Dynamic Delay and Maximal Dynamic Error in Continuous Biosensors

In real-time operation of sensors, a maximal dynamic error based on the maximal ..... It must be emphasized that this analysis accounts for error due ...
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Articles Anal. Chem. 1996, 68, 1292-1297

Dynamic Delay and Maximal Dynamic Error in Continuous Biosensors Dale A. Baker and David A. Gough*

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

When biosensors are operated continuously, a dynamic delay and a dynamic error relate the sensor signal to the changing analyte concentration. The dynamic delay is the temporal displacement of the signal, or the lag, and is specified solely by properties of the biosensor and external mass transfer. The dynamic error is the difference between the actual concentration and the simultaneous reported concentration and is the product of the dynamic delay and the instantaneous rate of concentration change. In real-time operation of sensors, a maximal dynamic error based on the maximal expected rate of concentration change must be employed to estimate the worst-case error because the actual instantaneous rate is not independently known. Values of dynamic delay and maximal dynamic error that are acceptable in particular monitoring situations can be used in the design of acceptable continuous biosensors. This analysis suggests experimental alternatives to the standard response time approach for sensor characterization that are particularly advantageous for continuously operated biosensors. The concepts are applied here to in vitro operation of a continuous glucose sensor. From one perspective, all biosensors operate in one of two modes, either discrete or continuous. In the discrete mode, biosensors are exposed to the analyte on an intermittent basis. For example, a sample of the analyte is injected into a chamber containing the biosensor, and a reading is made. After this brief exposure, a rinse solution or calibration fluid is introduced to prepare the biosensor for the next cycle. The reported concentration reflects the analyte concentration at the moment of sample collection, and the ability to capture rapid changes in concentration is limited by the frequency of sample collection and the speed of sample processing. The discrete mode of operation is amenable to automation, and most biosensors operate in this mode. In contrast, biosensors operated in the continuous mode are usually positioned directly in the analyte medium and are intended to continuously report dynamic concentration changes. The properties of the continuous biosensor must be such that the sensor can follow the maximal anticipated concentration fluctuations * Address correspondence to this author at the Department of Bioengineering, University of California, San Diego, La Jolla, CA 92093-0412.

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within a specified acceptable error, but sensors need not respond faster. How should linear biosensors be designed so that the error between the actual and reported concentrations during continuous operation is always within tolerable limits? And, how are biosensor properties best characterized for continuous operation? These questions are addressed in this article. BACKGROUND As a practical guide, most investigators consider the temporal response of a continuous sensor to be acceptable if it is much faster than the maximal anticipated concentration change.1-4 Rarely, however, is any information given by the investigator about the maximal expected concentration dynamics of the intended monitoring situation to validate this premise. More often, the common strategy has been to simply make the response time as short as possible by designing small biosensors with the thinnest possible membranes. Biosensors that have sensing regions a few square micrometers in size and membranes several micrometers thick have been produced with modern fabrication technology.1-3 These small sensors, or “microelectrodes”, may have certain advantages in addition to rapid response, including being small enough to be positioned close to biological structures and negligible consumption of analyte.4 These advantages are important in some applications for recording of rapid concentration changes directly in tissues or living organisms. But it may not be advantageous or even possible to use this strategy in certain biosensor applications. For example, the enzyme electrode-based glucose sensor we have developed5 for long-term implantation in people with diabetes requires a minimum electrode area of the order of 104 µm2 and membrane thickness of the order of 102 µm. This relatively large electrode area is necessary to provide sufficient signal current for low noise amplification by state-of-the-art portable or implantable instru(1) Turner, A. P. T.; Karube, I.; Wilson, G. Biosensors: Fundamentals and Applications; Oxford University Press: Oxford, UK, 1987. (2) Hall, E. A. H. Biosensors; Prentice Hall: Englewood Cliffs, NJ, 1991. (3) Blum, L. J.; Coulet, P. R. Biosensor Principles and Applications; Marcel Dekker: New York, 1991. (4) Scheller, F.; Schubert, F. Biosensors; Elsevier: London, 1992. (5) Armour, J. C.; Lucisano, J. Y.; McKean, B. D.; Gough, D. A. Diabetes 1990, 39, 1519-1526. 0003-2700/96/0368-1292$12.00/0

© 1996 American Chemical Society

mentation,6 and the relatively thick membrane contains reserve immobilized enzyme for long-term operation.7 The ideal biosensor design may therefore require a compromise in which rapid response is balanced against long service life and practical instrumentation demands. In addition, when biosensors are implanted in tissues, there exist lags due to analyte diffusion through tissues.8 These diffusional lags may be comparable to or greater than the intrinsic dynamic lag of the sensor. For these reasons, a more general analysis of the dynamic response is needed. Response Time. A standard method of sensor characterization is to record the response to a concentration step challenge.1-4 After a concentration step-up, the sensor signal typically rises in an S-shaped curve and asymptotically approaches the new steady state value. This response is characteristic of well-damped, inherent second-order systems. A single parameter from the response curve is usually reported, such as T95, the 95% response time, defined as the time required for the amplitude of the signal to change from its initial steady state value to 95% of the final steady state value. The value of T95 is often imprecise because of the difficulty in estimating the attainment of steady state. There is also no direct, general relationship between T95 and the time to steady state; there may exist different S-shaped curves and respective times to steady state for sensors with the same T95 values. This method is often used because of experimental convenience for characterization of discrete sensors, but it provides little information for sensor design and is of minimal relevance to operation of continuous biosensors. Models of Biosensor Transients. Models for specific biosensors have been developed to describe the transient response to concentration steps (e.g., refs 1, 2, and 9). These models are typically cast as initial value problems in which a single step or ramp challenge is presented to the sensor. Such models can be useful for design and operation of discrete sensors but do not indicate how a sensor would respond during a continuous and ongoing challenge. Distributed models of this type may also require a substantial number of iterations to predict the dynamic response to arbitrary concentration challenges. Convolution. Another approach is to use convolutional methods to relate temporal and spatial differences in the signal to the original time-dependent concentration input.10,11 A set of inverse functions are sought that relate output signals to known input concentration challenges for specific sensor geometries. Simulations based on such convolutions have been compared to experimental observations.11 This approach is computationally intensive, does not provide information that is directly useful for sensor design, and is difficult to adapt to continuous sensor operation. Time Lag. An alternative to these approaches is the time lag method.12,13 The S-shaped response to a step challenge is integrated with time to produce an upward-sloping curve that (6) McKean, B. D.; Gough, D. A. IEEE Trans. Biomed. Eng. 1988, 35, 526532. (7) Tse, P. H. S.; Gough, D. A. Biotechnol. Bioeng. 1987, 29, 705-713. (8) Ertefai, S.; Gough, D. A. J. Biomed. Eng. 1989, 11, 362-368. (9) Lucisano, J. Y.; Gough, D. A. Anal. Chem. 1988, 60, 1272-1281. (10) Wightman, R. M.; Wipf, D. O. Electroanalytical Chemistry; Marcel Dekker: New York, 1989; Vol. 15, pp 267-353. (11) Engstgrom, R. C.; Wightman, R. M.; Kristensen, E. W. Anal. Chem. 1988, 60, 652-656. (12) Gough, D. A.; Leypoldt, J. K. AIChE J. 1980, 26, 1013-1019. (13) Siegel, R. A. J. Membr. Sci. 1986, 26, 251-262.

approaches a steady state after a sufficiently long time. The intercept of that line on the time axis is the time lag, L. As an example, the time lag for a simple membrane in a homogeneous medium is given by

L ) δm2/6Dm

(1)

where δm is membrane thickness and Dm is the analyte diffusion coefficient in the membrane.12 (The Appendix contains some time lag expressions relevant to biosensors, including the effects of external mass transfer,12 and substrate partitioning coupled with an irreversible first-order chemical reaction within the membrane.14,15) The time lag has been used in conjunction with a membrane-covered rotated disk electrode system for the determination of membrane properties.12 A comprehensive analysis of the time lag approach has been given for generalized membrane systems.15 The time lag approach provides a single parameter from the response to a step challenge and avoids the difficulties of determining the completion of the asymptotic transient. The time lag method in its various forms is a useful approach to interpretation of the sensor response to a concentration step and facilitates sensor design,16 but still it provides little insight about continuous sensor operation. Dynamic Delay and Dynamic Error. The dynamic delay has been used historically to describe the delay, lag, or temporal displacement between physical transients and the sensor responses to those transients.17 Adapted to chemical sensors,18 the dynamic delay is specified solely by properties of the biosensor and external mass transfer. The dynamic error,17 a related term, is the difference between the actual value of the variable at a given moment and the value simultaneously reported by the sensor. The dynamic error is the product of the dynamic delay and the instantaneous rate of analyte change. The dynamic delay and dynamic error have been adapted for in vitro characterization of biosensors exposed to concentration challenges in the form of linear ramps.18 Equivalence of Time Lag and Dynamic Delay. In their original conceptions, the dynamic delay and dynamic error were derived from lumped parameter models of the sensor, in which analyte concentrations within sensor membranes were assumed to be uniform and gradients were approximated as linear. Although the dynamic delay and dynamic error are useful for characterization of the response of continuous sensors, no detailed information is provided by these terms for sensor design or for relating the effects of design parameters to the signal. Nevertheless, it has recently been shown19 and confirmed20 that the dynamic delay is equivalent to the time lag. This is important because values of dynamic delay and dynamic error that are acceptable for particular monitoring situations can now be used to specify values of sensor design parameters as indicated by the respective time lag expressions. Equivalence of the complemen(14) Leypoldt. J. K.; Gough, D. A. J. Phys. Chem. 1980, 84, 1058-1059. (15) Siegel, R. A. J. Phys. Chem. 1991, 95, 2556-2565. (16) Gough, D. A.; Leypoldt, J. K.; Armour, J. C. Diabet. Care 1982, 5, 190198. (17) Draper, C. S. Instrument Engineering; McGraw-Hill: New York, 1952; Vol. 3. (18) Baker, D. A.; Gough, D. A. Biosens. Bioelectron. 1993, 8, 433-441. (19) Baker, D. A.; Gough, D. A. J. Phys. Chem. 1994, 98, 13432-13433. (20) Siegel, R. A. J. Phys. Chem. 1995, 99, 17294-17296.

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Table 1. Apparent Maximal Rate of Change (Rmax) for Selected Analytes in Blood Rmax analyte pyruvate urea oxygen carbon dioxide glucose lactate potassium a

mM

min-1

% min-1 a

sampling interval (min)

monitoring situation

ref

9 3 150 12 3 130 60

1 10-15 b 15-30 30 3 1

newborns dialysis respiration hyperventilation/apnea diabetes exercise exercise

21 22 23 24 25 26 27

0.01 0.15 0.20 0.22 0.25 1.50 2.50

Rates expressed as percent of mean normal.28

b

Determined by a continuous electrochemical sensor.

tary time lag and dynamic delay approaches is the bridge between sensor design and continuous operation. Maximal Expected Rate of Concentration Change. The actual rate of concentration change can be controlled in sensor characterization experiments, but is not independently known in most real-time monitoring situations. Therefore, a maximal dynamic error applicable in a given monitoring situation can be specified from the maximal anticipated rate of concentration change to which the sensor is likely to be exposed, as judged from studies of similar monitoring situations. (Many biological systems have characteristic maximal rates of analyte change.) The resulting maximal dynamic error serves as a limit or bound on the actual instantaneous dynamic errors that correspond to actual rates of concentration change during a particular monitoring session. There must be an estimate of the rate of maximal concentration change in order to design a sensor that is always sufficiently rapid. Examples from the literature of estimates of the maximal rate of concentration change, Rmax, of certain analytes in blood are given in Table 1.21-27 The table shows the apparent rate of concentration change in absolute and normalized units and the sampling interval used in each study. All entries are based on discrete sampling rather than continuous sensor recordings to avoid artifacts related to the dynamic response of the sensors (except for oxygen, where appropriate discrete data are not available). The rates are estimated from the maximal concentration change between successive samples. The normalized rates in Table 1 are the absolute rates divided by the mean normal analyte concentration28 for purposes of comparison. Table 1 shows that the apparent maximal rate of change for different analytes in blood can differ by as much as 2 orders of magnitude, with some analytes changing rapidly on a percentage (21) Prentice, A.; Vadgama, P.; Appleton, D. R.; Dunlop, W. Brit. J. Obstet. Gynaecol. 1989, 96, 861-866. (22) Smirthwaite, P. T.; Fisher, A. C.; Henderson, I. A.; McGhee, J.; Mokhtar, N.; Simpson, K. H.; Whitehead, A. J.; Gaylor, J. D. ASAIO J. 1993, 39, 342-347. (23) Conway, M.; Durbin, G. M.; Ingram, B. A.; et al. Pediatrics 1976, 57, 244250. (24) Neumark, J.; Bardeen, A.; Sulzer, E.; Kampine, J. P. J. Neurosurg. 1975, 43, 172-176. (25) Kobayashi, M.; Shigeta, Y.; Hirata, Y.; Omori, Y.; Sakamoto, N.; et al. Diabet. Care 1988, 11, 495-499. (26) Schwaberger, G.; Pessenhofer, H.; Schmid, P. In Cardiovascular System Dynamics: Models and Measurements; Kenner, T., Busse, R., HingheferSzalkay, H., Eds.; Plenum: New York, 1982; pp 561-567. (27) Linton, R. A. F.; Lim, M.; Wolff, C. B.; Wilmshurst, P.; Band, D. M. Clin. Sci. 1984, 67, 427-431. (28) Braunwald, E., Isselbacher, K. J., Petersdorf, R. G., Wilson, J. D., et al., Eds. Harrison’s Principles of Internal Medicine, 11th ed.; McGraw-Hill: New York, 1987.

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Figure 1. Biosensor response to a concentration ramp challenge. Normalized signal current is plotted as a function of time. The input concentration ramp and sensor response are shown. The dynamic delay and dynamic error are indicated.

basis (e.g., oxygen, lactate) and others relatively slowly (e.g., glucose, pyruvate, urea, and carbon dioxide). The data indicate that acceptable biosensors for these analytes can have quite different dynamic responses and need not all be uniformly fast. Once an estimate of Rmax is established for a given analyte, sensor design can proceed systematically. EXPERIMENTAL SECTION Sensors. The glucose and oxygen sensors used here are potentiostatic amperometric electrodes for glucose described in detail elsewhere.29 Glucose sensors were operated in a linear range of physiologic oxygen concentration, and the response was oxygen-independent. Sensors were modeled with previously described methods.9,16 Sensor Characterization. An automated testing apparatus and method to characterize the in vitro response of the glucose sensor to step and ramp changes in concentration have been described.18 The sensors were connected to individual potentiostats, similar to those described previously.6 The analog signals from the sensors were multiplexed and digitized with a 12-bit analog-to-digital converter and then routed to the computer for analysis and storage. RESULTS AND DISCUSSION Concentration Ramps, Dynamic Delay, and Dynamic Error. The calculated response of a biosensor to a concentration (29) Lucisano, J. Y.; Armour, J. C.; Gough, D. A. Anal. Chem. 1987, 59, 736739.

ramp is shown in Figure 1. The normalized current in response to a concentration ramp challenge is plotted as a function of time. The normalized current is (i - ii)/(if - ii), where i is the transient current, ii is the initial steady state current, and if is the final steady state current. The abscissa is labeled on the bottom as time in seconds and on the top as dimensionless time τ, given by τ ) t/TS, where TS, the time to steady state, is equal to δm2/Dm. For illustration, a value of 500 s was chosen for δm2/Dm. External transport effects are considered negligible. As the ramp input rises linearly from the origin, the biosensor response initially lags for ∼0.2τ and then becomes parallel to the ramp at steady state, thereafter rising at the ramp rate. The dynamic delay defined in the figure is the steady state lag between the ramp input and the parallel response. The dynamic error is the difference between the input concentration and the response after the initial transient. The dynamic delay and dynamic error are proportional to the sensor properties. Specifically, the dynamic delay, δD, given in units of time is

δD ) Kδm2/Dm ) L

Figure 2. Effect of biosensor properties. The predicted responses of three biosensors (broken lines) to a ramp challenge (solid line) at a fixed ramp rate. The biosensors have different values of δm2/Dm, as indicated.

(2)

where K is a unitless proportionality constant equal to 1/6 in the most simple case12 but having other values depending on the sensor design and operating conditions, and L is the time lag. The dynamic error, D, given in units of concentration is

D ) RδD ) RL

(3)

where R is the absolute value of the rate of concentration change, given in units of concentration per unit time. If the ramp rate is normalized by a suitable reference concentration, as in the third column of Table 1, and expressed in units of % time-1, and the signal is normalized by the signal at the reference concentration, the dynamic error can be expressed as a percentage of the reference. In the case shown, the dynamic delay is ∼0.166τ or 83 s, and the dynamic error is 0.166, or 16.6% of the full concentration excursion used here. This approach to biosensor characterization provides new information. For example, the dynamic delay is a sensor time constant that indicates the extent to which the response is simply shifted in time from the actual instantaneous concentration value. This delay can be made small by using membranes with appropriate values for Dm and δm. The dynamic error is proportional to the product of the dynamic delay and the imposed ramp rate and is a measure of the instantaneous error due to the system dynamics. Although the actual instantaneous ramp rate in most monitoring situations is not independently known, the apparent maximum rate of concentration change, Rmax, for specific biological systems can be used to estimate the maximal dynamic error. Alternatively, in sensor testing situations where variable forcing functions are used and the instantaneous ramp rate is specified, a repetitive calculation of the instantaneous variable dynamic error is possible. The responses of amperometric and potentiometric sensors are analogous (not shown). The response of the amperometric sensor will be used in subsequent examples. Effect of δm2/Dm. Figure 2 shows the calculated normalized signals of three simple sensors with different values of δm2/Dm to a ramp challenge of specified rate. In practice, similar results

Figure 3. Effect of ramp rate on biosensor response. The predicted responses of a biosensor (broken lines) to concentration ramps (solid lines) at three ramp rates are indicated.

can be obtained with a series of sensors having increasing membrane thickness, as the value of Dm is characteristic of membrane material and more difficult to vary. The effects related to internal reaction and external mass transfer are not included in this simulation. For each sensor, the predicted response first rises slowly and then eventually becomes parallel to the ramp at a respective displacement. These results show that both the dynamic delay and the dynamic error can be reduced by decreasing the value of δm2/Dm. Effect of Ramp Rate. The calculated effect of ramp rate on the sensor response is shown in Figure 3. The normalized signal is plotted as a function of time and dimensionless time for one sensor challenged at different normalized ramp rates. The challenges, shown in solid lines, are ramps of rate R equal to 0.004, 0.002, and 0.001 s-1, respectively. The value of δm2/Dm is fixed at 500 s. The sensor response to each challenge is given by the corresponding broken line. As before, each response initially lags for ∼0.2τ (or 100 s) and then rises in parallel to the ramp challenge. The important feature is that, in each case, the dynamic delay is constant, being a property of the sensor. In contrast, the dynamic error increases with ramp rate, as given by eq 3. Quantitatively, the dynamic error for R ) 0.002 s-1 is 0.166, whereas the dynamic error for R ) 0.001 s-1 is half that value at 0.083. These results demonstrate that ramp rate has a strong effect on the dynamic error but no effect on dynamic delay. Applications. There are several ways to use these relationships to interpret continuous signals. First, in the case of sensor Analytical Chemistry, Vol. 68, No. 8, April 15, 1996

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testing, where the concentration forcing function is under control of the investigator, as simulated in Figures 1-3, the dynamic delay or time lag can be determined experimentally or calculated from known sensor properties and operating conditions, such as membrane thickness, diffusivity, reaction rate constant, external mass transfer resistance, etc. In a second mode of application, a knowledge of the maximal ramp rate, Rmax, of the biological system of interest and of the dynamic delay of the sensor allows a maximal value of the dynamic error to be calculated. These values of dynamic delay and error can then be compared with the maximally acceptable dynamic delay and error requirements of the data for the intended application, and adjustments in the sensor design can be made if necessary. It should be emphasized that, although the present analysis describes the response to a sustained ramp, the use of dynamic delay and dynamic error alone is not sufficient to reconstruct the detailed sensor output of a monitoring episode. This is because, in an actual monitoring situation, both the rate and the direction of concentration change can vary spontaneously. Additional information about the instantaneous rate and direction of concentration change would have to be available from independent measurements of the analyte concentration obtained in parallel to the biosensor response. This type of information is not typically available in monitoring situations for obvious reasons, and one must rely on the maximal dynamic error as an error bound. If the dynamic delay, the dynamic error based on instantaneous rates, and the instantaneous direction of analyte change are known in research studies, simple algorithms can be constructed for approximate reconstruction of sensor signals. Examples of multidirectional response are shown experimentally in Figure 4. Glucose sensors having time lags of 4 min in the upper panel and 6 min in the lower panel are exposed to rising and falling concentration ramps of the indicated rates. The solid circles connected by the solid lines represent glucose concentration, determined by discrete assay to within 0.1 mM. In each figure, the rising and falling rates of a given challenge are the same, and the ramp rate on the left is greater (0.27 mM/min) than that on the right (0.15 mM/min). The broken lines are the experimentally observed sensor responses. The dynamic error corresponding to each ramp rate and sensor is indicated. Other sensors (not shown) gave similar results. The ramp rate on the left in each panel is comparable to the maximal rate of blood glucose change in diabetes (0.25 mM/min) given in Table 1, and the dynamic error is therefore a maximum error bound for the assay as determined by the respective sensor. If an acceptable maximal error bound for the assay is known, a specific sensor design and dimension can be recommended. This example shows the utility of this analytical approach. These results also demonstrate transient effects. The sensor responses to the the more rapid challenges on the left of both panels of Figure 4 first rise with an increasing delay, followed by a period of rise parallel to the input from which the dynamic error is calculated. At one point, the input ramp direction is abruptly reversed, but the sensor signal continues upward to a delayed maximum before turning downward. Similar responses are observed to the slower ramps on the right of each figure. After the initial transient, the sensor signals parallel the input, and smaller values of dynamic error corresponding to the lower ramp rates are shown. When the ramp directions are reversed, the respective sensor signals pass through delayed maxima and then 1296 Analytical Chemistry, Vol. 68, No. 8, April 15, 1996

Figure 4. Response of two glucose sensors to concentration ramp challenges of various rates in both directions plotted as a function of time. Glucose concentration, determined by conventional assay to within 0.1 mM, is given by the solid circles connected by the solid line. The sensor signals are shown in the broken lines.

become parallel to the downward ramps. This process is repeated when the ramp directions are again reversed. This shows the use of the maximal dynamic error as an error bound. When the direction of the ramp abruptly changes, the sensor signal does not reverse directions immediately because of the lag but breaks from the parallel and continues to rise through the delayed maximum before turning downward. During this time, the difference between the sensor signal and the actual concentration is always less than the dynamic error and even the zero at the crossover point. Therefore, although the sensor cannot be expected to reproduce the detailed characteristics of the challenge, the difference between the actual and reported concentrations is always bounded by the maximal dynamic error corresponding to the maximal rate of concentration change. It must be emphasized that this analysis accounts for error due to the dynamics of the system but does not include constant or systematic errors, such as those related to sensor drift, offset, signal resolution, etc. Sensor Operation in Heterogeneous Media. Continuous biosensors may be required to operate in heterogeneous media, such as tissues or poorly stirred fluids, where external mass transfer resistances exist between the sensor and the well-mixed analyte medium. Under these conditions, the dynamic delay is equivalent to the time lag given by eq A1, and the ratio of mass transfer in the surrounding medium to mass transfer within the membrane is given by eq A2. When external mass transfer is comparable to or smaller than internal mass transfer (i.e., Bi e 1), features of the external medium, such as diffusional length δ or concentration boundary layer thickness and diffusivity, need

to be taken into account. Under these conditions, the time lag will be larger by as much as a factor of 3 for external mass transfer limitation, with a proportional increase in dynamic error. There will also be a reduction in the steady state signal. The extent of this effect remains to be determined in many applications, such as biosensors implanted in tissues, but the resulting increase in assay error may be detrimental. A simple reduction in membrane thickness would reduce the time lag and dynamic error but increase the dependence on external mass transfer. CONCLUSIONS The dynamic delay, dynamic error, and time lag can be combined to characterize the response of continuously operated biosensors. Together with the maximal dynamic error for specific monitoring applications, these terms provide a useful approach for the design of continuous biosensors and interpretation of the response. ACKNOWLEDGMENT This work was supported by grants from the Juvenile Diabetes Foundation and the National Institutes of Health. GLOSSARY Bi

dimensionless mass transfer parameter, eq A2

D

analyte external diffusion coefficient

Dm

analyte diffusion coefficient in membrane

i

current

K

constant

k

reaction rate constant

L

time lag, eqs 2, A1, and A2

R

instantaneous ramp rate

Rmax

maximal ramp rate for a given analyte and monitoring situation

TS

time to steady state

t

time

R

partition coefficient

δ

external diffusional length

δD

dynamic delay, eq 2

δm

membrane thickness

D

dynamic error, eq 3

φ

dimensionless reaction rate

τ

dimensionless time

ξ

equilibrium coefficient

APPENDIX: SOME TIME LAG EXPRESSIONS RELEVANT TO BIOSENSORS12,13,15 When external mass transfer is significant, the time lag expression12 is

L)

(

)

δm2 2 36Dm 1 + Bi-1

(A1)

where Bi is the dimensionless ratio of external to internal mass transfer, given by

Bi ) Dδm/RDmδ

(A2)

D is the analyte diffusion coefficient in the external medium, R the equilibrium partition coefficient in the membrane, and δ the external mass transfer boundary layer thickness or distance to a homogeneous analyte distribution. The value of Bi increases with external mass transfer, and, in the limit of no external resistance, the time lag expression of eq A1 reduces to eq 1. In the presence of a first-order irreversible reaction within the biosensor membrane, the time lag expression14 is

L)

(

)

δm2(1 + ξ) coth φ 1 - 2 2Dm φ φ

(A3)

where ξ is the equilibrium coefficient for the local free and bound analyte within the membrane and φ ) kδm2/Dm, in which k is a first-order reaction rate constant. When the reaction rate is zero and binding of the analyte within the membrane is negligible, the time lag of eq A3 reduces to 1. It has also been shown that the time lag is increased by finite electrode kinetics and reversible reaction within the membrane.14,15 Received for review January 15, 1996. Accepted January 22, 1996.X AC960030D X

Abstract published in Advance ACS Abstracts, March 1, 1996.

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