Anal. Chem. 1994,66, 1520-1529
Prediction of Pocket-Portable and Implantable Glucose Enzyme Electrode Performance from Combined Species Permeability and Digital Simulation Analysis Rathbun K. Rhodes,'ptp$ Mark C. Shults,tp* and Stuart J. Updiket Department of Medicine, University of Wisconsin Medical Center, 600 Hbhland Avenue, Madison, Wisconsin 53792, and Markwell Medical Institute, 1202 Ann Street, Madison, Wisconsin 537 13
Development of enzyme electrode sensors can be facilitated by the use of computer simulation modeling techniques. In this work, the model accounts for contacting solution effects, changes in species diffusion and partition coefficients across multiple membrane layer interfaces,cofactor limitationeffects, enzyme loading,enzymekineticsand decay, and other variatiolrs in electrode layer geometry. Experimentally determined single species membrane permeabilities are included in the digital simulations to predict the performance characteristics of a multilayered glucose oxidase based enzyme electrode. Information obtained includes range of sensor linearity under different substrate and cofactor concentration combinations, sensor time response and output as permeability parameters and enzymeloading are varied, sensorwashout and interference effects, and predictionof decay of sensorenzyme concentration and resultant estimated sensor lifetime as a function of contacting solution conditions and sensor permeability. In the development of amperometric enzyme electrodes, the need arises to quickly identify optimized sensor membrane materials and geometries with a minimumof experimentation. These sensor materials can serve one or more purposes, including the following: (1) tailoring of electroactive species flux and thus device output and time response, (2) preferential extracting of either substrate or cofactor to linearize signal in the desired measurement regime, (3) providing an immobilized enzyme reaction layer for conversion of substrate and cofactor to products (any of which may be monitored electrochemically), (4) discriminating against either electroactive interferences or other chemically reactive constituents which would degrade underlying materials, and ( 5 ) providing the requisite hydration and osmotic properties to maintain electrode stability. This process can be expedited by the development of a sensor simulation model which, with the inclusion of a few known or experimentally determined membrane material permeability parameters, can be iterated upon in different final geometries to predict transient and steady-state response curves. Thus, the experimenter can specify the type of material and appropriate dimensions required to fit the desired application. Previous digital simulation models of enzyme electrodes have been used to explain observed electrode transient behavior and to predict steady-state outputs for a variety of membrane parameters.1-8These papers explain extension of substrate University of Wisconsin Medical Center.
* Markwell Medical Institute. 1520
Analytical Chemistry, Vol. 66,No. 9, May 1, 1994
measurement linearity through membrane diffusion control,' the importance of the relative permeabilities of substrate and cofactor in defining the usable sensor measurement range," the idea of simulating speciesflux through multiple membrane layer^,^ transient matching of experiment to theory when enzyme activity is varied and simultaneous step changes occur for substrate and c0factor,6.~and decay of enzyme concentration over time as a function of contacting solution composition.8 In most of these simulations, the calculations were performed in dimensionlessunits and in all cases assumed no perturbation of concentrations in external contacting solutions. The sensor simulation model used in this paper solves the problem of multiple species diffusion through a diffusioncontrolling resistance layer with reaction in an immobilized enzyme layer. The mathematics to describe the simulation can be reproduced easily by reviewing the diffusional portion of solutions to prior simulations9and then adding in a reaction term to account for enzyme kinetics. The species concentration profiles and sensor outputs are solved by the Newton-Raphson method carried through iteratively to a desired degree of convergence.lo This model expands upon the previous enzyme electrode simulations by the inclusion of an unlimited number of interfaces, thus allowing for effects due to both membrane/ solution partition and partition from one membrane material to another. This permits changes in species diffusion coefficient as an interface is crossed and allows different element thickness from layer to layer. The model makes provisions for calculating complete transients where any relevant species is the limiting reactant and allows the typical calculations for response variations due to membrane thickness, enzyme activity, enzyme decay factors and kinetics, and species diffusioncharacteristics. It allows concentration profile carryover and parameter modification from one simulation to the next, thus making possible the description of significant but previously unreported phenomena, such as sensor washout Mell, L. D.; Maloy, J. T. Anal. Chem. 1975, 47, 299-307. Leypoldt, J. K.; Gough, D.A. Anal. Chem. 19&4,56, 2896-2904. Leypoldt, J. K.; Gough, D.A. Biotechnol. Bioeng. 1982, 24, 2705-2719. Leypoldt, J. K.; Gough, D. A. Applied Biochemistry and Bioengineering; Academic Press: New York, 1981; Vol. 3, pp 175-206. ( 5 )' Bergel, A.; Comtat, M. Anal. Chem. 1984, Sa, 29W2909. (61' Tse, P. H. S.; Gough D.A. Anal. Chem. 1987, 59, 2239-2344. (7j Lucisano, J.; Gough, D. A. A M I . them. I.WI,~O, 1272-1281. ( 8 ) Tse, P. H. S.; Gough, D.A. Biotechnol. Bioeng. 1987, 29, 705-713. (9) Brumleve, T. R.; Buck, R. P. J. Electroanal. Chem. 1978,90, 1-31. (10) Camahan B.; Luther H. A.: Wilkcs, J. B. AppliedNumerical Methods; John Wiley; New York, 1969; pp 274-321, 436540.
0003-27OOf 94/0366-1520%04.50/0
0 1994 Amerlcan Chemlcal Socletv
Tabk 1
paramete9
D (GLU) K D (02) K D (HzOz)
K D (ASC) K D (ACETA) K Thickness$ cm
solutionb
resistance
enzyme
interference
hydration
3.6 X 10-6 1 2.1 x 10-4 1 1.0 x 10-4 1 (2-4) X 10-6 1 (2-4) X 106 1 0.089
2.2 x 10-7 0.075 6.0 x 10-7 3.5 5.8 x 10-7 1.41 1.8x 10-7 0.15 5.2 X 10-8 8.8 0.0040
1.0 x 10-8 0.15 1.0 x 1od 1 5.0 X 10-8 1
nmc 0 1.0 x 10-7 1 3.3 x 10-8
1.0 x 10-8 0.15 1.0 x 10-6 1 5.0 X le
1
1
nmc
1.0 x 10-8 0.25 5.0 X 10-’ 1.5
1.0 x 10-8 0.25 5.0 x 10-7
1.5 O.OOO64
0
1.0 x 10-8 1 o.OOO1
O.OOO64
Units for D are centimeters squared per seconds and X is dimensionless. K is the ratio of the equilibrium membrane layer concentration relative to the solution layer concentration. b Note that literature solution layer diffusion coefficientshave been multiplied by 10 to account for mixed diffusion/convection effects. Not measurable. d Thicknesses are known to be within &20%for the thin interferencelayer and within +5% for all other layers. Q
transients and interference effects. We have chosen to carry out these simulations without normalizing to dimensionless units since we are working in a relatively small subset of parameter space, and in our experience, the results were easier to explain to nonspecialists who wished to evaluate our work. Several experimental methods exist for obtaining the necessary single-species diffusion and partition coefficients to include in the simulation model. One can assume that the concentration step technique commonly used with membranes contacting gas is valid and that only minimal perturbation of external concentration occurs.1 However, when membrane permeability approaches that of the contacting solution, this assumption is badly in error as speciesconcentration gradients extend well into solution. Even rapid stirring cannot completely alleviate this problem.12 A better approach to separate the membrane from solution permeabilities has been demonstrated by Gough and Leypoldt with a rotating disk technique.13J4Variations of limiting current by adjusting the electrode rotation rate allow extrapolation to the membrane permeability. Further combination of this technique with analysis of transients due to voltage steps15 or concentration steps16 allows an elegant separation of the membrane permeability into its diffusion and partition components. In the absence of rotating disk instrumentation, a further technique comes to mind and in fact is used in obtaining the permeation parameters for our model. A concentration step experiment is run and analyzed as for a gas contacting membrane to obtain initial diffusion and partition coefficients. These values, with the appropriate distance grid describing membrane and solution regions, are then included in the simulation model and iterated through several cycles providing updated values for these parameters until the experimental vs calculated transient is matched with respect to both output and time response. This resultant information for single species is then included as part of the complete enzyme electrode model. Since the sensor’s solution boundary layer is not as well-defined as in the RDE method, this technique does not provide the same degree of accuracy in calculating diffusion (11) Manoy, K. H.; Okun, D. A.; Reilley, C. N. J. Elecrroanal. Chem. 1962, 4, 65-92. (12) Malone, D. M.; Anderson, J. L. AIChE J. 1977, 23, 177. (13) Gough, D. A.; Leypldt, J. K. Anal. Chem. 1979,51,439-444. (14) Gough, D. A.; Leypoldt, J. K. Anal. Chem. 1980,52, 1126-1130. (15) Gough, D. A.; Leypoldt, J. K. AIChE J. 1980, 26, 1013-1019. (16) Gough, D. A.; Leypldt, J. K. J. Electrochem. Soc. 1980,127, 1278-1286.
and partition coefficients. However, for our purposes in describing an overall electrode system and in making relative comparisons of parameter variation on electrode performance, this technique proved useful.
EXPERIMENTAL SECTION The permeability parameters of the complete system are listed in Table 1 under generic headings which describe the contacting solution and membrane layer functions in sequence (Le., solution, resistance, enzyme, interference, and hydration). Four replicate samples of well-defined thickness for each membrane layer type were used to verify the permeability parameters obtained. Thicknesses for all layers were directly measurable by micrometer to within f5%. Membranes were given 24 h to hydrate fully prior to parameter measurement. Resistance and interference layer permeability parameters were obtained directly from single layer measurements. Enzyme and hydration layer permeability parameters were obtained indirectly from multilayer measurements where all other layer parameters had already been determined. In the complete four-layer membrane, the interference layer is 10 times thinner than the samples used for initially obtaining permeability parameters and thickness is calculated to within f20% from preparation volume, solution percent solids, and resultant membrane square footage. Enzyme layers of constant thickness, but with concentration of glucose oxidase (Sigma Type VII) varied from 0.78 pM to 0.2 mM, were prepared to study effects of enzyme loading. The0.2 mM concentration resulted in -4% enzyme by weight within the enzyme layer. This by weight percentage scaled proportionately with lowering of enzyme concentration. Glutaraldehyde (Sigma Grade I, diluted to 0.1%) served as the cross-linker. All four membrane layers were polyurethane in nature and prepared by standard coating techniques. Membrane samples for study are available from the authors. The complete four-layer membranes define a portion of the system used in the Direct 30/30, a pocket-portable electrochemical glucose sensing unit for diabetic patients.” Diffusion and partition coefficients for the species hydrogen peroxide, oxygen, glucose, ascorbic acid, and acetaminophen
-
~
~~~~
(17) Updike, S.J.; Shults, M. C.; Capelli, C. C.; von Heimburg, D.; Rhodes, R. K.; Tipton-Joseph, N.; Anderson, B.; Koch, D. D. Diobetes Care 1988, 11, 801-807.
Ana&ticaI Chemistry, Voi. 66,No. 9, May 1, 1994
IS21
Figure 1. Cross-sectional view of three-electrode cell for permeability measurements: (1) bottom case, (2) top case, (3) electrode insert, (4) electrodes, (5) snapon membrane holder, (6) membrane, (7) sample well, and (8) space for electronics.
in the polymer materials of interest were determined by the combination of permeability measurements with iterative digital simulations. All solutions used were prepared with distilled, deionized water (McGaw), reagent grade chemicals (J. T. Baker, Aldrich, Amend), and research grade gases (Matheson). Direct 30/30 top cases in standard threeelectrode c~nfigurationl~ were used as the electrode test platform (Figure 1). The working and counter electrodes were 0.020-in.-diameter Pt wire while the reference electrode was a 0.020-in.-diameter silver wire. All electrodes were potted in epoxy, and the final assembly cut to a 0.200 in.-radius dome, which was then polished at the surface. The silver wire reference electrode was then chemically chloridized.18 An eight-channel, three-electrode potentiostat (Markwell Medical) with switching box was used to direct sensor outputs for computer acquisition. Potentials were set to -0.6 V (vs AgC1) for oxygen reduction and +0.6 V for hydrogen peroxide, ascorbic acid, and acetaminophen oxidations. Glucose parameters were determined indirectly by monitoring the peroxide formed in a thin layer of laminated polyurethane/ glucose oxidase on the electrode side of the membranes. The transient responses for solution concentration steps were acquired with a Macintosh Plus using a Remote Measurement Systems A/D board and Quickbasic software. Since the three electrodes were situated directly beneath the membrane, concentration steps were made by blotting dry the top side of the membrane and then pipeting the solution of interest into a 50-pL well. Data acquisition synchronization with sample application was assured by applying the sample at the sound of a computer-generated tone. In cases where oxygen exclusion was required for extended times, a capped polycarbonate cylinder was positioned above the electrode to increase the solution volume to 0.5 mL. This allowed easy maintenance of a constant pOz. The concentration steps used to determine the permeability parameters included 0-5 mM for hydrogen peroxide, 0-1 1 mM for glucose, 0-160 mmHg for oxygen, 0-6 mM for ascorbic acid, and 0-0.8 mM for acetaminophen. A trace of catalase was included in the glucose solutions to scavenge any enzyme-produced hydrogen peroxide which diffused back through the membranes to the contacting solution. Additional concentration steps were performed on the complete membrane system to provide the data required for model verification. Linearity was checked by varying the glucose level from 0 to 55 mM in air-saturated solution. Further variation was made by checking the higher glucose level steps (22-55 mM) as a function of solution oxygen concentration (30-160 mmHg, verified by either Radiometer BMS3 Mk2 Blood MicroSystemor ColeParmer 551360oxygen monitor). (18) Sandifer, J.
1522
R.And. Chem. 1981, 53,312-316.
AnalyNcal Chemlsiw, Vol. 88, No. 9, M y 1, 1994
Exposure times for these specific glucose and oxygen concentration permutations were varied and then the electrode assembly was reexposed to storage solution (air saturated with no glucose). This allowed following of sensor washout curves. Interference transients were also checked by performing concentration steps using freshly prepared solutions of ascorbic acid or acetaminophen. Enzyme deactivation rates were measured using our lower loading enzyme recipes in a manner equivalent to that described by Gough.8 This information was then entered into our simulation model, the simulated steady-state species concentrations and enzyme fractions were calculated as a function of contacting solution exposure and membrane permeabilities, and sensor lifetime was simulated. To look at the kinetics of the possible decay mechanisms, contacting solutions were used which would leave glucose oxidase in either its resting state or with known fractions of the reduced forms. The starting enzyme concentration was */@ofthe standard recipe or lower so that the electrode output to a 11 mM glucose concentration step was approximately linear with enzymeconcentration. Theconcentration of active enzyme was then calculated from the ratio of sensor output after a defined decay period to the starting sensor output. The averages from four membrane samples were used to calculate remaining active enzyme concentration. To look at the deactivation of the resting-state enzyme, membranes were suspended in a phosphate-buffered saline (pH 7.05) contacting solution at a p02 of 150. Two sets of samples were tested at temperatures of 22 and 37 OC. Fresh samples were selected periodically over 4 months from these storage solutions and tested after mounting on Direct 30/30 cases. To study peroxide-induced decay pathways of the reduced enzyme fractions, the phosphate-buffered saline solution also included 27.5 mM glucose and 5 mM hydrogen peroxide. Decay measurements were made using membrane samples exposed to this solution for varying lengths of time. Solution p0z was set to either 0 or 30 mmHg by continuous degassing. This provided a high level of convection and allowed accurate calculation of oxygen and glucose concentration within the enzyme layer. These oxygen levels produced preferential decay pathways for the enzyme. At 30 min to 4-h intervals, membrane samples were removed, rinsed in phosphatebuffered saline to quench the decay, and tested as before for remaining active enzyme concentration. This process was repeated over four to six time intervals until at least a 60% decay in active enzyme concentration occurred. Simulations of electrode output and membrane layer concentration profiles were run on a Macintosh SE/30 computer using True Basic as the programming language. The time required to run any simulation is dependent on the number of species, elements, and time steps desired. The typical transients shown here used eight or less species, 10 elements per layer, and 120-480 time steps. The species included glucose, oxygen, hydrogen peroxide, interferent, oxidized enzyme, glucose-complexed enzyme, reduced enzyme, peroxide-complexed enzyme, and deactivated enzyme. An eight-species, 50-element, 120-time step simulation required -20 min to complete. In addition, the algorithm is designed in such a way that calculation times will scale directly with
changes in any of these threevariables. To verify the accuracy of the simulation, the time step size was varied from 0.05 to 5 s to find the times which showed no appreciable distortion of the simulated current curves due to diffusional/kinetic mismatches. In addition, mass balance checks were made on the resultant concentration profiles to ascertain that no distortion occurred. The simulation of enzyme decay as a function of contacting solution required an important approximation. This was related to the simulation time step that can be used without distortion of the concentration profiles. For sensor transient development, it took 10 min of computer time to simulate 1 min of real time. However, simulating months to years of enzyme decay with these small time steps presented a computational bottleneck. This was addressed by realizing that enzyme decay occurred slowly. To start the simulation, initial steady-state concentration profiles for specific contacting solution conditions were first established using a time step of 0.25-1 s. However, from this point on we scaled up the decay kinetics from seconds to days as long as two conditions were met. First, the product of the enzyme decay kinetics and the time step was sufficiently small so that no more than 1% of the hydrogen peroxide steady-state concentration was removed during any time step by reaction with enzyme subpopulations. Second, after each half-loading of simulated enzyme decay, the simulation was put into a "zero decay" mode until it was verified that glucose, oxygen, and hydrogen peroxide were indeed at the correct steady-state concentrations for the remaining active enzyme. These considerations guaranteed that enzyme subpopulations were correct and minimized the underestimation of decay that would occur at highest enzyme loading when the simulation was being based on hydrogen peroxide concentration that was a few percent low. Worstcaseerrors with this approximation resulted in underestimated decay times of 10-15%. Thus, as long as the concentration estimates of the enzyme subpopulation fractions are correct and hydrogen peroxide is close to the correct steady state, propagation of error in enzyme decay during the course of the simulation is reasonable.
-
RESULTS Determination of Model Parameters. The gas contacting model serves as the starting point and predicts steady-state and transient output currents of the form is, = (nFAD,K,C,)/X
(1)
and it = i,(l
+ 2~(-l)"e-"")
where z = (?r2Dmt)/X2
(2)
and C, equals the bulk solution concentration (mol/cm3), X themembrane thickness (cm), t the time (s), D, the membrane diffusion coefficient (cm2/s), and Km the dimensionless membrane partition coefficient and n, F, A , and ?r have their usual meaning. When expanded, eq 2 leads to it/iss= 1 - 2e-'
+ 2e4'
-2
+
~ ~2e-16' '
...
(3) Thus, by determining the factor z at different fractional currents, it becomes possible to solve for D, using the equation
D, = (zX2)/(?r2t)
(4)
where for example z = 1.37 at it/iss= 0.5. The partition coefficient then is calculated from Km = (issu)/ (nFADmCg) (5) We typically find the averages for D , and Kmby determining the times to fractional currents of 0.1, 0.3, 0.5, 0.7, and 0.9 using the corresponding z factors of 0.66,l .OO,1.37,1.89, and 3.00. These D / K pairs serve as the starting point in the iterative process to match experimental data to theoretical transients. After these values are plugged into the relevant distance grid which defines our experimental conditions, the concentration step is then simulated with literature values for solution diffusion coefficients in all solution-layer elements. Curve matching of simulated to experimental data for a more permeable species, such as hydrogen peroxide, is performed first to check for mismatches in transient curve shape due to solution permeability effects. In this experimental situation, where sample application is less well defined than with a RDE, the classical approximation of D,ff = D a t a sensor boundary layer of defined thickness and D,ff >> D in bulk solution is not likely to apply. We have used the approximation of D,ff = 1OD throughout the entire solution layer. This approximation is suggested through the high-quality curve shape match of experiment to simulation, but it should be recognized that this factor would be different if applied to a different system with different permeability parameters and geometry. Once this modeling of the solution layer is defined, then iterations in the membrane resistance layer D / K pairs with further matching of real to simulated data completes the parameter determination. This process is repeated for glucose, hydrogen peroxide, and oxygen as each successive membrane layer is added. The uncertainty introduced by this model of solution permeability has little effect on the reliability of experimentally determined membrane resistance layer permeability parameters for the less permeable species, which include glucose, ascorbic acid, and acetaminophen. However, for more permeable species such as oxygen and hydrogen peroxide, the tabulated values of the resistance layer permeability components are probably not better than 45-896 for D and f 1 0 12% for K . This does not imply equivalent errors in the total layer permeability so much as a lack of resolution between its D and K components. Representative fitted curves are shown for two- and fourlayer membranes for hydrogen peroxide and glucose in Figures 2 and 3, respectively. All subsequent simulation predictions are made for the full sensor system where four of the layers are membrane materials and the fifth layer is the contacting solution. The full permeability parameter sets for these layers are included in Table 1. Oxygen Cofactor ConcentrationLimitationEffects. Figure 4 shows the simulated oxygen concentration profiles obtained through 120 s for the glucose concentration step of Figure 3 (0-1 1 mM glucose, p02 = 160 mmHg = 0.000 25M). Notice that oxygen reservoirs are present in the membrane layers prior to the concentration step. Once the step occurs, new profiles are generated with the glucose profiles reaching a Analytical Chemism, Vol. 66, No. 9, May 1, 1994
1523
40
r
0 0
60
30 TIME (SECONDS)
0
30 TIME (SECONDS)
60
Flgwe 2. Slmulated(solid1ines)vsexperimental sensor outputtransients for 0.5 mM hydrogen peroxide (filled squares) and 11 mM glucose (open squares) concentratlon steps through two-layer membrane. cm2. Electrode area is calculated to be 2.0 X
20 r
0
11
22
33
44
55
mM GLUCOSE
0
120
60 TIME (SECONDS)
Flgure 3. Shnulated (solkl1lnes)vsexperknentalsensor output transients for 0.5 mM hydrogen peroxlde (filled squares) and 11 mM glucose (open squares) concentration steps through four-layer membrane. Electrode area is calculated to be 1.8 X lo3 cm2. Membrane Layers - Function and Thicknesses enzyme interference 0.0001 cm 0.00064 cm
hydration 0.00064 cm
resistance 0.0040 cm
solution 0.089 cm
Oxygen Concentration Profiles
0 D. E
1524
I
L
Analytical Chemistry, Vol. 66, No. 9, May 1, 1994
Flgure 5. (a) Simulated output transients for (1) 5.5, (2) 11.O, (3)27.5, and (4) 55 mM glucose concentration steps with a contactlng blood pO2 of 30 mmHg. (b) Simulated outputs at (1) 10, (2) 15, and (3) 20 s for the four blood glucose concentration steps at a pOzof 30 mmHg. Note that best linearity Is obtained at shorter measurement times.
When dissolved oxygen concentration decreases in the contacting solution, as is the case for many relevant matrices such as venous blood samples (0.1-0.025 mM), oxygen flux into the membrane very badly lags glucose flux. As Figure 5a shows for simulations of 5.5-55 mM glucose with 0.05 mM dissolved oxygen, membrane oxygen quickly becomes exhausted and the sensor converts from a glucose-dependent to an oxygen-dependent sensor. At the higher glucose levels, the current begins to decay as early as 15 s after sample application and current is now controlled by oxygen rather than glucose flux. Thus, steady-state current is no longer useful for analytical purposes. However, by simulating a family of output transients as a function of glucose concentrations at the lowest concentration of oxygen to be encountered, it is possible to identify a usable time window over which glucose linearity is maintained. This time window can then be incorporated into the analysis software, as was done for the Direct 30/30, to provide a high degree of measurement linearity. This is demonstrated in Figure 5b. Discrimination Against Interference Effects. The ability to discriminate against electroactive interferences is usually assumed to be due to the complete rejection (i.e., partition coefficient of 0) of the species at one of the membrane interfaces. However, as the species partition and diffusion coefficients of Table 1 show, there must be an additional mode of interference discrimination since the K for ascorbic acid is vanishingly small while that for acetaminophen is large. This second mode can be explained on the basis of the much slower diffusion coefficient of acetaminophen relative to the other species. As Figure 6 suggests, we can use the kinetic mode to measure substrate in the first 30 s before development of the interference signal at greater times. An analogous type of time domain discrimination has been demonstrated pre-
viously by Sandifer18 for chloride potentiometric sensors in the presence of bromide. Sensor Washout Effects. One phenomenon of significance which has not been reported in previous enzyme electrode simulation studies is that of sensor washout. This relaxation in sensor output occurs when the contacting sample solution is replaced with one containing no substrate and a normal level of cofactor. The speed of this relaxation or recovery process determines the fastest possible sensor cycle time, which is of great practical importance to the analyst trying to run a series of samples. Three classes of relaxation curves can occur during sensor washout. In the simplest case, when cofactor remains in excess during the entire measurement, the curve relaxes toward its zero level in a time comparable to the initial transient development time. In situations where the enzyme layer cofactor has been depleted due to extended exposure at high substrate concentration, the relaxation curve to wash solution is nonmonotonic (Figure 7a). When the contacting cofactor concentration is low to start with and then is brought back to normal levels, this transient can be quite striking. This occurs because increased cofactor flux reaches and reacts with the accumulated glucose in the enzyme layer. The resultant output moves to its highest level with cofactor flux maximization and holds this level until the residual substrate has been completely scavenged from the enzyme layer. At this point, the cofactor begins to replenish itself in the enzyme layer and a region of relatively fast decay occurs with removal of the last residual substrate and reaction product. The last class of relaxation curve occurs for a slowly diffusing interferent which has not achieved its maximum inward flux. Upon return to wash solution, the output transient appears unaffected and continues to grow in magnitude. However, at some point the bidirectional flux of species will begin to diminish the concentration profile near the electrode. At this point the output transient reverses and begins its slow decay back toward zero (Figure 7b). Samples of this type give an extremely slow decay and can increase the cycle time and apparent baseline of the sensor when the interferent concentration is high. Enzyme Loading Effects. In setting up enzyme loading permutations, enzyme was lowered by factors of 2 from the standard recipe to a ' 1 2 5 6 recipe. Assuming homogeneous distribution in the enzyme layer, no enzyme decay during preparation, and concentrations calculable from normal geometric factors, enzyme concentrations ranging from 2.00 X 10-4to 7.81 X M can be predicted. For the standard glucose oxidase mechanism,19 ki
E
+ G -,RL
R + 0,
k,
-..R + L ki
-+
(6)
k4
E*-H202 E
H202
(7) the room-temperature rate constants given in the literature20 are kl = 12 200 M d , k2 = 1220 s-l, ks = 2 000 000 Mss-', and kq = 660 s-l. Combining this information with the previously determined species permeability data gives the curve (19)Gibson. Q.H.; Swoboda, B. E. P.; Massey, V. J. Biol. Chem. 1964, 239, 3927-3934. (20) Duke, F. R.; Weibel, M.; Page, D. S.; Bulgrin, V. G.; Luthy, J. J. Am.Chem. Soc. 1969, 91, 3904-3909
2
'Or
0 0
30
60
TIME (SECONDS) Figure6. Simulated output response fora 1 1 mM glucose concentratlon step in the absence (1) and presence (2) of 0.4 mM acetaminophen interference.
fits shown in Figure 8a for transient current as a function of enzyme loading. The match is quite good at highest loading but appears to lag in time and underestimate current magnitude as loading is decreased. To try to obtain a better match, the different k values are individually increased. Increases in k2, k3, and k4 by up to a factor of 2 leave essentially the same transient as before. Increasing kl produces a faster development of the transient as well as an increase in magnitude. When kl is increased by a factor of 2, fairly good matches result for ail enzyme recipes, as shown in Figure 8b. Enzyme Decay Measurements. Prior work has demonstrated that glucose oxidase can decay through three primary pathways.8 An apparent "autoinactivation" can occur when the enzyme layer is submersed in fluid containing no glucose and no hydrogen peroxide. This resting-state enzyme shows a simple first-order decay of the form dxldt = k,,(a - x) where a = total and x = inactive enzyme (8) with the solution x = a(1 -
with a half-life of
r1,2 = 0.693/k,,
(9) In this work deactivation of the resting-state enzyme at 37 OC, as shown in Figure 9a, was fit to project a half-life for this pathway of 105 days. From this information, klt is calculated to be 7.64 X l e 8 s-l. This is 15% slower than that measured in previous work.8 This rate of decay did not change when externally added peroxide but no glucose was added to the contacting solution. The other two pathways of enzyme deactivation occur in the presence of hydrogen peroxide when contacting solution conditions force enzyme to the reduced or peroxide-complexed forms. The reduced form is exclusively formed in glucose solutions with zero oxygen. Both forms are present in significant fractions when glucose solutions are at low PO*. These decay pathways are "psuedo" first order and of the form
-
dx/dt = (kit
+ k2&)(~ - X) + k&)(c
- x)) ( a - X)
(10)
where c = peroxide, fR = fraction of reduced enzyme, and fc = fraction of peroxide-complexed enzyme, thus leading to the solution = a( 1 - e-(ki,)(t)-(kz,)(/~)(c)(f)-(kjr)(/~)(c)(r)) Analytical Chemistry, Vol. 66, No. 9,May 1, 1994
(1 1) 1525
1.0 s
K K
3
u
n
,0
,6
W
N
i ,4 a E K
0
z 90 TINE (SECONDS)
0
180
,2
.o 0
120
0
3
240 360 TIME (DAYS)
480
600
12
15
1.0
ii
w K
-8
K
I)
u
n
,6
W
N
.4
E
0
90 TINE (SECONDS)
180
Flgure 7. (a) Experimental (squares) vs simulated (sotid line) output transient for the case where the inittal contacting sdutlon ia 27.5 mM Inglucose and at a p02of 30 mmHg, while thewash solutionintroduced at 90 s contains 0 mM glucose and Is at a pO2 of 160 mmHg. (b) Simulated output transient for the slowly diffusing Interference acetaminophen Introduced at 0.4 mM concentration at t = 0 s with a return to an lnterferencafree wash solution at t = 60 s.
K
p
#2
.o 6 9 TINE (HOURS)
10
..................................................................................................
I i\
h
2
v
22OC
I-
E 5 K K
3
u
0
0 0
t
_I
60 TINE (SECONDS)
.
/
120
-
-
.I
60 120 TIME (SECONDS) Flgure 8. (a) Experimental vs simulated transients for 1 1 mM glucose step as a function of enzyme concentration. (1)2.0 X lo4, (2)3.13 X lO-O,and (3)7.81 X lo-' M. All kinetic parameters are at literature values. (b) Experimental vs simulated transients for 1 1 mM glucose step as a function of enzyme concentration: (1)2.0 X lo4, (2)1.25 X (3)3.13X and (4)7.81 X le7 M. k, istwkethe Herature value. 0
and
+ kZ,CfR)c+ k&&)
fl,2 = 0.693/(k1,
(12) The results for enzyme concentration decay in excess glucose, 1526 AnalyticalChemlstty, Vol. 66,No. 9, M y 7, 1994
5
10 15 TINE (HOURS)
20
25
Flgure 9. (a) Enzyme decay at 37 O C from restingstate enzyme. Exponential fit projects an enzyme half-life of 105 days. Points are the average value of four sample membranes. (b) Enzyme decay at 22 and 37 O C for a contacting solution of 0 mmHg pop, 27.5 mM glucose, and 5 mM hydrogen peroxide. The simulation predicts an equilibrium enzyme layer glucose concentration of 4.1 mM. (c) Enzyme decay at 22 and 37 O C for a contacting solution of 30 mmHg p02, 27.5 mM glucose, and 5 mM hydrogenperoxide. The sknulatkm predictsa steadystate enzyme layer glucose Concentration of 2.9 mM and an enzyme layer pop of 18 mmHg.
low oxygen, and externally added hydrogen peroxide are shown in Figure 9b,c. To determine rate constants from these decay curves, the enzyme fractions are first calculated. At a contacting solution p02 of 0, the enzyme is fully converted to the reduced form by excess glucose. At a contacting solution pOz of 30 mmHg (0.05 mM) with oxygen flux to the enzyme layer from both sides of the membrane, simulations using the previously defined permeability parameters and modified enzyme kinetics predict enzyme layer glucose and oxygen concentrations of 2.9 and 0.028 mM. This results in predicted enzyme fractions of 0.522 for the reduced form and 0.044 for the complexed form. Rate constants at 22 and 37 O C are then calculated to be 0.013 and 0.029 (M.s)-' for the reduced enzyme and 0.075 (Mas)-' and 0.132 (M.s)-' for thecomplexed
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Flgure 10. Simulated output transients for 22 mM glucose (filled symbols)vs11 mM glucose (opensymbols)for Direct 30130 membrane sensor covered with an additional 0.0025-cm-thick resistance layer. The glucose diffusion coefflcient for the extra layer is either (1) 5.50 X (2) 2.75 X or (3) 1.84 X cm2/s. Oxygen diffusion coefficient did not change from that for the Direct 30130 membrane.
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Figure 11. Simulated sensor output vs time for 0-1 1 mM glucose concentrationsteps using membranesof implantablepermeabiilty with varied enzyme loading. ID numbers represent the number of halfloadings below the standard recipe of 2.00 X lo4 M: (0) 2.00 X lo4, (5) 6.25 X (6) 3.12 X lo", (7) 1.56 X lo4, (8) 7.83 X lo-', (9) 3.92 X lo-', (10) 1.96 X lo-', and (11) 9.80 X M.
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enzyme. These experiments compare favorably with the 37 OC decay constants of 0.02 and 0.76 (M.s)-' measured previously.8 Simulationof Permeability Parameters for an Implantable Glucose Sensor. This model was used in predicting the required membrane permeabilities for an in vivo subcutaneous glucose sensor. The challenge here was to continuously maintain linear response to glucose in the useful clinical range (1 5 - 2 2 mM) with tissue oxygen tension as low as 30 mmHg. Thus, at equilibrium the flux of oxygen from contacting environment to the enzyme layer must always be greater than that for glucose. Simply increasing membrane oxygen permeability does minimal good since this will result in a depletion layer into the contacting solution and only a small increase in oxygen flux to the sensor area. The alternative is to lower glucose flux through the outermost resistance membrane. This can be done by lowering the partition or diffusion coefficients. The former would be preferred since this would not decrease the sensor response time. For us, finding materials with low partition coefficients that maintain reasonable diffusion coefficients has proved illusive. Thus, theoretical and experimental work has focused on materials with comparable partition coefficients and decreased diffusion coefficients. Membrane and solution permeabilities listed in Table 1 serve as the simulation starting point. The contacting solution p02 is set to 30 mmHg. Now a fifth membrane layer is interposed between this solution and the previous membranes. As the glucose diffusion coefficient is lowered by up to a factor of 12 in this fifth layer (with no change in glucose partition coefficient and oxygen permeability), the family of simulated curves shown in Figure 10 is generated. This desired set of parameters predicts the development of a linear signal from 0 to 22 mM. This strategy allowed us to develop membrane resistance layers which, upon continuous aqueous solution exposure, could provide in vitro glucose linearity through 22 mM when pOz is as low as 20 mmHg. In addition, these membranes have now been used in subcutaneous glucose sensors for periods of up to three months in a dog model.21 Estimation of Implantable Sensor Lifetime Limit Due to Enzyme Decay. A useful presentation of remaining active (21) Updike, S.J.; Shults, M. C.; Gilligan, B. J.; Rhodes, R . K.;Luebow, J. 0.; vonHeimburg, D. ASAIO Trans., in press.
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TIME (DAYS) Figure 12. Simulated active enzyme Concentration decay curves vs time for Implantable membranes continuously exposed to different glucose concentrations of 0.0, 11.0, 22.0, 24.8, and 27.5 mM at a contacting solution pOn of 30 mmHg. Starting active enzyme concentrationis 2 X lo4 M; (.I and (- -1 respectively represent the enzyme concentrations where 90 and 50% of the sensor output remains.
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enzyme concentration and its correspondence with sensor output is given in Figures 11 and 12. Figure 11 shows the simulated output transients to a 1 1 mM glucose concentration step for sensors with implantable membrane permeabilities and enzyme concentration lowered in 50% increments from 0.2 mM to 0.098 pM. Note that approximately six halfloadings of enzyme must be removed before signal drops more than 10% and that nearly nine half-loadings must be removed to drop the signal more than 50%. These serve as standard curves for predicted sensor output. In Figure 12, the active enzyme concentration vs time has been simulated for membranes with implantable sensor permeabilities. The contacting solution p02 was 30 mmHg and glucose was varied from 0 to 27.5 mM. Note that these plotted enzyme concentrations are the lumped sum average of remaining active enzyme throughout all enzyme elements. These remaining active enzyme concentrations can be compared to Figure 11 to predict the remaining sensor output to a standard glucose challenge. Two lines are drawn at the enzyme concentration levels where 90 and 50% of the standard loading sensor output remains. At 90% sensor output, only minimal changes in overall sensor performance will have occurred. However, at the 50% sensor output level, the sensor time response will be so slow (2-4 times slower) and the output so enzyme concentration dependent that the sensor is no longer useful. Analytical Chemistty, Vol. 66,No. 9, May 1, 1994
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There are several points to note from this plot. The first is that, even with no glucose exposure, the enzyme concentration will decrease from resting-state enzyme decay. This represents the best-case limit of electrode lifetime. The second is that as long as the glucose flux is lower than the oxygen flux into the enzyme layer, the enzyme decay will be only slightly faster than this slowest decay pathway. Finally, once glucose flux is greater than oxygen flux, glucose will accumulate in the enzyme layer, the fractions of reduced and complexed enzyme concentrations will increase, and enzyme concentration decay will accelerate. The contacting solution concentrations where this occurs will be very sensitiveto both initial membrane permeabilities and changes in permeability that may occur in the sensor membranes due to aging, foreign body encapsulation, or membrane pore occlusion. However, if as a starting point we are willing to believe this simulated time to 50% output, then we can speculate that glucose concentrations in the typical diabetic patient range can be in continuous contact with the sensor for 2 years without appreciably accelerating the speed of enzyme decay. Thus, enzyme lifetime is not likely to be the primary limiting factor in implantable sensor development.
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DISCUSSI ON In our system, there are several experimental situations which we need to be able to predict accurately before we can consider our model complete. These include the following: (1) single species permeability measurements corrected for contacting solution effects, (2) electrode output magnitude and time responses as a function of membrane thickness (not cofactor limited), (3) range of substrate linearity as a function of cofactor concentration, (4)output transients vs time for cofactor-limited cases, (5) output transients vs time for different enzyme loadings (at constant membrane thickness and solution conditions), and ( 6 ) washout transients vs time for a variety of solution cofactor/substrate concentration permutations and exposure times. We start first with the simplest case, that of determining single species diffusion and partition coefficients. For the directly electroactive species this is easy. However, when we get to the enzyme converted species measurements, a degree of uncertainty must initially be accepted in making the measurement. This uncertainty occurs since we are using a single output transient to try to explain several possibly counteracting effects including (1) the bidirectional flux of peroxide generated in the enzyme layer, (2) enzyme activity and kinetics, (3) cofactor concentration, and (4) the substrate and cofactor partition coefficients in resistance and enzyme layers. In fact, no single transient of this type is uniquely determined as tradeoffs in species concentrations, species DIK pair, and enzyme kinetics and loading can provide essentially indistinguishable curves. To assess our initial glucose permeability estimate we provide an experimental situation that will force a nonmonotonic output transient. For our system this is provided simply by setting up solution conditions where cofactor will quickly be depleted. Thus our simulation substrate/cofactor fluxes will have to be matched correctly to reproduce the experimental sensor output magnitude and more importantly the time when cofactor becomes the limiting reagent and sensor output decays 1528
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with loss of cofactor. The subset of parameter tradeoffs which can match this curve type is much smaller than that for the simple glucose transient with adequate cofactor, and thus we approach convergence on the final parameter set. A more stringent test is the prediction of the washout curve after cofactor-limited situations. In these situations, substrate will have accumulated in the enzyme layer after cofactor was depleted. Once the contacting solution is replaced with another containing no substrate and normal levels of cofactor, accumulated substrate will begin to either react or diffuse away. The shape and magnitude of the output washout curve and the time required for removal of residual substrate can be used to verify enzyme kinetic parameters and partition of substrate from the resistance into the enzyme layer. This is checked at several widely spaced permutations of substrate and cofactor concentrations to verify that some unknown artifact has not assisted in this match. The final test for the model is its ability to predict transient response curves as a function of enzyme loading. That we could simply take literature values for the enzyme kinetics as determined in solution and have them transfer so well to the immobilized membrane case is quite remarkable. The small increase in kl to match a 250-fold swing in enzyme loading implies that only minimal deactivation occurred in the production of these membranes. The question arises as to what is considered a good fit of simulation to experimental data. It is not possible to simply do a least-squares fit and project from the residuals which of the simulation parameters should be changed to get a better fit. For this work, the answer has been empirical in that if all relevant situations are matched without experiencing large errors in transient development time response or output, then we will be satisfied with the model. This seems reasonable since there are unquestionably certain assumptions in the model which are oversimplified, such as layer homogeneity, unidimensional diffusion, uniform convection in the contacting solution resulting in an average “effective” diffusion coefficient, no temperature or membrane stretching effects on DIKpairs, no time lags on solution exchanges, etc. Thus, the ability to predict all major trends as opposed to an exact fit for one transient is our criterion for “goodness of fit”. We have found many applications for this methodology in our work. This model serves us well in quality control of resistance membrane polymer permeability and subsequent calculation of required thickness prior to production. It has been used to follow enzyme decay over the shelf life of membranes. It proved valuable in defining the measurement time window in the software for the Direct 30130 and provided guidance in establishing the required permeability range for our implantable glucose sensor development. Finally, it was used in combination with enzyme decay measurements to estimate the possible lifetime theoretically obtainable for sensors in an in vivo environment. CONCLUSIONS A simulation strategy has been developed for modeling the glucose oxidase enzyme electrode. It provides for the calculation of sensor performance information, such as range of linearity, time response, output magnitudes, cofactor limitation regimes, and enzyme loading and decay effects. It
also allows one to establish different sets of contacting solution conditions (pure diffusion, RDE boundaries, convection, etc.) which can produce strikingly different performance patterns for the identical set of membrane parameters. Concentration profiles can be simulated under one set of solution boundary conditions and the decay then followed for a different set of conditions. This allows for visualization of sensor washout transients, which provides information related to sensor cycle time and answers questions about enzyme loading and reaction mechanisms. Any number of layers can be handled, providing single-layer parameters can be adequately measured. There is great flexibility in handling of input parameters, and in fact, if one wishes, these can be changed during the course of the simulation (Le., enzyme activity decay, growth of stagnant layer thicknesses, etc.). Finally, while the experimental verification of this simulation model was performed with the
glucoseoxidase system, this modeling strategy should be easily extendible to other enzyme electrode systems and has the potential to be used with minimal modification to describe the detection of neutral species by spectroscopic as well as electrochemical means.
ACKNOWLEDGMENT Markwell Medical Institute thanks the SBIR program within NIH for grant support (DK 4065743) pertaining to simulated and experimental implantable glucose sensors during the course of this work. Recehred for revlew October 1, 1993. Accepted February 8, 1994.' Abstract published in Advance ACS Absrracrs, March 15, 1994.
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