Theoretical and experimental studies of glucose sensitive membranes

267

Journal of Controlled Release, 6 (1987) 267-291 Elsevier Science Publishers B.V., Amsterdam - Printed

THEORETICAL MEMBRANES*

AND EXPERIMENTAL

G.W. Albin, T.A. Horbett”“,

in The Netherlands

STUDIES

OF GLUCOSE

SENSITIVE

S.R. Miller and N.L. Ricker

Department of Chemical Engineering BF- 70, Universiry of Washington,

Seattle,

Washington,

98 195 (U.S.A.)

A mathematical model has been developed to describe the steady-state behavior of two types of glucose sensitive membranes. Both membranes are synthetic hydrogels containing immobilized glucose oxidase enzyme (GluOx). The formation of gluconic acid from glucose and oxygen, catalyzed by GluOx, is the key to the functioning of either type of membrane since it causes a pH decrease within the membrane. In amine group-containing membranes, thepH decrease enhances the swelling (and permeability) of the gels, allowing control of insulin delivery in response to glucose concentrations. In membranes containing immobilizedpH indicator dyes thepH decrease causes a color change, thus providing the basis for a glucose sensor. The model’s predictions of the response of amine-containing membranes to glucose show that the pH decrease is often limited by 0, depletion and that the occurrence of 0, depletion is strongly influenced by enzyme loading and membrane thickness. At a given thickness, an optimal enzyme loading exists which results in the maximum response to glucose over a given range of glucose concentrations. The influence of buffer and amine concentrations, amine pK, solute diffusivities, and flowrate past the membrane also have been examined. Forpolyacrylamide membranes containingphenol red (but no amines), substrate turnover rates, oxygen depletion, and thepH decrease within the membrane have been calculated using the model and found to agree qualitatively with experimentally determined values of these parameters. Quantitative agreement is lacking, however. In particular, the model predicts that as GluOx loading is increased (all other parameters, including glucose concentration, held constant) the pH decrease should asymptotically approach a maximum value. However, it is found experimentally that as GluOx loading is increased, the measured pH decrease increases progressively over the full range of GluOx concentrations that we have studied. Possible reasons for the quantitative disagreements between simulated and observed results will be discussed. Implications of the model with respect to optimizing membrane designs will be presented. One finding of part icular interest is that a glucose sensor using immobilized GluOx will achieve a maximum response at sub-physiological concentrations of glucose, and not respond to higher glucose concentrations, unless the enzyme loading is made sufficiently low.

INTRODUCTION *Paper presented at the Third International Symposium on Recent Advances in Drug Delivery Systems, February 24-27,1987, Salt Lake City, UT, U.S.A. **To whom correspondence should be addressed.

0168-3659/87/$03.50

0 1987 Elsevier Science Publishers

Insulin-dependent diabetes is a major health problem worldwide largely because of the complications associated with the disease, such as retinopathy, neuropathy, and vascular disease.

B.V.

These complications may result from the poor control of blood glucose level that is provided by conventional insulin injection therapy [ 11. An “artificial pancreas” providing continuous feedback control of blood glucose via glucosedependent insulin release would therefore have many advantages. Also, a device that would reliably measure glucose continuously in uiuo would be useful for monitoring any type of intensive insulin therapy. In this paper we present mathematical analyses of two types of glucose-sensitive membranes, one of which could function as a barrier membrane in an implantable “artificial pancreas” to release insulin at rates dependent upon the concentration of glucose, and another intended for use as part of a glucose sensor. Membranes of the first type have been under study in our labs for some time [ 2-61 whereas the use of glucose-sensitive membranes as sensors is described here for the first time. As applied to the “artificial pancreas”, a glucose-sensitive membrane could control insulin delivery from a reservoir containing a saturated insulin solution. Membranes of this type are hydrogels containing pendant amine groups and entrapped glucose oxidase. Glucose diffusion into the gel and its conversion to an acid causes a pH drop and consequent increased protonation of the pendant amine groups. The electrostatic repulsion between protonated amine groups causes an increase in the degree of swelling and the permeability of the hydrogel to insulin. Thus, the membrane’s permeability to insulin is a function of glucose concentration external to the membrane, and insulin delivery is accelerated by an increase in glucose level. Previously published work demonstrates that membranes of this type do become more permeable to insulin in the presence of glucose [ 61. To serve as glucose sensors, membranes consisting of a polyacrylamide gel containing both entrapped glucose oxidase enzyme ( GluOx) and covalently-immobilized phenol red indicator dye have been developed. The incorporation of phenol red into polyacrylamide utilized a pre-

viously described protocol, modified to allow GluOx incorporation [ 71. As glucose diffuses into the gel, glucose oxidase catalyzes its conversion to gluconic acid, thereby lowering the pH within the gel microenvironment. The dye’s color is a function of pH and therefore pH (and in turn, glucose concentration) can be measured using a spectrophotometer. To facilitate their optimization, a mathematical model to describe the steady-state behavior of both types of glucose-sensitive membranes has been developed. Predicted profiles of substrate concentrations, product concentration, and pH within the membranes were calculated at various values of the external substrate concentrations, enzyme concentration, membrane thickness, boundary layer thickness, flowrate, and buffer concentrations. The predicted performance of membranes containing GluOx and amine groups that may be useful in an insulin release device has been analyzed under a series of simulated operating conditions. For membranes containing GluOx and phenol red dye that may be useful for glucose assay, the predicted results are compared with experimental data. The implications of the model in the further design of both types of membranes are discussed.

MATERIALS Preparation

AND METHODS of membranes

Polyacrylamide gel membranes incorporating glucose oxidase enzyme and phenol red pH indicator dye were prepared by polymerizing the monomer solutions listed in Table 1. The components of each monomer solution, except for the initiator ammonium persulfate and the accelerator TEMED, were mixed at room temperature. The initiator and accelerator then were added and the solution immediately was cast between glass plates separated by steel shims. Polymerization was allowed to proceed at room temperature for at least 24 h, after

269

TABLE 1 Formulations of pAAm/GluOx/phenol Compositions” of monomer solutions membranes were polymerized

red membranes: from which the gel

0.1777 g/cm” acrylamide monomer 0.00771 g/cm” N,N’ -methylene-bis-acrylamide crosslinking monomer 12.5 mg/cm” phenol red indicator dye 12.5 mg/cm” ammonium persulfate initiator 0.25 cm3/cm3 N,N,N’,N’-tetramethylethylenediamine accelerator 0.025 to 100 mg/cm” glucose oxidase enzymeb “Concentrations are given per cm” of CBS buffer in the monomer solution. bSigma Type X.

which the membrane was removed from the glass and allowed to swell in buffer. This means of covalently incorporating phenol red into polyacrylamide gels has been described previously [7]. Buffer

Glucose solutions were prepared in “CBS” buffer (0.0195 M citrate, 0.0972 M chloride, and 0.154 M sodium) which previously had been saturated with air (by sparging) and adjusted to pH 7.40 using NaOH. This buffer was chosen because it has a titration curve very similar to that of human blood over the pH range of 7.0 to 7.4, and also because it has the same ionic strength (0.154 M) at pH 7.4 as human blood. The membranes were allowed to reach their equilibrium swelling in CBS buffer before being used experimentally. Measurement of membrane tension using electrodes

pH and oxygen

A Plexiglas flow cell (see Fig. 1) was constructed so that solutions of glucose could be pumped past a membrane fixed in place. The cell has ports through which both a Clark O2 electrode and a pH electrode were inserted to measure pH and 0, tension at the side of the

FLOW

Ti

CROSS

SECTION

Fig. 1. Flow cell for measurement of membrane pH and 0, gradients using electrodes. A, 0, port seal; B, membrane; C, Dacron support mesh; D, Teflon gasket; E, stir bar; F, stirring rotor; G, stirring motor; H, pH port seal.

membrane opposite to that of the flow. When only one electrode was in use, the remaining port was closed with latex rubber to prevent leakage of air to the membrane from the outside. Latex tubing was also used to form an airtight sleeve around the electrode. To minimize stray electrical signals it was necessary to connect the flow cell, buffer reservoir, and electrode to a common ground and to make liberal use of antistatic spray in the vicinity of the cell. A stirring bar was used to minimize boundary layer thickness near the membrane surface in this flow cell. Experiments with the flow cell consisted of measuring pH and/or 0, tension while glucose in air-saturated buffer was pumped through the cell. Measurements were recorded continuously to ensure that steady-state data was collected at a given glucose concentration before any change was made in the glucose concentration. All experiments were carried out at room temperature, - 22’ C. To measure O2 tension in the buffer exiting the flow cell, an addition to the main flow cell was used, as shown in Fig. 1. This consisted of another Plexiglas cell with a port through which an electrode could be inserted into the flow stream. Since the buffer entering the first flow cell was known to be saturated with air, this measurement of 0, concentration in the effluent allowed calculation of the substrate depletion

in the cell and the overall reaction rate in the membrane. A Microelectrodes Inc. MI-410 micro-combination pH electrode connected to a Radiometer PHM 62 pH meter with a Radiometer REC 80 Servograph chart recorder were used to measure pH at one side of the membrane (see Fig. 1) . The system was calibrated using commercially-available buffers. Oxygen was measured using an Instech Laboratories, Inc. model 125/05 combination 0, electrode connected to a Diamond Electrotech Inc. model 1201 Chemical Microsensor and recorded by connecting the output from the Microsensor to a chart recorder. To calibrate the 0, measurement system, airsaturated buffer was pumped through the cell and the gain on the meter was adjusted to show 21% (the mole fraction of O2 in air). Spectrophotometric brane pH

measurement

of mem-

Since phenol red indicator dye was incorporated covalently into the membranes, measurement of pH in these gels was accomplished by recording their light transmission and then converting the recorded data to pH with an appropriate calibration curve. A special flow cell similar to the one described in the previous section was constructed for use in the spectrophotometer (see Fig. 2). The cell was constructed from Plexiglas, chosen for its transparency to visible radiation. It has a small volume on the flow-through side of the membrane and an expansion of the channel upstream from the membrane. These design aspects promote rapid wash-out (about 3 min) and good mixing. Experimental measurements were done at 22 “C. Only steady-state values are reported. Light transmission was measured using a Perkin-Elmer Lambda Array 3840 spectrophotometer connected to a Perkin-Elmer 7300 computer. A 0.318 cm-thick piece of Plexiglas was used as reference background. All other measured light transmission were divided by this reference transmission. Changes in trans-

AXIAL

VIEW

Fig. 2. Flow cell for measurement of membrane pH using a spectrophotometer. A, entry port; B, aluminium face plate; C, rubber gasket; D, Plexiglas entry window; E, membrane (cross hatching) or Teflon gasket (non cross hatched area shown in axial view only) ; F, flow channel; G, Plexiglas exit window.

mission due to changes in pH were measured simultaneously at each of five wavelengths: 510, 530, 550, 560, and 570 nm. To compensate for changes in transmission resulting from factors other than pH, transmission at 700 nm was also measured. The ratio of (% transmission at the given wavelength) to (% transmission at 700 nm) was used to indicate pH changes. Since the color change of the indicator dye does not affect transmission at 700 nm, fluctuations in transmission at this “reference” wavelength are the same as the non pH-related fluctuations in transmission at the other wavelengths. Calibration of the system was performed by recording transmission spectra at the known pHs of 7.4, 7.1, 6.95, and 6.8. Measurement

of turnover

rate

The substrate turnover rates caused by GluOx immobilized in membranes were measured at

271

pH 7.40 and room temperature by continuously titrating the reaction product, gluconic acid, with NaOH. These measurements were made with the membrane in buffer. Both sides of the membrane were exposed to the buffer, which was stirred continuously. The buffer contained glucose and was sparged with air throughout the experiment to maintain oxygen saturation. A Radiometer automatic pH stat/titration apparatus (PHM62 pH meter, TT80 titrator, ABU80 autoburette, REC80 Servograph chart recorder, and G2040C/K4040 electrode combination) was used in these experiments. Measurement

of permeability

to glucose

The permeability of glucose-sensitive membranes to glucose was measured using a membrane sample clamped between two stirred compartments in the transport cell described previously [ 61. At time t= 0 one compartment was filled with a solution containing tritiumlabeled glucose plus 100 mg unlabeled glucose in CBS buffer and the other compartment was filled with CBS buffer alone. Both compartments were sampled over time and the samples were assayed using a liquid scintillation counter. Permeability was calculated from the rate at which concentration changes occur across the membrane using previously developed equations [ 61. The solution added to each compartment was degassed under vacuum immediately prior to the experiment and the transport cell was kept under N, blanket during the experiment. These precautions were taken to exclude 0, from the system and thereby to suppress the turnover of glucose by GluOx. The glucose in these experiments was in great molar excess over the GluOx so that the initially oxidized enzyme is immediately converted to the inactive reduced form, preventing further turnover of the glucose. Membrane

titration

Titration curves of glucose-sensitive membranes were measured using the Radiometer

autotitrator apparatus. The gel samples were cut into small pieces for these experiments in order to reduce diffusion resistance and the titrant, NaOH, was added slowly. Both precautions were taken so that equilibrium would be maintained during titration between the interiors of the gel pieces and the exterior solution. Determining membranes

the volume fraction

of buffer in

The volume fraction of buffer in a gel was calculated from the weight fraction of buffer in the gel, the density of the swollen gel, and the density of the buffer. The weight fraction of buffer was calculated from the wet and dry weights of the gel sample. The wet weight was obtained by swelling the sample in CBS buffer at pH 7.4 and weighing it after briefly blotting excess liquid from the outer surface. The dry weight was obtained by re-swelling the same gel sample in deionized Hz0 to leach out buffer salts, ovendrying the sample, and weighing it. The density of the swollen gel was calculated from the weight and volume of a sample disk. The volume was calculated from the thickness and diameter of the disk, and the thickness was determined by using a micrometer to measure the thickness of a “sandwich” consisting of the sample between two glass cover slips and then subtracting the thickness of the cover slips.

MATHEMATICAL

MODEL

In this section the simplifying assumptions, the development of the differential equations which describe our system, and the means of solving these equations will be described and explained. The equations describe the diffusion of glucose and oxygen into the membrane from its exterior, their conversion to gluconic acid, and the resultant pH decrease. Any fluxes and gradients within the membrane are considered to exist only in the direction normal to the membrane surface, so the differential equa-

272

tions need be written for only one dimension (i.e., transport through the “edges” of the membrane is neglected). Boundary conditions are specified at the membrane surfaces and are different for the two surfaces, so that concentration and pH profiles inside the membrane are asymmetrical. The model has been used primarily to predict internal membrane pH as a function of a number of operating parameters. 1. Simplifying

(8)

assumptions

(1) Only steady-state operation is considered. (2) For those simulations in which the mem-

(3)

(4)

(5)

(6)

(7)

brane is considered to contain immobilized catalase, an excess of catalase is assumed. Catalase reduces hydrogen peroxide ( H,O,) , which is a byproduct of the glucose oxidase reaction, to 0, and H20. Enzyme turnover rates are considered to depend only on local substrate concentrations, not on pH, and enzyme kinetic coefficients are assumed to equal the intrinsic kinetic coefficients of non-immobilized enzyme. Passive diffusion (i.e., Fick’s Law diffusion) only is considered. Therefore, ion convection are and not exchange considered. In most calculations the diffusivity of each substrate (glucose and oxygen) inside the membrane is assumed to be equal everywhere to its diffusivity in bulk solution. Therefore, diffusivities are considered to be unaffected by swelling or pH gradients. Also, the diffusivity of gluconic acid is assumed to equal the diffusivity of glucose. The acid/base equilibrium of each buffer, including the amine groups on the polymer, is assumed to be described by the Henderson-Hasselbach equation with a single distinct PK. All acid/base reactions are assumed to be at local equilibrium. This assumption is justified because these reactions are much

faster than the glucose oxidase-catalyzed reaction. Resistance to transport between the bulk phase and the membrane is described by mass transfer coefficients which are given by k= Db”lk/6, where Dbulk is the bulk diffusivity and 6 is the boundary layer thickness, Thus, it is assumed that there is a linear concentration gradient between the bulk solution and the membrane surface over some distance S. The boundary layer thickness is assumed to be the same for all solutes and the same over the entire membrane area.

2. Reaction equations

stoichiometry;

differential

The stoichiometry of the reaction catalyzed by glucose oxidase is: glucose + O2 = gluconic acid + HzO,. Catalase catalyzes the reaction: H202 = 0.5 0, + H20. If an excess of catalase is immobilized with glucose oxidase, so that all H202 is reduced, the overall stoichiometry becomes: glucose + 0.5 0, = gluconic acid + HzO. Therefore, only one-half of an oxygen molecule is consumed per molecule of glucose when an excess of catalase is present. The equations describing the simultaneous diffusion and reaction of the substrates are statements of Fick’s Law (written for dilute solution) and the equation of continuity, as follows:

=J,

-D,(dC,/&)

(1)

- Q,, ( dG, lh ) = Jm

(2)

(dJ.&)

= -u

(3)

-u

(da)

(dJ,,/dx)= or, (dJ,,,/dx) where

= - 0.5~

(4b)

D,, D,, = diffusivities brane

within the memof glucose and oxygen,

273

respectively, cm2/s; C,, C,, = concentrations of glucose and oxygen, mol/cm3. J,,J,,,= fluxes of glucose and oxygen, mol/cm2 s; x = length parameter, cm; u = reaction velocity, mol/cm3 s. Either eqn. (4a) or (4b) may be appropriate, depending on whether or not catalase is included in the membrane formulation. The boundary conditions are as follows: atx=L,

Jg=Jox=O

at x=0,

J,=

(5)

-D,(dC,/dx)

=k,(C,b”‘k-Cg)

(6)

Jox= - Rx ( G, /dx 1 = k,, (C”/“‘k

where

(7)

-Co,)

Cgbulk,C,, b”1k= bulk solution concentrations of glucose and oxygen, respectively, mol/cm3; lz,, Iz,, = mass transfer coefficients for the diffusion of glucose and oxygen from bulk solution to membrane surface, cm/s; L = membrane thickness, cm. Here, the side of the membrane at which substrates enter (the “body side”) is labeled x = 0 and the opposite side (the “insulin reservoir side”) is labeled x= L. The boundary layer thickness determines the values of k, and k,, in eqns. 6 and 7 according to simplifying assumption number 8. If it is reasonable to assume that substrate concentrations in the bulk solution external to the membrane are fixed and are not affected by the presence of the membrane, then C bulk and coxbulk are constants. This assumptiin is used when simulating the performance of the amine-containing membranes which are discussed in the section on “Model Predictions”. However, during experimental measurements on glucose-sensitive membranes containing phenol red there was significant depletion of substrates in the bulk phase. For

this situation the predicted bulk concentrations can be calculated by mass balance: C gb”‘k=Cgin-(A/Fo)JgI,=O c

bulk ox

=

c

in ox

-

(A/F,)

(8)

Jox 1x=0

(9)

where

F,, = flowrate, cm3/s; A = area, cm2; Cc, Cop = concentrations of glucose and oxygen in the buffer entering the flow cell, mol/cm3. Equations 8 and 9 are used to calculate Cgb”lk and Coxb”lkwhen simulating the performance of the experimental,dye-containing membranes which are discussed in the “Experimental Results” section. The expression for reaction velocity is due to Nakamura and Ogura [ 81: C enzymeIv = Cenzyme/%ax +WC,+WC,,

(10)

where

u = reaction velocity, mol/cm3 s, V mar = maximum reaction velocity, mol/cm3 s, C enzyme= enzyme concentration, mol/cm3, k, and k2 are kinetic coefficients, mols/cm”. In the above dimensional units, u,,_ is proportional to the enzyme concentration. In this convention we follow the kinetic analysis of glucose oxidase given by Nakamura and Ogura (8)) who reported u,,, in mol/l-s. Since the reaction product gluconic acid is produced at the same molar rate as glucose is consumed, then the flux of gluconic acid is predicted always to be equal in magnitude but opposite in direction to that of glucose. It follows by mass balance that the gluconic acid concentration must everywhere be equal to Cacid~CacidIx=O+CgIx=O~Cg

(11)

where Cacid)x=-, may be obtained from a boundary condition similar to equations 6 and 7:

atx=O,

derson-Hasselbach equation with a mass balance on the buffer, as follows:

Jacid=-Jg= kacid(Cacid

=

Ix=0

kg( Cacid

(12)

-Cacidb”‘k)

I x ~0

-

Cacid

b”‘k

lx=0

(13)

It is assumed in all calculations that the concentration of gluconic acid in the bulk phase is a function of other operating parameters, notably the ratio of area to flowrate. 3. Computational

[base form] /[acid form]

)

and, if necessary, the bulk acid concentration may be determined by a mass balance similar toeqns. (8) and (9): Cacidb”lk=(A/Fo)Jg

pK, =pH - “log

methods

The system of ordinary first-order differential eqns. (l-4) with boundary conditions (5-7) is solved using the LSODE packaged program which uses the Gear method of numerical integration. A “shooting method” is used to satisfy the boundary values at both sides of the integration interval. In a shooting method, values of the dependent variables at x= 0 are adjusted based on how close the values at x = L are to their desired values, and this is done iteratively until the solution converges (i.e., all boundary values are satisfied). The converged solution contains values of the glucose concentration C, at values of X. These are converted to values of the gluconic acid concentration Cacidusing eqn.

(14)

[base form] + [acid form] = [buffer]

(15)

so that [H+

in buffer] = [acid form]

= [buffer](lO-PH)/(lO-pH+lO~PK)

(16)

where [buffer] designates the total concentration of buffer in both forms. The gluconic acid needn’t be considered a buffer in this system since its pK is too low. The concentration of unassociated protons is given simply by [free H+ ] = 10-pH

(17)

The total concentration of exchangeable protons at pH 7.40 (note that pH 7.4 corresponds to zero gluconic acid) can be calculated using the combination of eqns. (16) and (17) and will be designated here as [Hf]pH,7.4. Next, the total concentration of exchangeable protons in the system at a given glucocic acid concentration is determined simply by adding the gluconic acid concentration to [ H+ ] $= 7.4and will be designated as [ Ht 1. These operations can be summarized as follows: [H+] = [H+]pH=7.4+ [gluconicacid]

(11-13) *

The pH is computed from the acid concentration by the following procedure. First, “exchangeable protons” are defined as the protons that potentially can be involved in acidbase reactions. The exchangeable protons include those that are unassociated with any other species and those that are associated with a titrateable species, i.e. a buffer. The number of exchangeable protons carried by a buffer having a single titrateable group ( i.e., one pK,) is equal to the concentration of that portion of the buffer which is in the acid form at the pH of interest, and is found by combining the Hen-

= [free H+],u=,., + [ H+ in buffers] pH=7.4+ [ gluconic acid] =10-7~4+C([buffer](10-7~4)/(10-7~4 + 10VpK) ) + [ gluconic acid] (18) where the second term on the right side is a summation over all of the buffers in the system. Note that all of the pH-dependent calculations so far have been done at pH 7.4. The pH at the given gluconic acid concentration can now be

275

calculated from the value for [H+ ] computed above: [H+] = [free H+] + [H + in buffers]

1. Base values

=10-p” +C([buffer](lO-PH)/(lO-pH

(19)

+ lo-PK) ) The system’s pH is the one that satisfies eqn. (19) and can be calculated using the Newton-Raphson root-solving technique.

MODEL TAINING

PREDICTIONS MEMBRANES

simplification obviates the solution of partial, rather than ordinary differential equations.

FOR AMINE-CON-

This section contains predictions of the performance of membranes containing glucose oxidase, an excess of catalase, and amine groups. It is proposed that such membranes could function as part of an insulin delivery device (see “Introduction”). The values of several operating parameters are varied in these simulations in order to evaluate their relative importance in the design of membranes having improved performance. The function of catalase is to alter the overall reaction stoichiometry such that O2 consumption is reduced by 50% (see part 2 of “Mathematical Model”). The depletion of O2 in the membrane often limits the turnover rate, and consequently limits the pH decrease, at physiological glucose concentrations. The assumption is made in this section’s calculations that substrate concentrations outside of the membrane’s boundary layer can be fixed at arbitrary values and are not affected by the membrane itself. The concentration of the reaction product, gluconic acid, is not considered to be a fixed value; it varies as a function of the other operating conditions. Note that the acid concentration varies with distance from the membrane surface within the boundary layer, but is considered not to vary with distance in the direction parallel to the membrane surface. This

A “base case” is defined by assigning default values to all parameters. Usually, only one parameter is varied from its default value for a given computation, and families of predicted data points are generated in which the value of only one of the parameters is varied from point to point. The default values are as follows: bulk glucose concentration = 100 mg%, bulk O2 concentration = 0.274 mM, glucose oxidase concentration in membrane = 1.0 PM, membrane thickness = 0.2 mm, concentration boundary flowrate = 10 layer thickness= 0.1 mm, cm3/min, temperature = 25’ C, buffer: 0.0195 M citrate, pH 7.40 when gluconic acid concentration is zero, amine concentration in membrane = 0.005 M and area of membrane = 0.7854 cm2. The O2 concentration used here is the concentration present in H,O saturated with air at 25°C computed using the Henry’s Law constant [ 91. The membrane area is that of a 1 cmdiameter disk. The flowrate is typical for blood flow through smaller arteries, which is reported to vary between 10 and 30 cm3/min, depending on the artery diameter [lo]. The boundary layer thickness is a reasonable value for diffusion of glucose and oxygen to a flat plate simultaneous with laminar flow past the plate at the assumed flowrate. The boundary layer was calculated for a linear velocity of 10 cm/s, using the method of Pohlhausen as described in Schlichting [ 111. This linear velocity corresponds to the assumed volumetric flow rate of 10 cm3/min through a 0.15 cm-internal diameter artery, values that fall within the ranges for these parameters in humans [ 121. The value of the boundary layer thickness used for modeling the amine containing membranes is larger than the values used when simulating the experimentally observed behavior of the phenol red containing membranes since the latter were

276 1

N 0

r m 0

I

0.8

i

z

z

0.6

P 0

r 0

m e t e r v a

0.4

0.2

1 u .?

I_0

-

-

1---T---I 0

0.

1

1

0.2

0.

1

3

0.4 Normalized

1

1

0.5

0.6

I

I

0.7

0.8

1

0.9

1

Length

Fig. 3. Predicted response of an amine confining glucose sensitive membrane to glucose under base conditions. See text for parameter values for base conditions. Triangles, C&pik; squares, Cox/Coxb”‘k; diamonds, U/U,,; 10; stars, C,,id/Cg”“‘k;circles, ApH= (7.40-pH).

done in a stirred flow cell (see “Experimental results”, Part 2 ) . The concentrations are given per volume of liquid in the membrane, not per volume of membrane. 2. Physical constants The following values (all at 25 ‘C ) for physical constants are used in all calculations in this section: IIab”lk=Dgbulk= 6.75 X 10e6 cm’fs [ 13 ] Doxb”lk=2.29 X 1O-5 cm”/s [ 131 k, (ineqn. (10)) =2.8X10-7mols/cm3

[8,14]

k2 (ineqn. (10))=1.65~10~~mols/cm~

[8,14]

V mLTIRX =

in mol/cm3; mol/cm3 s 226 times Cenzyme

18J4f pKa of citrate buffer = 6.39,4.77, 3.08 [ 91 Nakamura and Ogura [ 81 report values of the kinetic coefficients kl, k,, and u,,, at pH 5.5 and 25°C. Their values have been adjusted to

pH 7.2 using data reported by Weibel and Bright [ 141 on the dependence of these coefficients on pH. Values at pH 7.2 are used because the model typically predicts the membrane pH to be 7.0 to 7.4. Dependence of the coefficients on pH is minor over the range 7.0 to 7.4. 3. Presentation

of simulated

results

Figure 3 contains plots of glucose concentration C,, oxygen concentration C,,, gluconic acid concentration Cacid, turnover rate, and ApH ( where ApH = 7.40 - pH ) as functions of depth in the membrane for the “base case”. All parameters appearing in this plot except ApH have been normalized for convenience. The length parameter has been normalized with respect to the total membrane thickness, so that length = 0 is the “front” side of the membrane, where the flow is, and length = 1 is the back side (in the analogous physiological situation, length = 0 would be the “body side” of the membrane and length= 1 the “insulin reservoir side”). The quantities C, and Cacidhave been

211 TABLE

where pHavg is the average pH inside the membrane and is defined as

2

Predicted

effects of substrate

concentrations

on membrane

pH

value

7.40 -average

pH

L

variable glucose concentration; = 0.274 mh4 0, concentration

25 mg% 50 100” 150 200 300

0.032 0.060 0.105 0.135 0.155 0.174

400

0.183

500

0.188

25 mg%

0.031

50

0.056

100

0.085

150 200

0.096 0.101

300

0.106

400

0.108

500

0.109

0, concentration;

0.02 mM

0.017

glucose concentration

0.06

0.047

equals

0.10

0.070

glucose concentration: 0, concentration=0.140

“‘base

mM

100 mg%

0.14

0.085

0.19

0.096

0.24

0.102

0.274”

0.105

case” calculation.

normalized with respect to the bulk phase concentration of glucose, and similarly C,,, has been divided by the bulk concentration of OZ. The turnover rate is given as a fraction of the maximum turnover rate u,,, and is multiplied by 10 for more convenient display. The remainder of the section will consist of predicted values of the average pH decrease within the membrane at various combinations of values of the operating parameters. Usually in a given calculation, every parameter but one is at its default value; if more than one parameter at a time is varied from its default value, this fact will be noted. The “average pH decrease” ApH,,, is defined as APH,,, = 7.40 -pHavg

where x is the length parameter and L is the total thickness, both in cm. Table 2 illustrates predicted response to changes in bulk glucose concentration CgbUlk and bulk oxygen concentration C,,Xb”1k.Predicted pH’s as a function of Cgbulkare presented for C bu’k= 0.274 mM and Coxb”‘k= 0.140 mM. The foymer is the O2 concentration in air-saturated H,O at 25’ C and the latter is the 0, tension in human arterial blood [ 151. At high glucose concentration, the predicted average pH’s are close together because the enzyme reaction is 02-limited at high C,. The reaction becomes 02limited at lower glucose levels when the O2 concentration is lower, as expected. The next set of data in Table 2 demonstrates the effect of external O2 tension on the pH decrease. Qualitatively, it appears that the reaction does not become glucose-limited at up to 0.274 mA4 O2 when the bulk C, is 100 mg% (its base value). Table 3 illustrates the predicted dependence of membrane pH on certain physical constants. The first data set indicates the effect of buffering. A 0.0195 M citrate buffer has approximately the same buffering strength as does human blood over the pH range of 7.0 to 7.4. Since the amine groups in the polymer are themselves buffering agents, the second data set in Table 3 shows the effect of their concentration on the pH profile. As one would expect, the amine concentration easily can be made high enough to prevent significant pH drop by buffering it out. At the default amine concentration of 0.005 M the major buffer in the system is citrate. The pKof the amine group is shown in the next data set to have a relatively small effect on pH because there is only moderate buffering by the amines at their default concentration. However, the pK does have a strong effect on the charge density in the gel, which is what

278 TABLE 3 Predicted

effects of values of physical constants

on membrane

pH

value

7.40 -average

pH

0

0.005 0.01 0.0195” 0.03 0.10

0.263 0.187 0.147 0.105 0.080 0.032

0 c.001 0.003 0.005” 0.01 0.02 0.10

0.160 0.145 0.122 0.105 0.078 0.051 0.014

pK, of amine groups

6.0 6.5 7.0” 7.5 8.0 9.0

0.145 0.126 0.105 0.100 0.118 0.153

solute diffusivities, given as values of D,, DC,,,and Dac,d, normalized with respect to their values in the bulk phase

1.0, 1.0,l.O” 0.8,0.8,0.8 0.6,0.6,0.6 0.4, 0.4,0.4 0.2,0.2,0.2 O.l,O.l, 0.1 0.5,0.5,0.5 0.5, 0.75,0.5 0.5, 1.0, 0.5 0.7, 1.0, 1.0 0.7, 1.0,0.7 0.7, 1.0,0.4 0.7, 1.0, 0.2 0.7, 1.0,O.l

0.105 0.113 0.124 0.144 0.180 0.201 0.133 0.136 0.137 0.103 0.119 0.158 0.238 0.377

variable

citrate buffer concentration

(M)

amine group concentration

(M)

“relative charge density” (see text)

0.0144 0.0324 0.0518 0.0560 0.0400 0.0071

*“base case” calculation.

directly determines swelling. Table 3 shows the “relat,ive charge density”, defined as relative charge density

where C + = concentration of charged amine groups, C&=,., = concentration of charged amine groups at pH 7.40, and Cam,ne= concen*

tration of amines, charged or neutral. Therefore, the relative charge density is the fraction of amine groups that is protonated because of the pH decrease caused by gluconic acid; this fraction does not include the amine groups that are protonated at pH 7.40. The last three data sets in Table 3 deal with the effect of the diffusion coefficients of glucose, 0, and gluconic acid within the gel. Note

279 TABLE 4 Predicted

effects of operating

variable membrane (mm)

value thickness

boundary layer thickness (mm)

flowrate” (cm”/min)

conditions

on membrane

pH

7.40 -average

pH

0.01 0.05 0.10 0.20 0.50 1.0

0.004 0.023 0.049 0.105 0.208 0.231

0 0.10” 0.20 0.40 0.70 1.0 5.0

0.048 0.105 0.148 0.201 0.232 0.242 0.253

0.0001 0.001 0.01 0.03 0.10 0.30 1.0 10.0”

0.085 0.088 0.105 0.120 0.115 0.109 0.106 0.105

““base case” calculation. bCalculated with the assumption of flowrate-dependent depletion of substrates in the bulk phase.

that only the diffusivities inside the gel have been changed; bulk solution coefficients, which are used when calculating the mass transfer coefficients, were not changed. In the first of these data sets D, and D,, are assumed to be the same fraction of their values in bulk solution. As usual, Dacid is assumed equal to D,. The pH drop becomes larger as the diffusivities are decreased, but the effect is moderate. Next, D, and D,, have been reduced by different factors. Here, the “reduced Dg)’ is always 0.5, that is, D, in the gel is 0.5 times its bulk solution value. The reduced D,, is either 1.0, 1.5 or 2.0 times the reduced D,. The model predict.s that there is a negligible effect of varying D,, independently of D, over this range. In the final data set Dacidis varied while holding D, and D,, constant. The predicted pH is reduced signifi-

cantly by reducing the assumed value of Dacid. In Table 4 the predicted effects on membrane pH of varying certain operating conditions are shown. The first two data sets demonstrate that an increase in thickness of either the membrane or the boundary layer on one side of it leads to a significant increase in the pH drop. The last data set in this table illustrates the effect of flowrate on the “body side” of the membrane. For these data we relaxed the assumption of fixed substrate concentrations in the bulk phase since this assumption is not reasonable at very low flowrates. The effect of flowrate is seen to be very moderate. Lowering the flowrate leads to increased substrate depletion in the bulk phase, which tends to reduce the size of the pH decrease in the membrane because the turnover rate is decreased; however, a lower flowrate also means that the gluconic acid is not swept from the membrane surface as quickly, which tends to increase the pH-drop. The opposition of these processes accounts for the slight minimum in the pH,,, vs. flowrate data. Table 5 concerns the effect of enzyme loading on predicted average pH. First, the predicted membrane pH is shown for various enzyme loadings in a 0.1 mm-thick membrane (vs. 0.2 mm default thickness). The pH drop increases with increasing enzyme concentration up to at least 20 ,LLM,above which the pH drop is limited by oxygen depletion. This limit would have been observed at a higher Cenzymeif the membrane was thinner, because less total enzyme would be present. The glucose response calculations of the type illustrated in Table 2 were repeated for a 50 pmthick membrane at various enzyme loadings, and the results are shown in Table 5. At 60 PM glucose oxidase there is little change in the pH as glucose concentration is increased above 50 mg%, because severe O2 depletion occurs at C, < 50 mg%. At 15 ,uM GluOx there is a progressive response to C, at least up to 150 mg%, and at 4.0 PM GluOx this progressive response extends at least to C, = 300 mg%. However, if C enzymeis reduced much below 4.0 PM, there is

280 TABLE 5 Predicted

effects of enzyme loading on membrane

variable

enzyme concentration at 0.1 mm membrane thickness and C, = 100 mg%

pH

value

(j&f)

0.30 0.50 1.0 2.0 4.0 6.0 8.0 20 50

100 glucose concentration (mg% ) at C,,, = 60 @f and 0.05 mm membrane thickness

10

7.40 -average

0.058 0.109 0.200 0.231 0.245 0.247 0.249 0.250

glucose concentration (mg% ) at C,,, = 15 @f and 0.05 mm membrane thickness

50 75 100 150 200 300 500

0.124 0.168 0.196 0.218 0.225 0.230 0.233

glucose concentration (mg% ) at C,,, = 4 ,uM and 0.05 mm membrane thickness

50 75 100 150 200 300 400 500

0.045 0.063 0.079 0.105 0.124 0.146 0.159 0.166

(@4)

1.0 4.0 6.0 8.0 10

12 14 16 20 30 50

“responsiveness” (see text)

0.016 0.026 0.049 0.089 0.148 0.187 0.210 0.241 0.250 0.253

20 30 40 50 75 100 150 500

enzyme concentration at 0.05 mm membrane thickness

pH

0.158

0.0184 0.0600 0.0783 0.0901 0.0961 0.0977 0.0961 0.0926 0.0832 0.0590 0.0263

281

very little pH drop because of low turnover, so that pH within the membrane is close to 7.4 at any glucose concentration (simulated results are not shown for such a case). Therefore, response to glucose is seen to be small at either relatively high or relatively low enzyme loadbut substantial at intermediate ings, concentrations. The final data set in Table 5 demonstrates that an optimal enzyme loading exists for response to glucose concentration near a given average C,, which here is 100 mg%. We choose to define the “responsiveness” as the difference in average pH in the membrane at C, = 150 mg% and C, = 50 mg%, that is, responsiveness

= pH,,, ( CgbUlk= 50 mg% )

- pHavg ( C, b”‘k= 150 mg% )

This quantity is tabulated at different values of Cenzymefor a 50pm thick membrane. An optimal enzyme loading is predicted to be about 12 PM. Note that a similar effect occurs with membrane thickness as the independent variable. A very thin membrane would lead to low turnover (as would a low Cenzyme) and a very thick membrane would lead to 0, depletion (as would a high Cenzyme) . In other words, an optimal thickness exists for each value of Cenzymejust as an optimal CL,,, exists for a given thickness.

EXPERIMENTAL PREDICTIONS MEMBRANES 1. Membrane characterization

RESULTS AND MODEL FOR DYE-CONTAINING

preparation

and

The formulations of the membranes used experimentally are listed in Table 1. All formulas were identical except for GluOx concentrations. Measurement of the fractional retention of glucose oxidase in the membranes was attempted but was not successful because of inability to radioiodinate this enzyme. How-

ever, for membranes having the compositions listed in Table 1 except that immunoglobulin G ( IgG) replaced GluOx, fractional retention ranged from 0.92 for the membrane having an initial loading of 0.025 mg/ml to 0.40 for the membrane with an initial loading of 25 mg/ml. The molecular weight of IgG is 168,000, which is very close to that of GluOx (160,000). Catalase enzyme was not included in any of the experimental gels. The phenol red indicator present in our gels acts as a buffer and must therefore be considered when using the model for predicting pH. The pK, of the immobilized dye has been reported as 7.57 [ 71. The measured titration curves of the membranes containing 0.25, 2.5, and 25 mg/ml GluOx were found to be well described by the Henderson-Hasselbach equation with a pK of 6.7. This value of the pK was used when predicting the performance of phenol red-containing gels. The diffusivity of O2 in these gels was assumed to equal its diffusivity in the bulk solution. The permeabilities to glucose of the membranes containing 0.25, 2.5, and 25 mg/ml GluOx were measured and averaged 4.05 x 10e6 cm”/s with no statistically significant dependence on GluOx loading. The partition coefficient for the distribution of glucose between bulk solution phase and gel phase was assumed to equal the volume fraction of buffer in the membrane, which for these same three membranes was found to be 0.83. The diffusion coefficient of glucose in the gel was calculated by dividing its permeability by the partition coefficient. The measured diffusivity was 0.72 times the reported bulk phase diffusivity of glucose. 2. Calculations response

of

predicted

membrane

A value of the boundary layer thickness must be assumed when using the model to make predictions. In the experimental measurements on membranes containing phenol red, the fluid flowing past the membrane was either stirred

282

Fig. 4. Effect of glucose oxidase depletion by pAAm/GluOx/phenol parison of theory and experiment. bols, solid lines. Theory: dotted near them. GluOx concentrations circles, 25. The membranes were

concentration on bulk 0, red membranes: ComExperiment: open symlines with solid symbols (mg/ml): triangles, 0.25; 0.307 mm thick.

mechanically or was subject to turbulence because of sudden expansion or contraction of the flow channel, and so a very thin boundary layer can be expected. In most cases, the membrane was positioned in the flow cell with a piece of 60pm-thick Dacron mesh sandwiched against its front side for mechanical support, and so the boundary layer at one of these membranes can be assumed to be at least 60 pm thick. (Inci-

0 0

20 Glucose

60

Concentration

80

100

imq%)

Fig. 6. Effect of thickness on turnover by pAAm/GluOx/phenol red membranes: Comparison of theory and experiment. Experiments: open symbols, solid lines. Theory: dotted lines with solid symbols near them. Thickness (mm): circles, 0.129; squares, 0.307; triangles, 0.307 with internal mesh. GluOx concentration was 25 mg/ml.

dentally, the membrane area was corrected for the presence of the mesh, which had a void space of 0.407. ) For a few of the membranes the mesh was incorporated directly into the gel during polymerization by casting the monomer solu-

Glucose

Fig. 5. Effect of glucose oxidase concentration on glucose turnover by pAAm/GluOx/phenol red membranes: Comparison of theory and experiment. Experiments: open symbols, solid lines. Theory: dotted lines with solid symbols near them. GluOx concentrations (mg/ml): diamonds, 0.25; squares, 2.5; triangles, 25. The membranes were 0.307 mm thick.

40

Ccmcentratlon (mg 46)

Fig. 7. Effect of glucose oxidase concentration on the oxygen gradient in pAAm/GluOx/phenol red membranes: Comparison of theory and experiment. Experiments: open symbols, solid lines. Theory: dotted lines with solid symbols near them. GluOx concentrations (mg/ml): circles, 0.25; triangles, 2.5; squares, 25.0. The membranes were 0.307 mm thick.

283

tion onto a piece of mesh. For the calculations in this section the boundary layer thickness was assumed to be 60 pm for those membranes with external mesh. 3. Comparison

of experimental

and theoreti-

cal results

Glucose Ccncentratw

hg %I

Fig. 8. Effect of thickness on the oxygen gradient in pAAm/GO/phenol red membranes: Comparison of theory and experiment. Experiments: open symbols, solid lines. Theory: dotted lines with solid symbols near them. Thickness (mm): circles, 0.129; triangles, 0.307. GluOx concentration was 25 mg/ml.

Glucose Concmtrotm

(mg %)

Fig. 9. Effect of flow rate on oxygen gradient in pAAm/ GluOx/phenol red membranes: Comparison of theory and experiment. Experiments: open symbols, solid lines. Theory: dotted lines with solid symbols near them. Flow rates (ml/min): circles, 0.16; triangles, 0.40; squares, 4.0. The membranes were 0.305 mm thick and contained 25 mg/ml GluOx.

Figure 4 shows both measured and predicted values of oxygen concentration in the external buffer, plotted against glucose concentration, for two membranes having different enzyme loadings. On the y-axis scale used here and in all other plots of O2 tension, 21% represents the solubility of 0, in water at the ambient temperature of the experiment. The measured O2 tenhigher than sions are always somewhat predicted; that is, measured O2 depletion is less than predicted. This implies that the substrate turnover was slower than predicted, since the bulk 0, concentration is an indirect measure of turnover rate. Figure 5 shows measured and predicted turnover rates, plotted against external C, for membranes having different enzyme loadings. Figure 6 is similar except that membrane thickness is varied instead of GluOx loading; also, one of the three membranes referred to in Fig. 6 contained internal Dacron mesh whereas the other two did not. As expected, higher enzyme concentration gave faster turnover. Actual turnover rates are always much lower than predicted. The discrepancy between predicted and observed reaction rates is much larger than in Fig. 4. However, turnover rates were measured with the membrane in a closed stirred reactor, which is a much different situation than when O2 tensions were measured, and this may make the two sets of results difficult to compare. The results in Fig. 6 for the two non meshsupported membranes show that increasing the membrane thickness did not greatly accelerate turnover, presumably because the enzyme loading was high enough that turnover rates in both membranes were limited by oxygen depletion. The model calculations, which predict no effect

284

72v 7.1-

zo 6.9. 1 0

8 I 20

I 40

I 80

I 60

I 100

Glucose COnCentrOtion

,/ ' ' hg

/ 250

500

%)

Fig. 10. Effect of glucose oxidase concentration on pH in pAAm/GluOx/phenol red membranes: Experimen~l data, collected spectrophotometrically. GiuOx concentrations (mg/ml f were: solid triangles, 0.05; solid circles, 0.125; open circles, 1.0; open squares, 4.0; open triangles, 10; solid, inverted triangles, 25; solid squares, 50. The membrane thickness was 0.129 mm.

0

I

I

I

T

I

20

40

60

60

100

Glucose Concentrations.

I

I

’ 500

m9%

Fig. 11. Effect of glucose oxidase concentration on pH in pAAm/GluOx/phenol red membranes: Predicted values. GluOx concentrations (mg/ml): triangles, 0.05; squares, 0.125; diamonds, 1.0;stars, 4.0, circles, 10.0;inverted triangles, 25.0. The membranes were considered to be 0.129 mm thick.

of thickness, agree with the observed data in this respect. An unexpected result was that turnover rates in the mesh-supported membrane were so high (for example, compare data in Fig. 6 from the two 0.307 mm-thick mem-

branes). The presence of mesh inside a membrane was expected to decrease the turnover rate simply because it occupied some of the volume that otherwise would be occupied by enzymeimpregnated gel. It is possible that polymeri-

285 Membrane PH

74 E:._

Glucose Ccoxntration

hg%l

Fig. 12. Effect of glucose oxidase concentration on pH in pAAm/GluOx/phenol red membranes: Comparison of theory and experiment. Experimental data collected spectrophotometrically. Experiments: open symbols, solid lines. Theory: dotted lines with solid symbols near them. GluOx concentrations (mg/ml): squares, 0.25; circles, 2.50; triangles, 25.0. Membrane thickness was 0.307 mm.

zation around this mesh caused channels to form in the gel and that these channels allowed the substrates to diffuse rapidly to the interior. Since the reaction rate is diffusion-limited, this might accelerate turnover substantially. No simulated data are included for the mesh-supported membrane because the model is not valid when the area available for diffusion is not constant. Figures 7 through 9 pertain to oxygen tension at the back side of the membrane. Figure 7 contains plots of both measured and predicted O2 tension against glucose concentration for membranes having different enzyme loadings. Figure 8 shows O2 concentration plotted against C, for membranes of different thicknesses, Finally, Figure 9 has plots of O2 tension vs C, for different buffer flowrates. The prediction of linear plots in Figs. 7-9 is verified by the experimental results. In general, the slopes of the experimentally-obtained lines are not as steep as the slopes of the predicted lines; that is, for a given C, less oxygen depletion was measured than predicted. This suggests that the actual turnover rate was lower than predicted, and so these data on 0,

7

101 0

20

40

60

80

100

5ct

,

Cc) ~.__.___~__~__________------._-.__

I 0

,,___ _.___

I

20

40

60

80

500

IO0

Glucose Cowentrollon

hg

J

“A)

Fig. 13. Effect of glucose oxidase concentration on pH in pAAm/GluOx/phenol red membranes: Comparison of theory and experiment. Experimental data collected with a micro pH electrode. Experiments: open symbols, solid lines. Theory: dotted lines. GluOx concentrations (mg/ml) : part A, 0.25; part B, 2.5; part C, 25.0. The membranes were 0.307 mm thick.

6.8

0

20

40

60

80

100

Glucose Concentrat!on (mg %)

Fig. 14. Effect of membrane thickness on pH in pAAm/ GluOx/phenol red membranes: Comparison of theory and experiment. Experimental data collected spectrophotometrically. Experiments: open symbols, solid lines. Theory: dotted lines with solid symbols near them. Thickness (mm) : squares, 0.080; triangles, 0.129; circles, 0.307. All membranes contained 25 mg/ml GluOx.

tension in the membrane follow a trend similar to that seen in the previous figures. Figures lo-14 pertain to pH in the membrane. In Fig. 10 the pH measured spectrophotometrically is plotted against glucose concentration for membranes having different GluOx loadings. Figure 11 has the predicted average pH’s corresponding to the measurements in Fig. 10. Figure 12 also has plots of experimental and simulated data for membranes of different enzyme loadings, but the thickness is different than for the membranes in Fig. 10. Figures 13 a-c contain plots of the pH at the back of the membrane (either predicted by the model, or measured using an electrode) vs. C, for three different enzyme loadings. There is good agreement between the results from the two techniques for measuring pH, which suggests that the data are correct. Figure 14 contains plots of pH (measured spectrophotometrically) against C, for membranes of different thicknesses. As expected, greater thickness leads to a larger pH decrease in the gel. ?‘he model predicts ( as shown in Fig. 11) that

the magnitude of the pH decrease should asymptotically approach a maximum value as GluOx loading is increased (all other conditions held constant). No further pH decrease can result from increased GluOx loading, according to the model, because of 0, depletion. However, the experimentally measured pH is seen in Fig. 10 to decrease progressively with increasing GluOx loading, over the entire range of GluOx concentrations that we have studied. At relatively low membrane GluOx concentrations ( < 1 mg/ml) the observed pH drops were lower than predicted. At relatively high GluOx loadings ( > 25 mg/ml) the measured pH drops were larger than the model predicts. The appearance of this asymptotic value of the pH drop in the simulated results and its absence in the observed data represent an important discrepancy between the experimental and predicted results. 4. Discussion

of results

The data in Fig. 10 at relatively low enzyme loadings ( < 1 mg/ml) are especially significant. Here there is demonstrably a progressive increase in pH drop with increasing C,, at least up to 500 mg% glucose, even though the pH drop is small. Inspection of Figs. 10 and 12-14 shows that each of the other membranes, which had higher enzyme concentrations and often were thicker, reached an asymptotic pH drop at some sub-physiological concentration of glucose. The data in Fig. 10 thus confirm the model’s prediction that the range of progressive response to C, can be extended by decreasing the total amount of enzyme, either by lowering the enzyme concentration or by making the membrane thinner. Data from the other membranes confirm that if the enzyme loading is too high, the device becomes unresponsive to further increases in C, at some low C,. This “saturation” effect results from 0, depletion in the membrane. Several other researchers, who are developing glucose sensors which employ immobilized glucose oxidase, have reported that

their sensors reach saturation at sub-physiological glucose concentrations. The explanation almost certainly is that their enzyme loadings are too high. The conclusion that Cenzymemust be sufficiently low to avoid the saturation effect holds true for any glucose sensor using immobilized GluOx, not just a sensor having the flat membrane configuration studied here. It is also possible to avoid the saturation effect by the somewhat more complicated means of using an additional permselective membrane outside the enzyme-loaded portion of the device. This membrane acts as a partial barrier to glucose transport and therefore reduces the depletion of 0, relative to glucose. The size difference between glucose and oxygen molecules allows such a barrier to operate simply by steric hindrance. Other researchers [ 16-201 have found various hydrophobic membranes to be useful in this application. There are several factors that may have contributed to the general lack of quantitative agreement between model and experiment. Firstly, it is possible that air leaked from the atmosphere to the back side of the membrane while O2 tensions were measured with the membranes in the flow cell. This might cause the measured O2 tension to be higher than predicted both in the membrane and in the bulk phase, which would explain the results shown in Figs. 4 and 7-9. More importantly, the more abundant 0, supply would allow a higher glucose concentration at a given enzyme loading - or, conversely, a higher enzyme loading at a given glucose concentration -before O2 depletion occurs and causes what we have called the “saturation” of the response. That is, higher glucose or enzyme concentrations would be required in order to closely approach the asymptotic maximum pH. Thus, an asymptotic pH is seen in the plots of simulated data in Fig. 11 but is not seen in the corresponding experimental data in Fig. 10, because air leakage might have caused the enzyme loading at which excessive 0, depletion would occur experimentally to

increase beyond the range of loadings that were dealt with experimentally. One avenue by which air might enter the cell is through the ports in which the electrodes were inserted. These were sealed with O-rings. The result of leakage here would be an artificially low pH or artificially high 0, tension at the point where the electrode touched the membrane surface, which of course is the point at which pH or O2 tension were measured. Figures 7-9 show therefore that there could not have been significant air leakage past the O2 electrode when these data were collected because essentially zero 0, tension was measured at the higher glucose levels. This does not preclude air leakage at other locations on either of the flow cells. A second possible explanation for the lack of quantitative agreement is the occurrence of significant convective transport through the membrane during experiments using the flow cell. Only passive diffusion of species into and out of the membrane is considered in the model, but some convective flow through the membrane should be expected since buffer was pumped under pressure past one side of it and since the gels had very open structures (their volume fraction of buffer was 0.83). This convection would increase the overall mass transport rate, thereby alleviating O2 depletion in the membrane. Therefore, convective transport is expected to have the same qualitative effect on membrane 0, tension and membrane pH as air leakage does and might therefore help explain many of our results. However, convective transport should also increase the turnover rate and thereby reduce the 0, tension in the bulk phase to a value less than what is predicted, and Fig. 4 shows that this is not the case. It is expected that significant amounts of GluOx enzyme leaked from the membranes prior to the experiments and that this leakage was more severe at higher initial enzyme loadings, based on our data for immobilized IgG. This would diminish the turnover rate and might therefore help explain the higher-than-

predicted measured O2 tensions and the lowerthan-predicted measured turnover rates. It would not explain the disagreements between measured and predicted pH’s. Note that enzyme leakage reduces the turnover rate only in the case when the rate is not limited by oxygen depletion, that is only at lower GluOx loadings. Therefore, it should not be expected that enzyme leakage affected the experimental results at very high initial enzyme loadings or at very high glucose concentrations. Some assumptions that were made concerning values of physical constants may have been in error. It was assumed that the kinetic constants of the immobilized enzyme were the same as those of non-immobilized enzyme. This probably is a good assumption since in our membranes GluOx was immobilized by simple steric entrapment rather than by covalent bonding. Predicted results could be significantly in error if the kinetic constants were changed such that the relative sensitivities of the reaction rate to glucose and oxygen were affected. Values for the diffusivities of gluconic acid and 0, were assumed without direct measurement, although D, was measured. The 0, diffusivity was assumed to equal its value in bulk solution, and this probably is a good approximation; Dacid was taken to equal D, and if Dacid actually was considerably smaller, it would help explain the lower-than-predicted pH’s that were observed in many of the membranes. Electrostatic interaction with the charged membrane can be expected to change Dacid from its value in bulk solution; however, the concentration of titrateable groups in the experimental membranes was not high (0.0034 M) and it is questionable whether the presence of charged groups at this concentration would significantly affect Dacid. Finally, it is possible that the buffer strength inside the gel matrix was significantly different than outside. This would be true if sodium citrate has a partition coefficient significantly different from unity for its distribution between the gel and the bulk phase. Weaker internal buffering would help explain why the

measured membrane pH was lower than predicted at enzyme loadings greater than 25 mg/ml. As noted in the Methods, some membrane pH’s were measured at the back of the membrane with an electrode and some were deduced from light transmission measurements. The % light transmission depends on the entire pH profile in the membrane. Calibration curves for converting % light transmission (%T) to pH were generated by measuring the %T through membranes which were swollen at various known pHs. Note that these membranes were uniform with respect to pH; they contained no pH gradients. It was found that the %T of these uniform gels was very nearly a linear function of pH. It can be shown (see Appendix) that if light transmission is linearly dependent on pH for membranes of uniform pH, then the light transmission of a membrane containing a pH gradient is the same as if the entire membrane were at the membrane’s average pH. In other words, when the numerical value of a membrane’s %T is substituted into the linear equation relating %T to pH, the calculated pH is equal to the membrane’s average pH. Therefore, we consider the spectrophotometric measurements to be indicators of average membrane pH, and accordingly we have compared these data with predicted values of the average pH.

CONCLUSIONS

The model of the steady-state performance of glucose-sensitive membranes containing phenol red is an accurate predictor of qualitative trends in membrane pH, turnover rates, and oxygen tension. For example, it correctly predicts whether the pH drop will increase or decrease when a certain design parameter is changed and it correctly predicts the shapes of the plots of membrane pH and membrane 0, tension as functions of C,. However, quantitative agreement is lacking. Several factors may have contributed to the disagreement, includ-

289

ing the possibility of convective transport, which was neglected in the model, the possibility of 0, leakage into the experimental flow cell from the atmosphere, the possibility that buffering within the gel matrix was different than assumed, and the certainty of enzyme leaching from the gel prior to any of the experiments. The model predicts lower oxygen tensions and higher turnover rates in the membrane than we have observed experimentally. It also predicts higher-than-observed pH decreases at GluOx concentrations of < 1 mg/ml and smaller-thanobserved pH decreases at GluOx loadings of > 25 mg/ml. The model may be extremely useful in the optimization of the design of amine-containing membranes. The most important property that is required of the membrane is that its steadystate pH changes progressively with glucose concentration over a physiologically useful range of concentrations, for example between 50 and 300 mg%. The term “progressively” is used to indicate that the pH changes continuously with C, and does not reach an asymptotic value at a value of C, that is below 300 mg%. The magnitude of pH drop is also important, although not equally so. The model demonstrates that, for a given membrane thickness, an optimum glucose oxidase concentration exists which provides the best compromise between these two properties (see Table 5). A rational means of selecting the “best” combination of parameter values may be to prepare a plot of the derivative d ( pH ) /d ( C,) vs. glucose concentration for each case under consideration. It is desirable that the magnitude of this derivative be as large as possible over the range 50 mg% < C, < 300 mg%. The important operating parameters that can be manipulated arbitrarily by the experimenter while still maintaining the physiological relevance of the experiment are membrane thickness, enzyme loading, and amine concentration. The model shows that for any glucose sensing device which uses immobilized glucose oxidase, progressive response to glucose

concentration in the physiological range is possible only if the enzyme loading is sufficiently low. This is an important finding which has not been recognized by several other workers developing glucose sensors, with the result that their devices have become “saturated” and cease to respond to glucose at sub-physiological glucose concentrations. Saturation can also be avoided, as others have shown, by increasing the ratio of O2 transport to glucose transport by adding an outer membrane that is more permeable to O2 than glucose. The response time of the device is just as important clinically as its steady-state performance. Future research on glucose sensitive membranes will focus on designs and methods to achieve optimal response time. It is expected that simply reducing the membrane’s thickness will prove to be the most effective means of reducing the response time.

ACKNOWLEDGEMENT The financial support of the National Institute of Arthritis, Diabetes, Digestive and Kidney Diseases through grant 2 ROl DK30770 is gratefully acknowledged. REFERENCES G.F. Cahill, D.D. Etzwiler and N. Freinkel, Blood glucose control in diabetes, Diabetes, 2 (1976) 237-238. T.A. Horbett, B.D. Ratner, J. Kost and M. Singh, A bioresponsive membrane for insulin delivery, in: J.M. Anderson and S.W. Kim (Eds.) , Recent Advances in Drug Delivery Systems, Plenum Press, N.Y., 1984, pp. 209-220. T.A. Horbett, J. Kost and B.D. Ratner, Swelling behavior of glucose sensitive membranes, in: S. Shalaby, A.S. Hoffman, T.A. Horbett and B.D. Ratner (Eds. ) , Polymers as Bioma~rials, Plenum Press, N.Y., 1984, pp. 193-207. J. Kost, T.A. Horbett, B.D. Ratner and M. Singh, Glucose-sensitive membranes containing glucose oxidase: Activity, swelling, and permeability studies, J. Biomed. Mater. Res., 19 (1984) 1117-1133.

290

5

6

I

8

9 10

11 12

13 14

15

16

17

18

19

20

B.D. Ratner and T.A. Horbett, Enzymatically Controlled Drug Release, in: K.Widder (Ed.), Drug and Enzyme Targeting, Methods in Enzymology, Vol. 112, Academic Press, N.Y., 1985, pp. 484-495. G. Albin, T.A. Horbett and B.D. Ratner, Glucose sensitive membranes for controlled delivery of insulin: Insulin transport studies, J. Controlled Release, 2 (1985) 153-164. J.I. Peterson, S.R. Goldstein and R.V. Fitzgerald, Fiber optic pH probe for physiological use, Anal. Chem., 52 (1980) 864-869. S. Nakamura and Y. Ogura, Action mechanism of glucose oxidase of Aspergillus Niger, J. Biochem., 63 (1968) 308-316. In: R.C. Weast (Ed.), CRC Handbook, 62nd ed., CRC Press, Boca Raton, Florida, 1981. A.J. Vander, J.H. Sherman and D.S. Luciano, Human Physiology: the Mechanisms of Body Function (4th ed.), McGraw-Hill, N.Y., 1985. H. Schlichting, Boundary Layer Theory, 7th ed., McGraw-Hill Co., New York, NY, 1979. H.L. Goldsmith and V.T. Turitto, Rhelological aspects of thrombosis and haemostasis: Basic principles and applications, Thromb. Haemostas., 55 (1986) 415-435. P.A. Johnson and A.L. Babb, Liquid diffusion of nonelectrolytes, Chem. Rev., 56 (1956) 387-453. M.K. Weibel and H.J. Bright, The glucose oxidase mechanism: Interpretation of the pH dependence, J. Biol. Chem., 246 n(1971) 2734-2744. B. Jacobson and J.G. Webster, Medicine and Clinical Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1977. S.J. Updike, M.C. Shults and M. Busby, Continuous glucose monitor based on an immobilized enzyme electrode detector, J. Lab. Clin. Med., 93 (1979) 518-527. T. Kondo, H. Kojima, K. Okhura, S. Ikeda and K. Ito, Trial of new vessel access type glucose sensor for implantable artificial pancrease in oiuo, Trans. Am. Sot. Artif. Intern. Organs, 27 (1981) 250-253. T. Kondo, K. Ito, K. Okhura, K. Ito and S. Ikeda, A miniature glucose sensor, implantable in the blood stream, Diabetes Care, 5 (1982) 218-221. M. Shichiri, Y. Yamasaki, R. Kawamori, N. Hakui and H. Abe, Wearable artificial endocrine pancreas with needle-type glucose sensor, Lancet (1982) 1129-1131. J.Y. Lucisano, J.C. Armour and D.A. Gough, IEEE/NSF Symposium on biosensors, 1984, p. 78.

APPENDIX Interpretation of the % light transmission through a dye-impregnated membrane containing a pH gradient

The pH’s in some dye-containing membranes were deduced by measuring their percent light transmissions ( %T) and comparing the data with calibration curves. These calibrations consisted of plots of %T versus pH for membranes of uniform, known pH. The calibration plots were found to be linear. It will be shown here that for a membrane containing a pH gradient, the relationship between %T and the average pH is identical with the relationship between %T and pH for a membrane at uniform pH, if and only if the calibration plot of %T vs. pH is linear. Define the absorption p as the fraction of incident light absorbed per unit length, cm- ’ and define j? as the total fraction absorbed over the entire length of the membrane, dimensionless. Then by definition

p=

j/Wdr

(Al)

0

where x is the length parameter and L is the total membrane thickness, both having the unit of cm. Since in general/3 is a function of pH and pH is a function of x, we can write L lJ=~B(pWdx

(A21

0

where pH, designates the pH at position x. For a membrane of constuntpH,P(pH,) simplifies to P(pH) and the total absorption becomes

291

We have found experimentally that the total fraction absorbed by a membrane at constant pH is a linear function of pH. Combining this result with eqn. (A3), we have f=

(k,~H+kp)

=JV(PH)

0

But = (l/L)(k,-pH+k,)

(A6)

0

(A4)

or, P(PH)

= (1/L)Sk,.pH,dx+S(1/L)k,~

(A51

definition,

the

average

pH

is

~Havg = WL)tkUn)dx 0

so

where k, and k, are the fitting constants and are thickness-dependent. For a membrane containing variable pH, the total absorption is given by substituting eqn. (A5) intoeqn. (A2):

by

eqn. (A6) becomes

@=kl.pH,,,+&

(A7)

This is identical to eqn. (A4) except that pH,,, is substituted for pH, and so the hypothesis is proved.