the degree of amplification and resolution needed to effectively record the signals. Both the absolute magnitude of the sensor signals and the difference between maximum and minimum during the excursion are greatly reduced as external mass transfer becomes larger. Implications for Sensor Design. These results can be used to show the required amplifier gain and degree of noise rejection in the recorded signal needed to achieve adequate concentration resolution. The results also indicate the need for independent experimental determination of the mass transfer parameters in order to achieve quantitative simulation. The additional difficulties related to the peroxide-based sensor, including the multivalued steady-state signal under certain transport and enzyme loading conditions and the direct electrochemical reaction of glucose and interferences, may explain some unexpected results reported from tissue implant studies.

Figure 3. Simulation of the response of an oxygen-based sensor implanted in tissues to blood glucose recordings taken from the literature.22 A: Normalized blood glucose (dark line) and normalized signal (light lines) for values of the glucose mass-transfer parameter, Big, that may be characteristic of tissue. B: Simulations of direct signal current corresponding to A, above. σ1 ) 0.1. Other parameter values are given in Table 1.

ing delay for each sensor design is shown in parts B and D of Figure 2. The frequency range easily includes the maximal anticipated biological frequency, 0.8 × 10-3 Hz. In each case, the mean pass-through ratio drops off with increasing frequency. The pass-through ratio is relatively constant at the physiological frequency and below that, but decreases substantially at higher frequencies, indicating that there may be minimal reduction of amplitude over the physiological range. The corresponding delay, however, may be substantial. The preservation of the wave shape was also studied, but differences in the pass-through waveform as a function of frequency were very small and similar for each case. Signal Delay and Amplification. Simulations of the response of an oxygen-based sensor to a quasi-continuous blood glucose recording obtained from the literature is shown in parts A and B of Figure 3. In the upper panel, the blood glucose challenge in the heavy line is a linear interpolation of frequently sampled blood glucose values determined by conventional analytical means from a diabetic patient.24 The light lines represent signals from oxygenbased sensors implanted in tissue, having the indicated values of the mass transfer parameter, Big, reflecting tissue mass transfer

CONCLUSIONS Simulation is a useful tool to understand glucose sensor performance. The dynamic forcing functions used here provide more information about the sensor response than do conventional initial-value experiments employing concentration steps. Simulation of sensor response was straightforward for the oxygen-based sensor, but additional information about electrochemical interference is needed for full simulation of the more complex peroxidebased sensors. Simulations suggest that blood glucose dynamics can be accurately recorded by certain sensor designs, but mass transfer effects in tissue must be accounted for. GLOSSARY A, electrode area; Bi, mass transfer Biot number, dimensionless transfer parameter; c, concentration; b, c dimensionless concentration; jc*, effective substrate concentration ratio; D, diffusion coefficient; h, external mass transfer coefficient; i, current; KM, Michaelis constant; k, k2, k3, rate constants; n, electron number; R, dimensionless reaction rate per unit volume; Vmax, maximum reaction velocity per unit volume; R, partition coefficient; κ, dimensionless rate constant; ν, stoichiometry coefficient; δm, membrane thickness; σ, dimensionless relative catalytic activity; φ, Thiele modulus; θ, active electrode area; e (subscript), electrode; i (subscript), respective substrate; 1,2,3 (subscripts), glucose, oxygen, hydrogen peroxide ACKNOWLEDGMENT This work was supported by Grants from the National Institutes of Health and the Juvenile Diabetes Foundation International. DAG is a scientific advisor to GlySens, Inc. Received for review September 3, 1999. Accepted January 27, 2000. AC991018Z Analytical Chemistry, Vol. 72, No. 8, April 15, 2000

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